Immunology and van Kampen's large N expansion Magic 042 ...

More immunology T cells Helper T cells and lymphocyte activation Large N expansion Hard niche case

Immunology and van Kampen’s large N expansionMagic 042 – Lecture 9 (or lecture 15)

Carmen Molina-Parıs

Department of Applied Mathematics, School of Mathematics, University of Leeds, UK

24th of November 2008


Outline of the talk

More immunologyLymphocyte recirculationB cells and antibodies

T cellsT cells and MHC proteinsT cell receptor (TCR)Effector cytotoxic T cellsEffector helper T cells

Helper T cells and lymphocyte activationHelper T cells and lymphocyte activation

Large N expansionLarge N expansion

Hard niche caseHard niche caseSymmetric clonotyes


Back to some more immunology


Lymphocyte recirculation through peripheral lymphoid organs I

• Pathogens generally enter the body through an epithelial surface, usuallythrough the skin, gut, or respiratory tract.

• To induce an adaptive immune response, microbial antigens must travelfrom these entry points to a peripheral lymphoid organ, such as a lymphnode or the spleen, the sites where lymphocytes are activated.

• The route and destination depend on the site of entry.

• Lymphatic vessels carry antigens that enter through the skin or respiratorytract to local lymph nodes.

• Antigens that enter through the gut end up in gut-associated peripherallymphoid organs such as Peyer’s patches.

• The spleen filters out antigens that enter the blood.


Lymphocyte recirculation through peripheral lymphoid organs II


Lymphocyte recirculation through peripheral lymphoid organs III

• Dendritic cells will carry the antigen from the site of infection to theperipheral lymphoid organ, where they play a crucial part in activatingT cells.

• Only a tiny fraction of the total lymphocyte population can recognise aparticular microbial antigen in a peripheral lymph organ (estimated to bebetween 10−4 and 10−5 of each class of lymphocyte).

• How do these rare cells find an antigen presenting cell displaying theirantigen?

• Lymphocytes continuously circulate between one peripheral lymphoidorgan and another via the lymph and blood.

• The continuous recirculation between the blood and lymph ends only if alymphocyte is activated by its specific antigen in a peripheral lymphoidorgan.

• The lymphocyte remains in the peripheral lymphoid organ, where itproliferates and differentiates into either effector cells or memory cells.

• Many of the effector T cells leave the lymphoid organ via the lymph andmigrate through the blood to the site of infection, whereas others stay inthe lymphoid organ and help activate B cells or other T cells there.


Lymphocyte recirculation through peripheral lymphoid organs IV

• Some effector B cells (plasma cells) remain in the peripheral lymphoidorgan and secrete antibodies into the blood for days until they die.

• Others migrate to the bone marrow, where they secrete antibodies intothe blood for months or years.

• The memory T and B cells produced join the recirculating pool oflymphocytes.

• Lymphocyte recirculation depends on specific interactions between thelymphocyte cell surface and the surface of the endothelial cells lining theblood vessels in the peripheral lymphoid organs.

• Lymphocytes adhere and then migrate out of the bloodstream into thenodes.

• The lymphocytes initially adhere to the endothelial cells via homingreceptors that bind to specific ligands (often called counter-receptors) onthe endothelial cell surface.

• Lymphocyte migration into lymph nodes depends on a homing receptorprotein called L-selectin.


Lymphocyte recirculation through peripheral lymphoid organs V

• The lymphocytes adhere weakly to the endothelial cells and roll slowlyalong their surface.

• The rolling continues until another, much stronger adhesion system iscalled into play by the chemoattractant proteins (chemokines) secreted byendothelial cells.

• This strong adhesion is mediated by members of the integrin family of celladhesion molecules, which become activated on the lymphocyte surface.

• The lymphocytes stop rolling and crawl out of the blood vessel into thelymph node.


Lymphocyte recirculation through peripheral lymphoid organs VI

• The T and B cells initially enter the same region of a lymph node, butthen different chemokines guide them to separate regions of the node.

• Unless they encounter their antigen, both T and B cells soon leave thelymph node via efferent lymphatic vessels.

• If they encounter their antigen, however, they are stimulated to displayadhesion receptors that trap the cells in the node.

• The cells accumulate at the junction between the T cell and B cell areas,where the rare specific T and B cells can interact, leading to theirproliferation and differentiation


B cells and antibodies I

• Antibodies defend us against infection by binding to viruses and microbialtoxins, thereby inactivating them.

• When antibodies bind to invading pathogens, they also recruit some of thecomponents of the innate immune system.


B cells and antibodies II

• Synthesised exclusively by B cells, antibodies are produced in billions offorms, each with a different amino acid sequence.

• Collectively called immunoglobulins (abbreviated as Ig), they are amongthe most abundant protein components in the blood.

• Mammals make five classes of antibodies, each of which mediates acharacteristic biological response following antigen binding.

• In this section, we discuss the structure and function of antibodies andhow they interact with antigen.

• All antibody molecules made by an individual B cell have the sameantigen-binding site.


B cells and antibodies III

• The first antibodies made by a newly formed B cell are not secreted butare instead inserted into the plasma membrane, where they serve asreceptors for antigen.

• Each B cell has approximately 105 such receptors in its plasma membrane.

• Each B cell clone produces a single species of antibody, with a uniqueantigen-binding site.

• When an antigen (with the aid of a helper T cell) activates a naive or amemory B cell, that B cell proliferates and differentiates into anantibody-secreting effector cell.

• Such effector cells make and secrete large amounts of soluble (rather thanmembrane-bound) antibody, which has the same unique antigen-bindingsite as the cell-surface antibody that served earlier as the antigen receptor.


B cells and antibodies IV


B cells and antibodies V

• Effector B cells can begin secreting antibody while they are still smalllymphocytes, but the end stage of their maturation pathway is a largeplasma cell.

• Plasma cells continuously secrete antibodies at the astonishing rate ofabout 5000 molecules per second.

• Although most plasma cells die after several days, some survive in thebone marrow for months or years and continue to secrete antibodies intothe blood, helping to provide long-term protection against the pathogenthat stimulated their production.


The antibody molecule I

• The simplest antibodies are Y-shaped molecules with two identicalantigen-binding sites, one at the tip of each arm of the Y.

• The basic structural unit of an antibody molecule consists of fourpolypeptide chains, two identical light (L) chains (each containing about220 amino acids) and two identical heavy (H) chains (each usuallycontaining about 440 amino acids).

• The molecule is composed of two identical halves, each with the sameantigen-binding site.

• Both light and heavy chains usually cooperate to form the antigen-bindingsurface.


The antibody molecule II


B cell development


T cells versus B cells I

• Like antibody responses, T cell-mediated immune responses are exquisitelyantigen-specific, and they are at least as important as antibodies indefending vertebrates against infection.

• Most adaptive immune responses, including most antibody responses,require helper T cells for their initiation.

• Unlike B cells, T cells can help eliminate pathogens that would otherwisebe invisible inside host cells.

• T cell responses differ from B cell responses in at least two crucial ways.

• First, T cells are activated by foreign antigen to proliferate anddifferentiate into effector cells only when the antigen is displayed on thesurface of antigen-presenting cells, usually dendritic cells in peripherallymphoid organs.

• T cells require antigen-presenting cells for activation because the form ofantigen they recognise is different from that recognized by B cells.


T cells versus B cells II

• Whereas B cells recognise intact protein antigens, for example, T cellsrecognise fragments of protein antigens that have been partly degradedinside the antigen-presenting cell.

• Special proteins, called MHC proteins, bind to the peptide fragments andcarry them to the surface of the antigen-presenting cell, where T cells canrecognise them.

• The second difference is that, once activated, effector T cells act only atshort range, either within a secondary lymphoid organ or after they havemigrated into a site of infection.

• Effector B cells, by contrast, secrete antibodies that can act far away.

• Effector T cells interact directly with another host cell in the body, whichthey either kill (as in the case of an infected host cell, for example) orsignal in some way (as in the case of a B cell or macrophage, for example).

• We shall refer to such host cells as target cells.


T cells versus B cells III

• Target cells must display an antigen bound to an MHC protein on theirsurface for a T cell to recognise them, they are also antigen-presentingcells.

• There are three main classes of T cells: cytotoxic T cells, helper T cells,and regulatory (suppressor) T cells.

• Effector cytotoxic T cells directly kill cells that are infected with a virus orsome other intracellular pathogen.

• Effector helper T cells help stimulate the responses of other cells: mainlymacrophages, dendritic cells, B cells, and cytotoxic T cells.

• Effector regulatory T cells suppress the activity of other cells, especially ofself-reactive effector T cells.


T cell receptor (TCR) I

• T cell responses depend on direct contact with an antigen-presenting cellor a target cell.

• T cell receptors (TCRs), unlike antibodies made by B cells, exist only inmembrane-bound form and are not secreted.

• For this reason, TCRs were difficult to isolate, and it was not until the1980s that researchers identified their molecular structure.


T cell receptor (TCR) II

• TCRs resemble antibodies.

• The three-dimensional structure of the extracellular part of a TCR hasbeen determined by x-ray diffraction, and it looks very much like one armof a Y-shaped antibody molecule.

• Various co-receptors and cellcell adhesion proteins greatly strengthen thebinding of a T cell to an antigen-presenting cell or a target cell.


Antigen presentation by dendritic cells I

• Naive cytotoxic or helper T cells must be activated to proliferate anddifferentiate into effector cells before they can kill or help their target cells,respectively.

• This activation occurs in peripheral lymphoid organs on the surface ofactivated dendritic cells that display foreign antigen complexed with MHCproteins on their surface, along with co-stimulatory proteins.

• Memory T cells can be activated by other types of antigen-presentingcells, including macrophages and B cells, as well as by dendritic cells.

• Dendritic cells interact with T cells to present antigens that either activateor suppress the T cells.

• Dendritic cells are located in tissues throughout the body, including thecentral and peripheral lymphoid organs.

• Wherever they encounter invading microbes, they endocytose thepathogens or their products.

• If the encounter is not in a lymphoid organ, the dendritic cells carry theforeign antigens via the lymph to local lymph nodes or gut-associatedlymphoid organs.


Antigen presentation by dendritic cells II


Antigen presentation by dendritic cells III

• The encounter with a pathogen activates pattern recognition receptors ofthe dendritic cell, which is thereby induced to mature from anantigen-capturing cell to an activated antigen-presenting cell that canactivate T cells.

• Dendritic cells have to be activated in order to activate naive T cells, andthey can also be activated by tissue injury or by effector helper T cells.

• Tissue injury is thought to activate dendritic cells by the release of heatshock proteins and uric acid crystals when cells die by necrosis rather thanby apoptosis.

• Activated dendritic cells display three types of protein molecules on theirsurface that have a role in activating a T cell to become an effector cell ora memory cell


Antigen presentation by dendritic cells IV

• (1) MHC proteins, which present foreign antigen to the TCR,

• (2) co-stimulatory proteins, which bind to complementary receptors on theT cell surface, and

• (3) cell-cell adhesion molecules, which enable a T cell to bind to theantigen-presenting cell for long enough to become activated, which isusually hours.

• Activated dendritic cells secrete a variety of cytokines that can influencethe type of effector helper T cell that develops, as well as where the T cellmigrates after it has been stimulated.

• Non-activated dendritic cells also have important roles.

• They help induce self-reactive T cells to become tolerant, both in thethymus and in other organs.

• Such dendritic cells present self-antigens in the absence of theco-stimulatory molecules required to activate naive T cells.


Antigen presentation by dendritic cells V


Effector cytotoxic T cells I

• Cytotoxic T cells protect vertebrates against intracellular pathogens suchas viruses and some bacteria and parasites that multiply in the host-cellcytoplasm, where they are sheltered from antibody-mediated attack.

• Cytotoxic T cells do this by killing the infected cell before the microbescan proliferate and escape from the infected cell to infect neighboring cells.

• Once a cytotoxic T cell has been activated by an infectedantigen-presenting cell to become an effector cell, it can kill any target cellharboring the same pathogen.

• Using its TCR, the effector cytotoxic T cell first recognises a microbialantigen bound to an MHC protein on the surface of an infected target cell.

• This causes the T cell to reorganize its cytoskeleton and focus its killingapparatus on the target.

• Focus is achieved when the TCRs actively aggregate, along with variousco-receptors, adhesion molecules, and signaling proteins at theT cell/target cell interface, forming an immunological synapse.


Effector cytotoxic T cells II

• A similar synapse forms when an effector helper T cell interacts with itstarget cell.

• In this way, effector T cells avoid delivering their signals to neighboringcells.

• Once bound to its target cell, an effector cytotoxic T cell can employ oneof two strategies to kill the target, both of which operate by inducing thetarget cell to kill itself by undergoing apoptosis.

• In killing an infected target cell, the cytotoxic T cell usually releases apore-forming protein called perforin.

• In the second killing strategy, the cytotoxic T cell activates adeath-inducing cascade in the target cell less directly.

• A homotrimeric protein on the cytotoxic T cell surface, called Fas ligand,binds to transmembrane receptor proteins on the target cell called Fas.


Effector cytotoxic T cells III


Effector helper T cells I

• In contrast to cytotoxic T cells, helper T cells are crucial for defenseagainst both extracellular and intracellular pathogens.

• They help stimulate B cells to make antibodies that help inactivate oreliminate extracellular pathogens and their toxic products.

• They also activate macrophages to destroy any intracellular pathogensmultiplying within the macrophage’s phagosomes, and they help activatecytotoxic T cells to kill infected target cells.

• They can also stimulate a dendritic cell to maintain it in an activated state.

• Once an antigen-presenting cell activates a helper T cell to become aneffector cell, the helper cell can then help activate other cells.

• It does this both by secreting a variety of co-stimulatory cytokines and bydisplaying co-stimulatory proteins on its surface.


Effector helper T cells II

• When activated by its binding to an antigen on a dendritic cell, a naivehelper T cell usually differentiates into either of two distinct types ofeffector helper cell, called TH1 and TH2.

• TH1 cells are mainly involved in immunity to intracellular microbes andhelp activate macrophages, cytotoxic T cells, and B cells.

• TH2 cells are mainly involved in immunity to extracellular pathogens,especially multicellular parasites, and they help activate B cells to makeantibodies against the pathogen.

• The nature of the invading pathogen and the types of innate immuneresponses it elicits largely determine which type of helper T cell develops.


Effector helper T cells III


MHC protein


CD4 and CD8 co-receptors


Processing of viral protein for presentation to CTLs


Processing of an extracellular protein


T cell development


Positive and negative selection


Helper T cells and lymphocyte activation


Activation of T cells


Activation of helper T cells


TH1 cells activation


Antigen binding to B cell receptors (BCRs) I


Antigen binding to B cell receptors (BCRs) II


B cell activation I


B cell activation II


Immune recognition molecules belong to the ancient Ig superfamily


Large N expansion I

• Let us consider two cell populations and their bivariate competitionprocess.

• Let n ≡ number of cells of type 1 and n′ ≡ number of cells of type 2.

• The master equation for the process is given by

∂P(n, n′, t)

∂t= λn−1,n′P(n−1, n′, t)+λ′n,n′−1P(n, n′−1, t)+µn+1,n′P(n+1, n′, t)

+ µ′n,n′+1P(n, n′ + 1, t)− (λnn′ + λ′nn′ + µnn′ + µ′nn′)P(n, n′, t) . (1)

• We introduce the difference operators:

Enf (n, n′) = f (n + 1, n′), (2)

En′ f (n, n′) = f (n, n′ + 1), (3)

E−1n f (n, n′) = f (n − 1, n′), (4)

E−1n′ f (n, n′) = f (n, n′ − 1). (5)


Large N expansion II

• The master equation can be written as:

∂P(n, n′, t)

∂t= (E−1

n − 1)λnn′P(n, n′, t) + (E−1n′ − 1)λ′nn′P(n, n′, t)

+ (En − 1)µnn′P(n, n′, t) + (En′ − 1)µnn′P(n, n′, t). (6)

• Define a suitable transformation of variables that will allow us to go fromthe discrete set of variables (n, n′) to the continuous set of variables (ξ, ξ′).

• The large N expansion allows us to go from a stochastic description to adeterministic one (plus fluctuations).


Expansion of the master equation I

• We expect n to consist of a deterministic part plus fluctuations.

• We introduce Ω – a parameter measuring the volume of the system, suchthat for large Ω the fluctuations are relatively small.

• We define the following transformation (n −→ ξ)

n = Ωx(t) + Ω12 ξ .

• We assume the fluctuations are of order Ω12 .

• We define the following transformation (n′ −→ ξ′)

n′ = Ωx ′(t) + Ω12 ξ′ .


Expansion of the master equation II

• Now, rather than a probability distribution P of n and n′, we have aprobability distribution Π of ξ and ξ′.

Π(ξ, ξ′, t) = P(Ωx + Ω12 ξ, Ωx ′ + Ω

12 ξ′, t) . (7)

• We can write

∂P

∂t=

∂Π

∂t− Ω

12dx

dt

∂Π

∂ξ− Ω

12dx ′

dt

∂Π

∂ξ′, (8)

where we have made use of the chain rule.

• Notice that in order to derive the previous equation, we have made use ofthe fact that

dξ

dt= −Ω

12dx

dt, and

dξ′

dt= −Ω

12dx ′

dt.


Expansion of the master equation III

• The translation operators can be written as follows

En = 1 + Ω− 12

∂

∂ξ+

1

2Ω−1 ∂2

∂ξ2+ . . . , (9)

E−1n = 1− Ω− 1

2∂

∂ξ+

1

2Ω−1 ∂2

∂ξ2+ . . . , (10)

En′ = 1 + Ω− 12

∂

∂ξ′+

1

2Ω−1 ∂2

∂ξ′2+ . . . , (11)

E−1n′ = 1− Ω− 1

2∂

∂ξ′+

1

2Ω−1 ∂2

∂ξ′2+ . . . , (12)

• We make use of these expansions in the master equation to write

∂P(n, n′, t)

∂t=

∂Π(ξ, ξ′, t)

∂t− Ω

12dx

dt

∂Π(ξ, ξ′, t)

∂ξ− Ω

12dx ′

dt

∂Π(ξ, ξ′, t)

∂ξ′(13)

= (E−1n − 1)λnn′P(n, n′, t) + (E−1

n′ − 1)λ′nn′P(n, n′, t) (14)

+(En − 1)µnn′P(n, n′, t) + (En′ − 1)µnn′P(n, n′, t) . (15)


Expansion of the master equation IV

• We know the birth and death rates depend both on n and n′.

• We need to express the translation operators in terms of the new variablesξ, ξ′.

• We need to express the birth and death rates in terms of the new variablesξ, ξ′.

• The previous equation is a power series in Ω12 .

• Identify terms of order Ω12 ⇒ equation for x(t) and x ′(t) (deterministic

part).

• Identify terms of order Ω0 ⇒ master equation for Π(ξ, ξ′, t) (fluctuations).


Hard niche case I

• In the case where νii′ 1, νi/i′ 1, νi′ i 1, νi′/i 1, we have thefollowing birth and death rates:

λnn′ =ϕ

n + n′(n + n′ − pn′), (16)

λ′nn′ =ϕ′

n + n′(n + n′ − p′n), (17)

µnn′ = µn, (18)

µ′nn′ = µ′n′, (19)

with ϕp = ϕ′p′.

• We make use of these birth and death rates and the variabletransformations (n −→ ξ) and (n′ −→ ξ′) to rewrite the master equationin terms of the new variables.


Hard niche case II

• The master equation (15) becomes

∂Π

∂t− Ω

12

dx

dt

∂Π

∂ξ− Ω

12

dx′

dt

∂Π

∂ξ′

=

24−Ω− 1

2∂

∂ξ+

1

2Ω−1 ∂2

∂ξ2+ . . .

35 Ωϕ

x + x′

"x + (1 − p)x′ + Ω

− 12

„ξ + (1 − p)ξ′ −

(ξ + ξ′)(x + (1 − p)x′)

x + x′

«#Π

+

24−Ω− 1

2∂

∂ξ′+

1

2Ω−1 ∂2

∂ξ′2+ . . .

35 Ωϕ′

x + x′

"x + (1 − p′)x + Ω

− 12

„ξ + (1 − p′)ξ −

(ξ + ξ′)(x + (1 − p′)x)

x + x′

«#Π

+

24Ω− 1

2∂

∂ξ+

1

2Ω−1 ∂2

∂ξ2+ . . .

35 µ(Ωx + Ω12 ξ)Π

+

24Ω− 1

2∂

∂ξ′+

1

2Ω−1 ∂2

∂ξ′2+ . . .

35 µ′(Ωx′ + Ω

12 ξ

′)Π .

• We have defined ϕΩ = ϕ and ϕ′Ω = ϕ′ in the previous equations.


Hard niche case III

• Terms of order Ω12 result in the following deterministic equations:

dx

dt=

ϕ

x + x ′(x + x ′ − px ′)− µx , (20)

dx ′

dt=

ϕ′

x + x ′(x + x ′ − p′x)− µ′x ′. (21)

• Terms of order Ω0 give a multi-variate linear Fokker-Plank equation:

∂Π

∂t=

"−

ϕ

x + x′+

ϕ

(x + x′)2(x + x′ − px′) + µ

#∂

∂ξ(ξΠ)

+

"−

ϕ′

x + x′+

ϕ′

(x + x′)2(x + x′ − p′x) + µ

′#

∂

∂ξ′(ξ′Π)

+

"−

ϕ

x + x′(1 − p) +

ϕ

(x + x′)2(x + x′ − px′)

#∂

∂ξ(ξ′Π)

+

"−

ϕ′

x + x′(1 − p′) +

ϕ′

(x + x′)2(x + x′ − p′x)

#∂

∂ξ′(ξΠ)

+1

2

"ϕ

x + x′(x + x′ − px′) + µx

#∂2Π

∂ξ2

+1

2

"ϕ′

x + x′(x + x′ − p′x) + µ

′x′#

∂2Π

∂ξ′2. (22)


Hard niche case IV

• The solution to the previous master equation is Gaussian.

• It is fully determined by its first and second moments.

• Thus, we then have the distribution of the fluctuations about thedeterministic value, to order Ω− 1

2 .

• This is referred to as the linear noise approximation.

• We introduce the following parameters:

K1 = x+x′(1−p)x+x′ , K2 = x′+x(1−p′)

x+x′ ,

K3 = 12(x + x ′(1− p)), K4 = 1

2(x ′ + x(1− p′)).


Hard niche case V

• From Eq. (22) above we have:

d

dt〈ξ〉 =

"ϕ

x + x′(1 − K1) − µ

#〈ξ〉 +

`(1 − p) − K1

´ ϕ

x + x′〈ξ′〉, (23)

d

dt〈ξ′〉 =

"ϕ′

x + x′(1 − K2) − µ

′#〈ξ′〉 +

`(1 − p′) − K2

´ ϕ′

x + x′〈ξ〉, (24)

d

dt〈ξ

2〉 = 2

"ϕ

x + x′(1 − K1) − µ

#〈ξ

2〉 + 2`(1 − p) − K1

´ ϕ

x + x′〈ξξ

′〉

+2ϕ

x + x′K3 + µx, (25)

d

dt〈ξ′2〉 = 2

"ϕ′

x + x′(1 − K2) − µ

′#〈ξ′2〉 + 2

`(1 − p′) − K2

´ ϕ′

x + x′〈ξξ

′〉

+2ϕ′

x + x′K4 + µ

′x′, (26)

d

dt〈ξξ

′〉 =

"ϕ

x + x′(1 − K1) − µ

#〈ξξ

′〉 +`(1 − p) − K1

´ ϕ

x + x′〈ξ′2〉

+

"ϕ′

x + x′(1 − K2) − µ

′#〈ξξ

′〉 +`(1 − p′) − K2

´ ϕ′

x + x′〈ξ

2〉. (27)


Symmetric clonotyes I

• We now consider the symmetric case where ϕ = ϕ′, p = p′ and µ = µ′.

• The deterministic equations (20) and (21) become:

dx

dt=

ϕ

x + x ′(x + x ′ − px ′)− µx , (28)

dx ′

dt=

ϕ

x + x ′(x + x ′ − px)− µx ′. (29)

• The stationary state of this system of equations is found to be

xs = x ′s =ϕ(2− p)

2µ. (30)

• Linear stability analysis shows that this stationary solution is stable.

• We now consider the fluctuations about this stationary state.


Symmetric clonotyes II

• Equations (23) and (24) (for the time evolution of the means of thefluctuations) become:

d

dt〈ξ〉 =

»µp

2(2− p)− µ

–〈ξ〉 − pµ

2(2− p)〈ξ′〉, (31)

d

dt〈ξ′〉 =

»µp

2(2− p)− µ

–〈ξ′〉 − pµ

2(2− p)〈ξ〉. (32)

• These equations have the following stationary solution

〈ξ〉s = 〈ξ′〉s = 0 .

• This stationary state is stable.


Symmetric clonotyes III

• With initial conditions 〈ξ〉|(t = 0) = 〈ξ〉0 and 〈ξ′〉|(t = 0) = 〈ξ′〉0, theabove system can be solved to give

〈ξ〉 =〈ξ〉0 + 〈ξ′〉0

2e−µt +

〈ξ〉0 − 〈ξ′〉02

e( pµ2−p

−µ)t, (33)

〈ξ′〉 =〈ξ〉0 + 〈ξ′〉0

2e−µt − 〈ξ〉0 − 〈ξ′〉0

2e( pµ

2−p−µ)t

. (34)

• It is clear that 〈ξ〉 → 0 as t → +∞ and 〈ξ′〉 → 0 as t → +∞.

• Note also that if at t = 0, 〈ξ〉 = 〈ξ′〉 = 0, then 〈ξ〉 = 〈ξ′〉 = 0 at allsubsequent times.

• Then in terms of the original variables, to the present order,

〈n〉 = 〈n′〉 =ϕ(2− p)

2µ.


Symmetric clonotyes IV

• Equations (25), (26) and (27) become:

d

dt〈ξ2〉 = 2

»µp

2(2− p)− µ

–〈ξ2〉 − pµ

2− p〈ξξ′〉+ ϕ(2− p),

d

dt〈ξ′2〉 = 2

»µp

2(2− p)− µ

–〈ξ′2〉 − pµ

2− p〈ξξ′〉+ ϕ(2− p),

d

dt〈ξξ′〉 = 2

»µp

2(2− p)− µ

–〈ξξ′〉 − pµ

2(2− p)〈ξ2〉 − pµ

2(2− p)〈ξ′2〉.

• These equations have the following stationary solutions:

〈ξ2〉s =ϕ(2− p)(3p − 4)

8µ(p − 1), (35)

〈ξ′2〉s =ϕ(2− p)(3p − 4)

8µ(p − 1), (36)

〈ξξ′〉s =ϕp(2− p)

8µ(p − 1). (37)


Symmetric clonotyes V

• The stationary solution is stable.

• Note that 〈ξ2〉s = 〈ξ′2〉S ≥ 0 and 〈ξξ′〉s ≤ 0.

• In terms of the original variables, the stationary variance of the number ofT cells of clonotypes i and i ′ is given by

σ2n = σ2

n′ = Ω〈ξ2〉 =(3p − 4)

4(p − 1)〈n〉 . (38)

• The covariance is given by

σ2nn′ = Ω〈ξξ′〉 =

p

4(p − 1)〈n〉 ≤ 0, (39)

so there is a negative correlation between n and n′.


Symmetric clonotyes VI

• With initial conditions 〈ξ2〉|(t = 0) = 〈ξ2〉0, 〈ξ′2〉|(t = 0) = 〈ξ′〉0, and〈ξξ′〉|(t = 0) = 〈ξξ′〉0, the system of differential equations (35)-(35) canbe solved to give:

〈ξ2〉 =ϕ(2− p)(3p − 4)

8µ(p − 1)+ c1e

−2µt + c2eµ(3p−4)

2−pt + c3e

4µ(p−1)2−p

t,(40)

〈ξ′2〉 =ϕ(2− p)(3p − 4)

8µ(p − 1)+ c1e

−2µt − c2eµ(3p−4)

2−pt + c3e

4µ(p−1)2−p

t,(41)

〈ξξ′〉 =ϕ(2− p)p

8µ(p − 1)+ c1e

2µt − c3e4µ(p−1)

2−pt. (42)


Symmetric clonotyes VII

• We have introduced the following parameters

c1 =3

2〈ξξ′〉0 +

1

4

`〈ξ2〉0 − 〈ξ′2〉0

´− 2ϕ(2− p)

8µ(p − 1), (43)

c2 =1

2

`〈ξ2〉0 − 〈ξ′2〉0

´, (44)

c3 =1

2〈ξξ′〉0 +

1

4

`〈ξ2〉0 − 〈ξ′2〉0

´− ϕ(2− p)2

8µ(p − 1). (45)

• Note that as t → +∞, 〈ξ2〉 → 〈ξ2〉s , 〈ξ′2〉 → 〈ξ′2〉s , and 〈ξξ′〉 → 〈ξξ′〉s .

Date post:	07-Nov-2014
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Immunology and van Kampen's large N expansion Magic 042 ...

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