Modeling the antigen and cytokine receptors signalling...

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João Tiago dos Santos Caldas de Sousa

Tese de Doutoramento Ramo de Bioquímica, especialidade de Bioquímica teórica

PhD thesis Branch of Biochemistry, speciality of Theoretical Biochemistry

Modeling the antigen and cytokine

receptors signalling processes and their

propagation to lymphocyte population

dynamics.

Faculdade de Ciências

Universidade de Lisboa 2003

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Faculdade de Ciências

Universidade de Lisboa

Tese de Doutoramento Ramo de Bioquímica, especialidade de Bioquímica teórica

PhD thesis Branch of Biochemistry, speciality of Theoretical Biochemistry

João Tiago dos Santos Caldas de Sousa

Modeling the antigen and cytokine receptors

signalling processes and their propagation to

lymphocyte population dynamics

Supervisors

Doutor Jorge Carneiro Instituto Gulbenkian de Ciência

Prof. Ruy Carvalho Pinto Faculdade de Ciências, Universidade de Lisboa /

Instituto Rocha Cabral

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Cover: Microscope photograph showing a cluster of lymphocytes in a culture. (Courtesy of Jocelyne Demengeot.

Photograph taken using IGC’s microscope facilities at José Feijó’s lab)

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i. Acknowledgements

I would like to express my deepest gratitude to the following people and institutions:

To Jorge Carneiro, for being an all-in-one supervisor, mentor, colleague and friend.

To Prof. Ruy Pinto, scientific mentor and role model since I was a student at

University.

I thank Instituto Gulbenkian de Ciência (IGC), for providing most of the scientific and

material infrastructure necessary for the realization this thesis, and Instituto Rocha Cabral

(IRC), for the support in the first months of this thesis. I also acknowledge the support of

the fellowship BD/13546/97 from Fundação para a Ciência e Tecnologia - Program Praxis

XXI.

To José Faro, for his friendship and for being always available to contribute, review

and discuss most, if not all, of the work of this thesis.

To Zvi Grossman, for discussions and suggestions in hypersensitivity, adaptation and

homeostasis, which transverse most of the work in this thesis.

To my friend Andreia Lino, for having the courage to work in the experimental

validation of parts of the TCR triggering model as part of her graduation thesis.

To Kalet Leon for his friendship, discussions, suggestions, Cuban coffee and rum.

To Jocelyne Demengeot for comments on the works of this thesis and for giving me

beautiful photographs of lymphocyte cultures for the cover. Also, I’m gratefull to the labs

of Jocelyne Demengeot and Paulo Vieira at the IGC, for being helpfull, supportive and

critic during my stay at that institution.

I thank all the people at the IGC and IRC, too many to name here.

To my friends, Isabel Abreu, Fernando Antunes, João Garcia, Rui Gardner and Rita

Lemos for their friendship, support and patience in dealing with me during the thesis.

Last, but not least, I am deeply thankful to Íris, for being wonderful and having a lot of

patience. Thank you for everything.

~ ~ I dedicate this thesis to my parents and my brother. I owe them more than I can say in

words.

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Table of contents

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ii. Table of contents

I. ACKNOWLEDGEMENTS............................................................................ VII

II. TABLE OF CONTENTS..................................................................................IX

III. ABBREVIATIONS AND NOTES TO THE READER ...........................XIII

IV. SUMÁRIO..................................................................................................... XV

V. SUMMARY..................................................................................................XVIII

1 GENERAL INTRODUCTION .......................................................................... 1

1.1 THE CLONAL SELECTION THEORY .................................................................... 3

1.2 T LYMPHOPOIESIS AND POPULATION DYNAMICS.............................................. 6

1.3 SIGNALS AFFECTING THE LIFE HISTORY OF T LYMPHOCYTES......................... 10

1.3.1 TCR engagement and triggering .............................................................. 11

1.3.2 Cytokine receptor engagement................................................................. 16

1.4 THE IMMUNE SYSTEM RANGES MULTIPLE ORGANISATION LEVELS................. 18

1.5 THIS THESIS. .................................................................................................. 21

1.5.1 Sharing cytokine receptor chains and their implications......................... 21

1.5.2 Mathematical analysis of TCR triggering. ............................................... 22

1.5.3 Adaptable activation thresholds and lymphocyte homeostasis. ............... 23

1.5.4 General discussion ................................................................................... 23

1.6 BIBLIOGRAPHY.............................................................................................. 24

2 IMPLICATIONS OF SHARING CYTOKINE RECEPTOR CHAINS........ 2

2.1 INTRODUCTION.............................................................................................. 41

2.2 MODELLING AND RESULTS............................................................................ 43

2.2.1 Coupling between IL-2 and IL-4 receptor signalling via the common

gamma chain ................................................................................................................ 43

2.2.2 IL-2 and IL-4 are potentially non-competitive inhibitors of each other but

the effect of IL-2 is quantitatively more important....................................................... 46

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2.2.3 Differentiation of ThP cells to committed Th2 cells — A population

dynamics model ............................................................................................................ 47

2.2.4 Inhibition of IL-4 driven differentiation by IL-2 allows for the persistence

of precursors................................................................................................................. 50

2.2.5 Modulating γc expression levels may be used to influence the extent to

which a clone is committed .......................................................................................... 54

2.3 DISCUSSION................................................................................................... 55

2.4 BIBLIOGRAPY ................................................................................................ 58

3 ANALYSIS OF TCR ENGAGEMENT, TRIGGERING AND DOWN-

MODULATION.................................................................................................................. 63

3.1 A MATHEMATICAL ANALYSIS OF TCR SERIAL TRIGGERING AND DOWN-

REGULATION...................................................................................................................... 65

3.1.1 Introduction .............................................................................................. 67

3.1.2 Results ...................................................................................................... 68

3.1.2.1 Experimental data and model variables ............................................. 68

3.1.2.2 TCR-dimerisation fails to explain the experimental data .................. 69

3.1.2.3 Non zero steady-state resulting from ligand independent TCR-

turnover 73

3.1.2.4 Insensitivity to changes in ligand density results from a transient

accumulation of a pool of triggered TCRs .......................................................... 73

3.1.2.5 TCR-triggering is ultrasensitive to ligand and TCR densities .......... 74

3.1.2.6 Two pools of membrane TCR with different triggering kinetics

explain biphasic kinetics of down-regulation...................................................... 75

3.1.3 Discussion ................................................................................................ 76

3.1.4 Methods .................................................................................................... 80

3.1.5 References ................................................................................................ 80

3.2 ON THE REQUIREMENT FOR HIGH COOPERATIVITY IN TCR TRIGGERING:

HYPOTHETICAL MECHANISMS OF TCR TRIGGERING. ......................................................... 85

3.2.1 Introduction .............................................................................................. 87

3.2.2 Results ...................................................................................................... 88

3.2.2.1 Kinetic analysis of TCR triggering.................................................... 88

3.2.2.2 Sequential oligomerization of TCR-ligand complexes...................... 89

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3.2.2.3 The cycle of TCR triggering.............................................................. 92

3.2.3 Discussion ................................................................................................ 95

3.2.4 Methods .................................................................................................. 100

3.2.5 Appendix................................................................................................. 100

3.2.6 References .............................................................................................. 102

3.3 ACTIVATION THRESHOLDS, ADAPTATION AND TCR TRIGGERING................ 105

3.3.1 Bibliography........................................................................................... 107

3.4 TOWARDS AN ANALYSIS OF THE EXPERIMENTAL SETTINGS TO STUDY TCR

TRIGGERING BY LIGAND................................................................................................... 109

3.4.1 Bibliography........................................................................................... 111

4 ADAPTABLE ACTIVATION THRESHOLDS AND LYMPHOCYTE

HOMEOSTASIS .............................................................................................................. 113

4.1 INTRODUCTION............................................................................................ 116

4.2 RESULTS...................................................................................................... 117

4.2.1 The AAT signalling model ...................................................................... 117

4.2.1.1 Activation threshold, adaptation and refractoriness ........................ 119

4.2.1.2 AAT and frequency of the stimuli ................................................... 122

4.2.2 Dynamics of a population of lymphocytes with AAT.............................. 124

4.2.2.1 Methods for simulation of many individual cells and quasi-steady

state analysis...................................................................................................... 126

4.2.2.2 Results of simulation and quasi-steady state analysis of the AAT

population model ............................................................................................... 127

4.2.2.3 Thymic influx of refractory and responsive lymphocytes ............... 133

4.3 DISCUSSION................................................................................................. 136

4.4 BIBLIOGRAPHY............................................................................................ 139

5 GENERAL DISCUSSION.............................................................................. 143

5.1 BIBLIOGRAPHY............................................................................................ 157

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Abbreviations and notes to the reader

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iii. Abbreviations and notes to the reader

This thesis includes four original manuscripts published or submitted in various

specialized journals, which are presented either as chapters or as sub-chapters, denoted as

sections for now on. Figures and tables are numbered sequentially within each section;

when adequate for proper reference, the section number always precedes cross-section

references. Each section contains its own bibliographic reference list.

Abbreviations and nomenclature are based in the guidelines for publication in the

European Journal of Immunology (year 2000 guidelines), together with the following

definitions that transverse all the sections of this thesis:

Abbreviation Definition

γc ........................................ Common γ Chain

AAT ................................... Adaptable Activation Threshold

Ab....................................... Antibody

Ag....................................... Antigen

APC.................................... Antigen Presenting Cell

APL.................................... Altered Peptide Ligand

BCR ................................... B Cell Receptor

BM ..................................... Bone Marrow

CLP .................................... Common Lymphocyte Precursor

DC...................................... Dendritic Cell

HSC.................................... Haematopoietic Stem Cell

IFN..................................... Interferon

Ig ........................................ Immunoglobulin

IL-2R.................................. IL-2 Receptor

IL-4R.................................. IL-4 Receptor

ITAM ................................. Immuno-receptor Tyrosine Activation Motifs

JAK .................................... Janus Kinase

MHC .................................. Major Histocompatibility Complex.

NK...................................... Natural Killer

PTK.................................... Protein Tyrosine Kinase

PTP..................................... Protein Tyrosine Phosphatase

Abbreviations and notes to the reader

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STAT ................................. Signal Transducer and Activator of Transcription

TCR.................................... T Cell Receptor

Th ....................................... Helper T cell

SAC.................................... Src-Associated Co-receptor.

EP........................................ Phosphatase

EK ....................................... Kinase

MIC.................................... Many Individual Cells

SMIC.................................. Simulation of Many Individual Cells

QSS .................................... Quasi Steady State

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iv. Sumário

A teoria da selecção clonal fundamentou a discriminação pelo sistema imune entre

antigénios próprios e antigénios estranhos (self/non self discrimination) no processo de

selecção de linfócitos com base na especificidade dos receptores para o antigénio expressos

dessas células. Após a aceitação da teoria da selecção clonal, os eventos que condicionam o

desenvolvimento linfocitário, a interacção entre linfócitos e outras células e as vias de

transdução de sinal tornaram-se da maior importância em Imunologia.

Durante os diferentes estádios de desenvolvimento de linfócitos são expressos

diferentes subconjuntos de receptores para os vários factores que determinam quais as vias

possíveis para a continuação desse desenvolvimento. Assim sendo, a dinâmica populacional

de linfócitos depende dos processos decorrentes ao nível de células individuais, como por

exemplo a regulação da expressão de receptores e a transdução de sinais. No entanto, esses

processos decorrentes em células dependem também da interacção directa ou indirecta com

outras células. Essas interacções podem ser mediadas quer por factores solúveis no meio

extracelular, quer por contacto directo com outras células. Logo os processos decorrentes a

nível celular e populacional são interdependentes.

A compreenção das interacções entre diferentes níveis de organização é fundamental

para o estudo de sistemas biológicos, para a compreensão da separação entre moléculas e

metabolismo, entre células e populações celulares, entre populações celulares e órgãos e

organismos e entre indivíduos e ecossistemas. Actualmente, as técnicas correntes em

biologia, bioquímica e biologia molecular têm uma forte componente reducionista. O

emprego dessas técnicas tem sido frutuoso na caracterização de muitos componentes e

interacções pertencentes a cada nível de organização, mas no entanto têm tido um sucesso

limitado na caracterização das interacções entre diferentes níveis de organização.

O objectivo desta tese é contribuir para a compreensão de como eventos ocorrentes ao

nível de uma única célula influenciam a dinâmica populacional dessas mesmas células. A

tese concentra-se na dinâmica populacional de linfócitos T CD4+ e nos receptores mais

importantes que condicionam o desenvolvimento linfocitário: o receptor para o antigénio

(TCR) e os receptores para as citocinas. Mais especificamente, os trabalhos aqui

apresentados estudam, recorrendo à modelação matemática, como a interferência entre

citocinas e a desensitização do TCR podem influenciar a dinâmica populacional dos

linfócitos T CD4+. A tese está dividida em três partes principais de resultados, com uma

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discussão final.

A primeira parte da tese aborda o estudo da consequência de receptores para citocinas

diferentes partilharem exclusivamente subunidades necessárias à transdução de sinal. Para

abordar esta questão foi usado um modelo prototípico da interferência entre os receptores

IL-2R e IL-4R, estendendo depois esse modelo à dinâmica de diferenciação de linfócitos

ThP a Th2. Os resultados do modelo da interferência entre receptores indicam que no

regime de parâmetros estimados experimentalmente, quando a IL-2 é saturante o receptor

da IL-2 é capaz de sequestrar a maior parte da cadeia partilhada (γc) entre o IL-2R e o IL-

4R, o que inibe a formação do receptor da IL-4 em aproximadamente 40% (em condições

de IL-4 saturante). A citocina IL-4 é secretada por células Th2 e é um importante factor de

diferenciação de linfócitos ThP em Th2, constituindo portanto um mecanismo de feed-back

positivo na diferenciação. A citocina IL-2 é um importante factor de crescimento

linfocitário, sendo um factor de crescimento das células ThP e sendo secretado por essas

mesmas células. Usando um modelo simples de diferenciação de ThP a Th2 demonstrou-se

que em determinadas condições a inibição da formação do receptor IL-4R em 40% é

suficiente para atenuar os efeitos do feed-back positivo mediado pela IL-4, impedindo a

exaustão da população de células precursoras ThP.

A segunda parte da tese é dedicada ao estudo da dinâmica da expressão e

desensitização por internalização do TCR em linfócitos T. Foi usado um modelo

fenomenológico para determinar as propriedades do mecanismo de activação e

internalização do TCR subjacentes aos dados experimentais da cinética da internalização do

TCR. O resultado mais importante deste modelo foi a identificação da necessidade de uma

elevada cooperatividade (ou hipersensibilidade) no mecanismo de activação do TCR. Esta

elevada cooperatividade é indicativa que o processo de activação do TCR é um processo

complexo, provavelmente envolvendo vários passos reaccionais. Assim sendo,

averiguamos qual dos mecanismos propostos para a activação do TCR seria o mais capaz

de descrever esta elevada cooperatividade assim como a cinética de internalização do TCR.

Os dois principais mecanismos propostos de activação do TCR são i) a dimerização do

TCR mediada pelo ligando, MHC-péptido apresentado por uma APC; e ii) a fosforilação do

TCR mediada pelo ligando e pelo coreceptor CD4 associado a lck. Os resultados dos

modelos estudados sugerem uma conciliação de ambos os mecanismos, dependendo da

dose de ligando apresentada pela APC. A baixas doses de ligando o mecanismo dependente

do CD4 poderá ser o mais significativo, mas a altas doses de ligando ambos os mecanismos

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de activação serão possíveis. Em aberto fica a questão se estes mecanismos transducem

sinais qualitativamente diferentes.

Finalmente, na terceira parte da tese estudou-se se a adaptação do limiar de activação

da sinalização do TCR é uma condição suficiente para a manutenção homeostática de uma

população periférica de linfócitos T auto-reactivos desensitizados. O modelo usado neste

estudo inclui dois níveis de organização: o nível de linfócitos adaptáveis com uma via de

transdução de sinal do TCR simples, com hipersensibilidade e com um limiar de activação

adaptável ao estímulo; e o nível da dinâmica da população de linfócitos adaptáveis.

Para se dividir, um linfócito necessita de receber um estímulo acima do limiar de

activação de uma APC. Um modelo simples de homeostase de linfócitos seria que o

crescimento de uma população de linfócitos é limitado pela competição desses linfócitos

por APCs. Neste modelo simples, não é possível que existam linfócitos anérgicos na

periferia, já que cada interacção com APC promove uma resposta que implica divisão

celular. Numa população de linfócitos adaptativos, a dinâmica depende não só do numero

de APCs, mas também do estado do limiar de activação de cada linfócito, que depende por

sua vez das interacções passadas com APC. Como cada linfócito tem um historial de

activação único, é necessário ter em linha de conta a contribuição de cada linfócito para a

dinâmica populacional. Neste estudo, usaram-se dois modelos para estudar a dinâmica de

linfócitos adaptáveis: Um modelo onde se simulou a população de linfócitos usando um

autómato celular; e um modelo onde se simulou a evolução de cada linfócito numa

população de linfócitos em estado estacionário. Os resultados dos modelos indicam que o

estado estacionário onde os linfócitos estão anérgicos, é instável. Pequenas perturbações em

torno desse estado estacionário podem levar a população de linfócitos a ficar cada vez

menos desensitizadas e a crescer até ficar limitada pelo número de APCs, tal como no

sistema simples sem adaptação, ou a decrescer até ao total colapso da população.

Consequentemente, a adaptação do limiar de activação do TCR não é por si só suficiente

para a manutenção homeostática de uma população periférica de linfócitos T auto-reactivos

em estado de desensitização (adaptados), no entanto é um mecanismo eficiente para a

deleção periférica dos mesmos se a população inicial for menor que o que o estado

estacionário instável onde os linfócitos estão anérgicos.

Esta tese constitui um passo inicial para a resposta à questão de como os processos

ocorrentes ao nível linfocitário se propagam até ao nível da dinâmica de populações e

influenciam o sistema imunitário. Este é um campo ainda numa fase embrionária, onde se

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antevê um forte impacto em Imunologia e biologia em geral.

v. Summary

The clonal selection theory provided a view of self/non-self discrimination based in

lymphocyte selection by the specificity of the antigen receptor expressed by the

lymphocyte. After the acceptance of this theory the characterization of lymphocytes, the

events that direct their development, the interaction with other cells and the signaling

cascades involved in the integration of signals by lymphocytes became central in

Immunology.

At different stages of lymphocyte development, different subsets of receptors for the

factors that condition lymphocyte development are expressed. This restricted expression of

receptors shapes the possible developmental paths of the lymphocyte; hence lymphocyte

population dynamics depends on the underlying molecular events, such as the regulation of

receptor expression and signal transduction, occurring in each lymphocyte. Conversely,

molecular single cell events also depend on the direct or indirect interaction with other

cells; hence single cell events depend on processes occurring at the cell population level.

The understanding of the interactions between different levels of organization is

fundamental for the study of biological systems, to understand the leaps from molecules to

metabolism, from cells to cell populations, from cell populations to an organ and organism

and from individuals to ecosystems. Current mainstream techniques in biology, molecular

biology and biochemistry rely on a prominently reductionist approach, providing plenty of

knowledge on the individual components and their interactions within the different

organization levels, but little or no information on how these components interact between

different organization levels.

The goal of this thesis is to contribute to the study of how do single cell events

influence the population dynamics of cells. We focus in CD4+ T cells and the receptors that

ultimately dictate lymphocyte fate: the antigen and the cytokine receptors. More

specifically we address, using mathematical models, the influence that molecular events at

the cell level, such as the competition between cytokines for the receptor subunits and T

cell antigen receptor desensitization, have on the dynamics of lymphocytes.

This thesis is divided in three major parts of results, with a final discussion. The first

part deals with the consequence for T cell population dynamics of cytokine receptors

sharing exclusively receptor subunits necessary for signal transduction. We study a

Summary

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prototypic model of IL-2R and IL-4R interference and extend the results of this model to

the population dynamics of ThP differentiation to Th2. The results of the model showed

that although this interference is about 40% inhibition of IL-4R signaling by IL-2R, it can

be significant enough to prevent Th2-mediated exhaustion of ThP precursor cells from the

system.

The second part of the thesis addressed the process of TCR engagement, triggering and

down-modulation. We used phenomenological models together with available experimental

data, to gain insight into the underlying mechanism of TCR triggering and down-

modulation. The most important of the results of this model is the requirement for high

cooperativity (hypersensitivity) in the mechanism of TCR triggering. We then explored if

either TCR cross-linking or TCR triggering via crosslinking CD4-lck could explain this

high cooperativity. Our results suggest that at low dose of ligand presentation CD4 cross-

linking might be the preferential mechanism of TCR triggering, but at high dose of ligand

presentation both TCR crosslinking and CD4 co-receptor crosslinking mechanisms should

be possible.

Finally, in the third part we addressed the question if adaptation of activation

thresholds in TCR signaling is a sufficient condition for the maintenance of a peripheral

pool of auto-reactive lymphocytes. The model used in this study includes two

organizational levels: The adaptable lymphocyte with a simplified TCR signal transduction

pathway displaying hypersensitivity and an activation threshold adaptable to stimulus; and

the dynamics of a population of adaptable lymphocytes, which reflects the state of the

individual cells in the population, which is in turn the consequence of all the past

encounters of each lymphocyte with APC. Our modeling results indicate that although

adaptation of activation thresholds in TCR signaling is not a sufficient condition for the

maintenance of a peripheral pool of auto-reactive lymphocytes, it is nevertheless an

efficient mechanism of peripheral deletion of self-reactive clones.

This thesis is an initial effort to answer the question of how do single cell events, such

as signal transduction and regulation of receptor expression, propagate to the population

dynamics of lymphocytes and influence the immune system. We barely scratch the surface

of what appears to be a most interesting and still mostly unexplored area of modeling in

Immunology and biology in general.

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General introduction

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1 General introduction

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3

1.1 The clonal selection theory

The clonal selection theory by Sir MacFarlane Burnet (Burnet 1957) bridged the

chasm between cellular and humoral Immunology by identifying a cell, the lymphocyte,

with a particular antigen receptor. Immunology became footed on the duality of the

specificity of antigen receptors and the life history of the cells expressing those receptors.

The properties of the immune system are the result of the individual contributions and

interactions of the cells that make it, thus under the clonal selection theory, to understand

the generation and maintenance of the immunoglobulin repertoire and immune response, it

is necessary to understand the life history traits of lymphocytes, i.e. the generation,

activation, differentiation and proliferation of lymphocytes, in addition to understanding

their interactions with other cells. Hence, the study of the events that control the transition

of lymphocytes between the various development stages became a fundamental problem in

Immunology.

The clonal selection theory proposed that during the early stages of development and

up to the neonatal period, antibody-producing cells are generated with random specificity

from a common precursor. During the development of these cells, antigen stimulation

arrests their development and causes their death (deletion). Since it is most likely the

antigen found during ontogeny originates from the host (self antigen), deletion of the

reactive cells during ontogeny selects only the cells that do not recognise self-antigen, thus

ensuring self-tolerance.

Since it's acceptance, the clonal selection theory has been extended and modified to

account for novel observations. Some of these extensions are the discovery that

haematopoiesis is a life-long process (Lederberg 1958); the positive selection of

lymphocytes (Zinkernagel and Doherty 1997); the elucidation of the molecular bases of the

generation of antigen receptor diversity (Tonegawa 1976; Brack, Hirama et al. 1978;

Tonegawa 1983; Wagner and Neuberger 1996; Fugmann, Lee et al. 2000), and the

discovery of the T helper, CTL and supressor subsets of lymphocytes. The two signals

model by Cohn and colleagues of the regulation of the immune response is a conceptual

elaboration of the clonal selection theory that incorporates these findings (Bretscher and

Cohn 1968; Langman and Cohn 1996). Briefly, the model proposes that activation of B

cells by monovalent antigens requires two signals: the antigenic signal (signal 1) and help


4

provided by T cells (signal 2). In the presence of only signal 1, B cells are rendered non-

responsive. Help for T cells is provided by effector T cells, whose activation also depends

on antigen (signal 1) and help from other T cells (signal 2). In the presence of only signal 1,

T cells are also rendered non-responsive. Mature T cells are also capable of becoming

helper effectors in the absence of signals 1 and 2. Self/non-self discrimination is possible if

T cell development is a slow process that ensures that self-reactive T cells capable of

delivering signal 2 do not mature or, more precisely, mature rarely.

An exception to the clonal selection theory is the existence, in normal healthy

individuals, of auto-reactive natural antibodies (Avrameas 1991) and of T cells that do not

elicit a destructive immune response (Sakaguchi, Fukuma et al. 1985; Pereira, Forni et al.

1986). This is paradoxical under the clonal selection theory, where recognition of antigen

either leads to deletion or to activation of the destructive effector function of the clone. This

phenomenon raises important questions on the regulation and organization of the immune

system. Several hypotheses have been proposed to solve this paradox, some of which are:

1. The Danger model, by Matzinger and colleagues (Matzinger 1994). The danger

hypothesis shifts the functional dependence of the immune system from self/non-self

recognition to danger/no danger recognition mediated by professional APCs. According

to this theory, the immune system reacts against danger, irrespectively of this danger

being self or non-self in origin. Danger can be many things, the spill of intracellular

contents, toxins, bacterial components, etc. The discrimination of danger and non-

danger is done at the APC, which are capable of presenting antigens with co-

stimulation, thus activating T cells and initiating the immune response (Cunningham

and Lafferty 1977).

2. The hypothesis of adaptable activation thresholds (AAT) of lymphocytes by Grossman

and colleagues (Grossman and Paul 1992; Grossman and Paul 2000; Grossman and

Paul 2001). This hypothesis postulates that the dynamics of lymphocyte activation is

akin to that of neuronal sensory systems. The main hypothesis is that lymphocytes are

only activated when the antigenic stimulus increases above a dynamic threshold that is

set by the background antigen present in the environment. Since self-antigens are (in

general) persistent and/or ubiquitous in the organism, it can be argued that self-antigen

constitutes a background stimulus. Hence, lymphocytes would adapt (tune) their

activation threshold and stop responding to the self-antigen background. The AAT

hypothesis has been used to explain some experimental phenomena (Nicholson,


5

Anderson et al. 2000; Smith, Seddon et al. 2001; Wong, Barton et al. 2001).

3. The hypothesis of regulatory or suppressor T cells. The clonal selection theory

postulates that tolerance is recessive, i.e. the only way to achieve tolerance is through

deletion of the clones with the potential to elicit an immune response. Experimental

evidence has been accumulated in which the acquisition of tolerance is mediated by the

adoptive transfer of a subset of cells from tolerant animals (Sakaguchi, Fukuma et al.

1985; Fowell, McKnight et al. 1991; Smith, Lou et al. 1992; Modigliani, Thomas-

Vaslin et al. 1995; Sakaguchi, Sakaguchi et al. 1995; Powrie, Mauze et al. 1997). These

“dominant tolerance” (Bandeira, Mengel et al. 1991) experiments demonstrate the

existence of a population of peripheral auto-reactive T cells, which might be important

for the regulation of the immune system. Suppressor T cells can “upgrade” the two-

signal model to a three-signal model, where the third signal is negative and suppresses

the second signal, which is provided by activator T cells; Suppressor T cells can also

extend the danger model by interfering in the APC T helper interaction. In the recent

years, the study of regulatory cells has gained considerable momentum, in the

understanding of their generation and phenotype and the mechanism of suppression

(Leon, Perez et al. 2000; Sakaguchi 2000; Leon, Perez et al. 2001; Shevach, McHugh et

al. 2001; Shevach 2002).

These hypotheses reflect the hierarchy of the organization of the immune system,

which depends on the interaction between populations of different cell types that

communicate via signals relying on the expression of receptors and respective agonists and

antagonists.

In this thesis, the focus will be on T lymphocytes, specifically in particular aspects of

T cell signalling related to life history traits of T lymphocytes, and the implications for the

population dynamics of these cells. The following introduction is therefore biased towards

T lymphocytes, making mention of B-lymphocytes sparsely and only for comparative

purposes. We introduce briefly the life history of lymphocytes, from the bone marrow

(BM) precursors to the differentiated peripheral effector classes, focusing on the

mechanisms of maintenance and differentiation of these cells. We also introduce the main

signals that influence the progression of lymphocytes through their life history, the TCR

and cytokine signalling, and introduce in more detail the AAT hypothesis, where the

history of cells influences the cell signalling apparatus.


6

1.2 T lymphopoiesis and population dynamics

Differentiation of lymphocytes is a spatial-temporal problem of population dynamics

bracing multiple organs, environments and specific factors. Great effort has been dedicated

to identify the receptors, master genes and patterns of expression involved in the transitions

in the developmental pathways of B (Henderson and Calame 1998; Hardy and Hayakawa

2001) and T lymphocytes (Kuo and Leiden 1999; Di Santo, Radtke et al. 2000; Murphy,

Ouyang et al. 2000; Quong, Romanow et al. 2002). In spite of all the progress made, the

molecular biology approach only provides a static picture of lymphocyte development and

differentiation. Thus little is known about the temporal and spatial components of the

dynamics of the different lymphocyte pools. What makes the regulation of the dynamics

and size of differentiating lymphocyte pools complex to understand is that each lymphocyte

clone has a unique antigen receptor (resulting from the somatic recombination of gene

fragments) and a characteristic expression profile of molecules (surface markers) at each

stage in the lymphocyte life history. These surface markers include membrane receptors,

adhesion molecules and co-receptors, and reflect the stage of development of the cell. They

can influence differentiation, migration, survival, proliferation, or death of a cell, as a

function of the surrounding environment.

All lymphocytes derive from haematopoietic stem cells (HSC), the common precursor

in haematopoietic organs. HSCs in the BM of mammals are the self-renewing precursors of

all the cells of the hematolymphoid lineages (Spangrude and Johnson 1990; Smith,

Weissman et al. 1991). One of the descendents of these pluripotent cells in the BM is the

common lymphoid progenitor (CLP) cell that can differentiate to B, T and NK cells

(Kondo, Weissman et al. 1997). Animals reconstituted with CLP cells lose cell numbers

after 4 weeks, which indicates that these cells have a limited self-renewal capacity (Kondo,

Weissman et al. 1997), in contrast to HSCs and memory peripheral lymphocytes (Tanchot

and Rocha 1995).

The thymus is where TCR genes of precursor T cells undergo somatic recombination

and selection of T cells takes place. Precursors originating from the bone marrow can

differentiate in the thymus to dendritic cells, NK cells and T cells, expressing either αβ

(CD4+ or CD8+) or γδ TCR. The T cells expressing γδ TCR represent only 1-10% of

peripheral T cells (Bank, DePinho et al. 1986). Compared to the αβ T cells, little is known


7

about the development and physiology of γδ T cells (reviewed in (Chien, Jores et al. 1996;

Hayday 2000)). In this thesis we focus only on αβ T cells; unless otherwise stated, we refer

to the αβ TCR simply as TCR.

The type of T cell precursors that enter the thymus from the BM is still a matter of

debate. These precursors maintain some degree of multipotency, being able to generate B,

NK and dendritic cells under appropriate conditions (reviewed in (Shortman and Wu 1996;

Di Santo, Radtke et al. 2000)). Notch1 signalling is necessary for differentiation of

precursor T cells entering the thymus and it might be required in the following stages for

αβ T cell commitment (Pui, Allman et al. 1999; Radtke, Wilson et al. 1999), reviewed in

(Di Santo, Radtke et al. 2000).

The most important feature of antigen receptors and antibodies is the variable region,

which confers the specificity of antigen recognition. The segments of the gene encoding the

Ig and TCR chains are assembled by somatic recombination of gene fragments; the V, D, J

(heavy and beta chain genes respectively in B and T cells) or the V, J (light and alpha chain

genes respectively in B and T cells) regions of the gene are recombined into an exon, which

encodes the antigen-binding domain of the molecule (Tonegawa 1976; Brack, Hirama et al.

1978; Tonegawa 1983; Wagner and Neuberger 1996; Fugmann, Lee et al. 2000). The

mechanism of somatic recombination and mutation of the Ig and TCR genes is the

molecular biology foundation of the clonal selection theory. B cells, but not T cells, can

still undergo additional stages of somatic mutation of immunoglobulin genes by single

point mutations during the germinal centre reaction (Weigert, Cesari et al. 1970; Wagner

and Neuberger 1996; Jacobs and Bross 2001).

The generation of antigen receptor gene diversity does not preclude the possibility of

generating out-of-frame rearrangements, receptors that are non-functional or have

potentially harmful specificities. The process of selection during the maturation of

lymphocytes provides a mechanism of quality control in the generation of lymphocytes,

preventing the maturation of cells bearing a non-functional or potentially harmful receptor.

T cell selection in the thymus comprehends two phases, taking place at the double

positive stage of T cell development: a positive and a negative selection stage (Sebzda,

Mariathasan et al. 1999; Hogquist 2001). During positive selection only the cells that

recognize the self-MHC complex are selected. During negative selection, the cells reacting

strongly to self-MHC complexes are deleted and the cells that do not interact with the self-

MHC complexes die from absence of survival signals.


8

Lymphocyte selection contributes to the high levels of apoptosis in the thymus (Surh

and Sprent 1994). Notwithstanding the high level of apoptosis, there is a fraction of auto-

reactive T cells that survive the selection process, which are then subject to mechanisms of

peripheral self-tolerance mentioned above.

T cell differentiation continues in the periphery, where T lymphocytes can acquire

different functional phenotypes, such as the Th1 and Th2 cells (O'Garra 1998), memory

cells (Dutton, Bradley et al. 1998), regulatory T cells (CD25+ (Sakaguchi, Fukuma et al.

1985) and others (Modigliani, Coutinho et al. 1996)), Th3 (Fukaura, Kent et al. 1996;

Kitani, Chua et al. 2000) and Tr1 cells (Groux, O'Garra et al. 1997).

The regulation of cell differentiation involves multiple factors, receptors and cells,

which may result in feedforward and/or feedback regulatory loops. Positive feedback loops

amplify perturbations and are characteristic of runaway processes, whereas negative

feedback loops dampen perturbations and thus are characteristic of homeostatic processes.

Cytokines, which are soluble factors that control multiple aspects of lymphocyte

physiology (Staudt and Brown 2000), are one of the main classes of factors mediating such

loops. The interaction between different cytokines and cytokine-secreting cells, which

express at different development stages different subsets of cytokine receptors, and in the

case of lymphocytes also a clone-specific antigen receptor, can conceivably be very

complex and may add to positive and negative feedback or feedforward loops. An example

of a positive feedback loop is the mutually exclusive differentiation of ThP cells to Th2 or

Th1, which is promoted respectively by Th2 and Th1 (Fishman and Perelson 1994;

Carneiro, Stewart et al. 1995; O'Garra 1998; Bergmann, van Hemmen et al. 2002). In this

thesis, we explore novel aspects of the control of Th2-dependent ThP differentiation, which

are somewhat more complex than this simple positive feedback loop.

In spite the heterogeneity of lymphocyte populations, the total number of lymphocytes

seems to be regulated by a homeostatic mechanism. Depletion of peripheral lymphocytes in

normal animals is not permanent. With some exceptions (Mackall, Granger et al. 1993;

Modigliani, Coutinho et al. 1996), lymphocytes tend to return to normal numbers via what

seems like a homeostatic regulatory mechanism, which is still not fully understood

(Tanchot, Rosado et al. 1997; Goldrath and Bevan 1999; Freitas and Rocha 2000; Almeida,


9

Borghans et al. 2001).

The complexity of the regulation of lymphocyte dynamics at the subpopulation level,

together with the importance of maintaining lymphocyte diversity, diminishes the

importance of classic negative feedback loops, such as the control erythrocyte numbers in

the blood mediated by erythropoietin production in response to O2 partial pressure in the

blood (Muta, Krantz et al. 1994), in the regulation of lymphocyte subpopulations.

The regulation of B and T cell numbers is independent (Kitamura, Roes et al. 1991;

Mombaerts, Clarke et al. 1992; Tanchot, Rosado et al. 1997; Bender, Mitchell et al. 1999)

and within the peripheral T cell compartment, memory and naïve cells are differentially

regulated (Tanchot and Rocha 1995; Goldrath and Bevan 1999).

Although in normal animals the CD4+/CD8+ cell ratio is maintained, in transgenic

animals there is a bias towards either CD8+ or CD4+, but even in these cases the total

population of T cells tends towards normal numbers (Rocha, Dautigny et al. 1989; Tanchot,

Rosado et al. 1997), suggesting that at least one homeostatic mechanism is counting the

total T cell population irrespective of MHC class. This mechanism is commonly referred as

“blind T cell homeostasis” because it does not depend on the MHC class of T cells

(Adleman and Wofsy 1993; Margolick, Munoz et al. 1995; Adleman and Wofsy 1996;

Mehr and Perelson 1997). Although the survival of T cells in the periphery seems to be

blind to the MHC class, it depends on MHC-TCR engagement. There is a correlation

between survival of peripheral lymphocytes and expression of self-MHC ligands (Benoist

and Mathis 1997; Freitas and Rocha 1997; Tanchot, Lemonnier et al. 1997). Conditional

knockouts of MHC class II (Witherden, van Oers et al. 2000) and of the TCRα chain

(Polic, Kunkel et al. 2001) show that in the absence of these molecules the pool of

peripheral T cells decays with time, following apparent first order kinetics, demonstrating

the need for the MHC-TCR interaction for the maintenance of peripheral T cells.

Memory T cells are less dependent on antigenic stimulation and once transferred in

various amounts to T cell deficient mice they expand by self-renewal to about the same

number, independently of the existence and number of naïve T cells (Tanchot and Rocha

1995). Naïve T cell recovery is dependent on thymic output, which indicates that these cells

are incapable or have a limited self-renewal capacity. With the increase in age, thymic

output to the periphery becomes less important (Mackall, Fleisher et al. 1995; Sempowski

and Haynes 2002), which is probably the reason why young animals recover faster from T

cell depletion than older animals (Mackall, Punt et al. 1998; Timm and Thoman 1999).


10

During homeostatic proliferation of naïve T cells in response to depletion, these cells bear a

partial and transient memory-like phenotype (Goldrath, Bogatzki et al. 2000). Once normal

numbers are restored, these cells regain the characteristic naïve phenotype.

A hypothesis is that homeostasis of T lymphocytes could be explained by competition

for antigenic niches in the periphery (Carneiro, Stewart et al. 1995; De Boer and Perelson

1997; Freitas and Rocha 2000). The mathematical model of De Boer & Perelson (De Boer

and Perelson 1997) presupposes that total T cell numbers expand to a fixed level (a fixed

carrying capacity), which is determined by the level of antigen presentation and the number

of APCs. According to the model, the T cell pool will expand to normal numbers whenever

lymphocytes have a sufficient cross-reactive epitopes and/or if the initial T cell population

is diverse enough (De Boer and Perelson 1997).

In summary, lymphocyte dynamics in the peripheral self-renewing T cell pool seem to

be regulated by mechanisms acting on two levels: a) At the clonal level to regulate the

generation, maintenance and extinction of a particular clone in a particular antigenic niche

(Carneiro, Stewart et al. 1995; De Boer and Perelson 1997; Freitas and Rocha 2000). b) At

the population level (blind T cell homeostasis) to regulate the total number of lymphocytes

(Rocha, Dautigny et al. 1989; Freitas and Rocha 1993; De Boer and Perelson 1997). The

dependence of the sustainability of T cell numbers in the periphery, on the TCR and MHC

(Freitas and Rocha 2000; Witherden, van Oers et al. 2000; Polic, Kunkel et al. 2001)

implies that these two levels of regulation are inter-dependent. How is this inter-

dependence mediated? It seems paradoxical how signalling via the TCR, a receptor

resulting from somatic mutation of gene fragments that can be considered specific of a

clone, mediates both T cell homeostasis and specific T cell activation and proliferation.

One of the proposed hypotheses to solve this problem is that depending on the stregth of

the agonist, T cells can be activated to effectors or engage in homeostatic proliferation

(Evavold, Sloan-Lancaster et al. 1993).

1.3 Signals affecting the life history of T lymphocytes

It is beyond the scope of this introduction to describe all the signals capable of

affecting the life history of lymphocytes. We are interested in lymphocyte homeostasis and

the regulation of lymphocyte differentiation. Hence we concentrate in the two classes of

stimuli which are more relevant for this: TCR and cytokine signalling.


11

1.3.1 TCR ENGAGEMENT AND TRIGGERING

TCR engagement by the MHC-peptide complex and signalling is an essential process

for T cell activation and immune response (Davis and Bjorkman 1988). The TCR is a

membrane receptor composed of two covalentely bound peptide chains of the

immunoglobulin super family, α-β for the αβTCR and γ-δ for the γδTCR and a non-

covalentely associated CD3 complex. The CD3 complex is commonly composed of six

peptide chains: γ-ε and δ-ε heterodimers, plus either a ζ-ζ homodimer or a ζ-ε heterodimer.

The αβTCR ligand is the MHC class II or class I molecule with a bound agonist

oligopetide (Katz, Hamaoka et al. 1973; Rosenthal and Shevach 1973; Zinkernagel and

Doherty 1974; Davis and Bjorkman 1988; Bennink, Peeters et al. 2001; Siemasko and

Clark 2001), generated through intracellular cleavage of proteins ((Babbitt, Allen et al.

1985; Townsend, Gotch et al. 1985), reviewed in (Bennink, Peeters et al. 2001; Siemasko

and Clark 2001)), and presented at the surface of antigen presenting cells (APCs).

The TCR lacks intracellular signal transduction domains, but the CD3 chains contain

ITAMs (immuno-receptor tyrosine activation motifs) that are phosphorylated upon TCR

engagement by the agonist MHC-peptide ligand (Reth 1989; Letourneur and Klausner

1992; Wegener, Letourneur et al. 1992) or as a consequence of cross-linking with antibody

(for example anti-CD3 antibody). The CD3ζ chains have three ITAM motifs whereas

CD3γ, CD3δ and CD3ε have one ITAM motif each. TCR trigering is the process of ITAM

phosphorylation that leads to downstream TCR signalling events.

The initial sequence of events that lead to ITAM phosphorylation is still not resolved.

Src kinases are known to be involved early in TCR triggering, but it is unknown if src

kinase recruitment to the TCR complex is indeed the first event following TCR engagement

by agonist. The activity of lck, the main src kinase implied in TCR triggering, is regulated

by CD45, a PTP expressed in haematopoietic cells, and T cells deficient in CD45 have

impaired TCR signalling, which suggests that it may have role in the initiation of TCR

triggering (Stone, Conroy et al. 1997). A proposed mechanism for the initiation of TCR

signalling is that CD45 maintains lck in a basal state in a resting T cell, but upon

engagement of the T cell by APC, CD45 is excluded from the TCR complex allowing lck

activation and ITAM phosphorylation (Thomas 1999). Co-receptor molecules CD4 and

CD8 are known to associate with the TCR-MHC-peptide complex, increasing both the

stability of the complex (Luescher, Vivier et al. 1995) and the cellular response to ligand


12

(Madrenas, Chau et al. 1997). These co-receptors are associated with lck, and it is probably

the complex CD4-lck (or CD8-lck) that is recruited to the TCR-ligand complex and

initiates the triggering process (Chan, Desai et al. 1994; van Oers, Killeen et al. 1996).

CD45 PTP activity on lck seems to target mainly the fraction of lck that is CD4 associated

(Biffen, McMichael-Phillips et al. 1994; Dornan, Sebestyen et al. 2002).

Phosphorylated ITAM domains recruit the Zap70 tyrosine kinase, which is then

activated by lck by phosphorylation of its activator domains (Iwashima, Irving et al. 1994;

Chan, Dalton et al. 1995). Activated Zap70 phosphorylates downstream adapter molecules,

which in turn initiate the different signalling pathways (Mege, Di Bartolo et al. 1996; Qian,

Mollenauer et al. 1996) that lead to the activation of the various responses of T cells to

antigen presentation (Germain and Stefanova 1999; Acuto and Cantrell 2000).

There is some evidence that fyn, a src PTK also expressed in T cells, can

complement, and in some conditions partially replace, lck in TCR triggering (Groves,

Smiley et al. 1996; van Oers, Lowin-Kropf et al. 1996).

The interaction of the TCR with ligand has low affinity compared to the BCR-ligand

interaction (see review (Davis, Boniface et al. 1998)). Considering that T cells are highly

specific for the antigen and respond to very low amounts of ligand presentation (Harding

and Unanue 1990; Valitutti, Muller et al. 1995), this “low affinity” seems paradoxical. To

address this paradox two non-exclusive models were proposed: the kinetic proof reading

model and the serial triggering model.

The kinetic proof reading model (McKeithan 1995) was inspired by the proof reading

mechanism of DNA transcription and replication (Hopfield 1974). The model proposes that

a high level of specificity can be obtained by coupling various intermediate steps in the

signalling pathway, all dependent on the TCR-ligand complex remaining associated. Such a

mechanism is very sensitive to changes in the dissociation rate of the TCR-ligand. High

dissociation rates cause only partial activation, since the complex only remains associated

on average time enough to activate some of the intermediate steps. Low dissociation rates

allow the activation of all the activation steps, thus producing the full response. This model

can explain the sensitivity of TCR triggering to the kinetic properties of the TCR-ligand

binding and proposes a minimum dissociation rate for the TCR-ligand complex capable of

activating the TCR. The ideas behind the kinetic proofreading model have been extended to

downstream signalling events (Germain and Stefanova 1999) and recently the model was

extended also to account for intermediate responses, which are relatively independent on


13

the kinetics of ligand binding (Hlavacek, Redondo et al. 2001; Liu, Haleem-Smith et al.

2001).

The serial triggering model is based on the observation that a reduced number of

MHC-peptide complexes can down-regulate many TCRs (Valitutti, Muller et al. 1995).

This model proposes that the MHC-ligand complex serially binds, triggers and dissociates

TCRs on the surface of the T cell. This model predicts a strong dependency of the

extension of TCR triggering on the dissociation rates of the TCR-ligand complex (Valitutti

and Lanzavecchia 1997). If the dissociation rate is high, then the TCR-ligand complex does

not associate for time enough for the TCR to be fully triggered. If the TCR-ligand

dissociation rate is low, then the MHC-ligand complexes will be long lasting and the

efficiency of serial TCR engagement and triggering is reduced. There is an optimal

dissociation rate that leads to an optimal yield of triggered TCRs and signal strength.

Both these hypotheses, on the dependency of signalling on the kinetic properties of the

TCR-ligand complex, are substantiated by studies of agonists, partial agonists and

antagonists of TCR, which have established a negative correlation between the TCR-ligand

dissociation constant (koff) and the agonistic properties of the ligand (Sloan-Lancaster and

Allen 1996; Bachmann, Speiser et al. 1998), independent of the TCR ligand association

constant (kon). This correlation supports the hypothesis that partial agonists do not associate

with the TCR complex long enough to allow full TCR triggering, resulting in partial ITAM

phosphorylation and signal transduction (Itoh, Hemmer et al. 1999). TCR antagonists are

able to induce negative signals to the T cell (Kersh, Kersh et al. 1999; Robertson and

Evavold 1999) and/or can merely act as decoy ligands, competing for free non-triggered

TCR (Viola, Linkert et al. 1997).

The stoichiometry of TCR-ligand binding required for the initiation of signal

transduction is controversial. Either the TCR is triggered by a homo cross-linking

mechanism, meaning that dimerization, trimerization or higher order oligomerization of

TCR-ligand are required for triggering; or a single TCR is triggered by a single ligand,

possibly with the collaboration of co-receptors (hetero cross-linking)(Figure 1). The homo

cross-linking models of TCR triggering are supported by the activation of T cells via

αCD3-mediated TCR cross-linking; by structural studies indicating that both TCR and

MHC can form dimers (Brown, Jardetzky et al. 1993; Fields, Ober et al. 1995; Garboczi,

Ghosh et al. 1996); and by several studies of the kinetics of binding (Reich, Boniface et al.

1997; Boniface, Rabinowitz et al. 1998). In favour of a monovalent mechanism of TCR


14

triggering, both the TCR and its ligand have monovalent binding sites, which is difficult to

conciliate with a homo cross-linking mechanism of TCR triggering similar to BCR. In

addition, there is evidence that a CD8+ T cell can elicit a response by engagement of very

few TCRs (Sykulev, Joo et al. 1996; Delon, Gregoire et al. 1998). Also, at very low ligand

densities (about 100 MHC-peptides seem to be able to activate a T cell (Valitutti, Muller et

al. 1995)) the probability of homo-dimerisation of MHC-peptide would be very low, hence

it is questionable that homo-dimers of MHC-peptide can be formed at physiological antigen

densities (Salzmann and Bachmann 1998). Bachmann et al. (Bachmann, Salzmann et al.

1998; Bachmann and Ohashi 1999) proposed a kinetic model of TCR serial triggering that

suggests the requirement of formation of TCR dimers or trimers for TCR triggering. We

have extended and generalised this model and found that it represents a particular case of

TCR triggering kinetics for high presentation of ligand on the APC (Sousa and Carneiro

2000).

Signalling

A B

APC

T-Cell TCR TCR

CD3 CD3CD3CD3

MHC-peptide MHC-peptide

Lck

CD4

TCR

CD3CD3

MHC-peptide

Lck

CD4

TCR

CD3CD3

MHC-peptide

Figure 1 – Two possible mechanisms of TCR triggering by class II MHC-peptide (also

applicble to Class I). A – TCR triggering by cross-linking of TCR complexes (homo cross-

linking); B – TCR triggering via co-receptor cross-linking (hetero cross-linking).

An additional level of complexity in TCR triggering is added by the immunological

synapse (Monks, Freiberg et al. 1998; Grakoui, Bromley et al. 1999; Anton van der Merwe,

Davis et al. 2000; Bromley, Burack et al. 2001). When the conjugate between a T cell and

an APC is formed, a complex structural re-organisation of membrane components on both

the T cell and APC takes place, forming the immunological synapse. Early on, the adhesion

molecules are mainly located in the centre of the interface with the TCRs forming a ring

outside this central area. As time passes, the TCRs migrate to the centre, exchanging


15

position with the adhesion molecules. This later stage was described as a “mature synapse”

and occurs in the later stages of the synapse formation (Grakoui, Bromley et al. 1999). All

these events correlate with the agonist strength of the MHC-peptide presented and are

dependent of cytoskeleton. Recently, the contribution of the T cell synapse to TCR

triggering has been questioned by evidence that TCR signalling precedes synapse formation

and maturation (Lee, Holdorf et al. 2002). Moreover, mathematical modelling suggests that

the formation of the synapse is possible simply by passive diffusion of membrane proteins

(Qi, Groves et al. 2001).

Rafts are membrane microdomains enriched in sphingolipids and cholesterol with

intense packing of lipids that are insoluble in non-ionic detergents at low temperatures

(Brown and Rose 1992). Membrane rafts compartmentalise GPI-anchored proteins, some

integral membrane proteins and alipid-cylated, cytosolically oriented, proteins (Srk kinases,

Ras kinases and heterodimeric G proteins). In a resting cell, rafts are enriched in Src

kinases (lck and fyn) (Montixi, Langlet et al. 1998; Xavier, Brennan et al. 1998; Janes, Ley

et al. 1999) and LAT (Zhang, Trible et al. 1998; Janes, Ley et al. 1999). The TCR seems to

be associated with rafts but it is easily labile by detergents (Janes, Ley et al. 1999). Upon

TCR activation by a cross-linking agent, rafts are enriched in phosphorylated CD3ζ. The

coalescence of rafts is enough to invoke signalling events mimicking TCR αCD3 cross-

linking (Janes, Ley et al. 1999; Valensin, Paccani et al. 2002).

The contribution of rafts, the immune synapse and diffusion for TCR triggering

involves modelling the spatial component of TCR engagement and triggering (see for

example (Salzmann and Bachmann 1998; Chan, George et al. 2001; Favier, Burroughs et

al. 2001; Wofsy, Coombs et al. 2001)), which was beyond the scope of this thesis, but

constitutes one of the possible extensions to the models presented here.

T cell activation is sensitive, among other factors, to the levels of TCR expressed in

the T cell (Viola and Lanzavecchia 1996). One of the responses of TCR engagement and

triggering is the down-modulation of the TCR by internalisation (Valitutti, Dessing et al.

1995). The internalised TCR is targeted to the lysosomal compartments for destruction

(Lauritsen, Christensen et al. 1998). TCR internalisation is dependent, among other factors,

on TCR engagement and therefore might depend on early events involved in TCR

triggering. Internalisation of the TCR and targeting to the lysossomes seems to be

dependent on Src kinases (D'Oro, Vacchio et al. 1997; Lauritsen, Christensen et al. 1998),

although there is some controversy regarding this observation (Cai, Kishimoto et al. 1997;


16

Salio, Valitutti et al. 1997). Another pathway of TCR internalisation is, at least to some

degree, PKC-dependent and leads the TCR to an internal pool where it can recycle back to

the membrane (Lauritsen, Christensen et al. 1998; Dietrich, Menne et al. 2002). The

recycling of TCR from the cytoplasmatic pool to the membrane is mediated by PKC

phosphorylation of specific residues in a leucine-based internalisation motif in the CD3γ

chain (Dietrich, Hou et al. 1994) and is important for cytokine-mediated fine-tuning of

TCR expression (Dietrich, Hou et al. 1994; Weiss and Littman 1994; Lauritsen,

Christensen et al. 1998). PKC-mediated TCR recycling does not seem to interfere with

ligand-mediated internalisation and degradation of TCRs (Salio, Valitutti et al. 1997;

Lauritsen, Christensen et al. 1998; Dietrich, Menne et al. 2002).

TCR down-modulation was used to study the kinetics of TCR engagement and T cell

responsiveness to very low ligand presentations (Valitutti, Muller et al. 1995). In this thesis,

these kinetic studies were used to gain additional insight into the mechanism of TCR

triggering and down-modulation with the aid of mathematical models. The objective is to

identify the properties of TCR triggering that will be relevant for lymphocyte physiology

and population dynamics. The models presented here for TCR triggering and down-

modulation, capture the main properties of TCR triggering and should be considered as the

first step stones towards more general and complete models of TCR engagement and

triggering.

1.3.2 CYTOKINE RECEPTOR ENGAGEMENT

If TCR engagement is the hallmark of T cell activation stimuli, the ligation of cytokine

receptors is key to the regulation of lymphocyte differentiation, proliferation and death.

Cytokines are soluble growth and differentiation factors acting in the cells of the immune

system (Kishimoto, Taga et al. 1994; Paul and Seder 1994; Schindler and Darnell 1995;

Decker and Meinke 1997).

The expression of the various cytokine receptors varies considerably during the life

span of the cells (Swain, Bradley et al. 1991; Kishimoto, Taga et al. 1994; Paul and Seder

1994; Schindler and Darnell 1995; Decker and Meinke 1997; Di Santo, Radtke et al. 2000;

Murphy, Ouyang et al. 2000). Changes in the quantity and type of cytokine receptors

usually reflect changes in cell physiology resulting from differentiation and or activation.

Cytokine receptors are multimeric transmembrane proteins compromising either two

or three receptor chains, which can be divided in two major classes (Schindler and Darnell


17

1995): Class I cytokine receptors includes the receptors for IL-2, IL-3, IL-4, IL-5, IL-6, IL-

7, IL-9, IL-11, IL-12, IL-15, Epo, PRL, GH, G-CSF, GM-CSF, LIF, CNTF. These

receptors share a cytosine-binding domain with a conserved WSXWS motif. Class II

cytokine receptors include the receptors for IL-10, IFN-α and IFN-β and, compared with

class I cytokine receptors; they are a more heterogeneous group from the structural point of

view. These receptor classes can be further divided into families according to structural

homology and to the receptor chains these receptors have in common.

Cytokine receptor activation requires cytokine-mediated engagement of receptor

chains and the consequent cross-linking and activation of protein tyrosine kinases. The

paradigm for cytokine receptor activation and signal transduction by the Jak-STAT

pathway was established for the IFN system (Darnell, Kerr et al. 1994; Schindler and

Darnell 1995). Jak-STAT signalling is initiated by the ligand-mediated cross-linking of the

cytokine receptor chains. Cross-linking of the receptor chains bearing intracellular Jak

kinase binding and phosphorylation domains is a sufficient condition for the onset of

receptor activation (Shuai, Ziemiecki et al. 1993; Lutticken, Wegenka et al. 1994; Stahl,

Boulton et al. 1994). More domains are present at the cytoplasmatic portions of the

receptors, which are used to initiate other signalling pathways (for example IL-2

(Sugamura, Asao et al. 1996; Gesbert, Delespine-Carmagnat et al. 1998) and IL-4 (Nelms,

Keegan et al. 1999)).

All cytokines rely on four Jak kinases and less than a dozen STAT proteins to regulate

gene transcription (Paul and Seder 1994; Schindler and Darnell 1995; Decker and Meinke

1997; Kisseleva, Bhattacharya et al. 2002). The specificity of the pathway results in the

combination of different signalling intermediates, on temporal expression of subsets of

cytokine receptors and on the kinetics of nuclear translocation. With such degree of

overlapping of signalling intermediates and receptor chains, it is not surprising that

cytokines have a high degree of functional redundancy, especially between members of the

same receptor family.

Some of the open questions in cytokine signalling are to what degree is a given

cytokine essential to a cell? Can it be compensated by other cytokines? How cytokine

signal pathways, which share signalling components, interact?

The answers to these questions might be achievable using mathematical models when

the pathways of cytokine signalling have been better described. At present there is only a

description of which STAT proteins are activated following receptor activation. Little is


18

known about the preferred stoichiometry of their dimerisation and the priority of nuclear

translocation (Zhang, Blenis et al. 1995). There is only incomplete data about the gene

products of the CIS family of genes that are activated early on cytokine-activated gene

expression and whose products inhibit cytokine signalling (Endo, Masuhara et al. 1997;

Naka, Narazaki et al. 1997; Starr, Willson et al. 1997). These gene products depend on

early events of STAT-activated gene expression and constitute a negative feedback loop on

cytokine signalling (Matsumoto, Masuhara et al. 1997). Cytokine receptor trafficking in the

cell membrane, in the presence or absence of ligand, is still not fully understood. There is

evidence that cytokine receptors are down regulated and degraded following activation by

cytokine engagement, but the quantitative data on receptor chain turnover is difficult to

analyse since it usually relies in radioactive markers, which lump several cellular

compartments in the same kinetics (for example (Duprez and Dautry-Varsat 1986;

Weissman, Harford et al. 1986; Gullberg 1987; Duprez, Ferrer et al. 1992; Ferrer, Hemar et

al. 1993; Hemar, Subtil et al. 1995; Subtil, Delepierre et al. 1997; Morelon and Dautry-

Varsat 1998; Friedrich, Kammer et al. 1999)).

In this thesis we approach the question of interference between cytokine receptors

sharing common signal transduction chains. To acomplish this, we study a prototypic

model of the interference between the IL-2 and IL-4 receptors (IL-2R and IL-4R) mediated

by the common gamma chain (γc), which is a signal transducing receptor chain shared

between the members of the common gamma chain cytokine receptor family. Using a

simple model, we also infer the possible implications of such interference at the level of

lymphocyte population dynamics. In this study, we explore a prototypic system of

differentiation of ThP to Th2, two T cell phenotypes with distinct cytokine receptor

expression profiles, as an example system of cytokine receptor interference via common

receptor chains.

1.4 The immune system ranges multiple organisation levels

With the clonal selection theory and the discovery of process of somatic rearrangement

of immunoglobulin genes (Brack, Hirama et al. 1978), the focus of interest in Immunology

shifted from antibody chemistry and serology to the molecular biology and biochemistry of

the antibody producing cells and related cells. Some of the emergent questions were: What

are the necessary and the sufficient conditions for lymphocyte activation that will lead to an

immune response? What are the factors and processes regulating this response? What is the


19

relationship and orchestration between cell biochemistry and immunity? The application of

molecular biology in Immunology has been fruitful in addressing, at least partially, these

questions by the mapping of the cell receptors and signalling pathway and by the study of

gene expression of the cells of the immune system.

The most severe limitation of the molecular approach to the study of Immunology is

the necessity of reductionism, with inherent simplifications such as assuming some degree

of independence of cells and clones, the assumption of constant conditions and the neglect

of indirectly pertinent factors.

Immunity is a phenomenon that bridges several levels of biological organisation, from

the basic molecular aspects, spawning beyond single cells to cell population dynamics, to

organ interactions and to organisms and their intractions. Four levels of organisation are

apparent: cellular and sub-cellular, cell population, organism and population of organisms.

The cellular and sub-cellular organisation level comprehends the single cell events

such as receptor activation, regulation of gene expression, signal transduction and cellular

differentiation. These events can be divided into two categories: fast and slow events. Fast

events are the signal transduction events that usually imply covalent and/or conformational

modification of proteins, which propagate throughout the signalling networks of cells.

These events occur generally in less than hours or days. The slow events comprehend the

de novo expression of proteins, which can either reinforce an already existing pool of

proteins or create a new pool, which can lead to a quantitative and/or qualitative change in

the cellular signal transduction network. These slow events compromise the expression of

new molecules and the differentiation of the cell and require, in general, more than hours or

days to take effect. Fast events usually produce functional quantitative modifications in the

cell, whilst slow events may result in functional qualitative modifications, which when

irreversible denote differentiation. Separating the fast and slow processes is, to some extent,

an artificial separation since these processes are inter-dependent.

The cell population events deal with the time evolution of populations of cells and the

interactions between cells, of the same or different type. Processes falling into this category

are the immune response, the regulation of the class of immune response the regulation of

lymphocyte differentiation, the germinal centre reaction, immune suppression and thymic

selection. Considering that each cell can be in a particular, almost unique, state defined by

the particular conditions it has encountered during it's life span, it might be misleading to

simply extrapolate the properties of a single cell, as enumerated by molecular biology, to


20

the properties of a population of cells. At this level, the study of the distributions of cells

and the interactions between cells is fundamental. Changes at the cell population level may

or may not reflect events occurring in the time scale of fast and slow cellular processes.

The immune system is a group of organs functionally distinct that are integrated in the

organism. Cell differentiation, selection and death, all take place at distinct specialised

locations in the body. Hence cell population dynamics becomes a problem with spatial-

oriented components. Processes like lymphocyte development and infiltration have an

important spatial component (Rossi and Zlotnik 2000). Moreover, depending on the

location of a challenge, the immune system responds differently to the same antigen

(Carvalho and Vaz 1996; Melo, Goldschmidt et al. 1996; Vaz, Faria et al. 1997). Another

problem within this organisational scope is the homeostasis of the cells of the immune

system. Very little is known about the mechanisms that regulate some pools of

lymphocytes independently and others inter-dependently. Some competition for spatial

and/or antigenic niches (Carneiro, Stewart et al. 1995; De Boer and Perelson 1997; Freitas

and Rocha 1997; Freitas and Rocha 2000) seems to be implied, but the problem is

somewhat more complex, involving the interaction between the primary and secondary

lymphoid organs. Another problem is the generation and maintenance of memory. The

generation of memory cells might be a single cell or a population level problem, but the

maintenance of memory cells should share regulatory components with the homeostasis of

the whole lymphocyte population.

At the organism level, the interactions with the environment and with other organisms

also have have consequences for immunity. Epidimiology studies the incidence of diseases

at the population level, using echology, sociology and population dynamics.

Due to the power of the techniques of molecular biology and their success in

Immunology, most of the attention of immunologists is directed towards sub-cellular and

cellular processes, which provides a limited, prominently reductionist view of immunity.

Moreover, this view is often biased by the choice of conditions that are not entirely

representative of the natural state of the organism, leading to a distorted notion of the

relevance of the object of study. Hence, simple extrapolation of molecular single cell events

to population dynamics and immunity is not trivial; the conceptual gaps between

organisational levels and the difficulties in linking and assessing the strength of their

interactions may lead to paradoxical conclusions. Perhaps the most illustrative cases are

some unexpected results from gene knockouts in mice. For example, IL-2 is an important


21

growth factor for lymphocytes, but IL-2 deficient animals still have lymphocyte

development and, what is striking, develop fatal immune pathology disease (Kneitz,

Herrmann et al. 1995). IL-2R-beta deficient produces a phenotype characterised by massive

lymphocyte activation, leading to death at 12 weeks of age (Suzuki, Kundig et al. 1995).

Another example is that the IL-4 and the Th2 bias in Balb/c mice does not seem to

influence the susceptibility to Leishmania major infection in those mice, as shown by IL-4

deficient Balb/c mice (Noben-Trauth, Kropf et al. 1996).

1.5 This thesis.

In this thesis, some of the interfaces separating different organisation levels in the

immune system are explored using mathematical modelling. We concentrate in what are the

main processes that determine the life history of a lymphocyte, namely cytokine and TCR

signalling. We aim to model these signalling events at a single cell level and extend these

models to the population level. The TCR is the hallmark of T cell activation (Davis and

Bjorkman 1988). It is essential for T cell development, survival, proliferation,

differentiation and the onset of effector functions (Davis and Bjorkman 1988; Germain and

Stefanova 1999). Cytokines not only influence T cell activation but also play an essential

role regulating the development, survival, proliferation and differentiation of the cells. For

simplicity, we restrict to CD4+ T cells.

This thesis is divided in four major parts:

1. Sharing cytokine receptor chains and their implications;

2. Mathematical analysis of TCR triggering;

3. Adaptable activation thresholds and lymphocyte homeostasis.

4. General discussion.

1.5.1 SHARING CYTOKINE RECEPTOR CHAINS AND THEIR IMPLICATIONS

In this part of the thesis we asked what are the functional consequences of cytokines

sharing a common receptor chain. To address this question, we used the prototypic system

of IL-2 and IL-4 receptor engagement and signalling. These two cytokines of the γc family

have been the target of intense study and although the available experimental data is still

missing some kinetic and quantitative aspects, it is nevertheless sufficient to construct and

study a model of cytokine receptor engagement that has biological relevance.

This model is divided in two parts: The first part deals with cytokine receptor binding


22

and the impact of sharing cytokine receptor chains in cytokine signalling. Our modelling

reached two conclusions: that at physiological or near physiological membrane densities of

cytokine receptor chains, γc is limiting for IL-2 and IL-4 signalling, and that the γc-

mediated interference between IL-2R and IL-4R is asymmetrical. IL-4R signalling is much

more sensitive to IL-2R engagement than IL-2R signalling is to IL-4R engagement.

The second part of the model addresses the possible impact of IL-2R and IL-4R

sharing γc on the dynamics of ThP differentiation to Th2. ThP depends on IL-2 for

proliferation and on IL-4 for differentiation into Th2. Our model shows that the apparent

non-competitive inhibition of IL-4 signalling by IL-2, together with the autocrine loop of

IL-2 production by ThP can explain the observed synergistic effect of IL-2 and IL-4 in Th2

differentiation. Moreover, this model illustrates how the balance between proliferation and

differentiation via autocrine loops can prevent positive feedback loops of differentiation

from depleting precursor cells.

1.5.2 MATHEMATICAL ANALYSIS OF TCR TRIGGERING.

In this part we try to answer what are the key mechanistic features of TCR engagement

and triggering by agonist MHC-peptide. This part is divided in four sections: The first

section builds a phenomenological model of TCR engagement, triggering and

internalisation by MHC-peptide using the available kinetic data on TCR down-modulation.

This model allowed us to identify the following key kinetic properties of TCR engagement,

triggering and internalisation: 1) The importance of basal TCR turnover on the membrane;

2) The existence of accumulated triggered TCRs in the surface of the T cell; 3) The

requirement for hypersensitivity in TCR triggering, which may reflect a complex reaction

process; and 4) The existence of two pools of TCR to explain the biphasic properties of the

kinetics of TCR down-modulation.

The second section takes from the phenomenological model and compares two TCR

triggering mechanisms for their ability to reproduce the high cooperativity of TCR

triggering. The two mechanisms are a generic TCR homo cross-linking model and a TCR-

CD4 cross-linking model inspired on the hypersensitive Koshland-Goldbeter mechanism.

The results of this comparison show that TCR homo cross-linking mechanisms are better at

high doses of ligand than at low doses of ligand, whereas the CD4-TCR cross-linking is

very efficient at low doses of ligand. We hypothesise that both mechanisms could be

possible at high doses of ligand, but that at low doses of ligand CD4-TCR cross-linking


23

should be the prominent triggering mechanism.

The third part of this section discusses the adaptation of activation thresholds of TCR

triggering and down modulation via the regulation of TCR membrane density. This part

demonstrates that TCR down modulation give the T cell an adaptable activation threshold.

The fourth part discusses the experimental setting being started to assess the

stoichiometry of TCR triggering based on the models we developed. The experimental part

is still being developed as a separate project, hence this section relates to a work in

progress.

1.5.3 ADAPTABLE ACTIVATION THRESHOLDS AND LYMPHOCYTE HOMEOSTASIS.

Tuning of activation thresholds in lymphocytes has been implied in several processes,

namely the induction of anergy and tolerance and the selection of lymphocytes (Grossman

and Paul 1992; Grossman and Paul 2000; Grossman and Paul 2001). The regulation of TCR

expression and down-modulation in T cells displays adaptation properties ((Viola and

Lanzavecchia 1996) and this thesis). Adaptation of activator signaling pathways can be

important for lymphocyte function and proliferation; hence it can have important

implications at the cell population level. In this part we asked what is the contribution of

adaptation of the TCR signal transduction pathway to the population dynamics of

lymphocytes.

We build a generic model of a adaptable lymphocyte that reproduces the qualitative

properties outlined by Grossman and colleagues, based in a minimalist TCR signal

transduction pathway. We then build a population dynamics model of adaptable

lymphocytes, which we discuss in the light of lymphocyte homeostasis in the periphery.

The results of the model suggest that AAT of lymphocytes provide a mechanism of

peripheral tolerance based in the deletion of self-reactive adapted cells. Only when the cell

numbers are brought up, possibly by external factors, are these cells able to persist in the

periphery. In the absence of those factors, the persistence of self-reactive cells is only

possible if these cells are not adapted or if there is a constant source of non-adapted cells to

the periphery.

1.5.4 GENERAL DISCUSSION

In this final part of the thesis, the results and discussions of the previous sections are

revisited and, when appropriate, reexamined under the light of new findings.


24

Modeling the integration between cytokine and TCR signaling pathways at the cellular

and cell population level is an ambitious goal, beyond the temporal scope of this thesis.

Starting with the work here present, this goal is now closer to be attained; hence in this final

part of the thesis, we discuss some of the possible paths and problems towards such an

integrative model.

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Implications of sharing cytokine receptor chains

37

2 Implications of sharing cytokine receptor chains 2

Implications of sharing cytokine

receptor chains


38


39

On the role of the common gamma chain

in ensuring functional pluripotency of peripheral Th lymphocyte clones

João SOUSA1,2 & Jorge CARNEIRO1

1. Instituto Gulbenkian de Ciência, Oeiras, Portugal

2. Grupo de Bioquímica e Biologia Teóricas, Instituto de Investigação Científica Bento da Rocha Cabral, Lisboa, Portugal.

Abstract

We raise the problem of the persistence of uncommitted precursor Th cells in peripheral lymphoid

organs and its relation with the capacity of the immune system to mount different classes of responses. We

argue that the pattern of “endpoint activation” of which IL-4 driven commitment to Th2 lineage is an

example, may lead to the full commitment of lymphocyte clones to this lineage. Using mathematical models,

we address the interference between IL-2 and IL-4 cytokine receptors by sharing γc and demonstrate that

under physiological conditions IL-2 may act as an inhibitor of IL-4 signalling. We explore the consequences

of this inhibition of IL-4 responsiveness in the context of Th cell differentiation and lineage commitment. We

demonstrate that if uncommitted Th precursors cells are proliferating and secreting IL-2 then they can resist

IL-4-driven differentiation and, in spite the pressure imposed by already committed Th2 cells, the pool of

precursor ThP cells persists. The model predicts that modulation of γc may affect the ability to sustain

uncommitted precursors, suggesting novel ways of manipulating the immune system.

Acknowledgements

We are grateful to Paulo Vieira, Jocelyne Demengeot, John Stewart, Vanessa Oliveira and Iris Caramalho

for helpful discussions while doing this work and to Thiago Carvalho for reading the manuscript. This work is

supported by Program Praxis XXI of the Ministério para Ciência e Tecnologia, Portugal (grant

Praxis/P/BIA/10094/1998). JS and JC were respectively supported by fellowships BD/13546/97 and

BPD/11789/97 of Fundação para a Ciência e Tecnologia - Program Praxis XXI.


40


41

2.1 Introduction

Hematopoietic development is a lifelong process in which uncommitted precursors go

through a series of lineage commitment events as they progress through branching

differentiation pathways. Progression through the branching points is tightly regulated,

since the pool of precursors must be replenished for the system to remain pluripotent. The

balance between differentiation, proliferation and survival of precursors is determined by

cytokines acting in autocrine, paracrine or juxtacrine modes. Animals deficient in one or

several of these cytokines or their receptors display significant ablation of lymphoid or

myeloid lineages (reviewed in (1)), which emphasises the importance of population

dynamics in hematopoiesis.

The discovery that peripheral Th lymphocytes can still undergo further lineage

commitment (1-4) raises the problem of what are the mechanisms ensuring the persistence

of their clonal uncommitted precursors. This is particularly relevant considering that loss of

uncommitted precursors and commitment of an entire clone to a single lineage may be

achieved, compromising the capacity of the immune system to mount qualitatively different

effector functions. The simpler explanation for the maintenance of uncommitted Th

precursor pools is their continuous replenishment by thymic output (5), much in the same

way that thymic lymphopoiesis relies on extrathymic stem cells with self-renewal capacity.

The plausibility of this explanation is minimised, however, by several observations:

Thymic output decreases in adult life (6) and peripheral Th populations can persist in the

absence of a thymus without major disturbances of repertoire diversity or capacity to mount

different classes of responses. Moreover, the actual exhaustion of lymphocyte clones has

been demonstrated for CD8+ T cells (7), and the same is expected for Th cells.

Alternatively, uncommitted Th precursors may be regarded as “stem cells” that persist and

compensate the loss due to differentiation and lineage commitment by “homeostatic

proliferation” (8). The persistence of these precursors becomes then a question of how is

the mechanism underlying homeostasis organised.

Differentiation of precursors of non-lymphoid hematopoietic cells seems to be

regulated by negative feedback loops (9). A complex mechanism of feedback inhibition

seems to be also involved in granulopoiesis and thrombopoiesis (9), and these regulatory

loops ensure a stable persistence of precursors. Feedback mechanisms have also been


42

suggested in the regulation of thymocyte populations (10). In the periphery, the

differentiation of uncommitted Th precursors cells seems to involve some kind of end-point

activation, defining a positive feedback loop in which committed cells produce factors that

promote the differentiation of precursor cells to the same lineage (11,12). Gamma

interferon promotes indirectly the differentiation of precursors to the Th1 lineage by

modulating IL-12 expression by the antigen presenting cells (APCs) (13) and enhancing IL-

12R expression (14). But the most striking example of end-point activation is provided by

IL-4, the hallmark of Th2 differentiated cells (15), which is a major driver of Th2 lineage

commitment (16). Minor populations of IL-4 producing cells are capable of driving the

differentiation of Th cell precursors to the Th2 phenotype in vitro (17) and in vivo (18,19)

Since the most straightforward expectation of endpoint activation is the exhaustion of

precursors, this pattern of regulation is desteleological considering the requirements for

diversity in the immune system in terms of both specificity and class of effector function.

In this article we propose and discuss one candidate mechanism namely the

interference, at the single cell level, between the signalling cascades triggered by different

cytokines of the common gamma chain (γc) family. The receptors for IL-2, IL-4, IL-7, IL-9

and IL-15 share a chain, γc, which is strictly required for signalling (20). While γc is

constitutively expressed throughout all the stages of T lymphocyte development (20-22),

including the late peripheral stages (23), the expression pattern of the other chains changes

as lymphocytes proceed in their differentiation pathway (21,22). In broad terms, these

cytokines are proliferation and survival factors, and at least IL-4 and IL-15 are

differentiation factors acting respectively on uncommitted Th precursors (24,25) and on NK

cell precursors (26). Therefore it is reasonable to anticipate that different patterns of

expression of receptor chains for these cytokines may govern the balance between

proliferation and differentiation of lymphocytes precursors in the central hematopoietic and

peripheral organs.

Using mathematical modelling, we show that at the relative membrane concentrations

of the cytokine receptor chains, IL-2 signalling may inhibit IL-4 responsiveness in

uncommitted Th precursor, because γc is limiting as compared to IL-2R α and β chains.

Furthermore, we show that this inhibition can be sufficient to allow the persistence of

uncommitted Th precursors, preventing the full commitment of a clone to the Th2 lineage,

and ensuring clonal pluripotency.


43

2.2 Modelling and Results

2.2.1 COUPLING BETWEEN IL-2 AND IL-4 RECEPTOR SIGNALLING VIA THE COMMON

GAMMA CHAIN

Similarly to Whitty et al (27), we study the coupling between IL-2R and IL-4R using a

steady state model (Fig 1). The aggregation of the various receptor chains for the IL-4

receptor was taken from the model they introduced with the following simplifying

postulates:

IL- 4IL- 4

IL- 2

IL- 4

αIL- 2 IL- 2

γ

γ

γ

β+

(i) (ii)

(iii) (iv)

(v)

α α αβ β β

αα α

IL-2R

IL-4R

Figure 1 - Model of IL-2 and IL-4 receptor association. IL-2 binds to the IL-2Rαβ heterodimer (i) forming

a complex that recruits the common γ chain (ii), forming the high-affinity IL-2 receptor. Similarly, IL-4

binds to the IL-4Rα chain (iii) forming a complex that recruits the common γ chain (iv) and forms the

high-affinity IL4 receptor. The IL-2Rα and IL-2Rβ chains are assumed to be pre-associated in the

membrane (v). See text for a detailed explanation.

Postulate i – That the binding of cytokine to the receptor chains is sequential (27,28).

The affinity of IL-4 for IL-4Rα is higher than for γc, so IL-4 will bind preferentially to IL-

4Rα and the resulting complex will then recruit γc, forming the full receptor complex

(27,28). Under this assumption, the equilibrium equations for IL-4 binding are:

4

444-ILKC

LAIL

×= , 1

KγIL-4 =

C4 ×γR4

, 2

where the variables γ, A4, L4, C4 and R4 are respectively the concentrations of free γc, IL-

4Rα, IL-4, IL-4-IL-4Rα complex, and IL-4–IL-4Rαγ complex. The constants K ILIL-4 and

KγIL-4 are the IL-4 and γc dissociation constants from IL-4R.


44

Unlike the IL-4R, the IL-2R has three receptor chains, which makes modelling IL-2R

binding more complex. Nevertheless, the problem of IL-2R binding can be simplified by

considering the following postulates:

Postulate ii – IL2-Rα chain is in excess relative to IL2-Rβ and γc. This postulate holds

for cells that have up-regulated the levels of IL-2Rα, such as activated cells (23,29);

Postulate iii – most of the IL-2Rβ is complexed with IL-2Rα. The postulate is

supported by the finding that IL-2Rβ can bind to IL-2Rα in the absence of IL-2 (30). The

postulate holds if the affinity of IL-2Rβ and IL-2Rα binding is sufficiently high and/or IL-

2Rα is in excess relative to IL-2Rβ, which is the case for activated T cells (31,32);

Postulate iv – IL-2 binds to IL2-Rαβ rather than to IL2-Rα or IL2-Rβ. This postulate

builds on postulates ii and iii and is supported by the finding that IL-2Rαβ heterodimer has

a higher affinity for IL-2 than IL-2Rα or IL-2Rβ alone (30,33). Postulates ii and iii have

the additional consequence of allowing us to neglect the contribution of lower and

intermediate affinity receptors of IL-2. Under these simplifying postulates, IL-2 binding is a

problem formally equivalent to IL-4 binding:

K ILIL-2 =

A2 × L2

C2

, 3

KγIL-2 =

C2 ×γR2

, 4

where the variables A2, L2, C2 and R2 are the concentrations of IL-2Rαβ, IL-2, IL-2–

IL-2Rαβ complex, and IL-2–IL-2Rαβγ complex, respectively. The constants K ILIL-2 and

KγIL-2 are the IL-2 and γc dissociation constants from IL-2R. Note that according to these

equations, γc binding to IL-2R increases the stability of the receptor-ligand complex. This

increase in the stability of the receptor-ligand complex is also present in IL-4R binding and

depends on the γc dissociation constantsK ILIL-2 and K IL

IL-4 .

The model is composed of equations 1-4 together with the following conservation

relationships for total IL-4Rα (αTIL-4 ), total IL-2Rαβ (αT

IL-2 ) and total γc (γT):

4444IL

T RCA ++=−α 5

αTIL-2 = A2 + C2 + R2 6

γ T = γ + R2 + R4 . 7

For the sake of simplicity, we are neglecting the influence of other cytokine receptors

in the numbers of free γc.


45

Table 1 – Parameters used in modelling IL2 and IL4 binding to receptors

Parameter Value Notes

K ILIL-2 120 pM (33)

KγIL-2 171 molecules/cell See derivation in text

K ILIL-4 850 pM (34)

KγIL-4 413 molecules/cell See derivation in text

K appIL-2 10 pM (33)

K appIL-4 130 pM (34)

αβTIL-2 2500 molecules/cell See derivation in text

αTIL-4 1200 molecules/cell (34)

γT 2800 molecules/cell (34)

The parameters of the model are based on the literature (Table 1), which originate

from in vitro studies using cell lines. There is considerable variability regarding the

expression of cytokine receptor chains between different cell lines, which together with the

scarcity of quantitative assessments, lead to some uncertainty regarding the generality of

the parameters listed in table 1. Notwithstanding these limitations, since the conclusions we

draw from the model are mostly qualitative and dependent on the relative numbers of

receptor chains, we use these reference parameters to illustrate the model properties.

Kondo et al (34) reports that at saturating IL-2 concentrations, there is little free γc,

which implies that the number of IL-2–IL-2Rαβ complexes is high enough to sequester

most of the free γc in the cell. Since there are available reports indicate that the level of

expression of IL-2Rβ is similar or higher than γc (31,32,35) we will assume, under

postulates ii and iii, that the number of IL-2Rαβ complexes pre-formed in the absence of

IL-2 is enough to sequester about 90% of the free γc at saturating IL-2 concentrations, thus

we assume 2500 IL-2Rαβ complexes per cell.

The dissociation constants KγIL-4 and Kγ

IL-2 were estimated by extending the model of

Whitty et al (27) to the IL-2R under postulates i-iv. IL-2R binding, the parameter KγIL-2 was

estimated from equations 3, 4 and 6, together with the conservation relation γ T = γ + R4 ,

assuming that the binding of IL-2 to IL-2R produces an approximately linear Schatchard

plot. Binding experiments support this assumption for both IL-4 (28,34) and IL-2 (36). The


46

apparent affinities of IL-2R and IL-4R, K appIL-2 and K app

IL-4 , were taken from Schatchard plots

of IL-2 (36) and IL-4 (34) binding experiments, respectively. The relationship between

KγIL-2 and the apparent affinity of IL-2R for IL-2 (K app

IL-2 ) is

( )( )

( ) ( )( )1K1KKKK1KK1K

K 2-ILapp

2-ILapp

2-ILIL2

2ILT

2-ILapp

2-ILIL

2-ILapp

2-ILIL

2-ILapp2-IL

++−

−+−= −

TL γαγ 8

The parameter KγIL-4 is estimated from K app

IL-4 using the same procedure.

We will assume (Postulate v) that signalling is proportional to the concentration of

fully assembled cytokine-receptor complexes (R2 and R4). The units of the model are such

that the maximal value of both IL-2 and IL-4 signals (i.e. the limit at infinite concentrations

of IL-2 and IL-4) is one.

2.2.2 IL-2 AND IL-4 ARE POTENTIALLY NON-COMPETITIVE INHIBITORS OF EACH OTHER

BUT THE EFFECT OF IL-2 IS QUANTITATIVELY MORE IMPORTANT

The results of the numerical simulations of the model, using the parameters in table 1,

are summarised in figure 2.

Comparing figures 2-A and 2-B, it is evident that although IL-2 and IL-4 can

potentially inhibit each other, IL-2 can inhibit IL-4 up to approximately 40% (figures 2-A

and C) while IL-4 can only marginally affect IL-2 signalling (figure 2-B). The difference

between IL-2 and IL-4 is due to two factors: the number of IL-2Rαβ complexes is higher

than the IL-4Rα chain and is also in excess compared to the γc; and the dissociation

constants K ILIL-2 and Kγ

IL-2 are lower than the constants K ILIL-4 and Kγ

IL-4 , respectively. Under

these conditions, and provided that enough IL-2 is present, the formation of IL-2 receptor

can sequester up to 40% of the free γc that is required for the formation of the IL-4R and

IL-4 signalling. The extent of inhibition of IL-4 signalling by IL-2 can be modulated by the

relative expression of the γc; sequestering of γc by IL-2 can be alleviated increasing the

levels of γc (figure 2-C) expression or by reducing IL-2Rαβ (not shown).


47

100 101 102 103 1040

0.2

0.4

0.6

0.8

1

100 101 102 103 1040

0.2

0.4

0.6

0.8

1

[IL-2]/pM

[IL-2]/pM

IL-4

sign

alIL

-2 si

gnal

100 101 102 103 1040

0.2

0.4

0.6

0.8

1

10 1 100 101 1020

0.2

0.4

0.6

0.8

1

[IL-4]/pM

γc

IL-4

sign

alIL

-4 si

gnal

( i)( ii)( iii)( iv )A B

DC

( i)

( ii)( iii)( iv )

Figure 2 - Results of the model of IL-2 and IL-4 receptor association. The IL-2 and IL-4 signals are relative

to the IL-2 and IL-4 signals at saturating cytokine concentration. A – IL-2 signal as a function of IL-2. The

curves correspond to various concentrations of IL-4 (i: 0, ii: 50, iii: 500, iv: 10000 pM). B – IL-4 signal as

a function of IL-4. The curves correspond to various concentrations of IL-2 (i: 0, ii: 50, iii: 500, iv: 10000

pM). C – IL4 signal at a saturating IL4 concentration as a function of IL-2 concentration. D – IL-4 signal as

a function of γc at saturating concentrations of both IL-2 and IL-4. The parameters are listed in table 1

2.2.3 DIFFERENTIATION OF THP CELLS TO COMMITTED TH2 CELLS — A POPULATION

DYNAMICS MODEL

We ask whether IL-2 inhibition of IL-4 driven differentiation can have a detectable

impact on the balance between self-renewal of uncommitted Th precursors and their

differentiation to the Th2 lineage. To address this question quantitatively, we propose the

minimal model of Th precursors and Th2 cells (illustrated in fig. 3) that involves the

following postulates:


48

ThP Th2

IL-2 IL-4

( ii)( iv)

( i)

(v)( iii)

( v ii)( v i)

De a t h De a t h... Figure 3 - Model of ThP and Th2 proliferation and differentiation. ThP differentiation to Th2 (i) is driven

by IL-4 (ii), which is secreted by Th2 cells, and is inhibited by IL-2 (iii), which is secreted by ThP cells.

Both ThP and Th2 cell proliferation is depends on autocrine/paracrine factors secreted by ThP and Th2

cells respectively (iv and v). Both ThP and Th2 cells die with a constant per-capita rate (vi and vii). See

text for details.

Postulate vi- A clone of peripheral Th cells is composed of a subpopulation of

uncommitted precursors (ThP) and a subpopulation of Th2-committed cells. For simplicity

we are ignoring cells committed to other ThP-derived lineages such as the Th1 cell lineage.

The numbers of ThP and Th2 cells are denoted by Tp and T2, respectively.

Postulate vii- ThP and Th2 subpopulations decay with a constant death rate (37),

which we will assume is identical for the two subpopulations. The number of ThP and Th2

cells that die per unit of time is then given by δ × Tp and δ × T2 respectively, with δ the per

cell death rate constant.

Postulate viii- The death of ThP and Th2 cells is compensated by interactions with

MHC that result in activation and proliferation. We assume that the per-cell proliferation

rate is limited by a carrying capacity (τ) which is independent for both cell types (38). Also,

the proliferation rate per cell of either ThP (ΠP) and Th2 (Π2) is enhanced by

autocrine/paracrine cytokines produced by ThP and Th2 respectively, whihc results in a

positive feedback loop. The per cell proliferation rate for the ThP subpopulation (for

example), growing under autocrine/paracrine stimulation mediated by an hypothetical

cytokine IL-P, is therefore:

ΠP =ρ P

TP + τP

×LP

LP + σ P

9

where ρP is the maximum per cell proliferation rate of ThP, LP is the concentration of the

cytokine IL-P and σP is the concentration of IL-P that induces half-maximum IL-P

signalling for ThP proliferation.

Postulate ix- Production and elimination of cytokines is a fast processes compared to

cell population dynamics. This assumption allows the simplification of the model using a


49

quasi-steady state approximation to the cytokine concentration, resulting in the

concentration of cytokine being proportional to the number of secreting cells. Applying this

approximation in equation 9 leads, after some rearrangement:

ΠP =ρ P

TP + τP

×TP

TP +θP

10

where θP is σP× rP/sP, and rP and sP are respectively the rate constants of cytokine IL-P

degradation and production. The same construction for Th2 yields an equivalent

proliferation rate:

Π2 =ρ2

T2 +τ 2

×T2

T2 +θ2

11

where θ2 is σ2× r2/s2, and r2 and s2 are respectively the rate constants of degradation

and production of the autocrine/paracrine cytokine controlling Th2 proliferation. Equations

10 and 11 have a quadratic dependence on the number of cells, which is a consequence of

the positive feedback loop of autocrine/paracrine cytokines controlling cell proliferation.

Postulate x- Differentiation of ThP into the Th2 lineage is a saturating function of IL-4

in the medium; IL-2 acts as an inhibitor of IL-4-driven differentiation of ThP decreasing the

maximum rate of differentiation. Since differentiation of cells requires activation of ThP,

which is concomitant with up-regulation of CD25 (39) then, following the results of

previous section, we expect a significant interference of IL-2 over IL-4 signalling.

Assuming that both IL-2 and IL-4 concentrations are in a quasi-steady state, then the per-

cell differentiation rate of ThP to Th2 is:

Φ = kT2

T2 +θL4

× 1 − χTP

TP + θL2

12

where the two terms represent respectively the IL-4 saturating response and the

inhibition of IL-4 response by IL-2, because of the quasi-steady state assumption both

terms are a function of the number of cells. The constants θL4 and θL2 are the number of

Th2 and ThP cells capable of excepting half-maximum effect. These constants are defined

as θL4 = σL4 × rL4/sL4 and θL2 = σL2 × rL2/sL, where σL4 and σL2 are the affinity of ThP for IL-

4 and IL-2 respectively, and rL4/sL4 and rL2/sL2 are the ratios between the degradation and

production rate constants for IL-2 and IL-4. The parameter k is the absolute maximum

differentiation rate and χ is the amount of inhibition due to IL-2. If χ=0, then IL-2 does not

interfere with IL-4 signalling. Setting χ=0.4 reproduces the situation in which IL-2 can

inhibit IL-4 signal up to 40%.


50

Following the previous postulates the dynamic of the subpopulations of ThP and Th2

cells are described by the following set of differential equations:

dTP

dt= ΠP − δ TP − Φ TP

dT2

dt= Π2 − δ T2 + Φ TP

13

The model was simulated using the parameters listed in table 2. According to these

parameters, about 1% of the total cell population is renewed per day, which is comparable

to previous estimates on peripheral lymphocyte turnover (40). The unit number of cells (N)

is arbitrary for the sake of generality of the model. The affinities of the ThP cells for IL-2

and IL-4 (σL2 and σL4) were assumed to respect the ratio between the apparent affinities of

the cytokine receptors (table 1). For simplicity, the ratio between the rate of production and

degradation of IL-2 and IL-4 were assumed to be the same and therefore the ratio θL4/θL2 is

equal to K appIL-2 /K app

IL-4 = 13.

Table 2 – Parameters of the model of ThP/Th2 differentiation

Parameter Value

ρ0 0.01 day-1

ρ2 0.01 day-1

τP 0.2 N

τ2 0.2 N

θL2 0.013 N

θL4 0.17 N

δ 0.01 day-1

κ 0.005 day-1

2.2.4 INHIBITION OF IL-4 DRIVEN DIFFERENTIATION BY IL-2 ALLOWS FOR THE

PERSISTENCE OF PRECURSORS

Two prototypic situations are of interest to assess the impact of IL-2 and IL-4

interference in ThP and Th2 population dynamics: one where there is no interference

between IL-2 and IL-4 and one in which IL-2 can inhibit IL-4 signalling up to 40%

(respectively χ=0 and χ=0.4). Figure 4 represents the phase diagrams of the model without

(4-A) and with (4-B) inhibition of IL-4 signal by IL-2. The horizontal axis represent the


51

number of ThP cells and the vertical axis the number of Th2 cells. The lines represent states

where either Th2 or ThP do not change in time and the circles in the intersections between

these lines represent the steady states of the system. The closed circles represent stable

steady states of the model, meaning that within some particular initial conditions, the

number of cells of Th2 and ThP will always converge to those numbers. The open circles

denote unstable steady states, meaning that the smallest disturbance will make the system

converge in the direction of a stable steady state. Figure 4-A shows two stable steady states,

one in which both ThP and Th2 populations disappear and another where ThP are

extinguished and Th2 persist. The first steady state denotes a situation where the number of

founding cells is so low that they are unable to produce enough cytokines to proliferate and

persist. The later stable steady state is attained with higher numbers of initial Th2 and ThP

cells and regardless how high the number of initial cells is, the system in these conditions

will always commit completely to Th2. This is akin to clonal exhaustion, as all the

precursors differentiate to effector cells and eventually die. There are four additional steady

states that are unstable in this system. One of these four states is worth mentioning since it

shows that in the absolute absence of Th2 cells the pool of ThP could be sustained.

However, an infinitely small number of Th2 cells would suffice for the system to progress

to full Th2 commitment.

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

1.2

0 0.20.2 0.40.4 0.60.6 0.80.8 1 1.21.20

0.20.2

0.40.4

0.60.6

0.80.8

1

1.21.2

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

1.2

0 0.20.2 0.40.4 0.60.6 0.80.8 1 1.21.20

0.20.2

0.40.4

0.60.6

0.80.8

1

1.21.2A B

ThP / N ThP / N

Th2

/ N

Th2

/ N

Figure 4 - Phase planes of the model of ThP and Th2 proliferation and differentiation. A – Phase plane of

the model without IL2 inhibition (χ=0). B – Phase plane of the model with IL-2 inhibition (χ=0.4). The

lines represent the null clines of the system, i.e. conditions where either the ThP population (dashed lines)

or the Th2 population (solid lines) do not change in time. Where the null clines intersect (circles), the

system is in a steady state. Open circles denote unstable steady states and filled circles denote stable steady

states. The parameters are listed in tables 1 and 2, see text for details.

Representing the phase plane for the model with inhibition of differentiation by IL-2


52

(figure 4-B) reveals three stable steady states. Two are qualitatively the same as before:

extinction of both ThP and Th2 populations and persistence of Th2 and extinction of ThP

populations. The new stable steady state corresponds to the persistence of both Th2 and

ThP populations and denotes a situation where the founding number of ThP cells is

sufficiently high to produce enough IL-2 to sustain the growth of ThP and significantly

inhibit differentiation to Th2. The balance between proliferation and differentiation of ThP

cells permits the persistence of ThP in the presence of Th2 cells. This model has

additionally five unstable steady states, four in common with the model without inhibition

and new unstable steady state, which lies in the separatrix between the two stable states

denoting full Th2 commitment and coexistence Th2 and ThP.

To illustrate how inhibition of IL-4 driven differentiation by IL-2 produces the

conditions necessary for the persistence of a pool of ThP cells, we will discuss in detail the

steady state of ThP, assuming that Th2 cells are in the steady state corresponding to co-

existence of ThP and Th2. Figure 5 shows the rate of ThP proliferation and of the rates of

ThP death plus differentiation with and without inhibition of IL-4 driven differentiation by

IL-2. Without inhibition (χ=0), the rate of ThP death plus differentiation is always larger

than the rate of ThP proliferation so the pool of ThP will shrink to exhaustion. With 40% of

inhibition of IL-4 driven differentiation by IL-2 (χ=0.4), the rate of ThP death plus

differentiation intersects the rate of ThP proliferation in three points, which correspond to

three steady states. Two are stable steady states, one corresponds to exhaustion and the

other corresponds to persistence of ThP. The third steady state is unstable and is located

between the two stable states. For ThP smaller than this unstable state, the rate of ThP

proliferation is smaller than the rate of death plus differentiation, so the pool of ThP shrinks

to exhaustion. For ThP bigger than the unstable steady state, the rate of ThP proliferation is

larger the rate of ThP death plus differentiation so the ThP pool grows towards the steady

state denoting persistence of ThP.


53

0 0.1 0.2 0.3 0.40

0.001

0.002

0.003

0.004

0.005 ( i)

( ii) ( iii)

ThP / N

Rat

e / N

day

-1

Figure 5 - Balance of ThP proliferation and the joint effect of differentiation and death, assuming that the

pool of Th2 cells is in the steady state corresponding to co-existence of ThP and Th2 (Th2 ≈ 0.71 N). (i)

Proliferation rate of ThP cells; rate of death plus differentiation of ThP cells, without (ii) and with (iii)

inhibition of differentiation by IL-2. Whenever the growth curve (i) and the death plus differentiation

curves (ii and iii) intersect (circles) there is a steady state. Stable steady states are denoted as full circles

and unstable steady states are denoted as open circles. See text for details. The parameters are listed in

tables 1 and 2.

This model predicts a synergy of the concerted action of IL-2 and IL-4 for Th2

differentiation. Comparison of the model with and without inhibition (figure 4), shows that

the maintenance of a pool of ThP cells increases the size of the Th2 population by

approximately 20% under the parameters of Table 2. This result provides a rational for the

well-known increased yields in Th2 differentiation when precursor cells are cultured with

IL-2 and IL-4 (41,42). In our model, this increase is because ThP cells and Th2 cells have

independent carrying capacities.

These results are obviously parameter dependent. Parameter regimes exist where in the

absence of inhibition the persistence of both ThP and Th2 cell pools is possible. These

parameter regimes imply for example, a lower differentiation flux from ThP to Th2 and/or

a larger growth rate or smaller death rate for ThP. Also, for the model with inhibition,

under some parameter regimes the inhibition by IL-2 of differentiation is not sufficient to

maintain a pool of precursor ThP cells, for example, when the death rate of ThP or the

proliferation rate Th2 is large or when the proliferation of ThP or the death of Th2 is small.


54

2.2.5 MODULATING γC EXPRESSION LEVELS MAY BE USED TO INFLUENCE THE EXTENT TO

WHICH A CLONE IS COMMITTED

As we have seen above, the levels of expression of γc modulate the maximal inhibition

of IL-4 signalling by IL-2 (fig. 2-D), so the levels of γc expression by precursor cells can

determine whether these cells can persist, by modulating IL-2-dependent resistance to IL-4

driven differentiation. This prediction is illustrated in the bifurcation diagram for ThP

population size as a function of the parameter χ (fig. 6).The main testable prediction of the

model is that the levels of γc can control the outcome of Th2-lineage commitment

experiments. Long-term cultures of naïve T cells in the presence of Th2-driving conditions

lead to irreversible Th2-commitment. Our model would predict that the actual time required

to fully commit a population of naïve cells could be shortened by forcing ThP cells to over-

express γc. Conversely, forcing ThP cells to down-modulate the expression of γc could

delay and maybe prevent exhaustion of the ThP pool.

ThP

/ N

χ0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

Figure 6 - Bifurcation diagram of the model showing the size of the ThP pool as a function of the extent of

inhibition of IL-4 driven differentiation by IL-2 (χ). The black lines represent stable steady states; the grey

lines represent unstable steady states. The parameters are listed in tables 1 and 2, see text for the details.


55

2.3 Discussion

We have raised the problem of persistence of uncommitted ThP cells and argued that

the pattern of “endpoint activation” of differentiation to Th2 pathway could lead to the full

commitment of lymphocyte clones to Th2 lineage. We have confirmed the plausibility of

this expectation by a quantitative analysis of a population dynamic model, but we have also

shown that exhaustion of uncommitted precursors may be avoided. The model, calling into

action cytokines whose receptors share γc, demonstrates that if uncommitted ThP cells are

proliferating and secreting IL-2 then they will resist IL-4-driven differentiation and, in spite

the pressure imposed by already committed Th2 cells, the pool of precursor ThP cells is

maintained.

In the minimal model of cytokine receptor dynamics and signalling presented here,

IL-2 interferes with IL-4 signalling by assembling a receptor complex that sequesters γc. In

reality, cytokine receptors are in constant turnover (43-45) and the engagement of cytokine

receptors increments the down-regulation and either degradation or recycling of receptor

chains (45,46). These processes were not considered in this work due to the lack of

sufficient experimental kinetic data. Nevertheless the model should be a good

approximation when the dynamic of the cytokine-receptor chains is practically in

equilibrium, which is probably the case if we consider the time scale at which the

populations of ThP and Th2 are evolving. Interestingly, IL-2-mediated down-modulation of

γc (43-45) might amplify the inhibition of IL-4 signalling, which may mean that our model

is underestimating the interference between the cytokines.

The cell population model makes also strong simplifications on cytokine signalling.

Signals leading to differentiation and proliferation were assumed to be proportional to the

amount of fully assembled receptors complexes, which should only reflect early signalling

events. A consideration of common downstream signalling components, such as Jak1 and

Jak3, homodimers and heterodimers of STAT1, STAT3 and STAT6, or the products of the

CIS and SOCS gene family (47), are expected to amplify the early effects leading to a more

pronounced interference between the cytokines.

The model only includes autocrine growth of ThP and Th2 cell populations, neglecting

for instance the effect of IL-2 on Th2 proliferation and the effect of IL-12 on the dynamic

of ThP and Th2 (1-4). IL-4 is required for the differentiation of ThP to Th2 (1-4), but it's


56

effects on the maintenance of the Th2 cells seems to synergize with IL-2 (48), as expected

from the model. To assess how cross-regulation would affect the results of our model we

performed simulations where growth of Th2 population is affected by both autocrine

factors and by paracrine factors produced by ThP (not shown). Only if Th2 cell

proliferation would be strictly dependent on a growth factor produced by ThP cells, would

the results change qualitatively. Under these conditions, Th2 pool persistence would be

impossible without ThP cells, and therefore precursor ThP cells would always persist

without any functional requirement for a putative inhibition by IL-2 of IL-4-driven

differentiation. These strict conditions are, however, unrealistic. IL-2 is the most obvious

candidate for a growth factor produced by ThP that affects the growth of Th2 cells,

however, this cytokine is not strictly necessary for Th2 development in vitro (49) and in

vivo (50).

In spite of the radical simplicity of our description of the inhibition of IL-4 signals by

IL-2, the model demonstrates that even a marginal effect at the level of early signalling

events, once placed in the context the non-linear population dynamics of Th differentiation,

becomes amplified to an all-or-none effect.

According to the model, precursors may persist in a situation that we interpret as

clonal pluripotency. Although there is no clearly identifiable molecular mechanism, the fact

that ThP persist in a stable steady state implies an effective “negative feedback” loop

embedded in the dynamic of ThP and Th2 populations. Hence, peripheral differentiation of

Th lymphocyte clones maybe regulated by a negative-feedback loop, functionally

analogous to those regulatory circuits that ensure the persistence of precursors for

erythrocytes and neutrophils.

Interestingly, within the multistable regime, where the population of ThP can either be

exhausted or grow to the homeostatic steady state, a perturbation could drive the system

from the state of co-existence of Th2 and ThP to the state of ThP exhaustion. This property

of the model is reasonable considering that the immune system must eliminate pathogens

and mount recall responses, and that the same pathogen may require different classes of

response depending on the context and route of its inoculation. In most sporadic challenges

it might be appropriate and sufficient to generate cohorts of Th2 cells, keeping nevertheless

precursors with capacity to produce cells of the same specificity but committed to other

lineage. In other situations, namely lifetime immunisation, it might be appropriate and

necessary to commit a clone once and for all to the Th2-lineage. In a nutshell, under some


57

circumstances specific clones may need to keep precursors by a functional feedback loop,

but under other circumstances this regulatory loop may be overruled, irreversibly

associating a clone with a class of effector function. Further theoretical elaboration on this

hypothesis requires defining experimentally whether or not clones remain pluripotent in

vivo and how different challenges affect the extent of clonal commitment. Populations of

naïve Th cells cultured under strong Th2 driving conditions will eventually loose

progenitors (1), which under less stringent conditions could persist longer. Analysis of Th

cell repertoire in animal models have shown the expansion of a restricted population of

cells expressing a defined Vα Vβ heterodimer which are in either a Th1 or Th2 phenotype

(51). Alternatively, several reports indicate that cells responding in vivo in Th1 or Th2

modes express different Vα or Vβ genes (52). One interpretation of these findings is that

these cells belong to different clones, which are fully committed to either Th1 or Th2

lineage, and are select under different pressures. However, this is not the only interpretation

and therefore more definitive experimental results are needed.

Mutant animals might allow assessing the realism of the hypothesis that interference

between IL-2 and IL-4 signalling contributes for clonal pluripotency. Animals lacking γc

have impaired peripheral compartments (53), making them uninformative in this context.

Animals deficient in IL-4 are unable to mount efficient Th2 responses (54) supporting a

model for IL-4 driven differentiation to the Th2 lineage but telling us little about the

quantitative interference that we have explored here. Animals lacking IL-2, IL-2Rα and IL-

2Rβ genes would be more appropriate to address our hypothesis on Th differentiation.

Interestingly, the level of γc expression level is increased in T cells from IL-2 deficient

animals (55), suggesting that continuous IL-2 signalling and IL-2 receptor down-

modulation in wild-type animals may reduce the availability of γc for other cytokines (43).

Our prediction would be that these animals should more easily loose clonal pluripotency

following Th2 stimuli. The bias towards IgG1 observed in IL-2 deficient mice (50), but not

in IL-2 × IL-4 double deficient mice (56) is in agreement with this expectation of our

model. In this context, it is tempting to speculate that the homeostatic abnormalities

observed in the three mutant animals (50,57,58) are due to loss of uncommitted Th

precursors and a subsequent inability to generate cohorts of further differentiated regulatory

Th cells involved in preventing the expansion of other cells (59).

Families of cytokines receptors share one or another chains making it possible for

dynamic coupling between the members of the same family. Here we called the attention to


58

a potential role of the interference between IL-2 and IL-4 in the commitment of peripheral

cells. Since the members of the γc family are involved at different stages of hematopoiesis,

both as differentiation and renewal factors, it is possible that similar mechanism may play a

role in other branching points of lymphocyte development. In other cytokines families,

quantitative interference between receptors might unfold into yet unknown functions that

deserve being investigated.

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Analysis of TCR engagement, triggering and down-modulation

63

3 Analysis of TCR engagement, triggering and down-

modulation

3

Analysis of TCR engagement,

triggering and down-modulation


64


65

3.1 A mathematical analysis of TCR serial triggering and down-regulation

A mathematical analysis of

TCR serial triggering and down-regulation

João SOUSA1,2 & Jorge CARNEIRO1

1. Instituto Gulbenkian de Ciência, Oeiras, Portugal

2. Grupo de Bioquímica e Biologia Teóricas, Instituto de Investigação Científica Bento da Rocha Cabral,

Lisboa, Portugal.

Summary

Despite the increasing knowledge on the pathways involved in TCR signal transduction and T cell

activation, the molecular mechanism of TCR triggering by ligand, MHC-peptide complexes, is still elusive

and controversial.

The present paper addresses the controversy on the early events of TCR engagement and triggering.

Mathematical modelling techniques are applied to experimental data to infer plausible molecular mechanisms

of TCR triggering and down-regulation. A similar approach has been followed by Bachmann et al. (Eur J

Immunol 1998, 28: 2571-9), who concluded that the TCR triggering requires the formation of MHC-TCR

dimers or trimers. We report here the failure to generalise this conclusion to the data reported by Valitutti et

al. (Nature 1995, 375: 148-51). We show that there are several kinetic features in these experimental curves

of TCR down-regulation that cannot be explained by the simple model proposed by Bachmann et al. unless

some phenomenological extensions are considered. These extensions are: 1) a ligand independent turnover of

the TCR; 2) a transient accumulation of triggered TCRs; 3) a high order of TCR triggering kinetics; and 4)

two pools of membrane TCRs in dynamic equilibrium.

Acknowledgements

The authors are indebted to Mlle. C. Brissac for many useful discussions at the beginning of this work. They

are grateful to J. Faro, K. Leon, and A. Coutinho for helpful comments and to J. Stewart for linguistic

revision of the manuscript. JC and JS are supported by Fundação para a Ciência e Tecnologia - Programa

Praxis XXI (fellowships BPD/11789/97 and BD/13546/97 respectively).


66


67

3.1.1 INTRODUCTION

Despite increasing knowledge of the signalling pathways involved in TCR signal

transduction and T cell activation (reviewed in [1]), the molecular mechanism of TCR

triggering by ligand is still elusive and controversial. Especially controversial is the

stoichiometry of TCR-ligand complexes that leads to TCR triggering by phosphorylation of

immunureceptor tyrosine activation motifs (ITAMs). Some authors suggest that the

formation of oligomers of TCRs (TCR cross-linking) is required for TCR phosphorylation,

while other authors suggest that TCR oligomerization is not required. Some structural

studies, in favour of TCR oligomerization, demonstrate the tendency of TCRs to associate

into clusters [2-4] and show that this association may be further stabilised upon MHC

ligation. Additionally, binding essays have shown that triggering of TCR requires dimers or

tetramers of ligand [5, 6]. Against the hypothesis of TCR oligomerization, it was shown

that a single MHC-peptide molecule of class I could trigger the TCRs on CD8+ T cells and

initiate signalling [7, 8]. Moreover, it was shown that T cells could be activated by fewer

than a hundred ligand molecules presented per APC [9-11]. For such a low density of

ligand, the formation of oligomers of TCR would be an improbable event [12] and

therefore the signal generated by TCR oligomerization would most likely be undetectable

by the cell. Another puzzling piece of evidence on TCR triggering was derived from

biophysical studies of the interaction of individual TCRs and MHC-peptide complexes.

These studies indicate that the affinities of the interaction are relatively low, namely in the

order of 10-5-10-6 M. Such low affinities are in apparent contradiction with the high

specificity and high sensitivity of T cells to low ligand presentation on APCs [9-11]. An

important contribution to understand this puzzle was provided by the model of serial

triggering of the TCR by ligand [11], which proposes that each ligand presented by an APC

is able to engage and trigger many TCRs, resulting in enough signalling events to activate a

T cell. The importance of TCR binding kinetics was further emphasised from comparative

studies of TCR agonist and altered peptide ligands, which have shown that the agonistic

properties of the ligands correlate with the affinity and off rates of binding to the TCR [13].

One of the limitations of the serial triggering model is that it neither proves nor disproves

the possible oligomerization of TCRs during the serial triggering process.

The goal of the present work is to contribute to the resolution of some of the


68

controversy surrounding the molecular mechanism of TCR triggering. The main

assumption here is that the pathways leading to full T cell activation and to TCR down-

regulation share a very early set of molecular events, which bifurcate later. This assumption

is readily compatible with the observations that TCR down-regulation and full activation

are frequently correlated but that they can nevertheless be uncoupled [11, 14, 15].

Mathematical modelling techniques are used in this paper to infer plausible

mechanisms of TCR triggering and down-regulation at the molecular level, based on the

experimental data. A similar approach was used before by Bachmann et al. [12], who

concluded that TCR triggering requires the formation of MHC-TCR dimers or trimers. We

report here the failure in the attempt to generalise this conclusion to the data on TCR down-

regulation reported earlier by Valitutti et al. [11]. We show that there are several kinetic

features in these experimental curves of TCR down-regulation that cannot be explained by

the simple model proposed by Bachmann et al. [12] unless some phenomenological

extensions are considered.

3.1.2 RESULTS

3.1.2.1 Experimental data and model variables

We have modelled the experimental data of Valitutti et al. [11], which was read off

from the figures in the publication. To our knowledge this is the most complete data set

available in the literature. Briefly, the data was obtained from the following experimental

system: T cells were incubated with APCs (B cells transformed with the Epstein-Bar virus)

pulsed with two concentrations of peptide, 25 nM and 20 µM, which correspond to

approximately 76 and 8700 ligand molecules (MHC-peptide complexes) presented per

APC, respectively (extracted from fig. 3A from Valitutti et al [11]). The decrease in TCR

expression over time was measured by CD3 staining using FACS. We refer to these two

data sets as the low- and high-density kinetic data set (fig. 1A). The CD3 staining after five

hours of incubation of the T cells with APCs pulsed with different peptide concentrations

was also giving the data of fig 3B and 3A in the report of Valitutti et al. [11]. We refer to

this as the 5 hours data set (fig. 1B); together with the kinetic data sets it makes up the

complete data set to which different candidate models will be fitted.

In the following sections we will propose several mathematical models and obtain

the best fit to the complete data set (see Methods). The models are set up as a system of


69

ordinary differential equations (ODE). The relative TCR and ligand densities (r.d.) are

calculated as density of molecules divided by the density of TCRs in a resting T cell. This

is appropriate because it allows a simple normalisation of the experimental data to the TCR

density at time zero. As in Bachmann et al. [12], we assumed that the TCR density per T

cell is proportional to the average intensity of the CD3 staining measured by FACS in the

population of T cells. There is more to be said about this assumption, which will be taken

up in the Discussion. For the sake of simplicity, it is assumed that the APC and the T cells

all have equal volumes and membrane areas.

3.1.2.2 TCR-dimerisation fails to explain the experimental data

Bachmann et al. [12] proposed a model for TCR down-regulation, which resulted from

an analysis of experimental curves obtained with relatively high ligand densities. Under

these conditions, when the rate of TCR down-regulation is plotted as a function of the

square of TCR density, an approximately linear relationship emerges [12]. Thus the down-

regulation of the TCR is an apparent second-order reaction. Bachmann et al. [12] derived

this second-order rate law using a simple model with mass action kinetics for the formation

of dimers of TCR-ligand complexes (TL), which upon productive dissociation would lead

to a triggered receptor (A) that would then be internalised.

T + Lkon →

koff← TL

2 TLkr → kd

← (TL)2ka→ 2 A + 2 L

A ki→

(1)

The second-order rate law was derived from this reaction scheme assuming that the

complexes of TCR and ligand are in a quasi steady state (for details see [12]). Thus, the

TCR dynamic is described by the following ordinary differential equation (ODE):

d [T]t

d t= −keff [T]t

2 L2 (2)

where [T]t is the concentration of membrane TCR at time t and keff is an aggregate rate

constant. Integration in time yields:

[T]t =[T]0

1 + t keff [L]2 [T]0 (3)


70

100

50

0100

50

0100

50

0100

50

0100

50

00 100 200 300 10.001 0.01 0.1

%C

D3

Stai

ning

Time /min [ligand]

A

B

C

D

E

Figure 1 – Best fit of different models to the complete data set of Valitutti [11]. The left column shows the

kinetics of TCR down-regulation. The symbols are the experimental points obtained with low (diamonds)

and high (squares) ligand densities; the solid lines are the corresponding predictions of the model. The right

column shows the level of TCR down-regulation after 5 hours of conjugation with APC as a function of

ligand density. The symbols represent the experimental data (the diamond and square represent the last

time-point the kinetic data on the left); the lines represent the predictions of the model at five hours (solid)

and at the steady state (dashed) (only the solid line is drawn if both coincide). The models are: A –

Bachmann model [12], B – Extension showing a non-zero steady state, C – Extension showing

insensitivity to changes in ligand density, D – Extension showing ultrasensitivity in TCR triggering, E –

Extension with two TCR pools. See text for details. The parameters of the fittings are listed in Table 1.

Anal

ysis

of T

CR

enga

gem

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trig

geri

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nd d

own-

mod

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71

Tab

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– B

est f

it pa

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s of t

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odel

s. Se

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r an

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s bet

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MO

DE

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mod

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[12]

MO

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L B

Non

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MO

DE

L C

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L D

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of t

he

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f TC

R

trig

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ng

MO

DE

L E

Tw

o po

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f TC

R w

ith

diffe

rent

trig

geri

ng

kin

etic

s

ka

839

r.d.-3

min

-1

4306

r.d.

-3 m

in-1

31

97 r.

d.-3

min

-1

1.4

x1012

r.d.

-9 m

in-1

1.

5 x1

012 r.

d.-9

min

-1

(2

.20×

101 —

2.02

×104 )

( 0 —

6.8

4×10

4 ) (1

.03×

103 —

1.32

×104 )

(2.9

5×10

11—

1.1

0×10

13)

(5.7

9×10

11—

4.6

8×10

12)

s /m

in-1

-

0.01

6 0.

0041

0.

0031

0.

0011

(0—

3.0

2×10

-1)

(1.8

6×10

-3—

6.5

4×10

-3)

(1.5

1×10

-3—

4.8

2×10

-3)

(3.3

7×10

-4—

1.9

2×10

-3)

k i /m

in-1

-

- 0.

043

0.03

6 0.

094

(2

.85×

10-2

— 7

.56×

10-2

) (2

.64×

10-2

— 5

.48×

10-2

) (5

.74×

10-2

— 2

.49×

10-1

)

h -

- -

5 5

( > 4

.7)

( > 4

.8)

φ /m

in-1

-

- -

- 0.

0055

(4

.14×

10-3

— 7

.80×

10-3

)

θ -

- -

- 61

%

(5

7%—

66%

)

χ2

2.11

1.

67

0.25

0 0.

142

0.04

4


72

The first question we ask here is whether this simple model is also able to fit the data

of Valitutti et al. [11], assuming that the TCR density ([T]t) is equal to the percentage of

CD3 ([T]t≡%CD3). The model can be appropriately fitted to the kinetics of down-

regulation induced by high ligand density, supporting the previous suggestion of Bachmann

et al. [12]. However, the attempt to describe the complete data set revealed two

shortcomings of the model (fig.1A and fig. 2). First, the apparent linear dependence of the

rate of TCR down-regulation on the square of the TCR density does not hold for the low

ligand density (fig. 2). The non-linearity emerges from the simple fact that the TCR down-

regulation stops before all the TCRs have been internalised (this can be inferred from fig. 2,

but also in fig. 1.A from [16]). This observation is in clear opposition with eqn. 2, which

predicts that for any positive ligand density, the down-regulation stops only when all the

TCRs have been internalised.

0.06

0.04

0.02

00 25 50 75 100

(%CD3)2 Figure 2 – Bachmann plot of the data of Valitutti et al [11]. The rate of TCR down-regulation, as a function

of the square of TCR density, is linear for the higher dose (squares) but not for the lower dose (diamonds)

of ligand. The solid and dashed lines are respectively the derivatives of the best fit kinetics of the

Bachmann model (fig.1A-left) and the last extension (fig.1E-left).

The second shortcoming of the model is that it predicts that the rate of TCR down-

regulation is proportional to the square of ligand density. This is in clear contrast with the

initial rates of approximately -0.01 and -0.03 r.d. min-1 observed for low and high ligand

densities, respectively.

We tried to identify features of TCR dynamics that will overcome these limitations. In

the following sections we derive and discuss four extension to this model.


73

3.1.2.3 Non zero steady-state resulting from ligand independent TCR- turnover

The T cell maintains a dynamic pool of membrane TCRs, by continuous de novo

expression and random internalisation and degradation. Mathematically, this can be

expressed as a simple turnover:

d [T]t

d t= s 1− [T]t( ) (4)

where s is the relative number of new TCRs expressed on the cell membrane per minute,

i.e. turnover rate. Adding the right hand side of eqn. 4 to the right hand side of eqn.2 we

obtain our first extension of the model of Bachmann et al. [12]

d [T]t

d t= s 1− [T]t( )− keff [L]2 [T]t

2 (5)

This model predicts a non-zero steady state TCR density in the presence of ligand.

Equation 5 was integrated numerically and fitted to the individual kinetic curves of TCR

down-regulation. However, because eqn. 5 still predicts a dependence of the rate of TCR

down-regulation with the square of ligand density, it still fails to fit the complete data set

(fig.1B). Nevertheless, this extension was sufficient to permit the model to fit the 5 hours

data set alone (data not shown).

3.1.2.4 Insensitivity to changes in ligand density results from a transient

accumulation of a pool of triggered TCRs

The second-order rate of TCR down-regulation results from assuming that triggered TCRs

are immediately internalised [12]. Our second modification to the model of Bachmann et

al. [12] results from relaxing this assumption. The process of internalisation of triggered

TCRs is now treated independently and not lumped together with the dimerisation and

triggering processes. This gives rise to a pool of triggered TCRs with dynamic described

by:

d [A]t

d t= keff [L]2 [T]t

2 − ki[A]t (6)

where [A]t is the concentration of triggered TCR at time t and ki is the internalisation rate

constant. The down-regulation of TCR is represented by the second term in eqn. 6, −ki[A]t .

The same reaction diagram is now translated into a system of two ODEs, composed of eqn.

5 and eqn. 6. The dynamic of total CD3 staining is represented by the sum of the two TCR

pools (obtained by adding eqns. 5 and 6), i.e. %CD3 ≡ [T]t + [A]t.

By the inclusion of a pool of triggered TCRs, the rate of TCR down-regulation will be


74

distinct from the rate of TCR triggering. The rate of TCR down-regulation will no longer be

proportional to [L]2. Under some parameter regimes in which it is limited by the rate of

internalisation, which can not exceed ki [T]0.

The fitting of this model to the complete data set (Fig 1C) is better than the previous

models. The fitting parameters (Table 1) are such that the rate of down-regulation is much

slower than the rate of triggering and consequently a significant pool of triggered TCRs

accumulates transiently in the cell membrane. For the high ligand density practically all the

available TCRs are triggered within a few minutes.

3.1.2.5 TCR-triggering is ultrasensitive to ligand and TCR densities

The kinetics of TCR triggering can be generalised by substituting the second-order kinetics

in both receptor and ligand densities with an analogous term with a kinetic order h.

d [T]t

dt= s (1 − [T]t) − keff [T]t

h[L]h (7)

d [A]t

d t= keff [T]t

h [L]h − ki[A]t (8)

The fitting of the model to the experimental curves progressively improves as the

kinetic order increases and is most noticeable in the initial phase of the kinetics, i.e. within

the first 50 minutes. The “best fit” is obtained with a rate law of kinetic order h equal or

greater than 5 (fig.1D); higher values do not cause a noticeable qualitative improvement in

the fitting. Also the greater the value of h the greater the value of keff, following the relation

keff ∝ kh, with k constant.

The sensitivity of the model to receptor and ligand densities increases as the kinetic

order of TCR triggering increases. Small changes in either ligand or receptor will lead to

large changes in the rate of triggering. Mathematically,

∂ log keff Lh [T]t

h( )∂ log X( )

= h With X = L,[T]t (9)

This logarithmic sensitivity to L or T means that increasing the ligand presentation in,

for example, 10% with a co-operativity of 4 will lead to an increase in the rate of TCR

triggering of 10000 %.The effect of this ultra-sensitivity of TCR triggering on the overall

rate of TCR down-regulation is most marked for low ligand densities, since for high ligand

densities the internalisation rate of triggered TCRs will mask it.

Close examination of the experimental data indicates that the rate of TCR down-

regulation is biphasic. Initially fast and dependent on MHC, the rate slows down after 50


75

minutes of incubation. In the slower phase the curves for high and low densities appear as

two parallel lines, suggesting that the later is virtually independent of MHC-density. The

second phase is not captured by any of the models we have presented so far, and has

prompted yet another extension to the model of Bachmann et al. [12].

3.1.2.6 Two pools of membrane TCR with different triggering kinetics explain

biphasic kinetics of down-regulation

A reasonable interpretation of the two phases in TCR down-regulation is that at the surface

of the T cell there are two pools of TCR. Only one of these pools, which we call the

‘interface’ pool, has access to the ligand. The second pool, which we call the ‘spare’ pool,

has no access to the ligand but is in dynamic equilibrium with the interface pool. The model

equations are:

d [S]t

dt= −λϕ [S]t − [T]t( )+ s(1− [S]t)

d [T]t

dt= ϕ [S]t − [T]t( )+ s(1 − [T]t) − keff [T]t

h [L]h

(10)

d [A]t

d t= keff [T]t

h [L]h − ki [A]t

where [T]t and [S]t are the densities of TCR in the interface pool and in the spare pool,

respectively; ϕ is the exchange rate constant and λ is a ratio between the areas of the

interface over the spare pool. The variables are now mapped to the experimental

measurements by %CD3 ≡ [S]t / (λ+1) + ([T]t +[A]t) λ/(λ+1).

The best fit to the complete data set obtained with this model is the only one that

explains both initial and late kinetics of TCR down-regulation (fig. 1E). The parameters

(Table 1) indicate that the exchange rate of TCRs between the interface and the spare pools

is slower than the TCR triggering and internalisation reactions. The rate of TCR turnover is

the slowest of the reactions, and is consistent with the observation by Valitutti et al. ([11];

Valitutti, personal communication) that the recovery of normal TCR levels takes several

days after the T cells are separated from APCs.

The biphasic characteristics of the kinetics of TCR down-regulation can be better

understood using two approximations to the model. In the first approximation we set to

zero the exchange rate φ obtaining results comparable to the previous model. In the second

approximation we assume that reactions in the interface are in quasi-steady state, setting

d[A]t/dt=0 and d[T]t/dt=0 in eqn. 10 and solving these equations numerically. In this case,


76

the internalisation of TCR is limited by the TCR exchange rate, and only the slow rate of

TCR down-regulation is present. These two limit conditions of the model actually

correspond to the asymptotic kinetics of down regulation (fig. 3).

100

50

00 100 200 300

Time /min

a

a

b

b

Figure 3 – Distribution of TCR molecules between interface and spare pools explains two phases in the

kinetics of TCR down-regulation (see text for details). Curves a assume that the exchange rate constant is

zero. Curves b assume that the all the reactions except the exchange of TCRs are in quasi-steady state. The

solid line is the same as in fig.1D-left.

3.1.3 DISCUSSION

The kinetics of TCR down regulation is complex and requires the formulation of complex

kinetic models to reproduce the experimental results. As was shown by Bachmann et al.

[12] a simple mass action kinetic model, without cross-linking of TCRs, cannot reproduce

the kinetics of TCR down-regulation. These authors proposed instead a simple TCR cross-

linking model that can explain their experimental data of TCR down-regulation.

Comparison of the results of the TCR cross-linking model with the experimental data of

Valitutti et al. [11], who measured TCR down-regulation using a broader interval of ligand

presentation, revealed some shortcomings of this model. The most important problem is the

prediction that the rate of TCR down-regulation is proportional to [T]2 and to [L]2, which is

not supported by the experimental data by Valitutti et al. [11]. Our attempts to resolve this

limitation lead us to introduce a series of phenomenologic extensions to the simple model

of Bachmann et al.

Although the four extensions were presented in order, they are almost independent in

respect to the contribution of each to the fitting of the data. This means that adding just a

subset of these extensions, for example hypersensitivity, to the original model of Bachmann


77

et al. does not produce a good fit to the experimental data. Several alternate extensions or

mechanisms were tested but the results were not satisfactory, thus we believe that this set of

extensions is the only one that is able to fit all the experimental data and is biologically

reasonable. The major criticism to this conclusion is the robustness of the experimental

data, which consists in few sparse points.. This problem may be minimised considering that

the kinetics of TCR triggering have been reproduced qualitatively in a number of different

experimental systems (see for example [16-19]).

The biology underlying the four assumptions, the fitting parameters and the

implications of the final comprehensive model deserves further discussion. T cells have a

basal turnover of membrane TCR that was modelled with a simple ligand independent term.

This is a simplification of the biological processes regulating TCR levels on T cells, the

most important of which is probably the phosphorylation by PKC of a leucine-based

internalisation motif in the CD3γ chain [20]. Although PKC can be activated following

TCR triggering [21], it does not seem to interfere with the extent of ligand-dependent TCR

down-regulation [22], so it was neglected in the models.

Contrary to Bachmann et al. [12], we did not assume that the rate of TCR

internalisation is practically instantaneous (i.e. much faster than the rate of TCR triggering).

According to the parameters of the final model E, in Table 1, the internalisation of triggered

TCRs has rate constants that imply a half-life of about 7 minutes. This result is supported

by the report that membrane TCRs in T cells constitutively expressing active lck (thus

presumably having constitutive triggering of the TCR) have an approximate half-life of 5-

10 minutes [23]. This constitutes evidence that triggered TCRs may accumulate transiently

in the membrane of T cells conjugated with APC. Interestingly, the model predicts that

triggered TCRs accumulate as a pool for about 50 min, providing a time window for

activation of down-stream signal transduction pathways.

Two pools of TCR with different access to ligand explain the two phases in the

kinetics of TCR down-regulation. The most straightforward interpretation for these pools

would be to assume that the interface pool contains the TCRs located in the contact area

between the T cell and the APC and the spare pool contains the remaining TCRs. If it were

assumed that the exchange of TCRs is simple passive diffusion then, according to our

results, more than 60% of the T cell membrane would be in contact with the APC, which is

an overestimate of the reported contact area of 20-30%. This discrepancy might be due to

the complexity of the dynamic of TCRs in the membrane, which involves cytoskeleton-


78

dependent active transport [24, 25]. It is possible that there is an area (greater than the

contact area between APC and T cell) that acts as a sink for TCRs being actively

transported to the contact area. This would be coherent with the recent study of the

“immunological synapse” [24] that shows that T cells are able to concentrate up to 100 fold

the ligand on the contact area.

Another interpretation of the interface and spare pools involves membrane rafts. Rafts

are stable membrane micro-domains containing co-receptor molecules and signal

transduction components [26]. Thus it is possible that the TCRs triggered and down

regulated in the first phase are already integrated in the rafts [27]. The second phase of the

kinetics of down-regulation would correspond to either diffusion or transport of the

remaining TCRs into rafts [27, 28]. Therefore, in T cells such as the ones used in Valitutti

et al., more than 60% of the membrane TCRs would be in rafts.

Direct comparison of the kon and affinities predicted by the model with the

experimental data in the literature is not possible. In the model, the binding of the TCR to

the ligand occurs between two molecules bound to two membranes, whereas in the binding

experiments, the binding reaction occurs in a liquid-solid interface. The importance of kon

to the triggering of the TCR is, however, minimised by the fact that agonist properties of

peptides are mainly defined by the koff [13].

The koff predicted by the model (100 min-1, Table 2) , which can be directly compared

to experimental measurements, is higher than the reported values of about 2-4 min-1 [13,

29]. This discrepancy may be due to steric constraints inherent to the APC-T cell contact

gap. The ligand and the TCR are relatively short compared with adhesion molecules and

thus it is conceivable that the stability of the complex is reduced [30]. Another, perhaps

more simple, justification for an overestimation of the koff by the models is that during TCR

triggering and down-regulation there might be some bystander down-regulation of non-

triggered TCRs [31].

For comparison between the model and the experimental results we assumed that TCR

density is proportional to the intensity of CD3 staining. However, several groups have

published data of TCR down-regulation as a function of measured ligand density [11, 15,

16]. In some of these reports, the amount of TCR down regulated after a fixed period of

time seems to be a linear function of the logarithm of ligand density [11, 15]. In other cases

this linearity is not observed [11, 16] and the curves are similar to the ones in fig. 1. This

may therefore represent a weakness of the models since none of them shows this linearity.


79

Clearly further investigation into this matter is required.

The requirement for a high kinetic order, or high co-operativity, that leads to ultra-

sensitivity in the TCR triggering process indicates that it might be a complex mechanism

involving several steps. The most straightforward candidate for this complex mechanism is

the formation of high-order TCR-ligand oligomers. Under certain conditions, namely the

ones where the oligomer concentration are in quasi-steady state, it is possible to map the

kinetic order of TCR triggering to the stoichiometry of receptor-ligand oligomers (not

shown). A general difficulty inherent to oligomerization mechanisms is that the formation

of the final triggering complex requires the several intermediates which introduces a delay.

As the lateral diffusion of complexes imposes an upper limit to the rate constants of

oligomerization [32], the delay in the formation of the oligomers must be lower bounded.

Thus, oligomerization may be incompatible with our result that most of the TCRs in the

interface are triggered within the first minutes after APC-T cell conjugation.

Oligomerization is not the only mechanism that could explain the ultra-sensitivity of

TCR triggering. The TCR is triggered by a complex mechanism that involves kinases and

phosphatases acting on the same substrate [1]. Interestingly, Koshland and Goldbetter [33]

have proposed a generic mechanism that can explain very high sensitivities in kinase-

phosphatase cycles. Several such cycles have been identified in early TCR signalling.

Examples are the phosphorylation/dephosphorylation of ITAMs in CD3 chains by lck and

possibly CD45 [34, 35] or are the regulation of Zap-70 by lck and a phosphatase [36, 37],

possibly SHP-1 [38]. Similarly to oligomerization mechanisms, kinase/phosphatase cycles

also imply several reaction steps that could introduce delays. However, at variance with

TCR oligomerization that requires lateral diffusion of the TCR-ligand complexes, these

reaction steps may be some kind of “intramolecular reactions” between signalling

molecules pre-assembled with the right stoichiometry in rafts. Eventually, oligomerization

and kinase/phosphatase cycles predict different kinetic properties of TCR triggering which

may be confronted with experimental data using a similar methodology to the one

employed here. We address this issue elsewhere (submitted) showing that the requirement

for high-order oligomers in TCR triggering is unrealistic at least for the lower densities of

ligand presentation.

In conclusion, the detailed kinetics of TCR down-regulation as a function of ligand

density impose tight constraints on any hypothetical mechanisms for the early events of

TCR signalling. These constraints provide an excellent source of information to test, and


80

eventually reject, alternative hypotheses. In this sense, the most promising quantitative

feature of TCR triggering revealed by our analysis is that the early events of ligand-

dependent triggering of the TCR are very fast and ultra-sensitive to changes in both the

densities of TCR and ligand at the interface between the co-operating cells. Candidate

models of TCR triggering should be confronted with these findings.

3.1.4 METHODS

Mathematical models are set up as systems of ODE. The ODE systems are integrated

numerically using the program Mathematica (R), except for a few particular cases that have

analytical solutions and were identified.

The different models were fitted to the complete experimental data set, containing both

the kinetic data and 5 hours data. The parameter space was explored using the routine

amoeba.c, described in Press et al. [39]. Numerical integration of the ODE systems for

the fits was done with the implicit Bulirsch-Stoer method using the routine bsstep.c,

described in Press et al. [39]. The criterion for optimisation of the parameters of the models

was the minimisation of χ2. All the data points in the complete experimental data set were

given equal weight in the fits. Many starting parameter sets were tested for each model to

avoid local minima.

There are no proper statistical procedures to estimate confidence intervals of

parameters in non-linear models. To estimate the confidence intervals of the fitting

parameters we adopted, pragmatically, the standard procedure for linear models. Let P be

the number of parameters in the model, N the number of experimental points, ρi is the best

fit value of this parameter. The 95% confidence interval defined by [ρi-∆-, ρi+∆+] fulfils the

following conditions:

F(0.05, N − P−1, N− P) =χρi +∆ +

2

χmin2 and F(0.05, N − P−1, N− P) =

χρi −∆ −

2

χmin2 (10)

Where F is the F distribution. We solve numerically equations 10 for ∆+ and ∆- for

each parameter (all others remaining fixed).

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by receptor clustering. Embo J,1995. 14: 3385-94.

37 Mege, D., Di Bartolo, V., Germain, V., Tuosto, L., Michel, F. and Acuto, O., Mutation of

tyrosines 492/493 in the kinase domain of ZAP-70 affects multiple T-cell receptor signaling

pathways. J Biol Chem,1996. 271: 32644-52.

38 Brockdorff, J., Williams, S., Couture, C. and Mustelin, T., Dephosphorylation of ZAP-70

and inhibition of T cell activation by activated SHP1. Eur J Immunol,1999. 29: 2539-50.

39 Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P., Numerical Recepies

in C: the art of scientific computing, 2 Edn., Cambrige University Press 1997.


84


85

3.2 On the requirement for high cooperativity in TCR triggering:

hypothetical mechanisms of TCR triggering.

Modeling the hypersensitivity of TCR triggering:

Coordination of alternate Triggering mechanisms.

João SOUSA1,2 & Jorge CARNEIRO1 1. Instituto Gulbenkian de Ciência, Oeiras, Portugal


Lisboa, Portugal.

Summary

Despite the increasing knowledge on the pathways involved in TCR signal transduction and T cell

activation, the molecular mechanism of TCR triggering by ligand, MHC-peptide complexes, is still elusive

and controversial. Central in this controversy is the stoichiometry of TCR-ligand complexes required for TCR

triggering. Using mathematical modeling we explore the qualitative and quantitative consequences for TCR

down modulation of some of the possible mechanisms of TCR triggering. We propose that both TCR

triggering by homo crosslinking of TCR-ligand complexes and TCR triggering by TCR-ligand crosslinking

with on co-receptor molecules are possible, albeit with different antigen dose dependences.

Acknowledgements:

José Faro for discussion and manuscript revision and Zvi Grossman for discussion of the work.


86


87

3.2.1 INTRODUCTION

The most important aspect of the activation of T cells is the recognition by the T cell

antigen receptor (TCR) of its ligand, the MHC-peptide complexes on the membrane of the

antigen-presenting cell (APC). Considerable progress has been made in the study of the

signalling events downstream of TCR triggering and engagement [1] but the mechanism of

TCR triggering still remains elusive and controversial. Central in the controversy is the

question if dimerisation (or formation of higher order complexes) is a necessary condition

for TCR triggering.

Indirect evidence in favour of the necessity of TCR oligomerization for triggering

came from structural studies suggesting the propensity of MHC molecules to dimerize [2-

4], from binding assays [5, 6] and from mathematical modelling [7, 8]. But the

interpretation of these observations is controversial [9], and ligand-dependent TCR

oligomerization is difficult to reconcile with the observations that T cells respond to very

low densities of their natural ligands (estimates range from 1 peptide per APCs for CD8 T

cells [10, 11] and 50-100 peptides for CD4 T cells [12, 13]), since under these conditions

multimolecular encounters should be highly disfavoured [14].

Elucidation of the initial events of TCR triggering has failed to resolve this

controversy. The earliest events identified in TCR triggering are ITAM phosphorylation by

Src kinases [15]. The Src kinases activity on the TCR complex might be the consequence of

the recruitment of these enzymes to eventual binding domains formed upon TCR

oligomerization, of the exposure of phosphorylation sites by ligand-induced conformational

change [16], of the exclusion of CD45 from the vicinity of TCR complexes [17], or of the

recruitment of CD4 or CD8 co-receptors [18-20] pre-associated with lck [21]. These later

hypotheses suggest that TCR oligomerization may be sufficient but not being necessary for

TCR triggering.

Mathematical analysis of the kinetics of TCR triggering and down-modulation as a

function of ligand presentation allowed the identification of a number of kinetic constraints

on the mechanism of TCR triggering [22]. The most salient of these constraints is the

requirement for hypersensitivity of TCR triggering to both receptor and ligand densities. In

this report we address the question if TCR oligomerization is a necessary condition to

describe these kinetic properties of TCR triggering by comparing a simple TCR-ligand

oligomerization model with a model of TCR triggering based in CD4-lck crosslinking with


88

single TCR-ligand complexes (the TCR-SAC crosslinking model).

3.2.2 RESULTS

3.2.2.1 Kinetic analysis of TCR triggering

A previous analysis of the kinetics of TCR down modulation [22] indicated that TCR

triggering is a highly co-operative process, a consequence of the ligand dose dependence

and sensitivity to low ligand presentations of TCR down-modulation [12, 13]. The

requirement for high cooperativity is indicative that TCR triggering is a complex

mechanism, possibly with several inter-playing factors and intermediates. The model was

formulated using several extensions to the simple Bachmann et al. model of TCR

engagement and down modulation [7]. The TCR in the surface of the T cell is free to

circulate between two pools in the cell membrane but only in one of those pools the TCR is

able to engage the ligand. Once engaged by the ligand the TCR can either dissociate non-

productively or be triggered. Triggered TCRs are then internalised and degraded. The

model includes a basal TCR production and loss from the cell membrane, which maintains

the resting levels of TCR expression in the cell. The model dynamic are:

TS →←φ 1

LALT trigr +→+ 2

→ ikA 3

→→ ss T 4

Where S is the density of TCR in a cell that is not directly accessible to ligand, T is the

density of TCRs directly accessible to ligand, and A is the density of triggered TCRs. All

densities in the model are relative to the TCR density of a resting cell (relative density –

r.d.), which was estimated by FACS as approximately 3×105 TCRs per T cell [13]. The

parameters in the model are λ, the ratio between the size of the compartments of the S and

T pools; φ, the per-capita rate of transport between the S and T pools; s, the per-capita rate

of TCR turnover in the membrane; koff is the per-capita dissociation rate of the TCR and

ligand; kon is the per-capita rate of TCR-ligand binding; rtrig is the rate of TCR triggering; h

is the kinetic order of TCR triggering and ki is the per-capita internalisation rate of

triggered TCRs.

The model assumes that TCR engagement is in a quasi steady state; hence the rate


89

of TCR triggering becomes a function of free TCR and MHC, which depends on the kon,

koff and ka (see appendix). One of the results of this model was that the rate of TCR

triggering (rtrigg) must be highly cooperative for the model to describe the experimental

data: hh

efftrig tLtTkr )()(= 5

Where keff is a constant dependent on kon, koff and ka and h is a hill coefficient (cooperativity

index). The minimal value of h required for the model to describe the kinetics of TCR

trigging is 4-5, which is relatively high. In the following sections, we will compare two

mechanisms of TCR triggering in their ability to describe the cooperativity of TCR

triggering and the kinetics of TCR down-modulation. One model is based on sequential

TCR homo-dimerization and the other on the interplay between a kinase and phosphatase

acting on the same substrate.

3.2.2.2 Sequential oligomerization of TCR-ligand complexes

Crosslinking mechanisms may, under particular circumstances, present

cooperativity of the type suggested for TCR triggering. Crosslinking models can be

relatively complicated and diverse due to the many combinatorial possibilities that yield

high order TCR-ligand complexes. One of the simplest models of TCR-ligand crosslinking

is the formation of a large TCR-ligand complex, with n TCR-ligand complexes, by

sequential addition of TCR-ligand monomers, following the mechanism:

on

off

k

k

T L TL→

+←

2

2

2( )TL TL TLγ

ρ

→+

←

3

3

2 3( ) ( )TL TL TLγ

ρ

→+

← 6

...

1( ) ( )n

a

n

kn nTL TL TL n A

γ

ρ−

→+ →

←

that replaces reaction 2 in the TCR triggering model. The parameter γ is the per-capita

rate of binding of an oligomer with a TCR-MHC-peptide monomer and ρ is the per-capita

rate of dissociation of that oligomer.

The kinetics of the model can be simplified assuming that each crosslinking step is

in equilibrium and that the product of each step can be neglected when compared to the one


90

of the reactants, i.e. that as the size of the oligomer increases, each additional

oligomerization step becomes less probable (ρj >> γj+1). Under this simplification, the rate

of TCR triggering (rtrig) by formation of an oligomer of TCR-ligand complexes of

stoichiometry n is approximated as: nn

efftrig tLtTkr )()('= 7

with keff’ a constant (see appendix). This equation shows a direct mapping of the

kinetic order of TCR triggering to the molecularity of the oligomer required for TCR

triggering. Increasing the size of the oligomer required for TCR triggering causes an

increase in the kinetic order of the triggering reaction. Considering the results of the

previous paper [22], it is trivial to prove that this quasi steady state model can fit the

experimental data for n greater than 5. Exploration of the parameter of the oligomerization

model, without assuming the equilibrium in the oligomerization steps, shows that as the

size of the oligomer increases from 2 to 4 (figure 1) the forward rate constants of

crosslinking increase (table 1) and the ability the model has to describe the experimental

data improves.

Figure 1 – Optimisation of the parameters of the homo-cross linking models. The models were fitted, as

described in methods, to the whole experimental data. Right column, the kinetics of TCR down-modulation

for two doses of ligand. Left column, TCR down-modulated for various ligand densities on the APC after 5

h incubation. The crosses in the right panel denote the 5h CD3 staining from the curves of the left panel. A:

Dimerisation model; B: Trimerisation model; C: tetramerisation model. The parameters used are in table 1.


91

Table 1: Parameters of the model of TCR triggering by

sequential dimerization of TCR-MHC complexes.

Model:

Parameter

Dimerization Trimerization Tetramerization

s / min-1 1.25×10-4 1.91×10-4 3.00×10-4

θ 1.79 2.06 1.78

φ / min-1 2.96×10-3 2.22×10-3 2.94×10-3

kon / r.d.-1 min-1 2.35×101 5.77×101 6.37×101

koff / min-1 2.48×102 2.50×102 2.50×102

γ1 / r.d.-1 min-1 1.42×105 8.38×104 4.45×104

ρ1 / min-1 1.63×10-3 4.39×10-1 1.28×10-1

γ2 / r.d.-1 min-1 - 1.63×106 1.17×106

ρ2 / min-1 - 1.93×10-1 6.29×10-1

γ3 / r.d.-1 min-1 - - 1.90×106

ρ3 / min-1 - - 1.05×10-1

ka / min-1 4.92×102 4.91×102 9.30×102

ki / min-1 9.38×10-2 4.19×10-1 9.80×10-2

χ2 0.23 0.16 0.15

Each crosslinking step adds a small delay to the kinetics of TCR triggering that

causes a loss of the sensitivity to low densities of ligand. To compensate the delays, the

forward reactions of crosslinking must be faster and irreversible as the size of the final

TCR-ligand oligomer increases (table 1). In these conditions the steady-state approximation

calculated above is no longer valid because the parameter sets required for the model to

describe the experimental data imply that 1−<< jj γρ . So the simple correspondence

between cooperativity and stoichiometry is lost. Moreover, the increase of the number of

intermediates implies a greater amount of MHC-peptide locked in the intermediate

oligomers, causing a decrease in the efficiency of serial triggering, which can be minimised

by fast rates of oligomerization and triggering. Forcing smaller crosslinking rate constants

of the oligomerization models yields increasingly poorer results for the low ligand dose


92

kinetics (figure 2) with little loss for the high dose of ligand presentation. Some slight

attenuation of these problems can also be achieved by other mechanisms of TCR homo

crosslinking, with fewer intermediate steps.

figure 2 - Effects of adjusting the crosslinking rate constant of the dimezation model, showing the

requirement of high crosslinking rates to reproduce the kinetics of low density of ligand presentation. Full

lines represent the model using the optimised parameter set in table 1. Dashed lines represent the model

using a crosslinking rate constant of 1×104; Dotted lines represent the model using a crosslinking rate

constant of 1×103.

3.2.2.3 The cycle of TCR triggering

Oligomerization is not the only mechanism that could explain the ultra-sensitivity of

TCR triggering. The triggering of TCR is a complex mechanism that involves kinases and

phosphatases acting on the same substrate [1] so it is possible that the ultra-sensitive nature

of TCR triggering is a consequence of a particular combination of activities of kinases and

phosphatases. Koshland and Goldbetter have proposed a mechanism, the zeroth-order ultra-

sensitive mechanism [23] that can explain very high sensitivities in biological systems that

have two enzymes operating with reverse activities as is the case of kinase-phosphatase

cycles.

We assume that the triggering of the TCR is dependent on the activity of two

enzymes, a kinase and a phosphatase. Enzyme EK, the CD4-associated src kinase, is

responsible for triggering of the TCR bound to ligand. Enzyme EP, the phosphatase, is

responsible for deactivation of the TCR once it dissociates from ligand. Both EK and EP

were assumed to follow simple Michaelis-Menten kinetics. Since the validity a quasi-steady

state approximation for the kinetics of EK and EP implies an instant conversion of substrates

into products, we relaxed this assumption by modelling explicitly the kinetics of enzyme

catalysis and binding and dissociation of substrate. The TCR engagement and triggering


93

mechanism is thus:

TLLToff

on

k

k

←→+

LAALCTLon

offaEK

offEK

KonEK

k

kk

Kk

Ek

+←→ → ←

→ ×

8

TCA aEP

offEP

PonEPk

Pk

Ek

→ ← → ×

where CK and CP are the complexes of TCR with enzymes EK and EP, respectively,

konEK, koffEK and kaEK, are respectively the enzyme EK TCR binding and dissociation and

TCR triggering by EK rate constants. The parameters konEP, koffEP and kaEP are respectively

the enzyme EP TCR binding and dissociation and TCR triggering by EP rate constants. The

rest of the parameters and variables are the same as in the crosslinking model.

This model capable of describing the experimental data (figure 3, table 2) provided

that EP is saturated even for the lower dose of ligand. Contrary to EP, EK does not need to be

saturated, and can be assumed as a first order reaction if the rate of TCR internalisation is

sufficiently low. The fastest reactions in the model are the enzyme binding reactions and

TCR phosphorylation and dephosphorylation. These reaction rate constants are

considerably slower than the crosslinking reactions of the previous model. The binding of T

and A to EK and EP can be assumed irreversible and the dissociation rate constants for the

enzymes have little contribution to the kinetics of the model provided they are small

enough.

The parameters of the model indicate that the TCR triggering cycle has properties

similar to the Koshland-Goldbeter mechanism [23], displaying hypersensitivity to low

ligand presentations. The model has a threshold near which a slight increase in the amount

of ligand density induces a strong response from the system (figure 4). As the ligand

density increases, the system becomes less sensitive to changes.


94

Figure 3 - Optimisation of the parameters of the TCR-SAC crosslinking model. The model was fitted, as

described in methods, to the whole experimental data. The parameters are listed in table 2. Right column,

the kinetics of TCR down-modulation for two doses of ligand. Left column, TCR down-modulated for

various ligand densities on the APC after 5 h incubation. The crosses in the right panel denote the 5h CD3

staining from the curves of the left panel.

Figure 4 - Sensitivity of TCR down-modulation to ligand presentation, measured by the average rate of

TCR down-modulation over the first 50 minutes of incubation. There is a threshold near 0.001, which is

equivalent to approximately 100 MHC-peptides. The rate of TCR down-modulation is most sensitive to

changes in ligand presentation between the threshold and 0.01, which are approximately 1000 ligand

molecules. The parameters of the model are presented in table 2.


95

Table 2: Parameters of the model of the cycle of TCR triggering

Parameter Estimated value

s / min-1 8.65×10-4

θ 1.62

φ / min-1 5.69×10-3

kon / r.d.-1 min-1 8.09×101

koff / min-1 2.99×102

konEK / r.d.-1 min-1 1.24×102

koffEK / min-1 1.46×10-8

kaEK / min-1 1.16×103

konEP / r.d.-1 min-1 4.55×103

koffEP / min-1 1.00×10-8

kaEP / min-1 1.37×100

ki / min-1 1.03×10-1

EK / r.d.-1 2.56×100

EP / r.d.-1 2.63×10-2

χ2 0.044

The model displays different parameter sensitivities for the two doses of ligands.

TCR down-regulation for the higher dose of ligand is mainly dependent on the turnover of

TCR, the TCR exchange rate constant, the activity of EK, the internalisation constant, and

the on and off rates of TCR-ligand binding. The down-regulation of TCR for the lower dose

of ligand is mainly dependent on the activities of EK and EP and on the on and off rates of

TCR-ligand binding. These results hold even is assuming a quasi-steady state for EK and EP.

3.2.3 DISCUSSION

A previous mathematical analysis of the kinetics of TCR down-regulation has

suggested that TCR triggering rate is hypersensitive to changes in both ligand and receptor

densities [22]. In the present work, we addressed if oligomerization of TCR-ligand

complexes is necessary to describe these kinetics properties of TCR triggering. To answer

this question, two candidate mechanisms were compared: an oligomerization mechanism

based on sequential crosslinking of TCR-ligand; and a mechanism based in the interplay

between a phosphatase and a co-receptor associated kinase, which was based on the

Koshland-Golbeter hypersensitive mechanism. The results show that both these models


96

describe the experimental data, albeit with different parameter requirements.

The limited size of the experimental data set used in this study, compared to the

number of parameters in the models, is very small and constitutes a weakness of this

analysis. As the complexity of the models and the number of parameters increase, the

significance of parameter estimation and model comparison decrease. For this reason, in the

this work the analysis and comparison of the more complex models of TCR crosslinking

and TCR oligomerization were kept strictly in a qualitative basis, using optimisation

techniques solely to explore the possible parameter regimes of the models. It is noteworthy,

however, that the fitting is based on very precise estimations of the variables. Hence, the

variables in the models are the average densities of different TCR species on the cell

membrane. The fittings minimize the difference between the sum of these variables as

predicted by the model and the average relative TCR density measured experimentally.

This average is based on the individual observations of the fluorescence intensity of

thousands of cells. Its value should be close to that of the true average value of TCR

density, according to the law of large numbers.

The most straightforward candidate for explaining hypersensitivity is the formation of

high-order TCR-ligand oligomers ([7], reviewed in [24]). For these crosslinking models to

describe all the kinetic data, they require high crosslinking rates, to minimise the delays, to

maximize the serial triggering efficiency and to compensate for the low density of TCR-

ligand complexes at low doses of ligand. This requirement violates the quasi steady state

assumption, invalidating the straightforward correspondence between the TCR oligomer

stoichiometry and the kinetic order. Another problem with crosslinking models of TCR

triggering is that diffusion of TCR-ligand oligomers in the interface between the two cells

is mostly likely limited. Lateral mobility of high order oligomers should be impaired by

their size and by the fact that they are made of several integral membrane proteins anchored

in the two cell membranes. If it is assumed that the formation of the first dimer of ligand-

TCR complexes is already diffusion limited (the theoretical result by Salzmann et al. [14]),

then the formation of the trimer or the tetramer, which are larger complexes with lower

diffusion rates, cannot be possible with a higher rate constant. In that case, diffusion

imposes an upper limit to the rate of formation of oligomers and therefore would limit the

rate of TCR triggering and down-regulation. These difficulties of the crosslinking

mechanism are more evident at the low ligand and TCR densities than at high densities.

The crosslinking model can easily describe the high dose of ligand with low crosslinking


97

rate constants; the requirement for non-realistic high crosslinking rate constants is imposed

by the kinetics of the low dose of ligand.

The model of the cycle of TCR triggering is based on a pair of enzymes with reverse

activity. This model also requires high binding rate constants with the enzymes EK and EP

with the TCR, which are nevertheless lower than the crosslinking rate constants of the

homo crosslinking models. The early events in TCR signal transduction are

phosphorylation of the TCR ITAMs and of enzymes such as src-family kinase lck, also

implied in TCR down-regulation [25, 26], and syk-family kinase ZAP-70. So it is possible

that the activity of EK might reflect the activity of lck or fyn on TCR ITAMs or on ZAP-70.

The phosphatase with reverse activity must act on the same substrate, also in early stages of

signalling. Possible candidates for this phosphatase are CD45 and/or SHP-1. CD45 is a

positive regulator of lck activity and was shown to dephosphorylate ITAMs of TCR-

complexes[27, 28]. SHP-1 is a negative regulator of TCR signalling that seems to target

preferentially ZAP-70 and syk [29]. The activity of ZAP-70 was demonstrated to be

dependent on the balance between phosphorylation by lck [30, 31] and dephosphorylation

by a phosphatase [31]. Simply by perturbing the balance between kinase and phosphatase

activity, Zap70 could be activated. This kind of regulation might be extensible to lck and

CD45, supporting the model of the cycle of TCR triggering presented here. The model only

makes predictions on the kinetics for EK and EP. EK or EP may be one kinase or phosphatase

or the combined activity of two or more kinases and phosphatases. Overall, we favour the

dependence of TCR down-regulation on CD45 and CD4-associated lck activity, both by the

regulation of lck activity by CD45 and by ITAM phosphorylation by lck and

dephosphorylation, possibly by CD45 [27, 28]. From the biological point of view, enzymes

EK and EP might be located in rafts, or closely associated with the TCR complex. The

preferential location of EK and EP in rafts would reduce the dependence of the triggering of

the TCR on the lateral diffusion of EK and EP in the membrane; therefore, it would facilitate

a finer regulation of TCR triggering.

The structure of TCR signalling complexes may support different triggering

mechanism that would operate under different conditions. Our results show that the TCR-

SAC crosslinking explains better the low dose curve of TCR down-regulation, whereas

both TCR-SAC crosslinking or TCR oligomerization models explain well the high dose

TCR down-regulation curve. TCR-SAC crosslinking could be very efficient at low ligand

densities, in conditions where diffusion would limit the rate of TCR oligomer formation,


98

and therefore it might be the favoured TCR triggering mechanism. At higher ligand

densities TCR oligomers may form and also trigger the TCR, which may or may not

interfere with triggering by TCR-SAC crosslinking. Moreover, since these two TCR-

triggers are different supramolecular complexes they could activate partially overlapping

downstream pathways. This possibility is not unlikely. Both anti-CD3 antibodies and

soluble MHC-dimers bearing agonist peptide can trigger the TCR, albeit eliciting different

T cell response patterns [32]. These different patterns could be partly explained by the fact

that anti-CD3 can trigger TCR oligomers only, whereas soluble MHC-dimers can elicit

both TCR dimmerization and TCR-SAC crosslinking. T cell responses to peptide

presentation is qualitatively affected by the presence or absence of CD4 co-receptor

molecules, and this qualitative effect is more evident in the response to partial agonists than

in the response to full agonists [33]. This difference can be interpreted if the affinity of

TCR for the full agonist, but not that of the partial agonist, is high enough to form TCR-

ligand complexes that reach densities at which TCR oligomers can be effectively triggered.

Under these conditions, the full agonist would be able to activate the cell by TCR-

oligomerization in the presence or absence of coreceptor, while the partial agonist would

predominantly activate the cell by TCR-SAC crosslinking, i.e. in the presence of the co-

receptor.

Depending on the steric hindrance of CD4-lck association with TCR oligomers, the

TCR triggering mechanisms could be or not mutually exclusive. Moreover the downstream

signals could also be qualitatively or quantitatively different. For simplicity, let us consider

that the triggering mechanisms are not mutually exclusive and that the formation of the

TCR triggers are independent. Several downstream signals could, conceptually, originate in

these conditions. Figure 5 shows the hypothetical ligand concentration range of response by

TCR-SAC crosslinking (RA) and TCR oligomer (RB) triggering mechanisms operating in a

nonexclusive manner. Some of the possible scenarios of the downstream responses for the

two mechanisms are: (i) If both RA and RB yield qualitatively the same downstream

response, then RA and RB could synergise (Figure 5-C) yielding together a greater response

than either one alone. (ii) If RB and RA are qualitatively different and do not interfere, then

at high ligand density, both responses would be present (Figure 5-D). (iii) If RA is inhibited

by RB, then at high ligand doses, RB inhibition would result in a bell-shaped response of RA

(Figure 5-E). (iv) If RA inhibits RB, then in the presence of signalling by TCR-SAC cross-

linking, RB would not be detectable (Figures 5-F). Scenario iii could explain, for example,


99

the regulation immune deviation [34, 35], where RA could favour Th2 differentiation and

RB, Th1 differentiation. Scenarios iii could also explain, for example, the postulated

responses of cells to antigen dose during negative selection [36-41], with the assumption

that RA promotes survival of negative selection and RB induces deletion or inhibits survival

signals.

Res

pons

e

Ligand dose

RA+ RB

RBRA

RA

RB

RA

RB

A

B

C

D

E

F

RARB

Figure 5 - Representation of the antigen dose-response curves possible with TCR crosslinking and TCR-co-

receptor crosslinking. A and B – Response to signalling via the TCR-SAC crosslinking (RA) and homo

crosslinking (RB) mechanisms, respectively. In the presence of both triggering mechanisms, some of the

possible scenarios are: C – Signals RA and RB are qualitatively the same and synergyse; D – Signals RA and

RB are qualitatively different and do not interfere; E – Signal RB inhibits RA; and F – Signal RA inhibits RB.

In conclusion, the mechanistic complexity of the TCR signalling cascades is such that

may allow the concurrence of alternative triggering processes. We suggest here that either

TCR oligomerisation or TCR-SAC crosslinking are sufficient to trigger at least some T cell

response, and thus neither of these mechanisms alone would be strictly necessary to explain

any observed T cell response. According to this view, the protocol the experimentalist uses

to induce activation of signalling cascades and score the response of the T cell may reveal


100

different aspects of this mechanistic complexity. This may be the source of controversy

surrounding the stoichiometry of the TCR triggering complexes and the role of co-receptors

in TCR triggering.

3.2.4 METHODS

The ordinary differential equations of the model were integrated numerically using the

implicit numerical method of Burlish Stöer [42]. The optimisation of the model parameters

was done using the amoeba method [42] combined with the Burlish Stoer integration

method. All parameter optimisations were performed using simultaneously the complete

data set: the data of the time course of TCR down modulation and the data of 5 h down

modulation of TCR for various ligand presentations. Confidence intervals were not

estimated because of the lack of adequate methods for complex non-linear models, such as

the ones presented here. Another reason is the small number of degrees of freedom

resulting from such a short sample of experimental points.

3.2.5 APPENDIX

The models presented are based in the previously published model of TCR

engagement and triggering, with equations:

( ) ))(1()()()( tSstTtS

tdtSd

−+−−= λϕ 9

( ) hh

eff L(t)tTktTstTtStdtTd )())(1()()()(

−−+−= ϕ 10

)()()( tAkL(t)tTk

tdtdA

ihh

eff −= 11

Where keff is defined as keff = ka (kon/koff)h, resulting from the assumption of a quasi

steady state for the conjugation of TCR with MHC.

For the sequential oligomerization model, the TCR triggering mechanism is expanded

as a sequence of crosslinking reaction steps. Consider Tj the oligomer complex of j TCR-

ligand complexes; the kinetics of formation and degradation of the jth oligomer are:

)1()()(

+−= jjdt

tdT j ωω 12

With )()()()( 11 tTtTtTj jjjj ργω −= − , where γ is the per-capita rate of binding of an


101

oligomer with a TCR-MHC-peptide monomer and ρ is the per-capita rate of dissociation of

that oligomer. The full model is thus:

( ) ))(1()()()( tSstTtS

tdtSd

−+−−= λϕ

( ) )()())(1()()()( tTLkL(t)tTktTstTtS

tdtTd

offon +−−+−= ϕ

∑

=

−+−=n

joffon jtTLktt)LTk

tdtTLd

2)()()(()( ω

1...2),1()(

)()(−=+−= njjj

tdtTLd j ωω

13

)()()()()( tTLkj

tdtTLd

nan −= ω

)()()()( tAktTLkn

tdtAd

ina −×=

∑

=

−=n

jj

Total tTLLtL1

)()()(

Assuming a quasi steady-state in the TCR crosslinking steps, together with

assuming that ρj >> γj+1, then the rate of formation of triggered TCRs via an oligomer of

TCR-ligand complexes of stoichiometry n is approximated as

nneff

nnn

j j

jnoff

non

a tLtTktLtTkkkn

dttdA )()(')()()(

2

=

××= ∏

= ργ

, 14

which maps the number of crosslinking steps to the cooperativity of TCR crosslinking.

The TCR triggering cycle is based in the interplay between a kinase and a

phosphatase, acting on the TCR. The model equations are:

( ) ))(1()()()( tSstTtS

tdtSd

−+−−= λϕ

( ) )()()())(1()()()( tCktTLkL(t)tTktTstTtS

tdtTd

PaEPoffon ++−−+−= ϕ

)()()()()(()( tCktEtTLktTLktt)LTk

tdtTLd

KoffEKKonEKoffon +−+−=

)()()()()( tCktCktEtTLk

tdtCd

KaEKKoffEKKonEKK −−=

15


102

)()()(()( tCktALktt)LAk

tdtALd

KaEKoffon ++−=

)()()(()()(()( tAktCktt)EAktALktt)LAk

tdtdA

iPoffEPPonEPoffon −+−+−=

)()()()()( tCktCktEtAk

tdtCd

PaEPPoffEPPonEPP −−=

)()()()( 2 tTtCtTLLtL KTotal −−−=

)()( tCEtE KTotalKK −=

)()( tCEtE PTotalPP −=

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105

3.3 Activation thresholds, adaptation and TCR triggering

An experiment by Valitutti et al. (Valitutti, Muller et al. 1996) shows how TCR down-

modulation and TCR levels can give T cells adaptable activation thresholds of the type

discussed by Grossman and colleagues (Grossman and Paul 1992; Grossman and Paul

2000; Grossman and Paul 2001). T cells were incubated with three cultures of APCs, two

where APCs were pulsed with a high dose of peptide and another where APCs were pulsed

with low doses of peptide. At a given time, one of the low dose of the T cultures with APCs

pulsed with low dose of peptide was washed of APCs and cultured with APCs pulsed with

the higher dose of peptide. The results show that these T cells undergo additional TCR

down-modulation; reaching TCR levels close to the cells cultured with APCs pulsed with a

high dose of peptide. Our mathematical model of TCR triggering is capable of fitting these

results and allows us to estimate the concentration of triggered TCRs in the surface of T

cells. A quantity not easily determined experimentally that is crucial for T cell activation

and signaling. The results of the model are summarized in figure 2.

0 100 200 3000

50

100

0

50

100

Time/min

%C

D3

stai

ning

Low

Low+high

High

A

B

Low

Low+high

High

Figure 1 – Prediction of the model together with data from (Valitutti, Muller et al. 1996). Briefly, the

experiment has the same setup as (Valitutti, Muller et al. 1995), hence all parameters and procedure in

(Sousa and Carneiro 2000) were used in these plots. T cells were incubated with APCs having either ~78

MHC-peptide complexes per APC (Low); with ~5282 MHC-peptide complexes per APC (High); or

initially with APCs having Low followed by APCs having High doses of MHC-peptide at 120 minutes of

incubation (Low+high). A – Predicted TCR down modulation in response to antigen presented by APC

measured by CD3 staining. B – Predicted density of triggered TCRs (measured by CD3 staining) in

response to antigen presented by APC.


106

For each addition of ligand presentation by APCs, the T cells respond accordingly by

triggering and down modulating more TCRs, and consequently activated TCRs accumulate

in the membrane and also (hypothetically) in intracellular compartments. For the sake of

simplicity we neglect in the following discussion the contribution of triggered TCRs in

intracellular compartments for signaling, based on the assumption that they reflect

qualitatively the dynamic of the triggered TCRs in the membrane.

How can the cell count the number of triggered TCRs? Looking at the kinetics of

triggered TCRs predicted by the model, it seems plausible that it reflects the requirement to

sustain signaling for at least 50 minutes, probably the time window necessary to accumulate

down-stream signaling intermediates. The peaks of triggered TCRs result from the balance

between the rate of TCR triggering and the internalisation and degradation of TCRs.

If sustaining triggered TCRs at the surface of the T cell is necessary for T cell

activation, then TCR signaling has an adaptable activation threshold resulting from the

regulation of the levels of TCR in T cells. A cell which has responded to a given ligand

density has consequently down-regulated a given number of TCRs. If TCR levels are not

allowed to recover, then the T cell will be unable to respond to the same density of ligands.

The cell will only be capable of responding to a higher ligand density, and will only be

activated if it has enough TCRs to sustain signaling for 50 minutes and if the ligand density

is high enough to allow this. Hence, the conjugation of TCR triggering and TCR down-

modulation gives T cells an adaptable activation threshold in accordance to the propositions

of Grossman and colleagues (Grossman and Paul 1992; Grossman and Paul 2000;

Grossman and Paul 2001). The significance of this adaptation mechanism for the

physiology of the T cell remains to be assessed.

The avidity model of thymocyte selection states that the primary selective pressure on

the lymphocytes is the avidity of their interaction with thymic APCs. Avidity is a broad

quality, which encompasses not only TCR-ligand interaction, but also other factors

conditioning T cell activation, such as co-receptor expression, kinase levels and adhesion

molecule expression. Hence, avidity depends on the affinity of the TCR to the ligand

presented on the APC, on TCR and ligand densities and on the general metabolic state of

the thymocyte and APC. All other things being considered equal, avidity in thymocytes at a

given differentiation stage can be assumed to be mainly a function of the net TCR-ligand

engagement and signaling. An adaptation mechanism that relies on the level of TCR

expression and controls the signal intensity might have important consequences for the


107

selection of thymocytes, as discussed by Grossman & Singer (Grossman and Singer 1996).

The TCR levels in thymocytes at a given stage would reflect the number of APCs engaged

per unit of time, giving the thymocyte a memory of previous stimulatory events. As

discussed later in this thesis, in the context of the maintenance of peripheral lymphocytes,

such a mechanism would act as a filter where cells that are easily and/or repetitively

stimulated are rendered unresponsive.

3.3.1 BIBLIOGRAPHY



Sci U S A 89(21): 10365-9.




Immunol 13(6): 687-98.

Grossman, Z. and A. Singer (1996). “Tuning of activation thresholds explains flexibility in the

selection and development of T cells in the thymus.” Proc Natl Acad Sci U S A 93(25): 14747-

52.

Sousa, J. and J. Carneiro (2000). “A mathematical analysis of TCR serial triggering and down-

regulation.” Eur J Immunol 30(11): 3219-27.



Valitutti, S., S. Muller, et al. (1996). “Signal extinction and T cell repolarization in T helper cell-

antigen- presenting cell conjugates.” Eur J Immunol 26(9): 2012-6.


108


109

3.4 Towards an analysis of the experimental settings to study TCR triggering by ligand

The Bachmann model (Bachman, Salzmann et al. 1998) is a very good approximation

to the TCR down-regulation kinetics with high dose of ligand, capturing only the most

significant aspects of the mechanism: TCR triggering and down-modulation. Our final

model is more complex, but approximates well the TCR down-regulation kinetics for both

the high and low doses of ligand. The model we propose can be seen as the unfolding of the

quasi-steady state approximation of the Bachmann model, increasing the domain of the

model to a larger set of experimental data. Fitting all the data published by Valitutti et al

(Valitutti, Dessing et al. 1995) is what uncovers the need for a high cooperativity in TCR

triggering.

These models are only approximations to the underlying mechanism of TCR

triggering. The criteria used to assess the validity, quality and scope of the models is the

ability to describe the experimental data and the biological correctness of the model

assumptions and properties. The ability to describe the data was assessed by fitting the

model using χ2 optimising techniques. The error in the χ2 fit was loosely used as a criterion

on how well the model was able to describe the experimental data. To complement this

quantitative analysis, the models also had to be biologically coherent. Models, which

clearly went against or did not reproduce what is known about the TCR down-regulation

mechanism, were discarded.

The most frequent criticism to our models relate to the robustness of the experimental

data used to compare the different models. The use of fittings to seven experimental data

points per kinetic curve and five data points for the five hours down-regulation seems, at

first sight, too feeble to attain a reliable comparison between the various models. The error

structure of the experimental data set is nevertheless peculiar. Each experiment has

associated a systematic error resulting from the initial set up of the cell culture, in addition

to the error associated with the measurement of time.

Typically each FACS data point is the average of thousands of single cell events.

Hence, each one of the 19 data points comprehends a large number of observations, which

means a very high precision in the estimated CD3 expression average. Classical statistics

are not adequate to characterize this data set, in which each data point is dynamically


110

related to the previous one. In light of this limitation, we compared the models using both

quantitative aspects provided by the statistics of the fittings and qualitative criteria based on

the biological aspects of TCR triggering. Clearly, robust methods to determine how many

experimental data points are necessary to determine the robustness of these models is

lacking.

The validity results and conclusions of our models depend on the precise estimation of

the amount of ligand presented on the APC. Valitutti et al (Valitutti, Muller et al. 1995)

show a linear relationship in the logarithms of peptide concentration and peptide loaded

(figure 1). A slope of about 0.7 indicates that peptide loading in that system resembles a

simple saturation curve; hence it should produce a fairly linear Schatchard plot. This linear

dependence might not be valid for different peptide ranges and experimental settings. Some

experimental studies of T cell activation and TCR triggering and down-modulation rely on

the assumption that there is a linear relationship between peptide in solution and peptide

presentation by the APC, which might not be entirely valid. The ligand presented to the

APC depends not only on the peptide in solution but also in the affinity of the MHC for the

peptide. Thus the peptide dose dependence of T cell activation cannot be correctly

interpreted without a study of peptide loading by the APCs.

Peptide concentration/µΜ

MH

C-p

eptid

e

100 1000 10000

100

1000

10000

y = m x + bm = 0.7b = 0.9r2=0.97

Figure 1 – Linear correlation (in log x log space) between peptide in solution and peptide presentation in

MHC context (data from (Valitutti, Muller et al. 1995)).

Another aspect pertinent to the conclusions of the model is the kinetics of T cell and

APC conjugation. Some results suggest that for a class II T cell to be activated, a minimum

conjugation time of about 6-12 hours is required (Dustin, Allen et al. 2001). Another


111

possibility is that T cells can be serially conjugated by different APCs (Friedl and Gunzer

2001). Hence, we cannot exclude that the kinetics studied by Valitutti et al (Valitutti,

Muller et al. 1995) are the result of both APC “serial” conjugation and TCR serial

engagement. The central limit theorem states that on average, both APC serial conjugation

and APC conjugation for 6 hours produce the same results. Therefore, our models cannot

be used to distinguish the APC engagement mechanism. To address this question we need

to model explicitly the distribution of peptide-loaded APCs and T cells, their conjugation

kinetics and the TCR kinetics. This would be a model compromising two organizational

levels: the molecular level with the fast kinetics of TCR down-modulation, and the cellular

level, with the slow kinetics of T cell and APC conjugation.

3.4.1 BIBLIOGRAPHY



Dustin, M. L., P. M. Allen, et al. (2001). “Environmental control of immunological synapse

formation and duration.” Trends Immunol 22(4): 192-4.

Friedl, P. and M. Gunzer (2001). “Interaction of T cells with APCs: the serial encounter model.”

Trends Immunol 22(4): 187-91.



577-84.




112

Adaptable activation thresholds and lymphocyte homeostasis

113

4 Adaptable activation thresholds and lymphocyte

homeostasis

4

Adaptable activation thresholds and

lymphocyte homeostasis


114


115

Adaptable activation thresholds and homeostasis of lymphocytes

João SOUSA1,2, José FARO1,3, Zvi GROSSMAN1,4,5 and Jorge CARNEIRO1 1. Instituto Gulbenkian de Ciência, Apartado 14 P-2781 Oeiras Codex, Portugal.


Lisboa, Portugal. 3. Departamento de Fisica Aplicada, Universidad de Salamanca, 37008. Salamanca, Spain.

4. Laboratory of Immunology, National Institute of Allergy and Infectious Diseases,

National Institutes of Health, Bethesda, MD 20892.

This mauscript is still under discussion for publication and was not reviewd by all collaborators.

Hence, this manuscript does not represent the consensus of all collaborators mentioned above.

Summary

Adaptation of activation thresholds is a possible explanation for the persistence of significant numbers

of auto-reactive T cells in the periphery of normal individuals in the absence of autoimmune disease. We

build a simple mathematical model of adaptable T cells and explore the implications of adaptation of

activation thresholds for homeostasis of auto-reactive lymphocytes. We study how the activation threshold

and homeostatic responses of autoreactive T cells depends on the frequency of interactions with APCs. We

show that a critical population size exists, bellow which, the cells are unresponsive and are prevented from

making appropriate homeostatic responses or from undergoing clonal expansion. Above this critical size the

population of autoreactive cells will grow until it is limited by APC availability. As long as T cell homeostatic

proliferation and T cell activation are dependent on the same factors and are adaptable, adaptation of

activation thresholds cannot explain per se the persistence of auto-reactive lymphocytes. Instead it offers a

means by which the immune system may purge autoreactive cells from the peripheral repertoire as long as

their production is relatively small.

Acknowledgements:

This work was supported by Program Praxis XXI of the Ministério para Ciência e Tecnologia, Portugal

(grant Praxis/P/BIA/10094/1998). JS, JC, JF and ZG were respectively supported by fellowships

BD/13546/97, BPD/11789/97, BCC/18972/98 and BCC/16442/98 from Fundação para a Ciência e

Tecnologia - Program Praxis XXI.


116

4.1 Introduction The main tenet of Clonal Selection Theory is that self-reactive lymphocytes must be

prevented from mounting deleterious autoimmune responses [1]. According to Burnet’s

original proposal this would be achieved by a putative mechanism that would delete auto-

reactive lymphocytes once and for all during embryonic development. The fact that the

generation of lymphocytes is a life long process in mammals invalidated this possibility.

After an early suggestion by Lederberg [2], deletion of potentially harmful specificities was

reformulated as a process of lymphopoiesis. Hence, following random rearrangement of

antigen-receptor genes lymphocytes that express an auto-reactive receptor would be deleted

at an immature stage of their development, before they could trigger destructive effector

functions. This hypothesis was later substantiated by observations in animals expressing

endogenous superantigens (reviewed in [3]) or expressing transgenic TCRs [4].

Intriguingly, absence of deletion has never been formally correlated with autoimmunity, in

contrast with some surprising reports that deletion of particular lymphocyte specificities

can actually enhance immune responses (eg. [5, 6]).

The major shortcoming of deletion models of self-tolerance is the well-documented

presence of autoreactive cells in normal healthy animals [7, 8]. Adoptive transfers into

syngeneic recipients reveal the potential for these autoreactive cells to cause autoimmunity

(reviewed in [9]). These observations raise the question of how the presence of these cells

is harmless in healthy individuals. Several hypotheses have been evoked notably that

regulatory cells control the responses of potentially harmful cells [8] or that autoreactive

cells become unresponsive or anergic [10]. While these two hypotheses are not mutually

exclusive [11, 12], the second hypothesis could offer a very simple cell-based explanation

for self-tolerance, if supported by a convincing mechanism for the generation of antigen-

specific unresponsiveness in peripheral cells. Among the possible explanations for

unresponsiveness or anergy in peripheral autoreactive cells, perhaps the simplest is the

hypothesis that lymphocytes adapt their activation thresholds to continuous stimuli [13-15].

According to this hypothesis autoreactive lymphocytes would become unresponsive by up-

regulation of their activation threshold because of continuous stimulation by peripheral

self-antigens. This hypothesis is supported by recent findings that the sensitivity of T cell

activation and TCR-signaling can be modulated in T-lymphocytes [16-19], that the


117

persistence of peripheral T-lymphocytes requires continuous TCR stimulation by self-MHC

ligands [20-23] (which may even be non-classical MHC ligands in the case of memory cells

[24, 25]), and that tolerance to peripheral tissues requires their continuous presence [26].

In this article we investigate the plausibility of the hypothesis that continuous

stimulation by self-antigens would be responsible for the persistence of a significant

number of auto-reactive T-lymphocytes, which become unresponsive following up-

regulation of their activation thresholds. We first propose a minimal model describing the

adaptation of activation thresholds (AAT) by single T-cells. Using this model we show that

T-cells could be more or less sensitive depending on the frequency of conjugation events

with APCs carrying TCR-ligands. We then place this single cell model in the context of a

population dynamics of interacting T-cells and APCs. In these ideal cells, the TCR signals

during conjugation with APC are responsible for activation, survival, proliferation and

adaptation of activation thresholds.

4.2 Results

4.2.1 THE AAT SIGNALLING MODEL

T-cell activation requires conjugation with an APC, expressing on its surface MHC-

peptide complexes able to ligate and trigger the TCR. The signal transduction processes

triggering T cell activation are complex and not yet fully understood [27, 28]. They involve

the interplay of intermediate molecules, such as Zap-70, lck, CD45, LAT, etc. [27, 29, 30].

The activation state of these signaling intermediates results from a dynamic balance

between positive signals and negative signals. Usually the positive signals reflect kinase

activities and the negative signals reflect phosphatase activities, as indicated by studies in

which T cells are more prone to activation by targeting phosphatase activities with

inhibitors [31-33].

In the present work, we try to capture these features of signal transduction cascades of

T cell activation dependent on TCR engagement and triggering, using the minimal model

illustrated in fig.1. This model describes a pathway of four molecular components: TCRs

bound by MHC-peptides which constitute the stimulus (S); two downstream enzymes, a

kinase (K) and phosphatase (P); and an adapter molecule that can be inactive (W) or active

(W*), acting as a molecular switch of T cell activation. The concentration of these

molecules in time will be measured as S(t), K(t), P(t), W(t) and W*(t) respectively.


118

W

W*

K P

RR

LL APC

T cell

+ +

Figure 1 - A minimal model of TCR signal transduction.

Upon TCR engagement by the specific ligand (L) (MHC-

peptide expressed in the APC membrane) the TCR is

activated. Activated TCR promotes the build-up of

intracellular kinase (K) and phosphatase (P), which regulate

the activation state of an adapter molecule (W). Depending

on the relative activities of K and P, the adapter is either

inactive or active. The later state leading to T cell activation

and proliferation.

We assume that the kinase and phosphatase (the main actors in our model) are in

constant turnover with a steady state dependent on the level of TCR binding. The basal

rates of kinase and phosphatase production in the absence of TCR binding are respectively

pk and pp. TCR binding accelerates the production rates of the two enzymes, being the

relative increase proportional to the amount of bound TCRs, with proportionality constant a

assumed to be identical for both enzymes for the sake of simplicity. Irrespective of the state

of TCR binding the kinase and the phosphatase are degraded with constant rates per

molecule dk and dp, respectively. The dynamic of these enzymes are therefore described by

the following differential equations:

( ) )()(1)( tKdtSapdt

tdKKK −⋅+= (1)

( ) )()(1)( tPdtSapdt

tdPPP −⋅+= (2)

The amount of bound TCRs is treated as an input variable in this model. Thus, the

concentration of bound TCRs is described as an arbitrary function of time. Heretofore we

will refer to the concentration of bound TCR as the stimulus. As to the adapter molecule,

we postulate that total concentration of the adapter is constant, and that the fraction of

activated molecules is in quasi-steady state, being a step function of the ratio between K(t)

and P(t):

<≥=

+=

1)(/)(01)(/)(1)(*

)(*)(

0

0

tPtKiftPtKif

WtW

tWtWW (3)

The adapter behaves as a molecular switch for T cell activation. Hence, the T cell will

be activated when the adapter is active and will rest when the adapter is inactive. This set-


119

up was inspired in the Koshland-Golbeter mechanism [34], which relies on the interplay in

the activities of a kinase and phosphatase, and can lead to ultra-sensitive molecular

response. From the point of modelling, other mechanisms could as well be responsible for

the hypersensitive activation of W as a function of K(t) and P(t) but the postulate of this

step function has the advantage of simplicity, without compromising the generality of the

qualitative results.

The model is rendered non-dimensional for the concentration variables by redefining

them as:

( )KK dptKt

/)()( =κ , ( )KK dp

tPt/

)()( =π , )()( tSat ×=σ and 0

)(*)(W

tWtw = (4)

where pK/dK is the basal steady-state of the kinase. In this way, the model simplifies to:

( )[ ])()(1)( ttdt

td κσφκ−+= (5)

( )[ ])()(1)( ttdt

td πσζλφπ−+= (6)

where φ ≡ dk is the relative turnover rate of the kinase, λ is the ratio between the basal

turnover rates of the phosphatase and kinase (dP/dK) and ζ is the ratio between the steady

state concentrations of the phosphatase and the kinase (pp/dp)/(pk/dk). Note that because we

assumed that the enhancement of the production rates of kinase and phosphatase due to the

stimulus is identical, the ratio between the steady states of the phosphatase and the kinase is

always constant irrespective of the value S(t) of the stimulus. The original variables, non-

dimensional variables and parameters are listed in Table 1.

4.2.1.1 Activation threshold, adaptation and refractoriness

According to the adaptation of activation threshold hypothesis, a cell under constant

chronic stimulus should not be activated. This property can be encoded in the model by

fixing the steady state of phosphatase above the steady state of kinase

1>ζ (7)

This means that at constant stimulus a steady state is ultimately reached in which the

adapter molecules are always switched off. Following a change in the magnitude of the

stimulus, the cell adjusts the levels of kinase and phosphatase to a new steady state. In order

to switch on the adapter in response to a change in stimulus, κ(t) must transiently be greater

than π(t), while the cell progresses to the new steady state. Whether the adapter will be


120

switched on depends both on the maximal intensity and rate of increase of the stimulus

changes.

Table 1 – Variables and parameters of the minimal model of TCR signal transduction

Variable Meaning Units Non-dimensional variable

K(t) Concentration of the kinase M κ

P(t) Concentration of the phosphatase M π

S(t) Concentration of ligated TCR / stimulus

M σ

W*(t) Fraction of activated

Adapter molecule

Non-dimensional -

Parameters Meaning Units Value

pk Production rate of kinase M s-1 -

dk Relative catabolic rate of kinase s-1 -

pp Production rate of phosphatase s-1 -

dp Relative catabolic rate of phosphatase

s-1 -

φ Baseline turnover rate of kinase (dk) s-1 0.025

λ Ratio between the baseline turnover rates of the phosphatase and the

kinase (dp/dk)

Non-dimensional 0.05

ζ Ratio between the baseline concentrations of the phosphatase

and the kinase (pp/dp)/(pk/dk)

Non-dimensional 2

Simulations of the AAT model with the reference parameters values listed in Table 1

are presented on fig. 2. Upon T cell conjugation with stimulatory APCs, a stimulus is

generated and one of two types of response are possible: either the stimulus is large enough

to lead to a super-threshold stimulation (κ(t)/π(t) ≥ 1) and switch the adaptor to the active

state (fig 2-left); or the stimulus is insufficient, resulting in a sub-threshold stimulation

(κ(t)/π(t) < 1) such that the adaptor remains inactive (fig 2-right). Both super- and sub-

threshold interactions change the kinase and phosphatase state of the T-cells and therefore

modify the activation threshold of the cell.


121

Time / mins0 500 0 500

0

0

0

0

1

3

15

11

W*/W0

κ/π

κπ

σ

κ

π

A

B

C

D

κ

π

Ι ΙΙ ΙΙΙ

Ι ΙΙ ΙΙΙ

Ι ΙΙ ΙΙΙ

Ι ΙΙ ΙΙΙ

Figure 2 - Dynamic of the minimal model of TCR signal transduction. The left panel shows a super-

threshold stimulation and the right panel a sub-threshold stimulation. A - Ligand concentration; B - Kinase

(solid lines) and phosphatase (broken lines) concentration in response to ligand; C - Ratio between the

kinase and phosphatase activities; D - Activation state of the adapter molecule (W*). The three phases that

characterise the κ and π response are: I – The response to an increase in stimulus from a basal level; II –

The cell begins to adapt to the increased level of stimulus; and III – The refractory phase, when the

stimulus decreases to basal levels and the threshold is augmented. See text for details.

The response of a cell to a super-threshold stimulus is triphasic (fig 2-B,C). There is an

initial phase (response phase - I) where the kinase grows and overshoots the concentration

of phosphatase, leading to the activation of the adapter if the kinase overshoots the

phosphatase. The second phase (adaptation phase - II) initiates after the concentrations of

kinase and phosphatase reach a maximum. In this phase, the cell becomes adapted to the

stimulus and the ratio between the kinase and phosphatase concentrations converge to the

basal level. The third and final phase (refractory phase - III) starts when the stimulus

decreases or disappears. During this phase the threshold for switching on the adapter is

augmented relative to the adaptation phase (phase II) because the kinase decreases faster


122

than the phosphatase, so a stimulus larger than the original one is necessary for the kinase

to overshoot the phosphatase. After the refractory phase, the cell returns to the initial state.

The memory a cell has of the previous encounters with antigen lasts for the duration of the

refractory phase.

4.2.1.2 AAT and frequency of the stimuli

The physiology of a T cell is such that it will have periods of conjugation with an

APC, which is loaded with stimulatory peptide and periods in which the T cell is detached

from APCs, receiving no stimulation. To simplify we will neglect interaction with APCs

that are not expressing the specific MHC-peptide. In order to have an analytic insight into

the behaviour the model in such conditions, lets assume that the cell is periodically

switching from receiving a stimulus σ(t)=σmax for an interval of time τs and receiving no

stimulus σ(t)=0 for an interval of time τ0. Under these conditions, a periodic oscillatory

state is attained where the kinase and phosphatase concentrations vary within a range that

depends on the intervals τ0 and τs, and the magnitude of the stimulus σmax. An analytic

solution for the orbit of the system can be obtained by piece-wise integration of equations 5

and 6.

Consider the nth period of stimulation and non-stimulation starting at instant tn =

n(τs+τ0). For an interval τS, the stimulus is positive (σ = σmax) and the analytical solution of

the system is:

( )

( ) ( )( )

+−+=+

+−−+=++−+

+−

)()(

)(

11)()(

)(11)(tt

maxtt

nn

ttmaxnmaxn

nn

n

eettt

etttφλφλ

φ

σζππ

σκσκ (8)

where κ(tn) and π(tn) are the concentrations of the kinase and phosphatase at the

beginning of the nth period (t = tn). After the interval τS, the stimulus ceases (σ = 0) for an

interval of time τ0 and the solution of the system is then:

( )

( )( )

−++=++

+−−=++++−++

++−

)()(

)(

1)()(

)(11)(tttt

snsn

ttsnsn

snsn

sn

eettt

etttτφλτφλ

τφ

ζτπτπ

τκτκ (9)

where κ(tn+τS) and π(tn+τS) are the values of κ(t) and π(t) when the stimulation ceases.

Because the system is periodic, the values of κ(tn), π(tn), κ(tn+τS) and π(tn+τS) are the same

for any nth period. Applying this condition, we obtain:

( )

( )

−−

−+=+ ++

++

sns

sns

n

n

ns een τττφ

τττφ

σττκ )(2

)(

max 1111))(( (10)


123

( )

( )

−−

−+=+ ++

++

sns

sns

n

n

ns een τττφλ

τττφλ

σζττπ )(2

)(

max 1111))(( (11)

( )sns

ns

n

n

sns een τττφ

ττφ

στττκ ++

+

−−

+=++ )(2

)(

max 111))(( (12)

( ) ,...2,1,0)(2

)(

max 1

111))(( =++

+

∀

−

−−+=++ nn

n

sns sns

ns

een τττφλ

ττφλ

σζτττπ (13)

With these solutions for the initial values of each stimulatory and relaxing period

together with equations 8 and 9, we fully define the time evolution of the cascade leading to

switching on or off the T cell activation. Additionally, we need an assumption that

determines whether a cell will be activated when it detaches from the APC and the

stimulatory interval τs ceases. We will assume, for the sake of simplicity, that the T cell

will be activated if the adapter is switched on when it detaches from the APC. The T cell

will not be activated if the adapter is inactivated when the cell detaches from the APC, even

if during the conjugation the adapter was transiently switched on.

The behaviour of the model in response to such a periodic stimulation schedule is

illustrated in fig. 3, for the particular case where τs=τ0. At low frequency stimulus (left)

there is little overlap between the refractory period and the period of stimulation, so the

system responds to each rise in the stimulus. For frequent enough stimulus (right) the

refractory period is longer than the interval when there is no stimulus, and the cells are still

refractory when the stimulus becomes positive again. The cell is adapted and recurrent

increases in the stimulus do not result in switching on the adapter molecule.

Due to the threshold of activation and to the adaptation, for each set of parameters,

there is an optimal duration of the stimulus τs for switching on the adapter and leading to

activation (fig. 3-bottom). There may be no activation if the period of the stimulus is too

brief, such that there is not enough time for the kinase to supersede the phosphatase. In the

other extreme, the duration of the stimulus may be too long such that adaptation occurs and

W is switched off after a transient activation.

In summary, the AAT model predicts that at low frequency of encounters between a T

cell and a stimulatory APC the T cell remains responsive. In contrast, when the frequency

encounters with a stimulatory APC is high enough the cell may become refractory and

unresponsive.


124

Time / mins

0 4000 0

0

0

0

0

1

2

100

55

4000

κ

π

κ

π

A

B

C

D

W*/W0

κ/π

κπ

σ

Figure 3 - Periodic stimulation of the minimal TCR signal transduction pathway. The left panel shows a

low frequency stimulation unable to fully adapt the system and capable of producing periodic super-

threshold stimulations. The right panel shows a high frequency stimulation, which is capable of inducing

adaptation in the system. A - ligand concentration; B - Kinase (solid lines) and phosphatase (broken lines)

concentration; C - Ratio between the kinase and phosphatase activities; D - Activation state of the adapter

molecule (W*).

4.2.2 DYNAMICS OF A POPULATION OF LYMPHOCYTES WITH AAT

De Boer and Perelson [35] have proposed previously a model for T cell population

dynamics, where activation, proliferation and survival of the cells are dependent on

conjugation with an APC site. Using this model they have shown that homeostatic

regulation of cell population sizes is a natural expectation from the density-dependent

proliferation/survival response that follows from competition among T cells for APC


125

accessibility. In this section, we use our AAT model to ask how a putative adaptation of the

homeostatic response threshold would change this model T cell population dynamics. The

postulated processes are illustrated in the following diagram:

*)1(

TCAT d

d

c

ff → ←

→+ ⋅

−

α

α

(14a)

TT p 2* → (15b)

→mfT (16c)

Here it is assumed that the T cells and APCs are homogeneously distributed in space

and that the APCs are all providing the same stimulus (σ). The total number of APC sites is

constant and denoted A. The numbers of free APC sites, free T cells, conjugated T cells and

total T cells change as a function of time and are denoted as Af(t), Tf(t), C(t), and T(t),

respectively. Free T cells form conjugates with free APCs with rate that is proportional to

their respective numbers, with rate constant c. Conjugated T cells detach from the APCs

with rate constant d. The probability that a T cell deconjugated from with super threshold

stimulation is denoted by α. Activated T cells (T*) divide with a rate constant p. Resting T

cells die with a death rate constant m, an assumption that is supported by observations in a

class II conditional knockout [20]. Since only resting T cells die, conjugation with APCs

and activation rescues cells from death.

In the De Boer & Perelson model [35] the probability that a cell gets a suprathreshold

signal is the same for all the cells, which means that the fraction of cells that are activated

during conjugation is a simple constant:

adef=α (17)

Embedding the AAT signalling model within these population dynamics framework

amounts to define α as a function of the history of stimuli received by the cells. If we keep

track of the kinase and the phosphatase values in each individual cell i respectively κi(t) and

πi(t), we can define α as:

)()1( drrfrdef

⋅⋅−= ∫θα with )()( ttr πκ= (18)

where θ is a step function returning 0 when the argument is negative and 1 otherwise,

and f(κ(t)/π(t)) is the frequency distribution of the ratio between kinase and phosphatase

activities in the population of T cells at instant t.

Despite being conceptually straightforward, the AAT population dynamics model,


126

formulated in this way, has an infinite state space that is not analytically tractable. To gain

insight into the properties and behaviour of this system, we followed to complementary

approaches. First, we simulated a population containing many individual T cells that realise

the Poisson processes above (model of Many Individual Cells – MIC model); second, we

derived a simpler quasi-steady state model (QSS model) describing the dynamic of the size

of the total T cell population. This quasi-steady state model uses simulation to obtain a

stationary frequency distribution of κ / π values and the corresponding value of α. The two

methods and their results are presented bellow. These results were obtained assuming, for

the sake of simplicity, that the rate of T cell proliferation is instantaneous (p = Infinity). We

also studied those cases where p is positive as well as implementations of the model where

the process of proliferation was described by an explicit delay, instead of a simple Poisson

process. The qualitative results presented bellow were systematically obtained with all

these versions and thus we believe they are general. Under these conditions we opted to

present only the results of the simplest case.

4.2.2.1 Methods for simulation of many individual cells and quasi-steady state

analysis

The simulations (see Appendix) follow the values of κi, πi and σi in a variable number

T of individual cells indexed i. A cell i has σi = 0 when it is free and σi = 1 when it is

conjugated with an APC, and the dynamic of κi and πi, followed according to eqns. 5 and 6.

The processes described in equations 14a-b take place at small discrete time steps τ. A free

cell has a constant probability τ × m of dying (being removed from simulation) and a

variable probability τ × c × Af of conjugating with any one of the Af free APCs. At the

instant of conjugation, we decrement by one the number of free sites Af. A conjugated cell

has a probability of dissociating from the APC and becoming a free cell of τ × c, per

iteration. When a cell i is released from the APC we reset the value σi = 0 and increment the

number of free sites Af. If, at the instant of T cell-APC detachment, κi / πi ≥ 1, then we

substitute the cell by two daughter cells which inherit the properties of its parent, i.e. they

have the same values of κ, π and σ. To simulate the case in which there is no adaptation,

we reset the values of κi and πi of each cell i at the instant a T cell-APC deconjugation.

To get better insight into this simulation we made a quasi-steady analysis of the

system. Following De Boer & Perelson [35] and Leon et al. [36] we write the dynamic of


127

the number of T cells T(t) as:

( )CTmCddt

tTd ˆˆˆ)(−×−××= α (19)

Note that the term Cd ˆˆ ××α corresponds to the assumption that the process of cell

division is practically instantaneous, where α is the quasi-steady state fraction of cells

being activated following deconjugation from the APCs and C is the quasi-steady state

number of conjugates obtained as:

( ) ( )( )

cdATccTAdATc

ATC2

4),(ˆ

22 +++⋅⋅⋅−++≅ (20)

The problem we face is to specify this value of α as a function of the value of T(t)

when cells adapt their AAT. To solve this we tabled this function using Monte-Carlo

simulation as follows. We produce chains of conjugation periods with APC τ0 and free

periods τs, representing the history of individual cells in a population of lymphocytes cells

at a quasi-steady state, by drawing them from the corresponding distributions:

0)( 0ττ ddef −= (21)

( )( ) ( )( ) sATCAcs eATCAcg ττ ,ˆ,ˆ)( −−−= (22)

We solved the AAT signalling model for each chain of free and conjugation intervals

setting σ respectively to 0 and 1, and obtained the value of the ratio κ / π in the last

conjugation. Iterating until we obtained a stationary distribution f(κ / π), from which we

obtain α according to eqn. 18. Having tabled α and T values we obtained a cubic-spline

interpolation function that was used for numerical integration of eqn. 19. The constant

value of α = a for the case of non-adaptable lymphocytes simply as:

s

t

td dea d

u

s ττ

⋅= ∫⋅−

1

(23)

Where τu and τd are the time where κ(t) and π(t) are identical, i.e. they limit the

interval of time where κ(t) > π(t).

4.2.2.2 Results of simulation and quasi-steady state analysis of the AAT population

model

The results of the MIC and QSS models are co-plotted in fig. 4a. The reference

parameter set (Tables 1 and 2) was chosen to illustrate that the kinetic of the two solutions

can differ significantly, but that the steady states obtained by the two approaches are the


128

same. With the MIC model it is possible to infer the presence of unstable steady states but

we cannot measure them (fig. 4A). Using the QSS model, it is easy to obtain phase spaces

and the bifurcation diagram of the model (fig 4B).

Figure 4 - Results of model simulations. A – Comparison between population growth with out adaptation

using the QSS model (black line) and the SMIC model (grey lines). B – Simulations of the QSS (dashed

lines) and SMIC models (full lines) with adaptation, which show that both predict the same steady states.

C & D – Predicted steady states, using the QSS model, without (C) and with (D) adaptation. Strong lines

denote proliferation rates and light lines death rates. Stable fixed points are denoted as closed circles

whereas unstable fixed points are denoted with open circles.

The population of non-adaptable lymphocytes shows logistic growth kinetics, reaching

a nontrivial steady state whenever the proliferation rate constant (α × d × (A - C)) is greater


129

than the death rate (m). Otherwise the population will always go extinct. The number of T

cells in the persistent steady state is a linear function of the number of APCs sites, which

reflects that the size of the T cell population is the actual limitation in the number of APCs

sites A. Table 2 – Variables and parameters of the model of a popula tion of adaptable lymphocytes

Variable Meaning Units

T Total number of T cells Cells

TF Number of non-conjugated T cells Cells

AF Number of non-conjugated APCs Cells

C Number of conjugates between APCs and T cells Conjugates

Parameters Meaning Units Value

A Total number of APCs cell 5000

σAPC Antigen stimulus provided by APCs Non dimensional 1100

c APC-T cell conjugation rate constant cell-1 hour-1 0.001

d APC-T cell deconjugation rate constant hour-1 1

m Death rate constant of T cells hour-1 0.002

We then asked how the dynamic of the AAT lymphocyte population differs from the

prototypic dynamic of a non-adaptable lymphocyte population. Both the MIC and the QSS

solutions reveal that for the reference parameters there are two possible stable steady states

(fig. 4A). Analysis of the phase-diagram of the QSS model indicates that there is an

additional saddle point (fig. 4B). this saddle point, most of the cess dissociating from

APCs are mostly adapted, whereas above this saddle point there is a significant fraction of

these cells in a responsive state (Fig 5).

The nature of this saddle point brought into existence by AAT is better understood by

the bifurcations introduced by changing the parameter that controls the rate of change of the

intracellular AAT signalling molecules, φ (fig 6A). As φ increases to values higher then the

reference the cells tend to relax faster to baseline κ and π values after deconjugation. Thus,

the behaviour of adaptable cell population tends asymptotically to that of the non-adaptable

population, and the saddle point tends to extinction steady state until it disappears and

extinction become unstable. As φ decreases to lower values than the reference the cells

relax slower becoming more and more unresponsive. Accordingly, the population size in

the stable steady state decreases progressively. The persistence stable state and the saddle

point eventually fuse and disappear, while the extinction steady state becomes stable.


130

Figure 5 - Frequency distributions of k/p values in cells after deconjugation according to the QSS model

for total T cell numbers bellow (100 cells, grey) and above (3000 cells, black) the saddle point. The

parameters are the same as in fig.4D.

The bifurcation diagram as a function of the number of APCs around the reference

parameters shows (fig 6B), by comparison with the diagram for non-adaptable

lymphocytes, that the contribution of AAT is the appearance of the saddle point that

stabilizes the baseline steady state. In the presence of AAT the stable state representing

persistence continues to increase linearly with the number of APC sites (fig. 6B), which

indicate that the size of the population is also limited by APC availability.

Figure 7 shows bifurcation diagrams for the remaining parameters that control

adaptation of the signalling pathway, λ and ζ. There is a maximum value of that allows the

kinase to overcome the phosphatase transiently, yielding two stable states in the AAT

lymphocyte dynamics (fig. 7A). In the bifurcation diagram of (fig. 7B), two bifurcation

points are apparent; the first originates a saddle from the steady state denoting the collapse

of the population, leading to the stabilization of this steady state. The second bifurcation

occurs when this saddle meets the stable state representing persistence of T cells, leading to

the existence of a single stable steady state denoting the collapse of the population. These

two bifurcation points limit the interval of where is sufficiently high for adaptation to

become significant, but sufficiently low to allow the activation of the cells for division.

Figure 7C shows the regions of the plane where the AAT cell population dynamics have

either a single stable steady state denoting the collapse of the population; a stable and an


131

unstable steady states, denoting persistence of the population unadapted; and two stable and

one unstable steady states, denoting either the persistence of lymphocytes in an unadapted

or the collapse of lymphocytes due to adaptation.

figure 6 - Bifurcation diagram of the cell population model as a function of the numbers of APCs A (A)

and the rate of the AAT signalling model φ (Β). The solid lines represent stable steady states and the

dashed lines represent unstable steady states as predicted by the QSS model. The dots represent average

solutions using the SMIC model. The remaining parameters are set to the reference values.

Adap

tabl

e ac

tivat

ion

thre

shol

ds a

nd ly

mph

ocyt

e ho

meo

stas

is

13

2

λ10

-410

-210

0

ζ

110

C

λ

10-6

10-4

10-2

100

0

2000

4000

Number of cells

ζ

0

2000

4000

Number of cells

110A B

λ

10-6

10-4

10-2

100

0

2000

4000

Number of cells

ζ

0

2000

4000

Number of cells

110A B

I II

III

Figu

re 7

- B

ifurc

atio

n di

agra

ms

of th

e ce

ll po

pula

tion

QSS

mod

el a

s a

func

tion

of ζ

(A),

λ (B

) and

ζ a

nd λ

(C).

A a

nd B

– S

olid

line

s re

pres

ent s

tabl

e st

eady

sta

tes

and

dash

ed li

nes

repr

esen

t uns

tabl

e st

eady

sta

tes.

C –

Reg

ions

I, II

and

III l

imit

ζ an

d λ

para

met

er v

alue

s w

here

the

mod

el h

as a

sin

gle

glob

ally

sta

ble

stea

dy s

tate

den

otin

g

the

colla

pse

of th

e po

pula

tion;

is in

a b

ista

ble

regi

me,

den

otin

g ei

ther

the

colla

pse

of th

e po

pula

tion

or th

e pe

rsis

tenc

e of

cel

ls; o

r has

a si

ngle

glo

bally

stab

le st

eady

stat

e,

resp

ectiv

ely.

The

solid

line

s rep

rese

nt th

e λ

and

ζ va

lues

in A

and

B.


133

The appearance of a saddle point under the AAT model is easily amenable to

qualitative interpretation. In a non-adaptable lymphocyte population, as the cell density

increases the average duration of the period the cells remain without interacting with the

APC τ0 increases (eqn 22). The frequency of stimulatory interactions with the APC

decreases and the death rate increases accordingly. This creates the negative feedback loop

responsible for the stability of the steady state. Adaptation of activator thresholds brings

into existence a positive feedback loop and the corresponding saddle point. As the average

period individual lymphocytes remain without interacting with the APC increases with

lymphocyte density, there is more time to reset the activator threshold. In other words, as

lymphocyte density increases the individual cells become, on average, less refractory to

activator signals. Reciprocally, the lymphocytes become increasingly more refractory as the

cell density decreases. Hence, the survival / proliverative responsiveness of adaptable

lymphocytes is the inverse of what would be expected from an homeostatic regulatory

mechanism, in contrast with competition for available APCs.

In conclusion, the embedding of an AAT signalling model within the dynamics of a

population of lymphocytes that respond to APCs with increased survival and proliferation,

indicates that adaptation brings in a positive feedback loop and a critical value. Bellow this

critical value the population becomes increasingly refractory and ultimately goes extinct if

not supplemented by an external source. Above this critical value the lymphocytes become

increasingly more responsive until they reach a steady state limited by external factors,

such as the availability of APCs as studied in the present case.

4.2.2.3 Thymic influx of refractory and responsive lymphocytes

The conclusion of the previous section was that according to the AAT population

model, sufficiently small populations of autoreactive T cells will be prevented from

expanding and will eventually go extinct. Hence, the model offers a mechanism for

preventing autoimmune disease. In this section we ask how robust is this mechanism of

tolerance concerning the magnitude of thymic influx.

We performed simulations of the MIC model which start with zero cells and allow for

a constant influx of new cells at a constant rate of s cells per unit of time (fig. 8A). We

explored two extreme situations, one in which the cells come out of the thymus adapted and


134

refractory to peripheral stimuli or fully responsive. In the first case, for sufficiently small

influx rates a non-zero steady state in which most cells are refractory/unresponsive to the

stimuli is reached. This baseline steady state corresponds to the extinction observed in the

absence of an influx. The number of cells in this steady state increases linearly with the

influx rate until a critical influx rate is reached (fig.8B). At influx rates higher than this

critical value the population of cells systematically grows to the steady state in which most

cells are responsive and the size of the populations is limited by APC accessibility (fig.8A

and 8B). This autonomous persistence state is the same as before as long as the influx is not

significant (~ 3535 cells). The critical influx rate corresponds to the saddle point created by

AAT, and thus is affected by the parameters that determine adaptation and refractoriness.

Cell refractoriness changes monotonously with the value of φ around the reference

parameters. When the influx is zero the saddle point converges monotonously to the

extinction state as φ increases and to the autonomous persistence state as φ decreases (fig.

4D). The value of the critical influx that determines that the system will switch from

baseline to autonomous persistence steady states changes inversely with that of φ (fig. 8C).

In the case all cells produced by the thymus are fully responsive the system will

systematically attain the autonomous persistence steady state in which most cells are not

refractory and in which the population size is limited by APC accessibility (fig. 8B-white

dots).

In conclusion, the implementation of AAT population indicates that the adaptation of

stimulatory thresholds by recurrent interactions with self-antigens in the periphery offers a

filter to delete self-reactive T cells produced by the thymus, i.e. to prevent clonal expansion

in response to self-antigens. However, this filter is only efficient if recent thymic emigrants

would be pre-adapted to self-antigens in the thymus, and if the contribution of the thymus

to the size of the periphery is relatively minor.

Adap

tabl

e ac

tivat

ion

thre

shol

ds a

nd ly

mph

ocyt

e ho

meo

stas

is

135

Figu

re 8

– Im

pact

of t

hym

ic in

flux

on th

e A

AT

popu

latio

n dy

nam

ics.

A. K

inet

ics o

f the

size

of t

he p

opul

atio

n fo

r the

indi

cate

d va

lues

of i

nflu

x of

refr

acto

ry c

ells

(s=1

).

B- S

tead

y st

ate

popu

latio

n si

ze a

s a fu

nctio

n of

thym

ic in

flux

of c

ells

that

are

fully

resp

onsi

ve (e

mpt

y ci

rcle

s) o

r ful

ly a

dapt

ed/re

frac

tory

cel

ls (b

lack

circ

les)

. The

arr

ow

indi

cate

s the

crit

ical

val

ue a

t whi

ch th

e ba

selin

e po

pula

tion

size

bec

omes

uns

tabl

e . C

- The

crit

ical

val

ue a

s a fu

nctio

n of

the

valu

e of

φ. T

he v

alue

s are

the

corr

espo

ndin

g

thym

ic in

flux

s of f

ully

refr

acto

ry c

ells

. A

ll th

e re

sults

wer

e ob

tain

ed ru

nnin

g th

e SM

IC m

odel

usi

ng th

e re

fere

nce

para

met

er v

alue

s.


136

4.3 Discussion

Recently, adaptation of activation thresholds has been discussed in the context of

peripheral tolerance and homeostasis [17-19]. We show here that a mechanism of

adaptation of activation thresholds once placed in the context of population dynamics of T

lymphocytes dependent on recurrent interactions with APCs results in the “deletion” of

cells by ignorance rather than in persistence of unresponsive anergic cells. In our model

lymphocytes that adapted to chronic stimulus by raising their activation threshold will fail

to respond appropriately to homeostatic challenges and thus are unable to persist.

Adaptation leads to proliferative responses of T cells that are inverse of those required by

homeostatic feedback control. This theoretical result shows a shortcoming of taking a

simple reductionist view of adaptation of cell responsiveness. On the other hand, it gives

clues for a comprehensive model of peripheral tolerance, where adaptation of cells to

chronic stimuli might be coordinated with other mechanisms controlling cell numbers.

Grossman & Paul [15] posited that a signalling pathway displaying adaptation of

activation thresholds requires at least two processes with different time scales. A fast

process, responsible for a positive activator signal, and a slower process producing a

negative signal, which ultimately brings the cell back to quiescence in the presence of

continuous stimulus. We engineered a model that shows these features. The parameters

were chosen such the kinase (positive signal) responds faster to stimulus than the

phosphatase (negative signal) but at the steady state the activity of the phosphatase always

supersedes that of the kinase. To our knowledge this is the simplest model featuring all the

properties postulated by Grossman & Paul.

The design of the TCR signalling cascade in our model was inspired in the regulation

of the activity of Zap 70, a protein tyrosine kinase downstream of the TCR. The balance

between the active and inactive forms of Zap70 influences the sensitivity of T cell

activation [37]. Zap70 is activated by lck phosphorylation [37, 38] and inactivated by a

tyrosine phosphatase, possibly SHP-1 [39]. Notwithstanding the similarities, the model is

too simple to be adequately identified with a particular molecular mechanism. However,

since it imposes particular relationships between the duration of adaptation and refractory

period and the duration of conjugation and relaxing periods of the cells, it may guide in the

choice of what could be the positive and negative process involved in adaptation of


137

lymphocytes. Thus, the plausibility of Zap70 being the adaptable switch of T cell activation

will critically depend on the dynamics of the regulation of its expression or the regulation

of its activation states in vivo, for which we have no kinetic data. Accessory molecules such

as CD4 [17] and CD5 [18, 19] can modify the sensitivity of T cells to antigen-mediated

activation in a lasting way and are thus also good candidates for adaptation mechanisms.

The down-modulation of TCR by specific ligand [40] is also a candidate mechanism for the

adaptation of the T cell signalling thresholds [16]. The refractory period in TCR down-

regulation corresponds to the recovery of normal levels of TCR in the T cell. The recovery

of normal levels of TCR expression may take about a week [21, 40, 41], which is in

agreement with the refractory periods in the simple signalling model of the AAT

lymphocyte.

Cell cycle has been shown to affect the expression or activity of several proteins [42-

44]. We assumed here that the daughter cells will display the same configuration of the

kinase-phosphatase cascade as their progenitor cell. This is a simplifying assumption and a

reflection of the absence of information on how cell division affects the adaptation state of

cells. Obviously, if the cell cycle induces a reset of the adaptation state of the cells, or if the

duration of the refractory period is shorter than the duration of cellular division, it would be

difficult to envisage the relevance of adaptation for T cell population dynamics.

As we have shown here, unresponsive cells will not be maintained if the homeostatic

response to T cell depletion is dependent on the availability of APC [35, 45, 46] and is

adaptable [13-15]. This shortcoming can be circumvented by complementary or alternative

assumptions that are worth discussing.

The trivial solution is that adaptation of activation thresholds has no impact in T cell

population dynamics because the later is independent of TCR signalling and interactions

with APCs. This is however difficult to reconcile with the observations that sudden ablation

of TCR or MHC expression results in loss of T cells [20, 21, 47].

In our model the signalling cascades leading to homeostatic responses of T cells are

downstream the TCR and display the same threshold as those responses leading to

activation to effector function. If the homeostatic response signals are not adaptable but

those for full activation are adaptable, then we could easily foresee a situation in which a

homeostatic feedback is responsible for an independent set point. In this homeostatic set

point the average frequency of conjugation with APC could be sufficiently high to maintain

a significant fraction of cells that are refractory to full activation. However, if the


138

homeostatic response curve is adaptable to recurrent interactions with the APC the problem

we pinpointed here is unavoidable.

Grossman et al. [13-15] have also proposed that adapted unresponsive T cells could

raise the thresholds of naïve cells in their neighbourhood. This mechanism offers a control

of T cell responses that is appropriate for homeostatic mechanisms. The more T cells the

more frequent are the interactions leading to adaptation and unresponsiveness, i.e. the more

T cells the lower the proliferation per cell. A population dynamic model in which a subset

of cells are irreversibly unresponsive and make naive cells become unresponsive has been

proposed and studied by Leon et al [36], who have shown that depending on parameter

values this can lead to homeostasis in a situation in which the loss of unresponsive cells is

compensated by a continuous source from naïve cells. However, in other regimes the

unresponsive cells may simply be lost. An explicit model in which naïve cells become

reversibly unresponsive by recurrent interactions with APCs or by interactions with other

unresponsive cells may not be trivial and remains to be done.

Despite the fact that our AAT population model cannot explain simultaneously both

the persistence and unresponsiveness of autoreactive T cells it offers nevertheless a means

by which recent autoreactive thymic immigrants could be prevented from undergoing

peripheral expansion and causing autoimmunity. Our results indicate that this mechanism

of peripheral tolerance is not absolutely efficient and would depend critically on the daily

production of new cells by the thymus. Only if autoreactive cells come out of the thymus in

a refractory state and their daily production is relatively minor, will peripheral expansion be

prevented. Otherwise, this mechanism of peripheral tolerance becomes unreliable.

Unresponsiveness of cells following expansion in adoptively transferred recipients has

been interpreted as the emergence of refractoriness in cell populations induced by recurrent

interactions with stimulating APCs [48]. The proliferative response curves in these

experiments comply well with a homeostatic mechanism since unresponsiveness increases

with cell numbers. However, our AAT population model predicts the opposite relationship

between cellular responsiveness and population size. Our theoretical analysis indicates that

caution should be taken when interpreting the former experiments since the formal

implication of AAT in these observations is not straightforward and might be unwarranted.

Indeed, in many of these adoptive transfer experiments, changes in whole population

responsiveness can alternatively be explained by changes in the composition of cell

populations [17-19].


139

Often simple molecular mechanisms offer parsimonious explanations to the behaviour

of a biological system. We have followed the implications of a molecular mechanism of T

cell activation into the domain of cell population dynamics. According to our model, based

on reasonable assumptions, adaptation of activation thresholds following recurrent

interactions with APCs leads to peripheral deletion in conditions where tuning is pertinent

to T cell population dynamics. We conclude that postulating adaptation of peripheral cells

to chronic stimulus by self-antigens is not enough to explain the persistence of a significant

pool of auto reactive T cells in the absence of autoimmunity. A convincing mechanism

explaining peripheral tolerance requires additional postulates about cell population

dynamics and will depend critically on those postulates.

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General discussion

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5 General discussion

5

General Discussion

General discussion

144

General discussion

145

Lymphocyte fate is determined by the intersection between the set of receptors

expressed in the cell and the set of stimuli present in the environment. Hence, the study of

the signalling pathways, the integration of signals and the interference between the

signalling pathways becomes crucial to understand lymphocyte population dynamics. The

coordination of the dynamics of signalling pathways and lymphocyte population spans both

temporal and physical scales. The models presented here deal with the problem of the

contribution of cellular processes to population dynamics processes. As part of this

ambitious goal, we built mathematical models focusing on T lymphocyte dynamics as a

function of underlying molecular processes.

To address what are the consequences of sharing cytokine receptor chains between

cytokine receptors, we built and studied a prototypic model of cytokine receptor

engagement and cross-linking. This model describes the engagement and cross-linking of

IL-2 and IL-4 cytokine receptors, which are members of the γc cytokine receptor family

that share the common gamma chain (Sugamura, Asao et al. 1996; Gesbert, Delespine-

Carmagnat et al. 1998; Nelms, Keegan et al. 1999). Using a steady state simplification of

the receptor engagement mechanism by IL-2 and IL-4, together with available experimental

data, we demonstrated that these two cytokines could interfere at the level of the receptor

via the common gamma chain. Such γc-mediated interference has already been suggested in

the IL-13 receptor system (Kuznetsov and Puri 1999). Our results suggested consequences

for the population dynamics of cells that rely on IL-2 and IL-4 for differentiation and

proliferation. Thus, the case of ThP and Th2 population dynamics was studied as a

prototypic model of the influence of cytokine receptor interference on cell population

dynamics. The results of the model show that IL-2 inhibition of IL-4 signalling can account

for the synergistic properties of IL-2 and IL-4 in Th2 generation (Morel, Burke et al. 1996;

Burke, Morel et al. 1997) and in addition, predict that the IL-2-dependent precursors for

Th2 are able to resist the pressure to differentiate the whole precursor pool to Th2. These

results can be summarized as a) competition for common cytokine receptor chains can have

an impact at the population dynamics level; and b) that positive feedback regulation loops,

like the kind that regulate lymphocyte differentiation, when combined with proliferation of

precursors and resistance to differentiation, leads to the preservation of precursors and to a

synergism in the number of differentiated cells.

General discussion

146

The IL-4R and IL-2R engagement model allowed us to derive the dose response curve

for IL-2 and IL-4, as well as the γc-mediated interference between IL-2R and IL-4R. The

experimental estimates of γc concentration are in a range where it is limiting (Kondo,

Takeshita et al. 1993); hence in the presence of high enough concentrations of IL-4 and IL-

2, the IL-2R and IL-4R will compete for available γc. Due to the respective concentrations

and affinities of the IL-2 and IL-4 receptor chains (this thesis, (Kondo, Takeshita et al.

1993; Roessler, Grant et al. 1994)), the IL-2R has a more prominent interference with IL-

4R than vice-versa; IL-4 interference with IL-2 can even be considered negligible.

Assuming that the concentrations of receptor chains are in a quasi steady state and that

cytokine signalling is proportional to the amount of fully assembled receptor-ligand

complex, it was possible to derive that IL-2 can inhibit IL-4 signalling. The model allowed

us to build adequate dose-response curves for combined IL-2 and IL-4 signalling, based on

parameters estimated experimentally (Robb, Greene et al. 1984; Kondo, Takeshita et al.

1993).

The model relies in strong assumptions of the mechanism of receptor-ligand

complexes, motivated both by the need of simplicity and the absence of enough quantitative

data. These approximations impair the quantitative interpretation of the results of the

model; hence only qualitative relationships such as the assymetry in the interference

between IL-2 and IL-4 under the relative scarcity of γc should be considered.

Different cells may express different levels of cytokine receptor chains as a result of

their differentiation status or of past stimulations. Unfortunately there is not enough

experimental data to provide a measure of such variability, thus the concentrations used in

the model should be considered only indicative of the possible concentration of these

receptor chains in the cells. Taking into consideration that receptor chains are internalised

and degraded (Duprez and Dautry-Varsat 1986; Duprez, Cornet et al. 1988; Galizzi, Zuber

et al. 1989; Hemar, Lieb et al. 1994; Hemar, Subtil et al. 1995), that shared cytoplasmatic

signal components might also be limiting (Decker and Meinke 1997; Leonard and O'Shea

1998), and that the γc might be used by other cytokine receptors (Sugamura, Asao et al.

1996), it can be argued that the model is minimizing the interference of signals by IL-2 and

IL-4. Further exploration on this issue as well as on the possible importance of the low

affinity IL-2 receptor is required to assess to which extent these arguments may be relevant.

The population dynamics model of cytokine receptor interference takes from our

finding that IL-2 can inhibit IL-4 signalling, and explores the possible consequences at the

General discussion

147

level of regulation of ThP differentiation to Th2. This population model is based in a mean

field description of both populations. The differentiation of cells requires several division

rounds (Hodgkin, Lee et al. 1996; Gett and Hodgkin 1998), producing a heterogenous

population of cells at various differentiated stages and/or in different stages of the cell

cycle. These assumptions allow us to use the mean field formalism to describe the cell

population dynamics of ThP and Th2 with relative simplicity. The model does not consider

the effect of ThP depletion by alternative differentiation pathways (O'Garra 1998) nor the

effect of cytokines secreted by other populations (such as Th1 cells, dendritic cells and

macrophages) on both ThP and Th2 population dynamics (Ma, Chow et al. 1996; Murphy,

Shibuya et al. 1996; Szabo, Dighe et al. 1997; O'Garra 1998). Considering these pathways

implies modelling the cells and cytokines involved (Fishman and Perelson 1994; Carneiro,

Stewart et al. 1995; O'Garra 1998; Bergmann, van Hemmen et al. 2002), which would taint

the relative simplicity of this model. Such an extension should nevertheless be added to this

model in the future, together with modelling further events downstream of the cytokine

receptor engagement.

The interference of IL-2 and IL-4 signalling might not be restricted to sharing γc. It is

not uncommon that different cytokines depend on the same Jak kinases and some of the

STATs (Decker and Meinke 1997; Leonard and O'Shea 1998). Interestingly, in the case of

IL-4 and IL-2, although Jak-STAT signalling depends on the same Jak kinases, it relies on

a different set of STATs (Leonard and O'Shea 1998). Other cytokine receptor systems of

the γc and other families also rely on overlapping elements of the Jak-STAT pathway, see

for example the reviews on signalling via the γc (Sugamura, Asao et al. 1996; Malek, Porter

et al. 1999), gp130 (Hirano 1998) and βc (Scott and Begley 1999) common cytokine

receptor chains.

A hypothesis is that sharing cytokine receptor chains is a means to organize the

hierarchy of cell responses to cytokines. IL-7 is a candidate cytokine for the regulation of T

cell homeostasis (reviewed in (Fry and Mackall 2001)). For example, in the same way as

IL-2 can inhibit IL-4-mediated differentiation, IL-2 could inhibit IL-7-mediated

homeostatic proliferation, shifting cell proliferation from a homeostatic mode to an immune

response mode.

An interesting possibility that this model illustrates is the existence of dynamic

mechanisms for self-preservation of peripheral naïve lymphocyte precursors. These

mechanisms might be clone specific, being important for the differentiation of effector

General discussion

148

classes, or clone unspecific, being important the preservation of self-regenerating precursor

cells. Figure 1 shows conceptually how such mechanisms could help preserve the

intermediate pools of differentiating cells, giving the immune system the ability to trace

back on lymphocyte differentiation (Bergmann, van Hemmen et al. 2002), channelling the

flow of differentiating cells to specific goals.

-

+

A

+

-

+

B

+

+

B

+

Figure 1 – Hypothetic differentiation pathway of lymphocytes illustrating how the maintenance of

precursors by balancing differentiation and proliferation can maintain open alternate differentiation

pathways. A – By balancing proliferation and differentiation the precursor cells at the differentiation

crossroad are maintained, hence the lymphocyte clone/population keeps multipotent. B – Without

balancing proliferation and differentiation, the drive to differentiation is such that the precursor at the

crossroad is extinguished leaving the clone/population completely committed to a single pathway.

This simplistic ThP and Th2 model illustrates how subtle differences at the level of

signalling can have important dynamic consequences at the cell population level. This type

of regulation of cell differentiation is often neglected from experiments of cell

differentiation since it dwells in the quantitative aspects of cell differentiation, which is

always more difficult to assess when compared with the possibility of clear-cut results

provided by the molecular biology approach. By using mathematical models it was possible

to assert the conditions that favour interference and assess the extension of the interference

between cytokine receptors sharing common chains. This is a very simple prototypic model

on interference between multi-chain membrane receptors sharing common chains, which

can be extended to other cytokine receptors and cytokine receptor families.

Cytokine receptors are important in modulating the activation and directing

differentiation of T cells, but the TCR is the crucial receptor controlling T cell activation

(Davis and Bjorkman 1988). The importance of the TCR reflects on the effort that has been

spent in the study of this receptor and of its downstream signalling pathways. In spite these

efforts, the stoichiometry of TCR-ligand engagement and the mechanism of activation of

General discussion

149

the earliest events of TCR triggering have been difficult to clarify experimentally. These

studies require quantification of molecular events occurring at the single cell level, which

are difficult to attain even with current state of the art techniques such as plasmon

resonance and flow cytometry. Mathematical models have been used to address some of the

questions in TCR engagement stoichiometry and internalisation (for example (Bachmann,

Salzmann et al. 1998; Bachmann and Ohashi 1999; Wofsy, Coombs et al. 2001)); in TCR

membrane dynamics and triggering (for example (Salzmann and Bachmann 1998; Chan,

George et al. 2001)); and in the early events of TCR signalling (for example (McKeithan

1995; Kaufman, Andris et al. 1999; Lord, Lechler et al. 1999; Hlavacek, Redondo et al.

2001)). The models presented here complement and extend some of these models.

In this thesis, we studied a series of phenomenological mathematical models, based on

TCR serial triggering (Valitutti, Muller et al. 1995; Bachmann, Salzmann et al. 1998)

mechanism, which were elaborated using available experimental data on TCR down-

modulation (Valitutti, Muller et al. 1995) to gain insight into the mechanism of TCR down

modulation. These models permitted the formulation of new hypothesis for the mechanism

of TCR triggering demonstrating how mathematical modelling techniques can be used

together with quantitative experimental data to gain insight into the underlying

mechanisms.

The TCR triggering models are based on a mean field phenomenological description

of the available experimental data on TCR triggering kinetics, hence they are representative

of the mean of a population of lymphocytes and peptide loaded APCs. Since the

experimental data uses a single population of T cells and peptide-loaded APCs, this

approach is valid.

The most important of the properties uncovered by the models is the requirement that

TCR triggering is a highly cooperative process, which stems from the sensitivity of TCR

activation to low concentration of MHC-peptide ligands. Although ultra sensitivity in the

kinetics of TCR down modulation and the sensitivity of TCR triggering to low doses of

ligand are tied in the model, they are distinct properties. Ultra sensitivity results from the

slope of the dose-response curves; it means that slight changes in ligand presentation

induce dramatic changes in the number of triggered TCRs. The sensitivity of TCR

triggering for low doses of ligand means that the most sensitive part of the dose response is

for low density of ligands, ranging from 100 to 1000 ligands, i.e. between 0.3% and 3% of

total MHC occupancy (estimated from (Valitutti, Dessing et al. 1995)). It is easy to

General discussion

150

speculate that in vivo the T cells are responding somewhere within this range of

concentrations since it is where the distinction between the quality and quantity of the

antigen, in the context of TCR triggering, is clearer.

In search for a possible mechanism to explain ultra-sensitivity of TCR triggering, two

candidate mechanisms were studied: Simple TCR cross-linking mechanisms (Bachmann,

Salzmann et al. 1998; Bachmann and Ohashi 1999) (homo cross-linking mechanisms) and

the TCR triggering cycle, which relies in the recruitment to the TCR complex of CD4 or

CD8 co-receptors associated with src kinases (hetero cross-linking mechanism). Our results

show that both mechanisms can reproduce the ultra-sensitivity of TCR triggering, but the

parameter regime required for the homo cross-linking mechanisms to describe all the

kinetics of TCR triggering requires very high cross-linking rate constants. Thus homo

cross-linking seems less robust from the biological point of view compared to the

parameter regime of the hetero cross-linking model. If only the kinetics of the high doses

of ligand is used, then the homo cross-linking model easily describes the data with a more

robust parameter set.

These qualitative results of the models suggested the hypothesis that both mechanisms

can mediate TCR triggering and internalisation. Hence, far from deepening the chasm

between the different hypotheses regarding TCR triggering stoichiometry, we propose a

conciliation of oligomerization models with co-receptor hetero cross-linking models by

proposing that both mechanisms are possible but have different ligand dose dependence. At

low density of ligand, TCR hetero cross-linking with CD4-lck or CD8-lck should be the

prevailing TCR triggering mechanism, whereas at high doses of ligand, both mechanisms

should be possible. The determination of dose-response curves for TCR triggering, together

with the assessment of which are the PTKs, co-receptors and adaptor molecules

phosphorylated for each dose would be valuable to determine the validity of this dual mode

hypothesis of TCR triggering. Some of the aspects of the present hypothesis are being

explored experimentally (Andreia Lino, Personal communication). It would also be

interesting to determine experimentally the influence of TCR antagonists and partial

agonists, since TCR antagonists and partial agonists usually require higher concentrations

to induce TCR signalling. The finding that week and partial agonists are more dependent on

CD4 for TCR triggering than agonists (Vidal, Daniel et al. 1999) supports the hypothesis

above.

Our analysis is based on TCR down modulation being dependent on TCR engagement

General discussion

151

and triggering, which seems to be the case for some experimental systems such as the

system on which we based the model (Valitutti, Dessing et al. 1995) but not in others,

where the extent of non-specific down-modulation is significant (San Jose, Borroto et al.

2000). The model also makes strong simplifications of the cycling of TCR/CD3 between

cytoplasmatic and membrane compartments (Liu, Rhodes et al. 2000). The possibility of

internal pools of TCR was modelled, but it was not significant for the kinetics of TCR

down modulation (not shown).

The duration of APC stimulation required for T cell activation seems to be between of

6-12 hours (Dustin, Allen et al. 2001), but in some conditions it can be much shorter (Lee,

Pasos et al. 2002). If the duration of T cell-APC engagement is shorter, then much like the

revised kinetic proof reading model (Hlavacek, Redondo et al. 2001), the T cell might elicit

different responses depending on the time it is conjugated with the APC. Perhaps a study of

conjugation times using partial agonists or non-agonists might be a fundamental experiment

to link the quantity and quality of signal transduction of the TCR with the duration of cell-

cell interactions.

One of the principal limitations of the models is that they do not include the spatial

component of the membrane dynamics of the T cell and APC. The interface between an

APC and T cell is structurally complex. Not only does it involve binding of numerous

molecules on the surface of both cells, it is also a dimensionally restricted space that may

condition both diffusion and reactivity of membrane molecules (Shaw and Dustin 1997).

The immune synapse is a complex structural pattern formed in the contact between an APC

and a T cell (Monks, Freiberg et al. 1998; Grakoui, Bromley et al. 1999; Anton van der

Merwe, Davis et al. 2000). The role of the immune synapse in TCR triggering and T cell

signalling is a matter of debate. The agonistic properties of the MHC-peptide correlate with

the extension and duration of the immune synapse and the formation of the immune

synapse have been implied in CD45 exclusion, a possible mechanism for lck activation.

Recently, some doubts were raised on the role of the immune synapse in TCR triggering

(Lee, Holdorf et al. 2002). Staining, using phospho-specific antibodies, of lck and Zap70

has shown that TCR triggering occurs mostly during the early 30 minutes of conjugation,

before the mature synapse has been formed, thus casting doubt that the immune synapse is

necessary for the initiation of TCR signalling (Lee, Holdorf et al. 2002). Still, immune

synapses may still play be significant for other responses (Revy, Sospedra et al. 2001; Lee,

Holdorf et al. 2002).

General discussion

152

One of the possible implications of membrane rafts is that the signalling molecules

associated with a raft might act as a single supramolecular signal transduction unit

(signalosome (Werlen and Palmer 2002)). Thus lipid rafts might play a function similar to

scaffold proteins, increasing the specificity and efficiency of interaction of the signal

components. The aggregation of membrane rafts can mimic an activation signal for the T

cell (Janes, Ley et al. 1999). This observation, together with raft heterogeneity regarding

the composition of signalling molecules (Gomez-Mouton, Abad et al. 2001; Schade and

Levine 2002) adds a larger complexity to the possible mechanisms of TCR triggering. It

seems that TCR triggering and signalling can be initiated by a number of experimental

protocols, but of all the triggering possibilities, the most likely to occur in vivo is still

controversial.

Our results indicated that TCR triggering is a mechanism with a threshold and very

steep response to agonists resulting from the hypersensitivity. We have also shown that

TCR down-modulation can confer adaptable activation thresholds to T cells, which means

that right at the early stage of TCR signalling by the cell, there are mechanisms capable of

filtering environmental noise from the TCR triggering, thus allowing the cells to perform

quality control on the stimulus received. How does the adaptation of TCR levels to APC

stimulation together with the hypersensivity of TCR triggering condition the physiology of

a population of lymphocytes?

The theory of adaptable activation thresholds of lymphocytes has been used to explain

immune phenomena based in the dynamics of signalling pathways of individual cells

(Grossman and Paul 1992; Grossman and Singer 1996; Grossman and Paul 2000;

Grossman and Paul 2001). This theory also provides an attractive conceptual framework to

explore the consequences of single cell events, such as the regulation of the activation

threshold of lymphocytes, at the cell population level. The work presented here constitutes

the first study of the implications of adaptable activation thresholds of lymphocytes in the

dynamics of populations of lymphocytes. We studied the adaptation of activation

thresholds at a single cell level and the population dynamics of adaptable cells using a

homeostatic model of T cell population dynamics. Each cell in the model has a particular

state, which depends on the particular history of antigen encounters particular to that cell.

In spite the unique structure of this model, it is simple enough to allow characterization of

its properties. This model constitutes a simple approach in dealing with the scaling problem

in biological systems. This model constitutes a step forward compared with standard mean

General discussion

153

field models of cell population dynamics, such as the ThP and Th2 model presented here.

The AAT model is based on an idealized simple TCR signal transduction pathway,

inspired in TCR hypersensitivity and on the model of TCR hetero cross-linking triggering,

capable of reproducing the qualitative traits that are the foundation of the lymphocyte AAT

hypothesis (Grossman and Paul 1992; Grossman and Singer 1996; Grossman and Paul

2000; Grossman and Paul 2001). The model couples T cell-APC conjugation kinetics with

the cellular TCR signalling pathway of a single cell. For the sake of simplicity, we

neglected the contribution of signalling in the kinetics of T cell conjugation with the APC.

Using this simple AAT model, the dynamic properties of a population of such

adaptable lymphocytes were explored from the point of view of homeostasis. Our results

showed that the maintenance of a peripheral pool of totally unresponsive lymphocytes is

not possible under the postulates of the AAT hypothesis. The stable steady states of the

population, in the absence of a source term, are either the collapse of the population

(peripheral deletion of the clone due to full adaptation and unresponsiveness), or a steady

state where most of the cells are responsive (low adaptation of the population).

Two factors determine the outcome of the population of adaptable lymphocytes in the

periphery: the initial number of cells (clone size) and the adaptation state of the cells. Low

numbers of cells together with high levels of adaptation favour the collapse of the

population whereas high numbers of cells with low levels of adaptation favour expansion to

the low-adapted state. Also important is the direction of evolution of the system. Since

adaptable lymphocytes have a memory of previous events, the same number of

lymphocytes might be expanding or collapsing depending on the current adaptation state of

individual cells. For a recent thymic emigrant, its persistence in the periphery is thus

dependent on its adaptation state, the size of the clone and the level of antigen in the

periphery. The clone will only be able to persist if peripheral stimulation and clone size is

high enough to allow the clone to expand above a critical population size.

AATs set a threshold on the ratio between the size of the clone and antigen for the

persistence of the clone. Hence, AATs may act as a filter, eliminating some clones that

would otherwise be able to persist on self-antigen stimulation.

If lymphocyte survival depends on TCR and MHC (Witherden, van Oers et al. 2000;

Polic, Kunkel et al. 2001) and TCR signal transduction is adaptable, then AATs cause the

deletion of the desensitised cells from the periphery. The results of our model show that

AATs alone cannot explain the persistence of lymphocytes fully desensitised to self-

General discussion

154

antigens in the periphery, but rather provide a simple mechanism of attaining peripheral

tolerance by deletion of a clone.

If TCR signalling is a necessary condition for T cell survival, it is difficult to imagine

that the organism produces a single survival factor, specific for each clone. Cells might rely

on self-MHC ligands and/or partial agonists as the means to transduce survival signals, but

not activator signals, via the TCR. This partial activation of the TCR signalling cascade by

specific ligands has been termed “TCR tickling” mechanism (Evavold, Sloan-Lancaster et

al. 1993). As far as the model presented here goes, promoting a simple linear increase of T

cell survival by “TCR tickling” would not change the qualitative properties of the model.

However, the results of the model indicate that the effect of such survival signals would

have to be non-linear to stabilize the persistence of unresponsive cells, i.e. the proliferation

and death curves in figure 4 of section 4 (page 128) would have to be such that a stable

steady state of desensitised cells is possible. The nature and existence of such an effect in

lymphocyte survival is completely speculative. Further modelling would have to be

performed to gain additional insight into the biological significance of the dynamic

constraints of such modification of the survival of lymphocytes.

These results do not diminish the possible importance of AATs in lymphocyte

physiology, or its possible importance in a more general mechanism of lymphocyte

homeostasis. The results simply state that AATs are not a sufficient condition, remaining to

be proved if AATs are or not involved in the persistence of peripheral auto-reactive

lymphocytes and also if they are necessary for the unresponsiveness.

Thinking of adaptable activation thresholds simply in terms of the TCR signal pathway

is too restrictive. The state of a lymphocyte and its response to stimulation is the result of a

combination of multiple signalling pathways. The adaptation of these pathways allows

lymphocytes to integrate and respond to stimuli, based on the past experiences. Adaptable

activation thresholds give lymphocytes memory of the last events experienced, casting on

the single cell cognitive properties. Looking at the similarities between lymphocytes and

neurons under the light of the AAT hypothesis and the myriad of receptors and ligands,

which regulate the physiology of lymphocytes and the dynamics of lymphocyte

populations, a parallelism between the cognitive characteristics of the nervous system and

the neuron and the immune system and the lymphocyte is unavoidable. The analogy can

even be pushed further if assuming that cytokines act primarily between interacting cells,

propagating information between the different cells in a manner akin to a neuronal network.

General discussion

155

The complexity of the cell-cell contacts, as is evident in studies of the immune synapse, and

the intricacies of cytokine signalling attest in favour of such cellular networks, even if for

no other reason than that there is place for such networks in the complex regulation of the

immune system.

Both the AAT and the ThP/Th2 models have states of stable collapse and persistence

of clones separated by a threshold that depends on the number of cells. Apparently these

models contrast. The ThP/Th2 model provides a mechanism for the persistence of cells

whereas the AAT model provides a mechanism for the elimination of cells. The models

deal with different aspects of lymphocyte population dynamics and depend on different

signalling pathways: the ThP/Th2 model relates to the regulation of lymphocyte

differentiation mediated by cytokines, whereas the AAT model relates to the persistence of

the whole clone, dependent on TCR signalling.

Integration of both models is missing in this thesis. At the single cell, the TCR and

cytokine pathways interfere and this interference should propagate to the cytokine and

antigen dependent population dynamics of lymphocytes. Extending the model of ThP/Th2

dynamics with AATs in cytokine and TCR signalling is not trivial. If only the TCR is

adaptable, then it is possible that the relationships within the ThP/Th2 model will still hold,

since both proliferation and differentiation of Th2 and ThP have an implicit dependence on

TCR signalling. If cytokine signalling is also adaptable, a plausible assumption considering

the complex regulation of cytokine receptor expression, then the dynamics of the ThP/Th2

model become non-trivial. Does IL-2 and IL-4 signalling adapt with the same dynamics as

the TCR? To which extent is adaptation dependent on the interference between the TCR,

IL-2 and IL-4 signalling pathways? How would adaptation influence the balance between

proliferation and differentiation of ThPs? How could antigen dose, which plays a role in the

polarization of helper T cells (O'Garra and Murphy 1994; O'Garra 1998), and adaptation

direct cells into alternative differentiation pathways? These are questions to which it is even

difficult to hypothesise answers with the current state of our models and knowledge.

This thesis addressed, respectively, the relationship between the regulation of receptor

expression and the effects of this regulation at the cell population dynamics.

Many open questions still remain in this work. The model of IL-2R and IL-4R barely

scratches the surface of the extent to which the signalling pathways interfere. The same for

the ThP and Th2 population dynamics model, which falls short of including other cell

General discussion

156

phenotypes and their differentiation and proliferation factors. Unfortunately, more detailed

models would still be limited by the available quantitative experimental data.

TCR triggering is a very complex phenomenon that spans from the molecular to the

cellular level. The models presented here mostly deal with the molecular aspects of TCR

engagement, triggering and down modulation. Modelling TCR downstream signalling as

well as adding a spatial component to the dynamics of the TC-MHC-co-receptor

interactions also remain to be explored.

The main problems faced with the AAT model were the inexistence of a formalism to

deal adequately with a population of heterogeneous lymphocytes with memory of past

stimulatory events. The model as it is now is open to extend to more detailed TCR-APC

dynamics. Remaining questions include the dependency of APC binding on the intensity of

stimulation of T cells, the competition between lymphocyte clones for APCs and/or

antigen, the inclusion of a more detailed TCR engagement and triggering model, and the

effect of compartmentalization of lymphocytes.

The broad aim of this thesis was to contribute to the study of the how do single cell

events influence the population dynamics of cells. Throughout this thesis, we presented

models that try to bridge the chasm between molecular events and the population dynamics

of lymphocytes. In spite the strong simplifications and abstractions used in the models

presented here, it is striking the complexity of the population dynamics of CD4+ T

lymphocytes when molecular events, such as the cytokine and TCR receptors expression

and signaling, are taken into consideration. A common and somewhat fair criticism to

these models is that they are too simplified, too abstract or omitting some (hypothetical)

important aspect. The disconcerting truth is that in spite of all the abstractions and

simplifications these models are non-linear, complex and difficult to understand, with some

of the results being non-intuitive at least to the collective intuition and understanding of the

researchers that were involved in this thesis. This complexity is not intentional, rather it

reflects the intricacies of the propagation of events between organization levels that

immunologists, biologists and biochemists try to understand relying (generally speaking)

solely in personal intuition and a few working examples.

Modeling the interface between organization levels is still in its infancy and hence an

open field, not only in immunology but also in biology in general. This thesis contributes

towards the goal of understanding how population dynamics properties arise from single

General discussion

157

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