J Neural Transm (2007) 114: 77–92

DOI 10.1007/s00702-006-0567-6

Printed in The Netherlands

A boolean network modelling of receptor mosaics relevanceof topology and cooperativity

L. F. Agnati1;�, D. Guidolin2;�, G. Leo1, K. Fuxe3

1 Department of Biomedical Sciences, University of Modena and Reggio Emilia and IRCCS, Ospedale San Camillo, Venezia, Italy2 Department of Human Anatomy and Physiology, University of Padova, Padova, Italy3 Department of Neurosciences, Karolinska Institutet, Stockholm, Sweden

Received: January 20, 2006 = Accepted: July 26, 2006 = Published online September 12, 2006

# Springer-Verlag 2006

Summary In the last five years data have been obtained showing that a

functional cross-talk among G Protein Coupled receptors (GPCR) exists

at the plasma membrane level where they can dimerise and are able to

generate high order oligomers. These findings are in agreement with the

receptor mosaic (RM) hypothesis that claims the existence of clusters of

receptor proteins at the plasma membrane level, where they establish

mutual interactions and work as ‘intelligent interfaces’ between the

extra-cellular and the intra-cellular environments. Individual receptor

dimers can be considered to have two stable conformational states with

respect to the macromolecular effectors: one active, one inactive. Owing

to receptor–receptor interactions, however, a state change of a given

receptor will change the probability of changing the state for the adja-

cent receptors in the RM and the effect will propagate throughout the

cluster, leading to a complex cooperative behaviour. In this study we

explore the properties of a RM on the basis of an equivalence with a

Boolean network, a mathematical framework able to describe how

complex properties may emerge from systems characterized by determi-

nistic local interactions of many simple components acting in parallel.

Computer simulations of receptor clusters arranged according to to-

pologies consistent with available experimental ultrastructural data

were performed. They indicated that RMs after a stimulation can achieve

a limited number of specific temporary equilibrium configurations

(attractors), characterized by the presence of receptor units frozen in

the active state. They could be interpreted as a form of information

storage and a role of RM in learning and memory could be hypothesized.

Moreover, they seem to be at the basis of very common ‘macroscopical’

properties of a receptor system, such as a sigmoidal response curve to an

extracellular ligand, the sensitivity of the mosaic being modulated by

changes in the topology and=or in the level of cooperativity among

receptors.

Keywords: Boolean networks, GPCR, engrams, topology, cooperativity

Introduction

The study of the dynamic behaviour of Boolean networks

(BN) has become an important area of experimental mathe-

matics in recent years. BN provide a mathematically

rigorous framework for a class of discrete dynamic systems

that allows complex, unpredictable behaviour to emerge

from the deterministic local interactions of many simple

components acting in parallel (Kauffman, 1993, 1995).

Such emergent behaviour in complex systems, relying on

a distributed rather than a centralised control (Wuensche

and Lesser, 1992), has become the accepted paradigm in

the attempt to model and to understand a number of biolo-

gical systems.

Von Neumann first proposed BN (cellular automata) to

model self-reproduction (von Neumann, 1966). Analogies

have been made between a BN rule table and a DNA

sequence (Li and Packard, 1990). Kauffman and co-

workers studied BN as models in genetics (Kauffman,

1984, 1990; Kauffman et al., 2004) and specific gene–

gene networks (such as those in E. coli and yeast) were

extracted and characterized by a BN approach (Huerta

et al., 1998; Lee et al., 2002).

Generally speaking, biological systems are formed by

networks, hence by nodes and channels interconnecting

the nodes. The communication in the biological networks

occurs via two main modes the Wiring Transmission (WT)

and the Volume Transmission (VT). WT is a point to point

transmission (one source one target) that occurs along a

well identifiable physical substrate (a ‘‘wire’’), the VT is

Correspondence: L. F. Agnati, Department of Biomedical Sciences, Section

of Physiology, Via Campi 287, 41100 Modena, Italy

e-mail: luigiagnati@tin.it� These two authors have equally contributed to the paper.

a diffuse type of transmission (one source many targets)

that occurs via the medium that embeds the nodes (Agnati

and Fuxe, 2000; Agnati et al., 2002, 2004). In the central

nervous system (CNS) as in any apparatus it is possible to

distinguish cellular networks from molecular networks and

these two types of networks are strictly intermingled, actu-

ally it has been suggested that a global molecular network

enmeshes the entire CNS that is both the extra- as well the

intracellular environment (Agnati et al., 2005b).

In previous papers (Agnati et al., 1980, 1984, 2003a;

Fuxe et al., 1983; Fuxe and Agnati, 1985; Franco et al.,

2000) we demonstrated that a functional cross-talk among

receptors at the plasma membrane level exists and we sug-

gested that high order complexes could be formed (Agnati

et al., 1982). In the last five years data have been obtained

showing that G Protein Coupled receptors (GPCR) can

dimerise and heterodimerise (Bouvier, 2001; Agnati et al.,

2003a; Fuxe et al., 2003) and are able to generate high order

oligomers (Carrillo et al., 2004; Strange, 2005; Milligan

et al., 2005; see Fuxe et al., this special issue). Thus, these

findings are in agreement with the receptor mosaic (RM)

hypothesis (Agnati et al., 1982, 2002, 2003b, 2004a, 2005b)

that claims the existence of clusters of receptor proteins

(G proteins and ion channel linked receptors) at the plasma

membrane level, where they establish mutual interactions

and work as input units in the horizontal molecular net-

works which represent ‘‘intelligent interfaces’’ between the

extra-cellular and the intra-cellular environments (Agnati

et al., 2004b).

The possibilities offered by a BN based approach to

describe the dynamics of RM and to shed some light on

their features were previously proposed and analysed by

Agnati and Fuxe in 1995 (see Fuxe et al., this issue and

Zoli et al., 1996; Agnati et al., 2002). The present paper

revisited the previous modelling of the RM as Random

Boolean Networks taking advantage of Gouldson’s and

Milligan’s data (Gouldson et al., 2000; Milligan et al.,

2005) and models which have been developed by Agnati

and Fuxe (Agnati et al., 2004a) as well on the basis of new

experimental evidence obtained by means of atomic force

microscopy, which is consistent with the ‘‘ring-assembling’’

of receptors, as proposed in a previous paper (Agnati et al.,

2004a). Computer simulations have been applied on these

receptor networks according to the BN theories and their

possible biological relevance has been investigated. The

three side-inputs to the RM that is from signals born

from extracellular, intracellular or membrane-associated

sources have been considered and, although not in a formal

way, the possible relevance of cooperativity among RMs

discussed.

The ‘receptor mosaic’ and its biological properties

Our attention has been focused on GPCR, since they are

both quantitatively and qualitatively very important. In

fact, they represent the largest superfamily of proteins in

the body (2–3% of the human genome encodes this type of

proteins) and they are capable of recognizing an astonish-

ing range of different signals, such as light, biogenic

amines, peptides, glycoproteins, lipids, nucleotides, ions,

proteases.

GPCR are composed of seven transmembrane helices

connected by extracellular and intracellular loops. The

neurotransmitter (first messenger) binding triggers the cou-

pling of the receptor to its effector molecule by a guanine

nucleotide-binding protein (the G-protein) (Gether, 2000).

Receptors, as all the proteins, can assume multiple con-

formations, each of which with potentially different bio-

chemical characteristics (Frauenfelder, 1991). The entire

spectrum of these possible conformations represents the

‘‘energy landscape’’ of the protein which describes the

range of possible conformations that the receptor can

assume and the relative stability of each conformation

(Kauffman, 1993; Strange, 1999).

It is now an accepted notion that GPCR have a certain

basal activity in the absence of agonists (Kenakin, 2005).

Thus, even slight chemical–physical influences can push

the receptor to random transitions through different states.

It is important to underline that GPCR span three distinct

microenvironments: the extracellular fluid, the membrane,

and the intracellular fluid. As indicated in the figure (Fig. 1),

a vast array of VT and=or WT intracellular, membrane-

associated and extracellular signals can affect the random

transitions of the receptor. However, it should be consid-

ered that the structural and functional plasticity of GPCR in

response to chemical–physical influences can be viewed not

simply as a random walking of the receptor in its energy

Fig. 1. Schematic representation of the multiple influences affecting

GPCR conformation and function including receptor–receptor interac-

tions. For further details, see text

78 L. F. Agnati et al.

landscape, but rather as a computation. In other words,

GPCR detects the relevant chemico-physical characteristics

of the three microenvironments in which it is embedded

and changes its conformation.

Proteins can interact with other molecules especially with

other proteins forming supra-molecular complexes. Also

GPCR interact (via VT and=or WT) with molecules present

in the three microenvironments in which they are embedded.

Hence, they can be part of molecular networks which are

located in the plane of the membrane (horizontal molecular

networks, HMN) and=or they are interconnected with molec-

ular networks that from the plane of the membrane protrude

towards the extra- and=or the intra-cellular environment

(vertical molecular networks, VMN) (see Fig. 2).

Of special importance are the receptor–receptor interac-

tions (Agnati et al., 1980; Fuxe and Agnati, 1985; Gozes,

2005; Fuxe et al., this special issue). As indicated by elec-

tron (Rash et al., 2004) and atomic-force microscopy stud-

ies (Liang et al., 2002), receptor molecules in RM are

arranged in quite ordered clusters, allowing specific in-

teractions between nearest neighbours (Milligan, 2001;

Carrillo et al., 2004). This means that of paramount impor-

tance for the RM function is its topology, i.e., the spatial

organisation of the monomers (tesserae) within the RM

where also dimers can be building blocks (Agnati et al.,

2005a). In fact, the relevance of topology is inherent to the

concept of Receptor Mosaic, since it implies that with the

same set of tesserae it is still possible to obtain substan-

tially different mosaics. A further aspect pointing out the

flexibility of the process of RM assemblage derives from

the morpheein model (Jaffe, 2005). Jaffe points out that

notwithstanding that conformational flexibility of proteins

(Huang and Montelione, 2005) is a well accepted concept

the idea still persists that protein sequence dictates a unique

secondary structure, a unique tertiary structure and, hence,

a unique quaternary structure. An alternative view is pro-

posed by the morpheein model which suggests that differ-

ent quaternary assemblies of the same primary protein

sequence are possible in which the stochiometry of the

complete assembly is dictated by different conformations

of the monomeric unit which are affected by the chemico-

physical influences of the microenvironment (Fig. 3). As

discussed in the cited paper, the function of the possible

different assemblies can be substantial. Thus, the term mor-

pheeins describes the differences in oligomeric multiplic-

ity, structure and function dictated by a conformational

change in the monomer. It is of substantial interest that

allosteric regulators can shift the equilibrium towards alter-

native structures, i.e., towards assemblies with a different

multiplicity and function (Jaffe, 2005). As mentioned

above, this model increases the degrees of freedom in the

RM assemblage starting from the same set of tesserae (see

Fig. 3).

As far as the biochemical mechanisms that allow recep-

tor assemblage two main modes of interaction have been

suggested for the formation of GPCR dimers: the ‘‘domain

Fig. 2. Schematic representation of the ver-

tical and horizontal molecular networks

(VMN, HMN) interacting with the receptor

mosaic (RM). Two main types of commu-

nications can occur in the molecular net-

works, the volume transmission (VT) and the

wiring transmission (WT). For further de-

tails, see text

Modelling of receptor mosaics 79

swapping’’ and the ‘‘domain contact’’ (Gouldson et al.,

2000). It is our opinion that both types of interactions

(domain swapping and domain contact) can also occur

simultaneously allowing the formation of high-order het-

ero-oligomers. The prevalence of one of the two ways may

depend on the receptor type and on the chemico-physical

environments in which the interacting receptors are

embedded that affect GPCR conformations and thus recep-

tor assemblage as predicted in the morpheein model.

Gouldson et al. (2000) found out that it is possible

to distinguish two pseudo-independent units in the GPCR.

Thus, the N-terminus and helices 1–5 constitute the

A-GPCR domain, while helices 6 and 7 through to the C-

terminus constitute the B-GPCR domain. A and B domains

are connected by a hinge loop (third intracellular loop:

ICL3), which is frequently the longest loop in GPCR and

therefore very well suited to allow reciprocal movements of

the two A- and B-GPCR domains. It has been suggested

that helices 5 and 6 form the dimerisation interface and the

5–6-domain-swapped dimer may be the active (R�) form

of the receptor that interacts with the G protein. Functional

sites have also been identified on the external face of helix 2

(Gouldson et al., 2000) which could be involved in the

formation of heterodimers, in the formation of high-order

hetero-oligomers as well as in heterodimerisation processes

with other proteins such as RAMPs and ion channels. Thus,

it could be of the highest relevance for the formation

of Horizontal Molecular Network (HMN, for a review of

these concepts and related findings, see Agnati et al., 2003,

2004a).

In a previous paper we have used the proposal of

Gouldson and colleagues considering contact 5–6 dimers

and swapped 5–6 dimers and on this basis suggested three

basic models: pure swapping, pure contact, and mixed

model (contact and swapping). Furthermore, by taking

advantage of both contact and swapping the formation of

high-order oligomers has been shown and also a closed-

loop oligomers has been hypothesised (Agnati et al., 2004a)

Fig. 3. Schematic representation of the assemblage of protein mosaics (hence also of receptor mosaics). The possible relevance of the morpheein model is

indicated. Furthermore, the possible existence of a protein (or of receptor) with a coordinating role (hub protein) in the mosaic assembling is also indicated.

For further details, see text

80 L. F. Agnati et al.

(Fig. 4). To this aim it is sufficient that free 6 helices of

the first GPCR takes contact with the free 5 helices of

the last GPCR (contact-dimerisation). As a further step,

it can be surmised that via helices 2 and via the N- and

C-termini of the GPCR (as was already postulated in

Agnati et al., 1995) high-order oligomers as well as, e.g.,

via RAMP (see above), complex HMN can be formed. This

is, obviously, only a hypothetical model but the ring-model

of RM has some experimental support as shown in the

figure (see Fig. 5) illustrating results obtained by means

of atomic force microscopy (for the protocol and details

on the procedure, see legend to the figure).

Summing up, a RM is a cluster of receptors working as

an integrated input unit according to the types of receptors

and the topology of the assembly. The RM is connected to

other protein molecules forming HMN and to Vertical

Molecular Networks (VMN, see Fig. 2, and Agnati et al.,

2005a, b). It is possible to give the following ‘‘operational

definition’’ (Bridgman, 1927) of a RM.

A receptor cluster works as a receptor mosaic if and only

if the cluster is such that any receptor modulates the bio-

chemical=functional features of at least one more receptor

of the cluster that is, if and only if receptor–receptor inter-

actions operate within the cluster.

Let us list some of the main biochemical features of

a RM:

� The fluctuation of each receptor (of the receptor mosaic)

among its possible conformational states is conditioned

by its intrinsic dynamics and by the intrinsic dynamics

of the other receptors in the mosaic (receptor–receptor

interactions).

� GPCR are very often allosterically regulated proteins

(Parmentier et al., 2002; Durroux, 2005) and RM, when

Fig. 4. A Possible assemblage into a receptor

mosaic with a closed-loop (ring-model) formed by

GPCR (Agnati et al., 2004). Interactions between

GPCR occur between 5 and 6 transmembrane do-

mains and via domain-swapping (see Gouldson

et al., 2000). For further details, see text. B Topology

of the Boolean network used to model the oligo-

meric complex of D2-receptors illustrated in panel

4a. The network is a cellular automaton composed of

12 units (N¼ 12), each receiving inputs from itself

and from the two nearest neighbours (K¼ 3). Two

possible arrangements of the interactions among

units are schematically shown. In the first one (left

panel) each unit receives equivalent inputs from the

neighbouring units (isotropic interactions). In the

second one (right panel) each unit establishes a dif-

ferent coupling with the neighbouring units (aniso-

tropic interactions)

Modelling of receptor mosaics 81

their monomers bind the same ligand (RM type 1) can

show cooperativity (Agnati et al., 2005a).

� An important role is played by topology in primis for

its possible interrelationships with cooperativity, if we

assume that rules analogous to the symmetry rule for

hemoglobin are also in operation at least in some types

of RM (Agnati et al., 2005a). As discussed above, RM

are part of both HMN and VMN and therefore they are

affected by signals arising from these molecular net-

works. As far as the intracellular signals (i.e., the in-

tracellular VMN) are concerned b-arrestin should be

mentioned for its importance since the normal transduc-

tion cascade is in part shut off and also altered by recep-

tor phosphorylation followed by b-arrestin binding with

b-arrestin exerting a scaffolding and important signalling

role (Lefkowitz and Sheney, 2005) leading to a different

transduction controlling e.g. cell motility and apoptosis.

Furthermore, available evidence suggests that b-arrestin

has one more functional meaning, namely of focusing

the classical transmitter action on some of the GPCR

present in the RM. This can be a widespread role in view

of the fact that b-arrestin may interact with virtually all

GPCR since all of them have phosphorylation sites even

if in different intracellular elements.

A different sensitivity characterises cooperative systems

with respect to the non-cooperative systems (Michaelis-

Menten) (Koshland and Hamadani, 2002; Changeux and

Edelstein, 2005). This feature can be applied to clusters

of RM. Thus, a RM1 may exist composed of two mul-

timeric receptor complexes (RM1þ and RM1�) formed

by different iso-receptors for the same transmitter. One

Fig. 5. Experimental evidence giving indirect support to the closed-loop model of a receptor mosaic shown in Fig. 4. Experimental procedure: CHO cells

were cultured as described in previous papers (see, e.g., Torvinen et al., 2005). CHO cells were stably transfected with the human dopamine D2L (long

form) receptor cDNA (2600 kb cDNA fragment cloned into the Plxsn-vector, which confers resistance to geneticin), and the clones resistant to geneticin

were selected (for further details, see Torvinen et al., 2005). As far as the immunogold staining cells were grown on glass slides (Chamber Slide Culture,

Labtek=Nunc, VWR International srl, Milano, Italy) coated with poly-L-lysine (Sigma, Milano, Italy). Cells were then rinsed in PBS, fixed in 4%

paraformaldehyde and glutaraldehyde 2% for 20 minutes and washed with PBS containing 20 mM glycine and subsequently treated with PBS=20 mM

glycine=1% BSA for 30 minutes at room temperature. Immunostaining was performed with the affinity purified mouse anti-D2 antibody (a kind gift of

Dr. Watson) in PBS, pH 7.4, supplemented with 1% normal serum at 4�C overnight. The cells were then rinsed three times for 10 min in Tris pH 7.4, three

times for 5 min in Tris pH 7.4þ BSA 0.2%, one time for 15 min in Tris pH 8.2þBSA 1% and incubated with gold particles (15 nm) conjugated anti-mouse

antibody (1:25) in Tris pH 8.2þBSA 1% for 1 h at room temperature. Cells were then rinsed twice for 10 min in Tris pH 7.4. Atomic force microscopy

(AFM, PARK Autoprobe CP instrument) was carried out on the D2 transfected CHO cells after immunogold labelling of D2 receptors with 15 nm

immunogold particles. An area of 700�700 nm was scanned by the AFM cantilever (no-contact mode). By means of this approach the effects the spatial

organisation of the D2 receptors could be visualised. The morphological results are shown in the two upper panels, while the quantitative evaluations of

the profile of the ‘‘hill’’ marked on the upper left panel is shown in the lower left panel. The lower right panel supports the specificity of our approach. As

a matter of fact, the modal value of the height of the ‘‘hills’’ present in the field is 15 nm, hence the size of the immunogold particles labelling the D2

receptors (Agnati, Fuxe et al., in preparation)

82 L. F. Agnati et al.

isoreceptor type, when aggregated in the multimeric com-

plex RM1�, shows negative cooperativity while the another

one, when aggregated in a multimeric complex RM1þ,

shows positive cooperativity (see Fig. 6). Such an arrange-

ment could very effectively detect and decode VT signals.

As a matter of fact, the RM1� could detect low levels of

VT signals and could trigger via allosteric interactions an

increase in sensitivity of the RM1þ (i.e., by affecting its

orthosteric binding sites) that could then give out the full

flagged response. Thus, the two multimeric complexes by

forming a single large RM gather the functional advantages

of negative and positive cooperativity.

Modeling the receptor mosaics as random

boolean networks

From the biological data introduced above, the following

assumptions can be deduced that will be used as a basis for

a computational model (see also Agnati et al., 2002) high-

lighting features of the receptor mosaic potentially relevant

for the information handling by the central nervous system

(CNS):

� A RM is modelled as a cluster of N units, each unit

representing a receptor formed by two domains (A and

B domain) linked via the hinge loop. These units should

be arranged in a two-dimensional space according to a

topology consistent with experimental data.

� As above described, a real receptor can assume multiple

conformations. From a functional point of view, how-

ever, it is generally thought (Agnati et al., 2002; Shi

and Duke, 1998) that they can be subdivided in two

broad classes:

a) Inactive conformations (R states, which will be scored

as conformation ‘0’), characterized by a ‘‘low affinity

state’’ for the macromolecular effectors.

b) Active conformations (R� states, which will be

scored as conformation ‘1’), characterized by a ‘‘high

affinity state’’ for the macromolecular effectors.

Thus, as a first approximation, the state of each receptor in

a computational model can be simply described by a binary

variable (the symbol S being used to identify it), assuming

only the values ‘0’ or ‘1’, indicating that the receptor is in

a R or R� state, respectively. At a given moment each

element in the mosaic is in one of these states and the

pattern of 0s and 1s across the whole mosaic is the

mosaic’s configuration at that moment in time.

� At each time instant the state of each receptor will be the

result of the interactions it establishes with the micro-

environment. They can be summarised as follows:

– Depending on the chosen topology, the state of each

receptor in the cluster is influenced by the configura-

tions of a number n of neighbouring receptors with

which it is coupled. Owing to this receptor–receptor

interaction, a state change of a given receptor will

affect the probability of changing the state for the ad-

jacent receptors and the effect will propagate through-

out the cluster.

Fig. 6. Possible functional meaning of a receptor

mosaic composed of two input units: RM1þ (the

receptors of this receptor mosaic interact according a

positive cooperativity behaviour) and RM1� (the

receptors of this receptor mosaic interact according a

negative cooperativity behaviour). The integrated

behaviour of these two units can represent an optimal

system to detect and decode low level intercellular

signals as present in volume transmission intercel-

lular communication. For further details, see text

Modelling of receptor mosaics 83

– The binding of molecular signals arising from the

extracellular environment (e.g. transmitters) will affect

the state of the bound receptor, strongly increasing the

probability for it to reach an active configuration lead-

ing to the activation or inhibition of some macromo-

lecular effectors.

– The state of each receptor will also be influenced by

the action of intracellular processes (such as phospho-

rylation and demethylation), which can inactivate the

receptors (Gurevich and Gurevich, 2004). Hence, the

system can return to the basal state after stimulation.

The evolution in time of the system, therefore, can be

modelled through a sequence of discrete time steps. At each

time step, each unit of the mosaic will change its state or

leave it fixed according to a boolean function (i.e. a function

providing as a result the ‘0’ or ‘1’ values only) involving the

effect of ligand binding, intracellular inactivation processes

and the states of the neighbouring receptors. All these con-

tributions will, in general, change with time. However, it has

to be observed that a large separation of time scales holds in

the system (Falke et al., 1997): ligand release, signalling and

processes aimed to return the receptor to its inactive state

occurs on a time scale of �0.1 seconds or more, while

changes in ligand binding and protein conformation occur

within milliseconds. The mosaic, therefore, once activated

has the time to achieve temporary local equilibrium before

coming back to a basal state. In the present paper we focused

the analysis on these quasi-equilibrium configurations.

Fig. 7. Examples of totalistic Boolean rules obtained from the Eq. (2) (see Appendix) for a receptor mosaic in which the state of a generic unit i at

time tþ 1 depends on three inputs (K¼ 3) at time t: its actual state and the states of two neighbouring units. A schematic representation of the assumed

wiring architecture is provided above each table, together with the assumed value for Ei (see text). The first rows of each table shows the 2K possible

configurations of the inputs. According to the Wolfram’s convention (Wolfram, 1986), they are ordered in descending values of the binary strings. The next

two rows in the table show the calculation of the corresponding argument of the function (2) on the basis of the parameters reported on the left and the last

row contains the obtained outputs. The string of outputs also provides a way to name each rule (Wuensche and Lesser, 1992). In fact, if the string of outputs

is regarded as a single binary number, we can simply use its decimal equivalent as an identifier (e.g.: the string of outputs of the rule in A forms the binary

number ‘11101000’, which is the number 232). This identifier is reported under each table. A The inputs to the unit i are isotropic (Jij¼ 1 8j) and Ei¼ 1.

This leads to the Boolean rule 232, stating that the unit i will assume the state ‘1’ if at least two inputs are ‘1’. B The arrangement of the interactions among

the units is as in A, but Ei¼ 0. The result is the rule 254, stating that the unit i will be active if only one of its input is ‘1’. C and D Ei¼ 1 as in A, but the

inputs to the unit i from the neighbouring units now are anisotropic: one of the interactions was assigned a coupling coefficient (J¼ 2) different from the

other (J¼ 1)

84 L. F. Agnati et al.

As detailed in the Appendix, under this condition a sui-

table function to describe the configurations changes the

mosaic undergoes shortly after stimulation can be written

as follows:

Siðt þ 1Þ ¼ f

Xj

JijSjðtÞ � Ei

!

with

f ðxÞ ¼ 1 if x>0;

f ðxÞ ¼ 0 if x � 0

where Si (t) indicates the state of a receptor at the time step

t, the Jij coefficients describe the strength of the coupling of

the unit i with the units j with which it interacts and Ei is a

constant (on the chosen time scale) accounting for the

effect of the intracellular processes.

As illustrated in Fig. 7, this equation simply corres-

ponds to a set of totalistic Boolean rules (Wuensche and

Lesser, 1992; Wolfram, 1986). Thus, the BN framework

(Kauffman, 1993, 1995) can be used to study the collective

properties of the receptor mosaics in the above specified

condition.

On this basis we performed an initial exploration of the

behaviour of receptor mosaic models characterized by a

topology (see Fig. 4) consistent with ultrastructural obser-

vations (Fig. 5). Details about the performed simulations

are reported in the Appendix. Albeit the model simplicity,

the results which have been obtained seem to capture some

core properties of RM. The major findings can be outlined

as follows:

� As illustrated in Fig. 8A, when started from random con-

figurations the system rapidly (the number of time steps

needed is on average 1.5) converges to one of a limited

number of short-cycle attractors (see Appendix for a defi-

nition), in general made of a single configuration.

� These configurations are always characterized by the

presence of elements fixed in the active state. As illu-

strated in Fig. 8B, when the attractors specific to a given

Fig. 8. A Examples of state transition graphs (basins of attraction) obtained when the rule 232 was applied at each unit of the mosaic illustrated in Fig. 8A.

Each graph represents all the mosaic states converging to the same attractor cycle. States are shown as dots linked to their successors and the direction of

time is inwards from the more external dots to the attractor cycle, located at the centre. The maximum number of steps to reach the attractor is the

parameter ml, an index of how quickly the system converges. The number of states belonging to each basin is reported at the bottom of each graph as

percent of the total number of states of the system (2N¼ 212¼ 4096). As shown, the observed attractors were formed by a single state (illustrated in the

thumbnail under the corresponding graph), each characterized by a number of units in the state ‘1’ (black spots in the thumbnail). B Two distributions of

the attractor states according to their number of frozen ‘1’ are shown. They correspond to the results obtained by applying the rule 232 or a mixture of the

rules 232 and 254, respectively (see text)

Modelling of receptor mosaics 85

mosaic are distributed according to the number of frozen

1s, they provide a spectrum of the possible activation

patterns of the mosaic.

� Such a set of activation patterns also influences the col-

lective response of the receptor system to an incoming

signal. As suggested by the simulations (see Appendix)

they seem to be at the basis of very common ‘macro-

scopical’ properties of a receptor system, such as a sig-

moidal response curve to an extracellular ligand. As

shown in Fig. 9, however, the sensitivity of the mosaic

can be modulated by changes in the topology and=or in

the level of cooperativity among receptors as indicated

by the different mid-point of the corresponding response

curves.

Discussion

The ‘‘philosophy’’ of modelling in biology

and some caveats for their use

Turner has proposed a simple classification of models, by

separating formal models from structural models. By struc-

tural models, he means those models whose entities in

some sense are more palpable than those we find in formal

models. Moreover, a formal model requires systematic

axiomatization. He observes that most of the models in

the area of neurophysiology are structural models (Turner,

1965; McCulloch and Pitts, 1943). As far as the formal

models are concerned Changeaux and Dehaene point out:

any model is a ‘‘representation’’ of a natural object or

process described in a coherent, non-contradictory and

minimal form, if possible mathematical. To be useful, the

way in which it is formulated must allow comparisons with

outside reality (Changeux and Dehaene, 1992). As far as

the structural models are concerned, recently attempts to

construct artificial networks mimicking neuronal network

operations have been carried out (Wasserman, 1990). These

models have been based either on silicon or on biomaterials

such as proteins (Rinaldi et al., 2003) or DNA (Seeman,

2005) or immunology molecules (Tarakanov and Dasgupta,

2000; Tarakanov et al., 2003). In any case, as Turner un-

derlines, any model possesses both positive analogy and

negative analogy, since the model is like the thing to be

modelled but not exactly, thus some aspects of the model

are inapplicable to the phenomenon and some aspects of

the phenomenon are either suppressed or idealised. How-

ever, models are often very effective heuristic instruments

(Turner, 1965).

The present model

The model introduced gives a highly schematic representa-

tion of the phenomenon of receptor activation and sig-

nal decoding in a RM. As far as receptor interactions are

concerned, formal models were proposed to describe homo-

tetramer formation (Powers and Powers, 2003) and the

cross-talk between ligand-binding sites within a dimeric

GPCR (Durroux, 2005) helping to understand the intrinsic

molecular dynamics of a receptor. In the present paper, a

higher level of receptor organization was addressed, i.e. the

mosaic that receptors can form at the cell membrane level by

establishing mutual interactions. The same problem was at

the basis of recent modelling efforts (Shi and Duke, 1998;

Duke and Bray, 1999; Shi, 2000) to describe the bacterial

sensing leading to bacterial chemotaxis. In these models a

Fig. 9. Response of the receptor mosaic to a ligand. As detailed in the text,

starting from a completely inactivated system (all the units were in the ‘0’

state) a number of units were switched to ‘1’ to simulate the binding of a

transmitter and the response was evaluated in terms of number of receptors

that were subsequently activated as an effect of the internal dynamics of

the mosaic. The curves were obtained by plotting such a response as a

function of the number of initially ‘bound’ units. The effect of changing

the interactions among the units is shown in A changes in the external

environment are shown in B

86 L. F. Agnati et al.

cluster of two-state receptors was considered in which a

cooperative coupling among receptors was present. The ap-

proach followed to model the local interactions in the cluster

of receptors was very similar to that used in the present

paper (i.e. equation (1) in the Appendix). Some constraints

on topology (a 2D array) and on the coupling between near-

est neighbours (Jij¼ Jji 8i; j; Jii¼ 0) allowed the reduction

of the model to the well known bidimensional Ising model

for magnetism, which can be solved analytically. In the

present model a different strategy was applied, based on

the observation that the equation describing the local inter-

actions corresponds to a specific sub-set of the Boolean

rules. Thus, no specific constraints were assumed either on

topology and on the coupling between receptor dimers and

a structural approach, based on the BN theory, was followed

to perform numerical simulations of the receptor cluster

behaviour in few specific cases, in order to capture some

core property of its dynamics (see Appendix).

The main object of the present structural modelling has

been the so called ring-model in view of the indirect experi-

mental evidence and also of some possible advantages the

closed-loop RM models possess versus the linear receptor

assemblies. To clarify this point, it is sufficient to mention

that several human diseases are due to GPCR mutations

(Sch€ooneberg et al., 2004; see also Fuxe et al., this special

issue) and any type of a closed model of RM (such as the

ring-model) has the advantage over linear models that the

presence of an altered GPCR (e.g., for inactivating muta-

tions) could have less dramatic effects since alternative

pathways of interactions are possible while this obviously

is not possible in a linear model of RM when the altered

receptor is placed inside of the linear chain.

Deductions from the present model

In a close-loop RM allosteric interactions allow a circula-

tion of the information in the entire cluster and this phe-

nomenon can occur in presence or even in the absence of a

ligand due to the possible constitutive activation of some

receptors. Thus, the conformations of the entire RM may

continuously walk in a complex energy landscape pre-

ferentially staying in some attractors. In particular, in

presence of cooperativity (negative or positive) marked

conformational changes affect orthosteric ligand (extra-

or intracellular) bindings especially when a ligand (extra-

or intracellularly) binds orthosteric sites of a monomer. It is

obvious the paramount importance to implement coopera-

tivity in a model of RM function, both as cooperativity

among monomers (single receptors) and among RM in an

HMN. It should be pointed out that highly cooperative and

allosterically regulated multimeric proteins have been

described as is the case of hemocyanins where changing

concentrations of several low molecular weight compounds

modulates hemocyanin oxygen binding (Menze et al.,

2005). This aspect is of the highest importance for RM

assemblage and function, since data from our group has

demonstrated that homocysteine can work as an allosteric

modulator of D2 receptors affecting both the binding site

and the internalisation of the A2^D2 heteromers (Agnati

et al., in preparation).

Even from the very simple model here presented some

deductions relevant to brain functions can be drawn. They

can be summarised and discussed as follows:

a) A RM submitted to the set of rules derived from the

interaction model of equation (2) (see Appendix) is

characterized by a specific and ordered set of activation

patterns (attractors) which represent the repertoire of

responses available to that RM. The possibility exists,

however, that some RM are Boolean networks charac-

terised by a critical value of � (Langton, 1986) and

hence are at the edge between order and chaos so they

can be in a highly ‘‘plastic state’’. As it has been demon-

strated in the present paper, these RM can be pushed

towards one of their several attractors. In other words,

they respond to impulses originated in one (or more

than one) of the three environments with which they

interact. In conclusion, the RM move towards an attrac-

tor which can be interpreted as a form of information

storage (engram) as suggested from our group already

decades ago (Agnati et al., 1982; Zoli et al., 1996; see

Agnati et al., 2003a, b; Fuxe et al., this special issue).

Thus, a role of RM in learning and memory can be

hypothesized and classical concepts such as that of

change of the ‘synaptic weight’ in learning (Hebb, 1949;

Kandel, 1979; Malenka and Nicoll, 1999) could be

founded on the existence of ‘‘plastic’’ RM at the synap-

tic membrane which, upon a specific set of signals,

move towards specific attractors.

b) The set of activation patterns defined by the local in-

teractions characterizing the mosaic also influences

the collective response of the receptor system to an

incoming signal. As suggested by the model, very

common ‘macroscopical’ properties of a receptor sys-

tem such as a sigmoidal response curve to an extra-

cellular ligand could simply emerge as a result of

the local set of interactions in the mosaic. Moreover,

changes in the level of cooperativity among receptors

or in the interaction of the mosaic with the intracel-

lular environment could modulate the sensitivity of the

Modelling of receptor mosaics 87

mosaic to the signals and, as a consequence, its func-

tional features.

It is interesting to note as a very simple Boolean approach

could capture (at least qualitatively) the main characteris-

tics of a protein ring discussed by Duke et al. (2001). In

their elegant and rigorous model of a protein ring the

authors showed that when coupling exists between neigh-

bouring proteins a conformational spread occurs giving rise

to regions in which all units have the same state. Moreover

such a conformational spread drives the system to a switch-

like, sigmoid response to changes in ligand concentration.

Thus, Boolean modelling, allowing an easier implementa-

tion of different topologies and interaction schemes, could

represent a useful tool for the exploratory analysis of the

dynamical properties of receptor systems.

It is also possible to discuss new possible functional

meanings of constitutive activation of receptors in the

frame of the RM hypothesis of the engram formation and

storage. As discussed by Kenakin (2005) GPCR sponta-

neously form active states that are capable of producing

elevated basal cellular activity and this activity can be

selectively blocked by ligands. It can be surmised that this

activity can also play a role in the maintenance of a mem-

ory trace by producing a low level but constant circulation

of the information in some HMN involved in the storage of

long-term memory traces. As a matter of fact, one of the

unsolved problems in the vast ‘‘mystery’’ of the process

of memory formation is the maintenance of long term

memory even if these engrams are used only after decades.

It may be speculated that closed-loop RM (e.g., a ring-

assembly of receptors) encoding long-term engrams are

continuously randomly activated since some of their tes-

serae (receptors) become constitutively active and rehearse

the engram via the activation of specific HMN and VMN. It

should be mentioned that such a close-loop RM can have

unwanted consequences since stably constitutively active

receptors can be one of the molecular triggers of diseases,

even of cancer (Sch€ooneberg, 2004). Thus, it has to be sur-

mised either that the process is self-limiting in the sense

that the circulation of the activity is slowly fading out and

needs either the ligand binding or a new random movement

of some receptors towards their constitutive active states.

Another possibility is a delayed activation of a feed-back

control (Shi, 2000) either from some VMN or from the

same HMN shutting off the system (Fig. 2).

It is likely that more than one RM is present in one and

the same HMN and sometimes these RM may work as

independent units fulfilling different but complementary

tasks (Fig. 6). This view is in agreement with the concept

of allosteric unit as suggested by Wyman (1969) and more

recently by van Holde et al. (2000).

It is also possible, even if not addressed in the present

paper, to apply a Boolean approach to describe the RM–

RM interactions on the basis of the concepts of ‘positive’

and ‘negative’ cooperativity described by Koshland and

Hamadani (2002). Thus, a complex picture is emerging:

intrinsic dynamics affects protein functions and some

sub-states are more suitable than others to form oligomers

(Jaffe, 2005). Specific interactions among sub-units are

then established leading to a ‘‘collective dynamics’’ affect-

ing the entire RM and probably affecting the functional

properties of neighbouring RM. Further developments in

computer simulation and modelling could be a useful

instrument to address such a complexity.

It is important to underline that at least some main

aspects of the present model can be used to formulate

heuristic hypotheses. As a matter of fact, it is technically

possible to extract post-synaptic membranes in some brain

regions crucial for learning processes (Li et al., 2005) and

to evaluate whether RM which are here located have spe-

cial characteristics of composition, stability and=or binding

of ligands.

About reductionism in biology

From a more general point of view some words of caution

should be spelt out on such a crude reductionism approach

to learning and memory. As discussed above, constitutive

activity is the result of the peregrination of receptors in

their energy landscape. Hence, to use a beautiful image

of Du Bois-Reymond, it results from the dance of the atoms

(Du Bois-Reymond, 1891) in a protein. Du Bois-Reymond

pointed out that even a precise description of the dance of

atoms in the CNS will not give us the key to have a re-

ductionism description of psychic phenomena. Du Bois-

Reymond’s strong statement is likely true. However, we

suggest that such a description will be a basic step to under-

stand not only conformations and reciprocal ties within and

between proteins hence, in particular, the shapes and func-

tions of receptors within a RM, but also how experiences

can be stored in some RM. Thus, it could shed light on

some still mysterious aspects of memory formation and its

continued consolidation.

Acknowledgements

The present work has been supported by Cofin (Ministero Ricerca

Scientifica). Dr. Massimo Tonelli (CIGS, Univ. Modena) carried out the

AFM acquisitions.

88 L. F. Agnati et al.

Appendix

A. Basic assumptions of the model

� A RM consists of a cluster of N units, each unit repre-

senting a receptor, arranged in a two-dimensional space

according to a topology consistent with experimental

ultrastructural data.

� Individual receptor are modelled as units characterized

by two stable conformational states with respect to the

macromolecular effectors: one active, one inactive. Thus,

as a first approximation, each receptor can be assumed to

be a binary element that can be in one of the states of the

set S¼ {1, 0}.

� The state of the receptor evolves in the time as a

function of:

– Its actual state and the state of the n receptors with

which it interacts;

– the binding of molecular signals arising from the

extracellular environment (e.g. transmitters);

– the action of the intracellular processes (e.g. phos-

phorylation and demethylation) aimed to inactivate

the receptor.

The evolution in time of the system, therefore, can be

modelled through a sequence of discrete time steps. At

each time step the state of each unit of the mosaic will be

defined by some function involving the effect of ligand

binding, intracellular inactivating processes and the

states of the neighbouring receptors. For a system in

which the state of each element has only two possible

values a natural and simple choice for such a function is

the expression proposed by Hopfield (1984):

Siðt þ 1Þ ¼ f

Xj

JijSjðtÞ � EiðtÞ þ TiðtÞ!

¼ f ðxÞ ð1Þ

with

f ðxÞ ¼ 1 if x>0;

f ðxÞ ¼ 0 if x � 0:

The new state of each receptor (i.e. the state at the time

step tþ 1) is defined on the basis of the sum of three

terms, expressing the effects at time t of the three above

mentioned processes acting on it:

– The first one (i.e. the summatory) gives the overall

effect of the receptor–receptor interactions, where

each Jij coefficient describes the strength of the cou-

pling of the generic unit i with a mosaic unit j with

which it interacts. It is also allowed that Jii 6¼ 0 to

consider the actual state of the receptor in the com-

putation of its state at the next time step. Thus, from a

computational point of view, each unit is considered

to be coupled to K¼ nþ 1 other units, the n neigh-

bours and itself. Furthermore in the present study,

we will assume that Jij>0 (8i; j), corresponding to a

situation of local positive ‘cooperativity’, in which

each unit tends to become active when its neighbours

are active (see Koshland and Hamadani, 2002).

– Ei is a value expressing the actions of the intracellular

phosphorylation and demethylation processes on the

receptor under scrutiny. Since these processes tend to

make the unit inactive (see Gurevich and Gurevich,

2005), Ei appears as a negative contribution in the

equation (1).

– The Ti value results from the transmitter action.

As pointed out by Shi and Duke (1998) the approach

described by (1) can also have a simple physical inter-

pretation, which can be stated as follows: JijSj, Ei and Ti

represent ‘‘forces’’ due to the energy exchange with the

close by receptors, with the extra-mosaic environment

and with the ligand, respectively.

� As indicated in the equation (1) all these quantities are in

general time-dependent. However, it has to be observed

that a large separation of time scales holds in the system

(Falke et al., 1997), changes in protein conformation and

ligand binding being faster (�10�3 sec) than processes

such as ligand release, signal transduction and receptor

inactivation (�10�1 sec). As a consequence, once acti-

vated the mosaic has the time to achieve temporary con-

formational equilibrium before coming back to its basal

state. If we focused the analysis on these quasi-equili-

brium configurations, the following additional assump-

tions can be applied:

– On the chosen time scale Ei can be considered as a

constant and acts as a threshold value for the transi-

tion of the receptor to an activated state.

– The transmitter will be present (Ti6¼0) only at specific

time instants and when Ti6¼0, the Ti value is positive

and large enough to make the unit i becoming active.

Thus, in the absence of the transmitter the equation (1)

becomes:

Siðt þ 1Þ ¼ f

Xj

JijSjðtÞ � Ei

!ð2Þ

It will be used to characterize the collective behaviour of

the system shortly after stimulation.

� As illustrated in Fig. 7, the equation (2) simply corre-

sponds to a set of totalistic Boolean rules (Wuensche

and Lesser, 1992; Wolfram, 1986) in which the result

Modelling of receptor mosaics 89

depends on the weighted sum of 1s in the K controlling

elements plus a threshold value. Thus, the BN frame-

work (Kauffman, 1993, 1995) can be used to study the

collective properties of the receptor mosaics and some

general properties of the discrete Boolean Networks (see

Kauffman, 1993; Wuensche and Lesser, 1992) can also

be applied to them.

In particular, we can define a state-space of the mosaic

as the set of all possible configurations it can have. For a

binary mosaic of size N (i.e. with N receptors) there are

2N unique configurations. Starting from some initial con-

figuration and repeatedly applying a Boolean rule the

system will move through a succession of configurations

which can be seen as a trajectory in the state-space.

Because the state-space is finite, sooner or later the tra-

jectory must encounter a state that occurred before.

When this happens, because the system is deterministic,

the trajectory becomes trapped in a cycle of repeating

configurations, or attractor. The number of time-steps

between the repeats of a configuration is the attractor

period, which could be just one if the system is stable in

a fixed configuration or could be very large if the system

is characterized by a chaotic behaviour. The same attrac-

tor can be reached starting from many different initial

configurations and the set of trajectories that flow in it is

called the basin of attraction. The whole set of basins of

attraction of a specific mosaic is known as the basin of

attraction field. Since it partitions, categorises, the whole

state-space into a limited number of attractors, the basin

of attraction field provides an explicit global portrait of a

mosaic entire repertoire of behaviour.

B. Numerical simulations of the receptor mosaic

All the simulations were performed by using the DDLab

software (Wuensche, 2003) and routines specifically devel-

oped by the authors.

The present simulations have been carried out on the

mosaic structure illustrated in Fig. 4a. It is simply the

structure of the oligomeric complex of D2-receptors as

proposed by Agnati et al. (2004a). The ring-shaped to-

pology derived from this structure and considered for the

analysis is shown in Fig. 4b. The Boolean rules describing

the internal dynamics of the corresponding BN were

derived from the equation (2).

Two characteristics of this mosaic model were explored:

1. Configurations at equilibrium (attractors)

The system under investigation is a linear cellular autom-

aton with N¼ 12 in which each unit interacts with the two

adjacent units and with itself (i.e. K¼ 3) and the analysis

consisted in the characterization of the basin of attraction

field following the method proposed by Wuensche (2003;

Wuensche and Lesser, 1992).

A few simple instances were considered:

� In a first simulation the effect of the intracellular pro-

cesses was the same at each receptor (Ei¼ 1 8i) and a

single Boolean rule corresponding to Jij¼ 1 (8i; j) was

operating at all the mosaic units. As illustrated in Fig. 4b

(left panel), this condition corresponds to a situation in

which each unit interacts in the same way with all its K

neighbours (isotropic interactions) and the correspond-

ing rule (rule 232 of Fig. 7A) states that a receptor will

be active if at least two of the receptors interacting with

it are active.

Two changes to this condition were then applied:

� The possibility exists that not all the K interactions that a

receptor establishes with its neighbours are equivalent.

In other words, the strength of some coupling can be

different from the others. This condition could occur in

a variety of situations, as, for instance, in mosaics com-

posed by receptors of different types. Thus, in a second

simulation, the architecture of the interactions each unit

establishes with its neighbours was changed. In particu-

lar one of the three interactions was assigned a coupling

coefficient (J¼ 2) different from the others (J¼ 1). This

situation (anisotropic interactions) is illustrated in Fig. 4b

(right panel). It was modelled by using two different

Boolean rules: alternate units along the ring were

assigned the rule illustrated in Fig. 7C (rule 236), while

the remaining units were submitted to the rule of Fig. 7D

(rule 234).

� As far as the effect of intracellular processes is con-

cerned, the possibility exists that they affect in a differ-

ent way the units composing the mosaic, for instance by

a mechanism involving a b-arrestin binding process (see

text). To mimic a condition in which the action of the

intracellular processes is not the same at all the mosaic’s

units a third simulation was then performed. In this case

locally isotropic interactions among receptors were con-

sidered (Jij¼ 1 8i; j), but 50% of the units were randomly

assigned the value Ei¼ 1 (i.e. the rule 232 of Fig. 7A),

while the remaining 50% of units were submitted to the

rule corresponding to Ei¼ 0 (rule 254 of Fig. 7B), stating

that the receptor will become active if just one of the

interacting receptors is active.

In all cases, the properties of the system were characterized

by estimating for each attractor in the basin of attraction

90 L. F. Agnati et al.

field a series of parameters (Wuensche, 2003). They in-

cluded the attractor cycle length, the number of frozen

elements (i.e. elements that don’t change their configura-

tion in the time), the percent of the state space made-up by

the basin and the maximum number of levels (ml) from the

attractor, an index of how quickly the system reaches the

attractor.

2. The response of the mosaic to an incoming signal

Let us now introduce the action of a transmitter. Thus,

starting from the state of inactive RM (i.e. from the mosaic

configuration where all the units are in the state ‘0’), an

increasing number (nb) of receptors was ‘bound’ (i.e.

switched to the state ‘1’). The RM response was recorded

in terms of number of receptors (na) that are subsequently

activated as an effect of the applied Boolean rules, expres-

sing the coupling between receptors. Since there are many

ways to choose nb receptors over N¼ 12 available units, all

the possibilities corresponding to a given nb were explored

and the obtained na’s averaged to provide the final estimate

of the mosaic response.

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