J. theor. Biol. (1996) 182, 513–529

0022–5193/96/200513+17 $25.00/0 7 1996 Academic Press Limited

A Model of the Immune Network with B-T Cell Co-operation.

I—Prototypical Structures and Dynamics

J C,†§ A C,† J F‡ J S†

† Unite d’Immubiologie, CNRS URA 1961, Institut Pasteur, Paris, France and ‡ Departamentode Fisica Aplicada, Universidad Salamanca, Spain

(Received on 17 November 1995, Accepted in revised form on 13 June 1996)

Hitherto, ‘‘second generation’’ network models of the immune system have all been restricted toB-lymphocytes and the Ig molecules they produce. These models have not so far been able to providea convincing mechanism for the distinction between a ‘‘Central Immune System’’ (CIS) composed ofa connected network of lymphocyte clones which couple with ‘‘self’’ antigens in a tolerant mode, anda ‘‘Peripheral Immune System’’ (PIS) composed of clones with little or no supra-clonal organizationand which produce classical immune responses when interacting with ‘‘non-self’’ antigens. Here, wepresent a new network model which explicitly incorporates B-T cell co-operation. In this model, B-cellactivation is dependent on T-cell help, and activated T-cells are down-regulated by engagement of theirTCRs by soluble Ig. We discuss the underlying biology on which we base the system of ordinarydifferential equations which defines the present network model. We then illustrate some basic featuresof the model by examining several prototypical situations with a small number of clones. Dependingon the idiotypic connectivity structure, the model exhibits two distinct modes of coupling with antigens:an ‘‘immune response’’ mode in which T- and B-cell clones grow exponentially; and a ‘‘tolerant’’ modein which T-cell clones are controlled by inclusion of all TCRs in the repertoire of an idiotypic B-cellnetwork. Finally, we discuss the simplifying assumptions of the present model and argue that its rangeof validity is indeed the region of the state-space of the system where the discrimination between theCIS and the PIS takes place.

7 1996 Academic Press Limited

1. Introduction

It has been suggested (Huetz et al., 1988a; Coutinho,1989) that the immune system (IS) is composed of twocompartments, a Central system (CIS) and aPeripheral system (PIS), which differ essentially intheir supra-clonal organization. The CIS is conceptu-alized as a connected network of activated lympho-cyte clones which are dynamically constrained tofrequencies or concentrations that preclude theirengagement in typical immune responses. Antigenswhich are available throughout ontogeny and whichinteract with the components of the CIS are

integrated in its repertoire, and participate in its stabledynamics. The PIS is the set of lymphocytes which aredecoupled from the CIS and show shallow or nosupra-clonal organization. The individual clones inthe PIS are free to grow exponentially anddifferentiate when stimulated, and are thus able tofollow classical immune responses when interactingwith available antigens.

The conceptual distinction between a CIS and aPIS is based on a large body of experimental evidence(Coutinho, 1989), and has already proven its heuristicvalue. However, the fact is that, at present, thisdistinction is essentially descriptive; to date noconvincing mechanism has been proposed which couldgive rise to the CIS/PIS distinction as an emergentself-organized property of the IS. The ‘‘second

§ Author to whom correspondence should be addressed at: United’Immubiologie. Institut Pasteur, 25, rue du Dr. Roux, 75724 ParisCedex 15, France.E-mail: carneiro.pasteur.fr

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generation’’ network models of the immune system(Varela & Coutinho, 1991) designed to date have allbeen restricted to B-lymphocytes and the Ig moleculesthey produce (Varela et al., 1988; De Boer & Perelson,1991; Faro & Velasco, 1993b). The results of thesestudies raise the real possibility that such models maynot be capable of illustrating the interplay betweenthe CIS and the PIS. The inherent problems are dual:(i) the developing network tends to expand by thecontinuous recruitment of new clones until it becomesvirtually complete, so that there is no appropriatepartitioning of the potential repertoire into networkand disconnected fractions (De Boer & Perelson,1991; Stewart & Varela, 1991; Detours et al., 1994);and/or (ii) the network is not reliably stable whencoupled to antigens that are continuously available(optimally stimulated clones tend to produce immuneresponses and to disconnect from the network)(Calenbuhr et al., 1995; Detours et al., 1994). It is truethat the available models have not been explored totheir fullest potential, and we cannot rigorouslyexclude the possibility that they may yet manage toshow how the CIS/PIS distinction can be achieved.However, we have developed the conviction that byrestricting the models to a single lymphocytecompartment we may be over-simplifying and missingessential features afforded by co-operation with othercomponents of the IS. This conviction is based notonly on the behaviour of current ‘‘second generation’’models, but also on compelling experimental evi-dence.

The active participation of ab T-lymphocytes in theselection of the immune B-cell repertoire is very wellestablished, as they are strictly required for efficientimmune responses (Hill & Chapel, 1993), and/orautoimmune diseases (either ‘‘spontaneous’’ or theirexperimental models) (Traugott et al., 1983; Botazzoet al., 1985; Holmdahl et al., 1985; Londei et al., 1985;Zamvil et al., 1985; Bendelac et al., 1987; Lindstromet al., 1988; Singer & Theofilopoulos, 1990;Marguerie et al., 1992). Even when autoimmunepathology is mediated by antibodies, the active role ofT-lymphocytes in the process is made evident by thehigh frequency of class-switch and somatic mutation(Marion et al., 1992; Randen et al., 1992; Zouali,1992). Similarly, the co-participation of B- andT-lymphocyte compartments in the establishment ofthe pre-immune repertoire, and, thus, the establish-ment of the CIS, is suggested by the ‘‘autonomous’’activity of the IS in the absence of external challenges.Non-immunized specific pathogen-free, germ-free orantigen-free mice have near-normal levels not only ofnatural antibodies and activated B-cells (Hooijkaaset al., 1984) but also of activated T-cells (Pereira

et al., 1985, 1986). Such ‘‘background’’ activity in theB-cell compartment is not independent of that in theT-cell compartment, as is shown by the recursiveselection of the actual repertoires of activated B- andT-cells (Marcos et al., 1988; Pereira et al., 1989), andthe T-cell dependence of natural B-cell activation(Huetz et al., 1988b).

In the present article we therefore question theassumption, underlying previous ‘‘second generation’’network models, that T-cell help is never a limitingfactor for B-lymphocyte proliferation or Ig-pro-duction. In an extended version of the modelproposed by Varela and co-workers (Varela et al.,1988), we make the activation of B-lymphocytesexplicitly dependent on co-operation with activatedT-lymphocytes. We explore the major qualitativechanges in the behaviour of the model introduced bythis amendment, paying special attention to themodes of coupling between the system and antigens.Many previous immune network models started bystudying network properties in the complete absenceof antigenic perturbation (De Boer & Perelson, 1991;Stewart & Varela, 1991); by contrast, the presentmodel includes antigens as one of its variables fromthe outset. Understanding the distinction between theCIS and the PIS starts by understanding how thenetwork develops and co-exists stably with thesomatic antigens.

In this article we formulate a minimal model of thedynamics of a normal ‘‘pre-immune’’ lymphoidsystem. We discuss the underlying biology (Section2.1) on which we base a system of ordinarydifferential equations (Section 2.2). We then presentsome basic features of the model by illustrating themin simple prototypical situations with a small numberof clones, paying special attention to the innovativefeatures of the model as compared with previous‘‘second generation’’ immune networks. In particu-lar, we show that depending on the idiotypicconnectivity structure, the present model is capableof both CIS and PIS modes of coupling withantigens. Finally, we discuss the simplifying assump-tions of the present model and argue that the domainin which these assumptions are valid is indeed theregion of the state-space of the system where thediscrimination between the CIS and the PIS takesplace.

In a companion article (Carneiro et al., 1996) wegive substance to these considerations by showingthat an appropriate distinction between a CIS and aPIS can indeed be achieved in numerical simulationsof a model with continuous generation and metady-namical recruitment of new clonotypes. The results

515

of these simulations lead to a reconsideration ofsome classical concepts such as the ‘‘immunologicalself’’ and the selective pressure for generation ofdiversity.

2. The Model

2.1.

The components of the model presented here are:peripheral B-lymphocytes and the Ig molecules theyproduce; peripheral T-helper lymphocytes; andavailable antigens in the body. B- and T-lymphocytesare continuously produced from precursors in bonemarrow and thymus respectively. The fate of eachperipheral B- or T-lymphocyte depends on its clonalreceptor ligation and its interactions with other cellsin the IS: it will either be activated (to proliferate andperform effector functions), or else it will die withina few days.

B-lymphocytes are rescued from death if they areactivated following productive ligation of theirmembrane Ig-receptors (mIg). This process involvestwo steps: firstly, crosslinking of mIg leads toadequate expression of membrane proteins (adhesionmolecules and co-receptors) that promote and enableco-operation with activated T-cells (the inductionstep); secondly, full activation of the B-cell followsactual co-operation with activated T-cells (theactivation step). Once activated, B-lymphocytes bothproliferate (resulting in clonal expansion) and secretesoluble Ig-molecules. Induced B-cells that are notfully activated will revert to a resting state.

T-cell activation depends on the specific TCR-de-pendent interaction of resting T-cells with antigenpresenting cells (APC). Once activated, T-cellsproliferate, and they can also trigger and modulatethe activation of B-cells, both by specific cell-to-cellco-operation and by cytokine production.

Multivalent Ig-molecules, secreted in soluble formby activated B-cells, attain a homogeneous concen-tration in the body fluids, and contribute to theinduction of B-cells and/or the regulation ofactivation state of both B- and T-lymphocytes bybinding their receptors. Similarly, antigens availablein the body can contribute in their native confor-mation to the induction of B-cells; once processed andpresented by the major histocompatibility complexprotein as (MHC)+peptide assembled complex onthe membrane of unspecific APCs, they can also driveT-cell activation.

In addition to these general assumptions, the modelwe present here is based on a further set of sixqualitative postulates that are stated below and

illustrated in Figs 1 and 2; the biological justificationfor these postulates is presented in the Appendix. Wewould like to emphasize that this set of simplifyingassumptions is only meant to be valid whenconsidering variable region-specific interclonal inter-actions during a ‘‘pre-immune’’ steady state in anormal IS. In conditions in which the immune systemis rapidly and strongly perturbed (for example byantigenic challenges that lead to clonal expansion anddominance) some of these simplifications almostcertainly fail to hold. Moreover, mechanisms foractivating Ig-production by B-cells in the absence ofT-lymphocytes exist and may operate in theautonomous activities in germ-free animals. To datethere is no conclusive evidence that naturallyactivated antibody repertoires are different in T-cellcompetent and deficient mice (Freitas et al., 1989;Malanchere et al., 1995).

(i) There are no specific interactions between restinglymphocytes. Productive interactions occur onlybetween ‘‘co-operation prone’’ activated T-cellsand induced (or activated) B-cells.

(ii) B-lymphocyte induction and T-cell activationrequire the cross-linking or polymerization oftheir clonal receptors.

(iii) Primary activation of resting T-lymphocytes istriggered by co-operation with unspecific APCs,involving the specific recognition of a set ofdominant and frequent antigenic peptides in thecontext of MHC molecules. The contribution ofB-lymphocytes is irrelevant.

(iv) Primary induction of resting B-lymphocytesfollows cross-linking of mIg by multivalentligands: these are either common antigens, orsoluble idiotypic Ig molecules.

(v) In order to engage and sustain co-operation withan activated T-cell, an induced B-cell mustsomehow engage its TCR either by presentingMHC+peptide complexes that are specificallyrecognized by that T-cell, or by direct mIg-TCRinteractions.

(vi) Soluble Ig-molecules are inhibitory for T-cells.

It is worth pointing out explicitly that in the presentmodel, this inhibition by anti-TCR soluble Ig is theonly possible down-regulatory influence on activatedT-cells. This is of course a major simplification.

2.2.

The translation of the qualitative model outlinedabove into an explicit set of quantitative differentialequations which can serve as the basis for computersimulations intrinsically involves a second set of

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simplifying assumptions. We will comment on theseadditional postulates in the presentation of theequations themselves. As in any modelling work, weaim at relating just two levels of description: anunderlying ‘‘micro’’ level of cell physiology andmolecular interactions, and an emergent ‘‘macro’’level of the global behaviour of the system (tolerance,immune responses and so on). The differentialequations and functional forms used below todescribe the underlying ‘‘micro’’ level are purely adhoc and phenomenological; they do not even attemptto reflect the ‘‘sub-micro’’ mechanisms which in turngive rise to the properties of the underlyingcomponents as emergent phenomena in their ownright.

2.2.1. The variables

The basic variables in our model immune systemare the size of the T-lymphocyte clones (T ); the sizeof the B-lymphocyte clones (B) and the concentrationof Ig-molecules they produce (F ); and the effectiveconcentration of the available antigens (A). At anygiven time the state of the system is defined by itscomposition:

Tl l=1, . . . , NT

Bi , Fi i=1, . . . , NB

Ak k=1, . . . , NA

In the present model, the units of the clonal sizes(T ) and (B) are ‘‘cell numbers’’. It is assumed thatcell-cell interactions and lymphocyte activation occurlocally in the lymphoid organs; and furthermore, thatthe induction of B-cells and the activation of T-cellsis the limiting factor for their productive encounter.The effective concentrations of Ig-molecules (F ) andavailable antigens (A), which diffuse freely, are thosein peripheral blood and lymph. This model thus takesinto account, to some extent, the spatial organisationof the immune system. A more fully realisticmodelling of spatial organisation, including thepossibility of stochastic effects, may or may not affectthe dynamics and stability of the interactions in themodel. This is clearly a potential area for future workin the field not only theoretical but also experimental;in this first approach we have adopted the approachof maximal simplification in order to establish a pointof reference.

In the presentation below we will use the followingnotation: (i) the main variables are identified byindexed roman capital letters; (ii) intermediatevariables are identified by small case greek letters; (iii)parameters are identified by small case roman letters(with indicative suffices); (iv) finally interaction

coefficients between the main variables Xi and Yj aredenoted as MXY

ij where XY identifies the nature of theinteracting variables and i and j identify the particularclonotypes.

2.2.2. The basic equations

The size of a given T-lymphocyte clone (Tl )decreases exponentially with a constant Deathrate (kDT ), and increases as a function of its averageproliferation rate and thymic output. The effectiveProliferation rate is proportional (kPT ) to the numberof activated cells (function aT ), which is determinedby the combined effect of both stimulatory (pl )and inhibitory (hl ) signals (see below). The termj(l, t) corresponds to the thymic output rate forcells of the clone l; j(l, t) is a function which describesthe result of processes of gene rearrangementand thymic selection, which may vary with time(t).

dTl

dt= − kDT ·Tl + kPT ·aT (pl , hl , Tl )+ j(l, t) (1)

The size of a given B-lymphocyte clone (Bi )decreases exponentially with a constant Death rate(kDB ), and increases as a function of its averageproliferation rate and de novo bone-marrow pro-duction. The effective proliferation rate is pro-portional (kPB ) to the number of activated B-cells inthe clone (function aB ), which is determined by theamplitude of induction signals (si ) and the number ofspecific activated T-lymphocytes available for co-op-eration (ti ) (see below). The contribution of thebone-marrow, resulting from random gene rearrange-ment and possible local selection, is represented by anappropriate function z(i, t).

dBi

dt= − kDB ·Bi + kPB ·aB (si , ti , Bi )+ z(i, t). (2)

The concentration of soluble Ig-molecules (Fi )produced by clone i decreases at a rate proportionalto their removal either as free molecules (kDF ), or asthe complexes they form with available ligands(kDC ·si ), and increases proportionally (kSF ) to thenumber of Ig-producing (activated) B-cells [the samefunction aB as in eqn (2)].

dFi

dt= −(kDF + kDC ·si )·Fi + kSF ·aB (si , ti , Bi ). (3)

Although the proliferating and Ig-secreting sub-populations of a B-lymphocyte clone are not alwayscoincident, and this distinction was included inprevious mathematical models (Varela et al., 1988),the present model assumes that they are identical (aB )

RestingB

RestingT

ActivatedT

InducedB

TCR

CoRec

CAM

CA

M

CoRec

CAM

p+MHC

mlg

CA

M

p+MHC

mlg

TCR

p+M

HC

p +M

HC

mlg

TC

R

CoR

ec

CA

M

CoR

ec

CA

M

TC

Rm

lg

517

in order to reduce the dimension of the parameterspace.

2.2.3. Antigens

The present model essentially aims at understand-ing the structural and dynamical features ofclonally specific B- and T-lymphocytes, that bringabout a stable network mode of coupling with thesomatic antigens in the body, and neverthelessmake available a significant repertoire of discon-nected clones (that, at least potentially, could bedriven by other antigens). Since the body has arather constant composition of endogenous somaticantigens, we will assume that these have a fixedconcentration in time (Ak ), and constitute the poolof available antigens. Additional antigens can beintroduced at various times in the simulatedontogeny of the system.

2.2.4. Stimulatory and inhibitory signals: T-cells

The activation state of T-lymphocytes is deter-mined by the combined effect of both stimulatory andinhibitory signals.

As we have argued [postulate (iii) above], in asteady-state IS the principal stimulatory signals arethe dominant and frequent MHC+peptide com-plexes presented by unspecific APC. The process bywhich the list of somatic antigens ‘‘maps’’ to a set ofMHC-presented peptides is very complex anddepends (at least) on the following factors: the relativerepresentation of the antigens in the body and theirefficiency in being captured by APCs; the differentialpeptide composition of each antigenic molecule andthe nature and efficiency of degradation; the relativeefficiency with which the different peptides areprocessed, assembled with MHC molecules andtransported to the cell surface; and finally the relativehalf-life of each MHC+peptide pair on themembrane. Consequently, the terms pl in eqn (1) are,in principle, functions fl of at least the antigenconcentrations Ak and the interaction coefficientsMTA:

pl =fl (Ak ; MTAlk ). (4)

The terms pl are thus akin to intermediate variables,and for this reason they are identified notationally bya Greek letter. However, in line with our previousassumptions concerning the pool of availableantigens, the loading of the somatic antigens on theMHC molecules on the membrane of unspecific APCsmust be constant in time (as long as we assume thatthe dynamics of lymphocytes does not significantlyinterfere with that function). In these conditions, thestimulatory signals perceived by a given T-cell clone

1 may also be considered constant. We thereforepropose to bypass the establishment of preciseequations for all these processes by simply assuming,for each and every T-cell clone 1, that the terms pl

in eqns (1) and (4) are arbitrary constants withvalues between zero and one which correspond tothe proportion of T-cells in clone 1 that can bepotentially activated. The proportion of cells whichare actually activated will be modulated by theinhibitory signals hl .

Following postulate (vi), the inhibitory signals hl

are calculated as a linear sum of the anti-TCR

F. 1. Interclonal interactions occur only between co-operationprone activated T-lymphocytes and induced (or activated B)lymphocytes. Constitutively expressed clonal receptors (mIg andTCR) are depicted in bold type; (MHC+p)eptide is depicted initalics to emphasize that although MHC molecules are expressedconstitutively, membrane representation of a particular peptide iscontingent. ‘‘CoRec’’ stands for pan-clonal co-receptors (such asCD40 in B-lymphocytes or B7 family in T-lymphocytes) whoseexpression is activation or induction dependent; ‘‘CAM’’ stands forcell adhesion molecules (such as CD44 in B-lymphocytes) which areupregulated upon receptor engagement. Constitutive co-receptorsor adhesion molecules (such as CD28 in B-lymphocytes or CD40Lon T-lymphocytes) are not depicted. The respective sets of CoRecand CAM upregulated in induced B and activated T-lymphocytesare complementary, and thus enable and stabilize B-T cellco-operation. The pattern of CAM expression also allowsappropriate interactions with other cells and the extracellularmatrix such that co-operation prone lymphocytes follow ‘‘conver-gent’’ pathways of migration. The down-regulation of CoRec andCAM in resting lymphocytes prevents them from consistentlyco-operating with other lymphocytes; the lack of such molecules isindicated by the sign ‘‘X’’.

Available Antigens

RestingB

RestingT

ActivatedT

ActivatedB

InducedB

Effective Antigenscontinuous

epitopes peptides

(J)(M)

(N)

(K)

(L)

Idiotypes

SolubleIg-molecules

–

Available Antigens

RestingB

RestingT

ActivatedT

ActivatedB

InducedB

(d)

Idiotypes

SolubleIg-molecules

Idiopeptides

virtualpeptides

virtualepitopes

(c)

(a) (b)

(e)(f)

(b)

(b)(a)

. E T A L .518

F. 2. (a): the components and interactions included in the immune network model. B-lymphocytes: induction of resting B-lymphocytesdepends on the engagement of the mIg by native epitopes on antigens (J), and/or by soluble Ig molecules produced by other B-lymphocytes(K); full activation requires co-operation with activated T-lymphocytes (L) mediated either by specific MHC+peptide complexes or byanti-TCR mIg molecules; activated B-lymphocytes can (i) produce soluble Ig molecules, (ii) divide, and/or (iii) revert to the resting state.T-lymphocytes: resting T-lymphocytes are activated following co-operation with APCs which present dominant antigenic peptides (M);this activation event can be inhibited by soluble anti-TCR Ig molecules produced by B-lymphocytes (N); activated T-cells can (i) activateinduced B-lymphocytes, (ii) divide, and/or (iii) revert to the resting state. (b): putative components and interactions which have beenexplicitly omitted from the model on account of their low probability in vivo. Induction of resting lymphocytes by specific interactions(a) with other B-lymphocytes or (b) with T-lymphocytes. Specific activation of T-lymphocytes by interactions (c) with T-lymphocytes, or(d) with B-lymphocytes. Activation of resting T-lymphocytes by interactions (e) with anti-TCR Igs polymerized on the membrane of APCsor (f) with MHC+idiopeptides presented by APCs.

Ig-molecules weighted by their respective affinities:

hl = sNB

j=1

MTFlj ·Fj . (5)

We assume that these inhibitory signals onlybecome effective when they are above a giventhreshold concentration. We employ a one-sided log-normal function for this inhibitoryresponse curve [eqn (6b)], by analogy with thesuppressive side of the cross-linking curvepreviously postulated for B-cells [see eqn (7)below].

Thus, the number of activated T-cells in our modelcan be calculated as:

aT (pl , hl , Tl )= pl ·Tl ; if log(hl )Q aT1 (6a)

aT (pl , hl , Tl )=

exp$−0log(hl )− aT1

aT2 12

%·pl ·Tl ; if log(hl )e aT1 (6b)

where pl is the l-specific constant obtained accordingto simplification in eqn (4), and aT1 and aT2 areabsolute constants.

This activation function is depicted in Fig. 3(a).

2.2.5. B-cell activation

The activation of B-lymphocytes is a two-stepevent requiring first the induction by cross-linking ofthe mIg, and then co-operation with specific T-cellclones.

Step 1. The degree of cross-linking as a functionof the concentration of soluble (multivalent) ligandhas been previously well studied in biophysical

B

σ

αT(π

,η,T

)

B/2

0

(a) Activation Function for T-Lymphocytes: αT(π,η,T)

exp(b1)

πT

η

πT/2

0 exp(aT1)

B

B/2

0B/2

B

B/2

0

β

τ

α B(σ

,τ,B

)(b) Activation Function for B-Lymphocytes: αB(σ,τ,B)

0

519

models, and follows a general bell-shaped curve.According to Faro & Velasco (1993a, 1994), theactual value of the kinetic constant of dissociationhas a strong impact in the precise shape andpositioning of the activation curve. However, ina maximal simplification, we consider here (asin previous models) that the bell-shaped curveis independent of the affinity and dissociationconstants. Thus, the number of induced cells in agiven B-cell clone i is calculated as:

bi =exp$−0log(si )− b1

b2 12

%·Bi (7)

where si is the sum of the concentrations of all themultivalent ligands for the Ig of the clone i [solubleIg-molecules produced by other lymphocytes (Fj ) or

antigen (Ak )], weighted by their respective pairwiseaffinities:

si = sNB

j=1

MBFij ·Fj + s

NA

k=1

MBAik ·Ak . (9)

Step 2. The fraction of fully activated cells in B-cellclone i is then calculated as a function of bi , thenumber of induced cells in the clone, and ti , thenumber of activated T-cells that are ‘‘available’’to co-operate specifically with them. In general,the bigger the size of the B-cell clone the moreT-cells will be required to activate the sameproportion of cells, since a single T-cell canonly co-operate with a limited number of B-cells at atime.

aB (si , ti , Bi )=ti ·bi

ti + bi. (10)

F. 3. Top: the inhibition function of T-lymphocyte clones as a function of h, the summed inhibitory signals from anti-TCR Ig.—Bottom:the activation of B-lymphocytes as a function of the field s and specific T-cell help t. (See text for full definitions of variables.)

. E T A L .520

This formula is comparable to that derived by DeBoer & Perelson (1995) for T-cell proliferationfunctions.

It may be noted at this point that when ti�bi wehave aB 1 bi which renders the eqns (2) and (3)formally identical to the ones proposed by Varelaet al. (1988); in other words, when specificT-lymphocyte help is freely available and notlimiting, the dynamics of B-lymphocytes [eqn (2)]and Ig-molecules [eqn (3)] reduces to that of atypical ‘‘second generation’’ network model, in acoherent general scheme. Innovative features canbe expected in the present network model whenti E bi , since in this case competition between B-cellsfor T-lymphocyte co-operation will become signifi-cant.

The induction and activation functions forB-lymphocytes are depicted in Fig. 3(b).

Finally, the number of ‘‘available’’ activated T-cells(ti ) depends necessarily on the other clones that canreceive specififc ‘‘help’’ from them: all the candidateB-lymphocytes will compete amongst each other forthe co-operation sites, and their relative fitness mustbe proportional to the interaction strength of theparticular pair Bi–Tl .

Thus, the quantity ti in eqn (10) can be calculatedas:

ti = sNT

l=1

$BTil

m·

$BTil ·bi

sNB

j=1

$BTjl ·bj

·v ·aT (pl , hl , Tl ) (11)

where the term v takes into account the number of‘‘co-operation sites’’ per activated T-cell; i.e. itdetermines how many B-cells can co-operate simul-taneously with a single activated T-cell, and m is aconstant which normalizes the affinity coefficients $BT

il

(we assume these terms to be fixed parameters). $BTil is

the interaction strength of the particular pair Bi–Tl ;notationally, the cross-bar on the $ indicates that thisterm has two distinct forms depending on whether theco-operation is achieved and maintained by directmIg-TCR recognition, or by MHC+Ag-peptidepresentation.

In the first case the interaction strength will beessentially the affinity of the two molecules:

$BTil =MBT

il . (12a)

In the second case the situation is more complex,since the affinity of the mIg for the native antigen, theefficiency of processing and presentation of thepeptides, and the concentration of the antigen

itself (both in absolute terms and relative to itsantigenic or peptidic competitors) all influence thestability of the co-operation Bi–Tl . In a maximumsimplification we will assume that the strength of theinteraction is proportional to all these quantities, andthus we have:

$BTil = s · s

NA

k=1

MTAlk ·MBA

ik ·Ak (12b)

where s is a non-dimensional constant that bringsthe two modes of B-T co-operation into the samescale.

2.2.6. Parameter settings

As a consequence of trying to stay as close aspossible to the biology, the model proposed here isfairly complicated, and has a considerable numberof parameters. A systematic exploration of theparameter space is not the issue in this first ap-proach; we aim at identifying certain qualitativeemergent features of the model (in particular, theCIS vs. PIS issue), and with that in mind we havechosen a reference set of parameter values thatexhibit the relevant behaviour. The present modelis an extension of the model of Varela et al.(1988), and this guided us in the choice of many ofthe parameters. Thus, the reference parameters forthe dynamics of B-lymphocytes and Ig-moleculesare in typical ranges previously proposed for thatmodel [kPB =0.3, kDB =0.1, kDF =0.04, kDC =0.008,kSF =4.0, b1 = log(80.0) and b2 = log(2.1) (Detourset al., 1994)].

For T-lymphocyte clones, we chose values thatwould make their intrinsic dynamics reasonable whencompared to those of B-lymphocytes [proliferationand decay rates comparable to those of B-cells; thereference values were kPT =0.2, kDT =0.15,aT1 = log(80.0), aT2 = log(2.1)]. This amounts toassume that the Ig-mediated inhibition of T-lympho-cytes follows a curve as a function of ligand thatis in the same range of the ‘‘suppressive’’ regionof cross-linking curve in B-lymphocytes; thisassumption is reasonable, since a concentration of Igwhich is sufficient to saturate the membrane Igs in aB-cell on a ‘‘one-to-one’’ basis should also besufficient to ‘‘block’’ the TCR molecules in aT-lymphocyte. Also, in the initial investigations to bereported here, the terms pl are all simply set at themaximal value 1.0.

Finally, the most difficult parameters to estimateare those that regulate B-T co-operation, namely vand s, since we have no good experimental estimationof their actual values. We will only suggest for the

521

moment a reference range. v is the parameter thatdetermines how many B-cells can co-operate simul-taneously with a single activated T-cell; clearly, atleast for obvious geometric reasons, this value mustbe finite and small (we have assumed that it is inthe interval [1, 10]). In fact, this parameter amountsin practice to a scaling factor between the size ofB- and T-lymphocyte clones, so it can be expectednot to influence the general qualitative behaviour ofthe system. In our initial investigations, it is set at1.0. The parameter m, which normalizes the affinitycoefficients $BT

il , is set at 1.0 since the ‘‘affinities’’employed here are typically of this order, incoherence with the parameters governing the dy-namics (Detours et al., 1994; Calenbhur et al.,1995).

The parameter s, which corresponds to therelative efficiency of the two modes of T-cell help,is assumed to be in the interval [sE exp(−b1)].There is some experimental evidence whichsuggests that anti-TCR B-cells may have somecompetitive advantage over the MHC-presentingones (Tite et al., 1986; Martinez-A. et al., 1988).That is in fact an a priori expectation since the‘‘valency’’ of the B-lymphocyte (mediated byeither MHC or mIg) for the T-cells must bethe major determinant of the value of s. It isknown that the total number of mIg bindingsites on the surface of a B-cell [about 105

molecules per cell, ref. (Resch, 1976)] will usuallybe some orders of magnitude higher than thenumber of individual peptides actually presentedby available MHC molecules [potentially between1 and 105 peptide copies per cell, but usually inthe order of a few hundreds (Harding & Unanue,1990)]. Moreover, as already discussed, mIgmolecules are constitutively expressed by theB-lymphocytes, and tuned according to theirintrinsic cell physiology, while the surface represen-tation of a particular MHC-peptide complex isessentially contingent and dependent on othercandidate peptides in the process of processingand presentation (Manca et al., 1991; Harding& Geuze, 1993; Kroon & McDonnel, 1993; Lehmanet al., 1993; West et al., 1994). Qualitatively, thevalue s=exp(−b1) implies that a B-lymphocytethat is optimally induced by cross-linkingantigen and presents its peptides to specific T-cells,will have a B-T interaction strength which iscomparable to that of an anti-TCR B-lymphocytewith unitary affinity; smaller values imply thata B-lymphocyte needs higher concentrations ofantigen to present peptides with the sameefficiency.

3. Results: Preliminary Studies in Small Prototypical

Systems with Fixed Connectivity Structures

3.1.

– -

In this section we describe some basic dynamicalproperties of a pair of co-operating B andT-lymphocyte clones that derive rather directly fromthe inbuilt postulates of the model. The two clones ofsuch a pair can couple in two prototypical modes: onthe one hand through (MHC+peptide)—TCRinteractions, on the other through mIg—TCRinteractions (Fig. 4).

In the case where the members of a B-cell clonetarget the members of a given T-cell clone bypresenting ‘‘specific’’ peptides on their membraneMHCs, there is no particular relationship between theclonal receptors of the two types of lymphocytes. TheIg-molecules produced by the B-cell followingactivation will not interfere directly with the TCR ofthe T-cells, and therefore will not interfere with thestate of activation of that clone [Fig. 4(a)]. Theconsequence of this mode of B-T co-operation isimmediate: both lymphocyte clones will growexponentially as long as the other stimuli they requirefor activation are maintained.

In order to analyse this mathematically, weconsider a prototypic system (prototype I) consistingof one B cell clone (B1), the Ig it produces (F1) andthe T-cell that it co-operates with (T1). Theinteraction coefficient between F1 and T1 is null. Wekeep both s1 = exp(b1) and p1 =1.0, i.e. constant andoptimal stimuli. To render the system as simple aspossible, the interaction coefficient $BT

il [definedaccording to eqn (12b)] and v are unitary, and thesource terms j(1) and z(1) are null. Equations (1–12)in this simplified system reduce to:

dT1

dt=(kPT − kDT )·T1 (13)

dB1

dt= − kDB ·B1 + kPB ·

T1·B1

T1 +B1(14)

dF1

dt= −(kDF + k'DC )·F1 + kSF ·

T1·B1

T1 +B1(15)

where: k'DC = kDC ·exp(b1).It is easy to see mathematically that this system

either collapses or is unstable. In Fig. 4(a) weillustrate the behaviour of this prototypical systemusing the standard set of parameters. In the bottomgraph, several ‘‘explosive’’ trajectories in state-spaceare depicted.

T1

T1

B1

F1103

103 103

10–3

10–3

10–3

T1

B1

F1103

103 103

10–3

10–3

10–3

(σ1)

B1A1F1

I

T2

(σ2)

B2A2

F2

II

(a) (b)

. E T A L .522

F. 4. Top panel: two prototypical modes of B-T co-operation. I—A B-cell clone (B1) and a T-cell clone (T1) which are both specificfor antigen A1 are stimulated, and their co-operation is mediated by MHC+peptide presentation by the B-cell clone. The immunoglobulins(F1) produced by B1 do not recognize T1 directly, so the latter are not controlled and grow exponentially. II—The B-cell clone (B2) andthe T-cell clone (T2) are both specifically stimulated by a common antigen A2 as in I, but their co-operation is now also mediated by directmutual recognition. The immunoglobulins (F2) produced by B2 recognize directly T2, and inhibits its growth, defining a negative feed-backloop that renders the system stable. Bottom panel: several trajectories in phase-space from numerical simulations of the two models, leading(I) to explosion and (II) to a fixed-point attractor. These representations were obtained using the package GRIND by R. J. De Boer; theparameter values are those defined in the text.

We will now discuss the second major prototype ofinteraction, where a B-cell clone co-operates with a

T-cell clone through direct mIg-TCR interaction[Fig. 4(b)]. The Ig molecules produced by theactivated B-lymphocytes specifically recognize theTCR of their T-cell counterparts. Since freeanti-TCR Ig molecules constitute inhibitory signalsfor the T-cell, the B-T pair will be dynamicallystabilized by a negative feedback loop. Thus, evenwhen the other stimuli they require are optimallystimulatory, the B-T pair will evolve towards asituation in which the anti-TCR Ig molecules (bothfree and membrane-bound) produced by the B-cellsmaintain a certain level of T-cell activity, which inturn is just sufficient to sustain the B-cells. Thisequilibrium is stable, because if the B-cell activitywere to increase, the additional anti-TCR Igs wouldinhibit the T-cells, and the reduced T-help woulddecrease the B-cell activity back to the equilibriumpoint.

Mathematically, we may consider as a minimalsystem with heuristic value the prototype IIcomposed of a single B cell clone (B2), the Ig itproduces (F2) and the T-cell that it co-operates with(T2), but now the interaction coefficient between F2

and T2 is not null (MFT22 q 0). We keep both s2 and p2

constant and optimal. To render the system as simpleas possible, the interaction coefficient MBT

22 [in thiscase idiotypic affinity, according to eqn (12a)] and vare unitary and the source terms j(2) and z(2) arenull. Equations (1–12) in this simplified systemreduce to:

dT2

dt=(kPT ·a'T (F2)− kDT )·T2 (16)

dB2

dt= − kDB ·B2 + kPB ·

a'T (F2)·T2·B2

a'T (F2)·T2 +B2(17)

dF2

dt= −(kDF + k'DC )·F2 + kSF ·

a'T (F2)·T2·B2

a'T (F2)·T2 +B2(18)

where:

k'DC = kDC ·exp(b1); a'T (F2)=aT (p2, h2, T2)

p2·T2.

This system has a single non-trivial fixed point whenthe following conditions are satisfied:

kDB Q kPB and kDT Q kPT and a'T (F2)= kDT /kPT .

B,F

0Time

T

100 200 300 0Time

100 200 300

987

6

5

4

3

2

2

100

0Time

100 200 300

987

6

5

4

3

2

2

107

106

105

104

103

102

0Time

100 200 300

101

100

10–1

10–2

10–3

10–4

102

A3 T3 B3 F3

A4 T4 B4 F4

100

523

F. 5. Prototype III. T-cell clones T3 and T4 are stimulated by specific antigens A3 and A4. B-cell clones B3 and B4 co-operate by directmutual recognition with clones T3 and T4 respectively. The induction of the two B-cell clones is ensured by mutual recognition throughtheir immunoglobulins (F3 and F4). The structure is symmetrical. The fate of this system is strongly dependent on the initial conditions:the system can either fall into a locally stable oscillatory regime in which all the variables are non-null (left); alternatively, a failure inestablishing a dynamical equilibrium between the two B clones results in collapse of the variables B3, F3, B4 and F4 with concomitantexponential growth of the T3 and T4 components (right). The two situations are illustrated by time-plots of numerical simulations of themathematical models described in the main text. First row: time-plots of B3 (dashed thick), F3 (dashed thin), B4 (plain thick) and F4 (plainthin); Second row: time-plots of T3 (plain) and T4 (dashed). Simulations were performed using GRIND; the parameter values are thosedefined in the text.

10

Time

T

100 200 300

100064

2

10064

2

1064

2

10

Time100 200 300

100064

2

10064

2

1064

2

10

Time100 200 300

100064

2

10064

2

1064

2

10

Time

B,F

100 200 300

100064

2

10064

2

1064

2

10

Time100 200 300

100064

2

10064

2

1064

2

10

Time100 200 300

100064

2

10064

2

1064

2

A5 T5 B5 F5

A6 T6 B6 F6

. E T A L .524

F. 6. Prototype IV. Two pairs of B- and T-cell clones [(B5, T5) and (B6, T6)] are specifically stimulated by antigens A5 and A6 respectively.B5 also recognizes clone T6 directly; symmetrically, B6 directly recognizes clone T5. Clearly, F6 will inhibit T5 and F5 will inhibit T6. Thestructure is symmetrical (diagram on top). The system regularly attains a fix point; however, the equilibrium composition depends on theinitial conditions. This behaviour is illustrated by time-plots of numerical simulations in which the symmetry in the initial compositionis progressively broken by increasing the initial value of B5 (left: B5=0.1; center: B5=10; and right B5=100), while keeping the initialvalues of other variables constant at 0.1). First row: time-plots of B5 (plain thick line), F5 (plain thin), B6 (dashed thick) and F6 (dashedthin); Second row: time-plots of T5 (dashed) and T6 (plain). Simulations were performed using GRIND; the parameter values are thosedefined in the text.

The equilibrium values at the fixed point are thus:

F2 = a'−1T 0kDT

kPT1 (19)

B2 =kPB ·(kDF + k'DC )

kSF ·kDB·F2 (20)

T2 =kPT ·kPB ·(kDF + k'DC )kDT ·kSF ·(kPB − kDB )

·F2 (21)

According to the Routh–Hurwitz criterion (Murray,

1989), the prototype II equations are locally stable atthe non-trivial fixed point as long as the inequalitykPB q kDB + kDT is satisfied.

In Fig. 4(b) we illustrate the behaviour of thisprototypical system using the standard set ofparameters. In the bottom graph, several stabletrajectories are drawn in the 3D state-space.

The two basic examples just presented suggest thatthe present model of B-T lymphocyte networks canshow some clear-cut correlations between thestructure of the interactions and the dynamicalbehaviour. Prototypes I and II constitute the basicbuilding blocks of the model which by anticipation

525

correspond to the putative ‘PIS’ and ‘CIS’ respect-ively. However, extrapolating from these situations tomore complex systems is certainly not trivial. Twoadditional prototypical systems will illustrate thispoint clearly.

Consider a simple model (prototype III) with twoB-lymphocyte clones (B3 and B4) which induce eachother through the Ig-molecules they produce (F3 andF4), and get to be activated by co-operating with twooptimally stimulated T-cell clones (T3 and T4) thatthey recognize through idiotypic interactions (Fig.5-top). In order for this system to demonstrateself-sustaining behaviour, the initial concentrations ofF3 and/or F4 must be in the inductive range; inaddition, the system can only maintain all itscomponents in a concentration above the virgin oneif it is dynamically stable. Illustrative examples ofstable and unstable dynamics obtained with thismodel are depicted in the left and right panels ofFig. 5.

Finally, let us consider a fourth situation(prototype IV) represented again by two B-lympho-cyte clones (B5 and B6) which are optimally inducedby different antigens and which do not interactdirectly with each other (Fig. 6). Nevertheless, theyinteract indirectly because each B-cell clone obtainshelp from the same two antigen-specific T-lympho-cytes (T5 and T6). Thus, B5 presents antigenic peptidesto clone T5 but also recognizes clone T6 by directmIg-TCR interaction; conversely B6 presents anti-genic peptides to clone T6 and recognizes specificallyclone T5. Clearly, F6 will inhibit T5 and F5 will inhibitT6. For simplicity let us assume that the structure ofthe model is perfectly symmetrical. The behaviour ofthis symmetric model is particularly interesting: itregularly approaches a fixed point. However, thesystem has a ‘‘metastable’’ character: there are anindefinite number of fixed points, and the precisecomposition of the asymptotic equilibrium dependson the initial conditions.

4. Discussion

The model of the normal non-immunized ISpresented here is based on a restricted set ofmechanisms of lymphocyte co-operation; this set isfar from exhausting all the possibilities that have beenproposed, or actually demonstrated in particularexperimental settings (Mazel et al., 1990). We havemade a strict selection of those mechanisms that haveexperimental support, and conform to observationsof clonal sizes, repertoire diversity and ongoingactivity in the peripheral lymphoid compartment of

normal unmanipulated individuals. In this section wewill discuss how this simplified model, which has nopretention to be complete, may nevertheless accountfor that level of organization where the distinctionbetween CIS and PIS actually comes about.

As the simple prototypic systems II–IV illustrate,the articulation of this restricted set of interactionscan potentially lead to a stable self-regulating immunenetwork. Such dynamic stability effectively ensuresthat clonal dynamics will evolve below the thresholdsof frequency and dominance that may reasonably besupposed to be necessary for an entire panoply ofadditional mechanisms of intercellular interaction orco-operation processes to come into play. In otherwords, under certain conditions (notably pertainingto the connectivity structure of the network), theorganization of the model can by itself ensure that itwill remain within its own range of validity.

In other conditions (illustrated by prototype I),exponential expansion of clones can proceed in ourmodel with no upper limits. Clearly in the real IS,clonal expansion during an immune response does notcontinue indefinitely: it is regulated, and can give riseto immunological memory, by processes made

F. 7. A ‘‘phase-space’’ diagram illustrating the domain ofoperation of the immune network model as a function of twoco-operating components X and Y (for example a B- andT-lymphocyte clone). The domain of validity of the model isillustrated by the dashed line that imposes a limit on the magnitudeof the components (for example clonal size or frequency), such thatthe simplifying assumptions detailed in Fig. 2 (right) arereasonable. We postulate that the operation of the network duringnormal physiology progresses below such limits (inside dark greyring). The internal dynamics of the immune system will defineunder which conditions the trajectories of the components, (a), willeither: (b) reach the domain of the network and enter into its stabledynamics (d); or alternatively (c) escape and give rise to a typicalimmune response dynamics (e). Note that although the immuneresponse dynamics (outside the light grey ring) is not included inthe simplified model, its initiation is explicitly taken into account.

. E T A L .526

possible and viable only when clonal frequencies ordominance are high enough. These processes do notform part of our model. However, we would like toemphasize, that the initiation of the ‘‘unlimitedgrowth’’ that makes them possible, is a part of ourextremely simplified model. In the course of theirexpansion from a single bone-marrow precursor, theclones responsible for immune responses traverse thedomain of operation of the network; only if theyescape the potential ‘‘attractor’’ represented by the‘‘stable’’ mode will they depart from the domain ofvalidity of the present model. In this way, our modelincludes the critical domain in which the differen-tiation between two distinct modes of coupling toantigen occurs (Fig. 7).

At this point a question naturally arises as towhether the fully-fledged model system, endowed withfree metadynamical recruitment of new clones,develops in such a way that these two modes bothappear spontaneously; and if so, whether these twomodes can co-exist and give rise to a meaningfuldistinction in the repertoire corresponding to thatbetween a CIS and a PIS. These questions areaddressed in the companion article (Carneiro et al.,1995).

We would like to thank all our colleagues at the PasteurInstitut for providing us with the unique interdisciplinaryenvironment that made this work possible. P. Pereira, A.Nobrega, and A. Freitas worth individual mention for thecritical and stimulating discussions during the design of themodel presented here. JC acknowledges the financialsupport of ‘‘Junta Nacional de Investigacao Cientıfica eTecnologica-Programa Ciencia’’, Lisbon (grant BD/2319/92-ID) for which he is most grateful. The work wassupported by grants from ANRS (France) the EuropeanUnion, and the PICASSO Programme for co-operationbetween France and Spain.

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APPENDIX

The Biological Justifications for the Six Major

Simplifying Postulates of the Model

(i) There are no specific interactions between restinglymphocytes. Productive interactions occur only

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between ‘‘co-operation prone’’ activated T-cells andinduced (or activated) B-cells.

Following induction by adequate engagement of mIgby ligand, B-lymphocytes modulate the expression ofa battery of adhesion molecules and co-receptors.Similarly, activation of T-lymphocytes also changestheir patterns of membrane protein expression. Thenewly expressed molecules in both cells endowthem with the potential to engage in cell-to-cellco-operation; a process that requires interactionswith adequate affinity, ‘‘valency’’ and duration,involving a concerted engagement of clonal receptors,non-clonal co-receptors and adhesion molecules, anda harmonious co-evolution of the physiology of bothcells (Dustin & Springer, 1991; Clark & Ledbetter,1994). In addition, activated lymphocytes alsoproduce chemotactic factors (Kornfeld et al., 1985;Clinchy & Moller, 1994) which target other sensitivecells to them. Finally, the particular set of adhesionmolecules expressed in induced or activated lympho-cytes modifies the nature of interactions with theirenvironment (other cells and the extracellular matrix)such that they home to the lymphoid organs, wherethey follow selective pathways and kinetics ofmigration, and participate in special processes ofmorphogenesis in lymphoid organs (Dustin &Springer, 1991; Foy et al., 1994).

Resting lymphocytes do not participate actively inany of these processes. We conclude that in thenormal steady-state IS, interclonal interactionseffectively only take place between “co-operationprone” lymphocytes, that is, activated T-cells andinduced or activated B-cells. Due to their “homing”capacities, increased mobility and “convergent”migration pathways, the probability of a meetingevent is increased; and due to their increasedintercellular complementarity and avidity, the prob-ability of productive engagements in any suchencounter is also increased. Hence, the overall rate ofproductive encounters between activated T-lympho-cytes and induced B-lymphocytes is postulated to beincreased by several orders of magnitude over thatbetween resting cells. Conversely, it is assumed thatspecific interactions involving resting lymphocytes arerelatively improbably (both because of a reducedfrequency of encounters, and because of a reduced

probability and efficiency of engagement), and aretherefore negligible from a network point of view.†Simplifying it is assumed that the probability ofproductive encounters between ‘‘co-operation prone’’cells is 1.0, and between resting cells is 0.

(ii) B-lymphocyte induction and T-cell activationrequire the cross-linking or polymerization of theirclonal receptors.

The model assumes that activation of lymphocytes isstrictly dependent on ‘‘cross-linking’’ or ‘‘polymeris-ation’’ of their clonal receptors (B-cell mIg or T-cellTCR). B- and T-cells, however, are known to displayqualitatively different activation pathways and ligandrequirements. Actually, in view of the differentdegrees of valency for antigenic-determinants of mIgand TCR molecules, B- and T-cells can a priori beexpected to be sensitive to qualitatively differentstimuli. While any ligand with valence higher thantwo can potentially lead to the polymerisation ofbi-valent mIg on B-lymphocyte membrane (Monginiet al., 1989; Faro & Velasco, 1993a), the extensiveaggregation of monovalent TCR effectively requires aligand which is already ‘‘polymeric’’. Hence, B-lym-phocytes can be induced in vitro by anti-mIg Ab eitherin soluble (free) or plate-coated forms. By contrast,T-cell activation cannot be achieved in vitro withanti-TCR Ab in solution; it can only be triggeredwhen the ligand is ‘‘polymerised’’ on coated plates oron the membrane of an APC [for example,MHC+peptide complexes, mIg on specific anti-TCRB-cells (Tite et al., 1986), or anti-TCR IgGpolymerised by binding Fcg receptor (Ceuppens et al.,1985; Clement et al., 1985; Stohl et al., 1987) orcovalent attachment to cell surfaces (Kranz et al.,1984)]. Concomitantly, it is to be noted that solubleanti-TCR Abs are excellent inhibitors of T-lympho-cyte responses (Reinherz et al., 1980; Hoffman et al.,1985; Shaw et al., 1985; Pantaleo et al., 1987), a pointto which we shall return.

(iii) Primary activation of resting T-lymphocytes istriggered by co-operation with unspecific APCs,involving the specific recognition of a set of dominantand frequent antigenic peptides in the context ofMHC molecules. The contribution of B-lymphocytesis irrelevant.

We assume that T-cell activation in vivo is triggeredby TCR aggregation induced by MHC+peptidecomplexes (with the concomitant engagement ofnon-clonal receptors and adhesion molecules) duringcell-to-cell co-operation with antigen presenting cells(APC) (Grusby et al., 1993). As mentioned inpostulate (i) and in spite of the fact that specific

† Two corollaries of this postulate are that T-cell activation isvirtually B-cell independent and that B-cell induction is virtuallyT-cell independent. Note that some immediate consequences of thisassumption are similar to those of ‘‘two signal models’’ oflymphocyte activation [see Cohn (1994) for a review]; noticehowever that at variance with ‘‘two signal models’’, we pay specialattention to the frequency of specific encounters in vivo.

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activated B-lymphocytes seem to be very proficientAPCs (Lanzavechia, 1985), they are in such lowfrequencies, both in absolute terms and as comparedto other ‘‘unspecific’’ APCs (dendritic cells, macro-phages, etc.), that their contribution to the activationof resting T-cells is likely to be insignificant (Lassilaet al., 1988; Ronchese & Hausmann, 1993). Similarly,induced or activated B-cells are just frequent enoughto co-operate with activated T-cells, but are unable toactivate resting T-cells. Again, the frequency ofindividual specificities in serum Ig is so low that thepotential stimulatory effect of anti-TCR Ig when‘‘polymerized’’ on the surface of an APC (followingbinding by the Fc-receptor) is also insignificant.Finally, for the reasons given above, the modelassumes that ‘‘idiopeptides’’ from clonal receptors(Mazel et al., 1990) are also negligible. In short, themodel postulates that T-lymphocytes are activatedprimarily when they recognize a set of frequent anddominant antigenic peptides presented by unspecificAPCs.

(iv) Primary induction of resting B-lymphocytesfollows cross-linking of mIg by multivalent ligands:these are either common antigens, or soluble idiotypicIg molecules.

Soluble multivalent Ig-molecules can contribute tothe induction of a resting B-lymphocyte because, first,their normal concentrations are not negligible and,second, they can polymerize the bivalent mIgreceptors. However, according to postulate (i), thecontribution of clonal membrane receptors of B- andT-lymphocytes to the primary activation of restingB-lymphocytes is not significant. It is worth notingalso that if induction occurs through the cross-linkingof receptors, as postulated, this automaticallyaccommodates the finding that high concentrations ofligand do not lead to clonal maintenance (Nemazee &Buerki, 1989), since high concentrations of ligandresults in monovalent binding.

(v) In order to engage and sustain co-operation withan activated T-cell, an induced B-cell must somehowengage its TCR either by presenting MHC+peptidecomplexes that are specifically recognized by thatT-cell, or by direct mIg-TCR interactions.

According to postulate (i), only activated T-cells havea significant probability of meeting and engagingcomplementary induced or activated B-cells. Pro-ductive cell-to-cell co-operation between B- andT-cells can only take place, moreover, if the B-cell is

able to aggregate the TCR complexes of theco-operating T-lymphocyte at the intercellular inter-face.

An induced B-lymphocyte can ligate TCR mol-ecules of an activated T-lymphocyte usingMHC+peptide complexes (as classical restrictionelements), as long as the specific peptides aresufficiently represented on its membrane. Sinceinduced B-cells do not themselves activate theT-lymphocytes [postulate (iii)], they must present thesame antigenic peptide(s) that drove the activation ofthe T-cell by unspecific APC (or potentially othercross-reactive peptides); in practical terms this impliesrecognizing the corresponding antigenic protein,processing it and presenting its peptides. Clearly, thesurface representation of a particularMHC+peptide complex on a B-cell is essentiallycontingent, and dependent on the metabolism andtotal proteins being processed.

A B-lymphocyte can also aggregate the receptormolecules of a T-lymphocyte if its mIg-molecules arespecifically complementary to the TCR. As in theprevious case, the B-cell in order to co-operate musthave been induced and must recognize a previouslyactivated T-lymphocyte. Hence, its mIg mustcross-react both with a frequent multivalent ligand(either antigenic protein or soluble Ig-moleculesproduced by other lymphocytes [postulate (iv)] andwith a TCR molecule that recognizes a dominantand frequent MHC+peptide complex [postulate(iii)].

(vi) Soluble Ig-molecules are inhibitory for T-cells.

Since TCR-molecules are monovalent they cannot bepolymerized by soluble Ig. Moreover, solubleanti-TCR Ig not only fails to activate restingT-lymphocytes, but it actually inhibits (in vivo orin vitro) both the proliferative response and effectorfunctions of T-lymphocytes (Reinherz et al., 1980;Hoffman et al., 1985; Shaw et al., 1985; Pantaleoet al., 1987). T-lymphocyte inhibition by solubleanti-TCR Igs is due to a variety of mechanisms,namely: blocking the interaction of the TCR with theMHC+peptide complex on the APC; uncoupling thesignals transduced by the TCR complex and othersignals required for full T-cell activation (Muelleret al., 1989); or by down modulating the expressionof TCR molecules on the membrane (Pantaleo et al.,1987; Moretta et al., 1989). As already discussed, thefraction of Ig-molecules polymerized on the surface ofan APC (by Fc-receptor ligation) is negligible due tothe diversity of serum Ig-molecules.