J. theor. Biol. (1996) 182, 531–547

0022–5193/96/200531+17 $25.00/0 7 1996 Academic Press Limited

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

II—The Simulation of Ontogenesis

J C,† A C J S

Unite d’Immunobiologie, CNRS URA 1961, Institut Pasteur, Paris, France

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

This paper is based on a new model of the immune network which explicitly incorporates B-T cellco-operation. A major feature of this model is the simplifying assumption that inhibition by anti-TCRsoluble Ig is the only possible down-regulatory influence on activated T-cells. This model is capable ofcoupling with antigens in both an ‘‘immune response’’ mode and a ‘‘tolerant’’ mode. In the presentpaper, we simulate the ontogenesis of the immune system by metadynamical recruitment of T- and B-cellclones from the thymus and the bone marrow, seeking to identify the conditions under which each ofthese modes of antigen coupling occurs. Achieving the tolerant mode depends principally on fourparameters: a high value of SB, the rate of bone-marrow production of B-cells; a relatively high efficiencyof T-help through mIg-TCR recognition compared with (MHC+peptide)-TCR interaction; and arelatively high value of the product PR.NA, where PR is the average probability that an Ig recognizesanother molecule and NA is the number of antigens which are present throughout ontogeny. Analysisof the conditions under which these two modes can coexist, shows that this is possible when a sufficientlynumerous set of founder antigens couple in a tolerant mode, whereas isolated antigens first presentedonce development is completed couple in an immune response mode. The present model thus providesa possible mechanism for the distinction (hitherto purely descriptive) between a Central Immune Systemorganized as a network and responsible for tolerance, and a Peripheral Immune System responsible forimmune responses.

7 1996 Academic Press Limited

1. Introduction

The ‘‘second generation’’ network models of theimmune system (Varela & Coutinho, 1991) designedto date have all been restricted to B-lymphocytes andthe Ig molecules they produce. The results of thesestudies raise the real possibility that such models maynot be up to the task of illustrating the interplaybetween the central immune system (CIS) and theperipheral immune system (PIS) (Carneiro et al.,1995). Calenbuhr et al. (1995) and Sulzer et al. (1994)have put forward the notion that natural toleranceand CIS dynamics may correspond to specialtopologies of an idiotypic B-cell network. Theseproposals are not fully satisfactory as an account of

the nature of the CIS and its relationship to naturaltolerance. These T-independent models require thatthe potential idiotypic repertoire has a rather rigidtopology, and are certainly not robust to themetadynamical generation of new specificities. De-tours et al. (1994) have addressed this question bystudying the ontogenesis of a (T-independent)idiotypic network model in a two-dimensional shapespace. They have shown that under certain parameterranges, ‘‘natural tolerance’’ to founder antigens and‘‘vaccination’’ to complementary antigens canemerge. However, this mature network has anessentially complete repertoire, where the ‘‘vacci-nated’’ and ‘‘tolerant’’ clones are both activated andare complementary subsets with equal sizes. This is incontradiction with the observation that the reper-toires of naturally activated and resting lymphocytes(understood to correspond respectively to CIS and

† Author to whom correspondence should be addressed.E-mail: carneiro.pasteur.fr

531

. .532

PIS compartments) differ in both VH gene usage andspecificity (Portnoi et al., 1986; Huetz et al., 1988b;Viale et al., 1992). It is possible, therefore, thatrestricting the models to a single lymphocytecompartment is an over-simplification that missesessential features afforded by co-operation with othercomponents of the immune system.

In the companion paper (Carneiro et al., 1995), wehave presented a new model of the immune networkin which B lymphocyte activation is dependent onT-cell help. A major feature of this model is thesimplifying assumption that inhibition by anti-TCRsoluble Ig is the only possible down-regulatoryinfluence on activated T-cells. We showed that thismodel was able to couple with antigens in twoqualitatively distinct modes: (i) an ‘‘immune re-sponse’’ mode in which T- and B-cell clones growexponentially; and (ii) a ‘‘tolerant’’ mode in whichT-cell clones are controlled by inclusion of the TCRsin the repertoire of an idiotypic B-cell network. In thepresent paper, we simulate the ontogenesis of theimmune system by metadynamical recruitment of T-and B-cell clones from the bone marrow, seeking toidentify the conditions under which each of these twomodes of antigen coupling occurs. In particular, weexamine whether it is possible for these two modes toco-exist, and thus whether the present model providesa mechanism accounting for the differentiation of theimmune system into two compartments, a ‘‘CentralImmune System’’ (CIS) and a ‘‘Peripheral ImmuneSystem’’ (PIS) as previously postulated (Huetz et al.,1988a; Coutinho, 1989).

2. The Model and Implementation

2.1.

The design of the equations representing thedynamics of the lymphocyte network was the mainissue in the companion article (Carneiro et al., 1995).Briefly, the basic variables in the model immunesystem are the effective concentration of the availableantigens (A), the size of the T-lymphocyte clones (T )they drive, as well as the size of the B-lymphocyteclones (B) and the concentration of Ig-molecules theyproduce (F ). At any given time the state of the systemis defined by its composition:

Ak , Tk k=1, . . . , NA (=NT )

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

Antigens are introduced with a fixed concentrationcorresponding to the stimulatory peak in B-lympho-cyte induction:

Ak =0 or Ak = b1 (1)

Once antigen k is introduced it drives acorresponding T-lymphocyte clone k, whose concen-tration Tk is from then on described by:

dTk

dt= − kDT ·Tk + kPT ·aT (pk , hk , Tk )+ kST . (2)

The size of the B-lymphocyte clone, Bi, and theconcentration of Ig-molecules it produces, Fi, aredescribed by:

dBi

dt= − kDB ·Bi + kPB ·aB (si , ti , Bi ) (3)

dFi

dt= −(kDF + kDC ·si )·Fi + kSF ·aB (si , ti , Bi ). (4)

These are the main equations governing thedynamics. The intermediate quantities they requirecan be found in Appendix A.

2.2. ()

As in previous models which simulate theontogenesis of the immune network (De Boer &Perelson, 1991; Stewart & Varela, 1991; Detourset al., 1994), a key feature is the generation of thematrices of pair-wise affinities between the com-ponents. In the present case, there are basically twosuch matrices to be generated: (i) affinities betweenIg-molecules and other molecular entities, be theyother Ig-molecules (idiotypic interactions), nativeepitopes (i.e. the binding domain in a native proteinantigen), or TCRs on T-lymphocytes; (ii) affinitiesbetween TCRs and MHC-peptide complexes. As amatter of principle, we postulate that interactionsbetween Igs, TCRs and MHC-peptide complexes alltake place in a common domain since they are allspecific protein-protein interactions.

The details of the procedures we have employed togenerate these affinity matrices, and their biologicaljustifications, are specified in Appendix B. Qualitat-ively, the major points to note are the following:

(i) We postulate that the diversity of TCRs andMHC+peptide complexes as Ig determinants is suchthat they are effectively distributed over the wholerepertoire of Ig clonotypes.(ii) In the present model we suppose that each antigengives rise to one and only one MHC+peptidecomplex, which can be recognized by one and onlyone TCR molecule. This simplification of theT-lymphocyte realm allows us to focus our attentionon the B-T interface and its involvement with theidiotypic network.(iii) In order to capture the fact that the pairwiseaffinities may be virtually unpredictable from thecharacteristics of the two isolated molecules, affinity

533

matrices have been generated by a procedure basedon random assignment (which implies no intrinsictopology in the resulting matrices) (Fig. 1). In orderto test the genericity of the results obtained, we havecompared them with simulations employing theshape-space concept to generate affinity matrices(Perelson & Oster, 1979; Perelson, 1989). The resultsof this comparison, which are not reported here,confirm that the results presented here are indeedgeneric.

2.3.

2.3.1. B-lymphocytes

In each iteration SB new clones are introduced andtheir interactions with all other components in thesystem obtained according to the GAC chosen. Thecorresponding state variables are initialized typicallyto Bi =1.0 and Fi =0.0. Any clone in the system willbe maintained as long as the state variables asdetermined by dynamics are above some lowerthreshold, typically Bi q =0.5 or Fi q 0.5. In thesimulations reported here the parameter SB wassystematically explored between one and eight.

2.3.2. T-lymphocytes

The metadynamical process in T-cells are reducedto their introduction whenever a new antigen appearsin the system (see below), with a constant source termwhich is intended to reproduce the average prob-ability that a clone with that specificity is produced bythe thymus. This constant source term was set to 0.1.

When each new T-cell clone is introduced itsconnections with all the B-cell clones already in thesystem and with the corresponding antigen areestablished according to the GAC.

2.3.3. Antigens

Antigens are instantiated in the system with a fixedconcentration for B and T cells (respectively, theircontribution to the sums si and the value pk). Theirinteractions with the B-cells clones already in thesystem and with their specific T-cell clone areestablished.

2.4.

The parameters concerning the dynamics of B andT lymphocyte clones are the same as in thecompanion paper in which the model was presented(Carneiro et al., 1995). The parameter ranges for thegeneration of affinity matrices and for metadynamicalprocesses have been specified in Sections 2.2 and 2.3above. The aim of the present study is to describequalitatively the types of behaviours that result fromthe metadynamical articulation of idiotypic B-cellinteractions both with each other and with antigen-driven T lymphocytes, and the parameters have beenset accordingly. As we shall see, the behaviour of thesystem depends crucially on four parameters: SB, therate of bone-marrow production of new B-cellsrelative to the dynamics of proliferation andsecretion; the parameter s which brings the helpprovided by (MHC+peptide)-TCR interaction into

F. 1. The random generator of affinity coefficients (GAC) mimics the experimental distributions of antibody-ligand interactions strengthas measured by ELISA. (a) Cumulative frequency distributions of the positive (significant) affinity coefficients generated by the randomGAM procedure are plotted; the parameters were aM =0.1 and dM =0.4, 0.6, 0.8 and 1.0; the heavy dots correspond to the value dM =1.0used in simulations reported here. (b) Cumulative frequency distributions of significant interactions (normalized optical densities) asmeasured by ELISA between a collection of 435 hybridoma antibodies and actin (—), spectrin (· · ·) and myoglobulin (– – –) [data fromFreitas et al. (1986)].

. .534

the same scale as that obtained through mIg-TCRrecognition; PR, the ‘‘average’’ probability that an Igrecognizes another molecule (such that for membraneor soluble Ig-molecules we have aP =PR; and forTCRs and antigens we have aP =PR/3); and NA, thenumber of antigens which are simultaneously presentwhen the system starts developing. We have thereforeconcentrated on studying the effects of systematicvariation in these four parameters.

3. Results: The Ontogenesis of the Model Immune

System

The model immune system fulfils the expectationsraised in the previous paper, as it can develop eitherby exhibiting stable tolerance to all the antigens, or byproducing immune responses to one or more antigens.In the next two sections (3.1 and 3.2) we will discussthe parameter dependence of these two prototypicalbehaviours. A qualitative explanation for thesefeatures will be presented in Section 3.3. In Section 3.4we will use this qualitative understanding to predictand illustrate how the stable and unstable behaviourscan be found during the same simulation in differentsubsets of the idiotypic repertoire.

3.1.

The focus here is to identify those behaviours whichare afforded by supraclonal organization in the modelimmune system. It is thus important to distinguishthem from the prototypical behaviour triviallyexpected by the ‘‘local rules’’ of lymphocyteactivation. We therefore need to identify the‘‘background’’ behaviour in the absence of anysupraclonal organization. By not allowing theIg-molecules to induce other B-cells or to inhibitT-cells, we eliminate supraclonal organization so thatthe only activation of both T and B cells that remainsis direct antigenic stimulation. The behaviour of thesystem in these conditions can be easily predicted andhas been discussed in the companion paper: acontinuous exponential growth of antigen-specific Band T lymphocytes resulting simply from thedynamics of clonal activation and proliferation. Thismonotonic behaviour is always the case when theIg-molecules do not recognize either the TCR of theantigen-driven T-cells or the mIg-receptor of otherB-lymphocytes.

As a parenthesis, we should make it clear thatallowing ‘‘half-way’’ network organization (i.e.idiotypic interactions between Igs but no Ig-TCRconnectivity) rather systematically results in explosivebehaviour. Thus, in the absence of any control,

antigen-specific T-cells grow exponentially and arethus not a limiting factor for B-cell growth anddifferentiation. In these conditions the present modelreduces to a typical ‘‘second generation immunenetwork’’ where T-cell help is not a limiting factor(Varela et al., 1988). The development of such modelsin the presence of a constant antigenic compositionhas been systematically explored by others (Detourset al., 1994), and we are just referring to theirconclusion.

3.2.

If supraclonal organization is rendered possible, byallowing the affinity coefficients $BF and $BT to takepositive values (according to the rules in the matrixgenerators), the present model immune system is nowable either to organize and develop into a stable stateor to grow exponentially, depending on the antigeniccomposition and other parameter settings. Typicalsimulations of such stable and unstable developmentsare depicted in Fig. 2. An exploration of theparameter space in the model has been performed inorder to identify the conditions required for the shiftfrom unstable to stable development in the presenceof constant antigenic stimuli. As already mentioned,the most crucial parameters that determine theresulting mode are SB, s, PR and NA. Stabledevelopment is associated with a critical value of thesource term SB: for standard values of s, PR and NA,values of SB lower than two lead to explosivebehaviour; only values of five or more can lead to fullstability [Fig. 3(a)]. The transition from stable tounstable development can also be seen whenexploring the impact of changing the value of s alone:the lower the value of s, the greater the relativeproportion of help provided through mIg-TCRinteraction, and the lower the probability of fallinginto unstable explosive behaviour [Fig. 3(b)].

Maintaining all other parameters constant, whenPR is progressively increased there is a given thresholdabove which we observe a rather abrupt transitionfrom full explosion (all antigen-driven clones growcontinuously) to full stability (all antigen-drivenclones show low concentration), with a restrictedintermediate interval (some but not all antigen-specific clones explode). A similar transition can beseen when exploring the impact of increasing NA thenumber of available antigens. Indeed, the twoparameters NA and PR are intimately related (asdiscussed below) and show a synergistic effect toensure a stable development: the higher the value ofNA, the lower the value of PR that is needed to achievea stable state (and vice versa) [Fig. 3(c)]. Empirically,

535

F. 2. Two examples of simulations in which the immune network model develops (a) in an unstable mode and (b) in a stable mode.(a) The C simulation was carried out using NA =2, PR =0.004, s=0.006, and SB =2; for each antigen the number of activated cells inthe specific T-clones and the sum of the concentrations of antigen specific Ig weighted by their affinity are plotted against time (respectivelyon left and right). (b) The simulation was carried out using NA =20, PR =0.04, s=0.006, and SB =3; for some representative antigensthe number of activated cells in the specific T-clones and the sum of the concentrations of the Ig molecules weighted by their affinity forthe antigen are plotted against time (respectively on top and bottom).

. .536

the critical value of the product (NA.PR) is slightly lessthan one.

3.3.

The results just presented in Section 3.2, which arecentral to the present paper, are amenable to aqualitative interpretation, supported by the proba-bilistic approximations presented in Appendix C,which strengthens an understanding of the processesinvolved. The ontogenesis of the mature stable systemfalls into two distinct stages: an initial phase ofrepertoire diversification; and a subsequent phaseduring which the activated part of the system‘‘focuses’’ on a restricted repertoire. The transitionbetween these two stages is illustrated in Fig. 4 where

we plot the time-course of the quantity (PR.NB) whichis a measure of the ‘‘completeness’’ of the idiotypicnetwork. We see that in several simulations (all ofwhich develop stably) this measure initially rises to avalue of the order of one, and subsequently declines.

3.3.1. The initial phase of clonal expansion andrepertoire diversification

The model immune system starts developing in astate where only antigens are present, so that the firstevent is necessarily the antigen-driven activation andexpansion of T-lymphocyte clones [Fig. 5(a)]. Thiswave of selection of T-cell clones is immediatelyfollowed by the antigen-driven activation andexpansion of B-cell clones [Fig. 5(a)]. These B-cells areinduced by recognition of the available antigens andare fully activated as a result of presenting theantigenic peptides to the available activated T-cells.These antigen-driven lymphocyte clones will continueto expand exponentially, unless they are down-regu-lated by cross-reactive Ig-molecules. It follows that afully stable development will only occur if all theantigen-activated T-cell clones in the system arecontrolled by Ig produced by activated B-cell clonesin the system [Fig. 5(b) and (c)].

At this initial stage, where only a first generation ofantigen-driven clones is available, it is already clearwhy there is a scaling relationship between NA and PR:the higher the values of NA.PR, the greater will be theprobability that the resulting Ig-molecules willcross-react with the antigen-specific repertoire ofTCRs and hence the smaller the probability that oneor more T-cell clones will escape recognition.However, this does not fully account for the efficiencyof the system in controlling all the TCRs. The Igmolecules produced by antigen-driven B-cells willcontribute to the sensitivity of cross-reactive B-clono-types. The latter will be induced and can then get tobe fully activated if they also cross-react with any ofthe activated members of the T-cell clones, which arewidely available at this stage. The Ig-moleculesresulting from this second wave of B-clonotypes willagain contribute to the recruitment of a third wave,and so on [Fig. 5(b) and (c)]. It is to be noted thatthese additional waves of B-cell clones in the‘‘idiotypic cascade’’ are particularly effective sincethey get their T-help by directly recognizing TCRs,and thus necessarily contribute to the down-regu-lation of these T-cell clones. In fact, as long as T-cellhelp is not a limiting factor, our model behaves likea classical ‘‘second generation’’ immune networkwhere B-cell activation percolates and tends to reachrepertoire ’completeness’ (De Boer & Perelson, 1991;Stewart & Varela, 1991; Detours et al., 1994). If the

F. 3. The fate of the immune network model depends mainlyon four parameters: SB, the daily production of new B-clones bythe bone-marrow; s, the scaling factor that brings B-Tco-operations by peptide presentation (MHC+peptide-TCRinteractions) to the same scale and as that by direct mIg-TCRinteractions; NA, the number of available antigens; and PR, theaverage probability that an Ig molecule (either soluble ormembrane bound) recognizes any candidate ligand. (a) Frequencyof simulations that develop stably as a function of SB. Frequencyof stable solutions as a function of s. (c) Frequency of stablesolutions as a function of −Log(PR) and NA: the white-gray-blackgradient corresponds to frequencies smaller than 0.2, between0.2–0.8, and greater than 0.8 respectively. In the three graphs eachdata point is calculated after ten randomized runs. Unlessspecifically indicated, parameters were set at the reference valuesSB =3, s=0.006, PR =0.01, and NA =10.

537

F. 4. The stable development of the immune network model always implies a shift from a phase in which the B-cell repertoire diversifiesby recruitment of new clones to a phase in which the repertoire focuses and becomes restricted to a fraction of its potential. The timecourse of the product PR ×NB is shown for several simulation runs that develop stably. SB =3, s=0.006, PR =0.02, and NA =3, 5, . . . , 15.

recruitment rate is fast enough this leads to theinclusion of all the antigen-reactive TCRs in theeffective repertoire of the idiotypic network and theirdown-regulation by soluble Ig-molecules (Fig. 6).

At this crucial point there is a significant ‘‘phasetransition’’ in the global behaviour of the system, asT-cell help rather suddenly becomes a limiting factorfor B-cell growth and differentiation.

3.3.2. Phase transition from co-operative recruitmentto ‘‘T-cell node focusing’’

When antigen-driven T-cells are down-regulatedand become a limiting factor for full B-cell activation,induced B-cells must now compete with each other forlimited ‘‘T-cell co-operation sites’’ in order to obtainhelp. The relative ‘‘fitness’’ of a B-cell clone in thiscompetition for ‘‘co-operation sites’’ is the product ofits size and its interaction coefficient with aco-operating T-cell. The B-clonotypes that are mostfit will thus compete all the other ones. The selectionfor higher coefficients of B-T interaction that followsthe decrease in the availability of ‘‘help’’ is illustratedin Fig. 7. This is the process where the value of sbecomes determinant for the fate of the system: sinces balances the efficiency of getting help through(MHC+peptide)-TCR vs. mIg-TCR recognition, itdirectly influences the relative capacity of the twomodes to obtain help. Only mIg-TCR recognitioncontributes to the stable regulation of the T-cellclones.

This stage of competitive exclusion results in areduction of the number of species and connections in

the idiotypic network, which ‘‘focuses’’ on the nodesanchored by antigen-activated T-clonotypes. In allthe simulations performed, there is always one or atmost a very small number of B-clonotypes that endup getting help from each T-cell clonotype, all othercandidates being progressively excluded. The effectiverepertoire of the idiotypic network becomes restrictedto just a fraction of the potential repertoire of theB-cells. The result is a partition of the available B-cellrepertoire into two compartments, represented re-spectively by the lymphocytes in the idiotypicnetwork, vs. all the other resting naive cells which aretransiently available in the system (freshly producedby the bone-marrow, but not being activated, they arejust waiting to die). These two compartmentscorrespond to the CIS and PIS as previouslyproposed (Huetz et al., 1988; Coutinho, 1989).

A particular implication of the restriction of therepertoire size reported here should be emphasized. Ageneric property of typical ‘‘second generationnetwork models’’ is their tendency to maximize therepertoire size which leads to progressive counterse-lection of multireactive clones (De Boer & Perelson,1991). It is easy to understand that in the present‘‘T-anchored’’ network such counterselection ofmultireactive species in late stages of simulationscannot be observed.

3.4. -

An organizational distinction between twocompartments—the B-T network coupled to

. .538

developmentally available antigens on one hand, andnaive lymphocytes on the other—is thus an emergentconsequence of the development of the modelimmune system. The aim of this section is to examinewhether this distinction can be rendered functionallyrelevant.

As seen in the previous section, a large number ofantigens in conjunction with a sufficiently elevatedvalue of PR reliably leads to the formation of a stableCIS network which is ‘‘tolerant’’ to all availableantigens. On the other hand, a small number ofantigens in conjunction with a relatively low value ofPR leads to an unlimited ‘‘immune response’’. Thequestion now is whether these two modes of couplingwith antigens can co-exist in the same immune system.If the initial set of antigens is sufficiently numerous,the critical value of NA.PR can be reached so that thesystem develops stably. If a small number of newantigens are introduced after maturation and

F. 6. The impact of the recruitment rate on the final stabilityof the model. 200 simulations were performed exploring aregion of the parameter space: NA =1, . . , 100; Log(PR)=−3.5, . . ,−1.0; s=0.6, . . , 0.8; SB =3, . . , 5. (a) The time courseof product PR ×NB is plotted for several individual runs; the seriesare sampled regularly and stop at the peak value to avoid overlapin the figure: the time points corresponding to runs that developstably or not are marked respectively by (w) and (+). (b) Thefrequency distributions of the product PR ×NB at time=50 inunstable (gray) and stable (black) runs are depicted.

F. 5. Scheme of the early development of the B-T lymphocytenetwork. Available antigens (A) will drive the exponential growthof antigen specific T- and B-clones. The activation of this first‘‘layer’’ of B-clones (B) is ensured by co-operation with the antigendriven T-clones mediated by antigen presentation. Subsequent‘‘layers’’ of B-clones (C) can be activated and therefore recruitedinto the network, if and only if they are specific for both theimmunoglobulins (Ig) produced by other B-clones and for theT-clones which are already activated. Since the newly recruitedB-clones have the potential to control the T-clones that sustainthem, the process of their co-operative recruitment can thereforelead to stability of the entire system, when all the antigen drivenT-clones are controlled by feedback loops.

restriction of the idiotypic repertoire, the associatedvalue of NA.PR can now be insufficient to result instability. This does of course require that the TCRswhich recognize the new MHC+peptide complexesshould fall outside the repertoire of the ‘‘focused’’idiotypic network. In fact, the interactions betweenthe previously established network and the new T-and B-cell clones that are recruited followingpresentation of a new antigen are very complicated,and not fully amenable to simple qualitativeexplanation. In most of the simulations that we haveperformed, if conditions are such that the initial setof antigens is entirely incorporated in the network,then antigens presented subsequently are also

539

incorporated. However, in some simulations this isnot the case: multiple founder-antigens are coupled inthe CIS-network mode, and yet a single antigenpresented once the network has gone through thefocusing process, drives the selection and exponentialexpansion of specific clonotypes. The new antigen-specific clonotypes rapidly reach frequencies andconcentrations that make them unable to couple intypical CIS modes. This situation is illustrated inFig. 8.

Qualitatively, it is clear that a distinction of thissort between ‘‘founder’’ antigens which couple in theCIS mode and ‘‘late’’ antigens which couple in theimmune response mode depends on a reduction in theproduct NA.PR. We attribute the relative scarcity ofthis distinction in our simulations to the fact that, forcomputational reasons, we have not been able toincrease the number of initial antigens above 100, sothat the ‘‘new’’ antigens are only one or two ordersof magnitude smaller in number than the ‘‘founder’’antigens. We anticipate that if the number of newantigens were several orders of magnitude lower thanthe number of founder antigens, the coexistence ofexplosive responses to the new antigens with stable

incorporation of the founder antigens would be muchmore frequent. Whether this is indeed the case is amatter for further investigation and a clear limitationof the present work.

There are however additional grounds for opti-mism concerning the capacity of the present model toaccount for differential coupling with ‘‘founder’’ and‘‘late’’ antigens. It is known empirically that duringthe development of the real immune system, there aresignificant changes in some of the parameters thatwere so far taken as invariant with time in oursimulations. In particular, it is well established that inthe peri-natal period the peripheral Ig-repertoire isessentially dominated by genes in germ-line configur-ation and molecules with highly degenerate specificityand a correspondingly elevated value of PR; in theadult, by contrast, these multireactive germ-line genesare counter-selected in the bone-marrow and thevalue of PR for new clones is correspondinglydecreased (Holmberg et al., 1984, 1986). In oursimulations, when the average PR of newly generatedclonotypes is lowered after the system has reached aninitial maturity, a system capable of concomitantlycoupling with different antigens in stable and unstablemodes is rather systematically obtained. This is easyto understand since PR and NA have a synergisticeffect. Indeed, the simulations implementing adecrease in PR can be interpreted as qualitativelyequivalent to simulations in which the numbers ofantigens in initial and later stages are reduced by thesame order of magnitude.

3.5.

The preceding presentation was systematicallyillustrated using simulations based on the generationof affinity coefficients by random assignment. This isthe best way that we found to recreate the‘‘continuous epitope’’ nature of the Ig bindingdomain that we and many others advocate (Westhofet al., 1984; Tainer et al., 1985; Colman, 1988;Carneiro & Stewart, 1994). It should be emphasized,however, that the present results and their qualitativeinterpretation are quite robust and apparentlygeneric, regardless of whether the GAC is the randomprocedure (described in Section 2.2.1.) or a shape-space generator (unpublished results). The onlyrestriction observed so far concerns the requirementfor a multiple epitope (points) representation of eachIg-molecule. Indeed, under a shape-space GAC inwhich any each molecule is represented by a singleepitope (point), it is a priori difficult to envisage the(i) spontaneous engagement of the system in stabledynamics (CIS establishment), and (ii) a dynamical

F. 7. The plots are time courses of (a) the average value of B-Tcell interaction strength ($TB) and (b) the average ‘‘T-help’’obtained per B-cell clone (the quantity ti) during a representativesimulation run in which the model develops in a stable mode. Notethe free availability of ‘‘T-help’’ per B-cell clone in the early phaseof development; a drop of the amount of ‘‘help’’ available resultsin a progressive increase in the average average B-T interactionstrength.

. .540

F. 8. The immune network model can couple differently with multiple antigens that are present from the onset of the simulation (founderantigens) and a small number of antigens that are introduced after it has gone through the repertoire focusing process (maturation). Weillustrate typical results of introducing a single new antigen (and its corresponding T-cell clone), at the time indicated by the arrow, i.e.after the network has matured stably in the presence of a relatively large number of founder antigens. (a) The biologically interestingsituation in which the new antigen triggers the exponential growth of specific T- and B-lymphocyte clones which are disconnected fromthe mature network; note that the network maintains stable coupling with the ‘‘founder antigens’’. Other results are, however, also possible.(b) The clones directed against the new antigen connect with other clones already in the network, and follow a stable dynamic. (c) Finally,the specific clones driven by the newly introduced antigen sometimes break the stability of the network. The graphs are timeplots of theaffinity-weighted sum of the antigen directed Ig-concentrations during simulations; the dashed lines correspond to some representative‘‘founder antigens’’; the full line corresponds to the newly introduced antigen. In simulations (a) and (c) NA =20, PR =0.015, s=0.006,SB =3; in simulation (b) NA =80, PR =0.01, s=0.006, SB =3.

distinction of CIS and PIS compartments. It is fair toremark, however, that a single epitope shape-space isquite unrealistic.

4. Discussion

4.1.

The conceptual distinction between a ‘‘Central’’and a ‘‘Peripheral’’ immune system has up until nowbeen essentially descriptive. The present paperfurnishes a possible mechanism whereby this distinc-tion can be established and maintained by thedynamics and metadynamics of the immune systemitself. We have enlarged the concept of a CentralImmune System (CIS) to include those TCRs thatinteract with the idiotypic network. The CIS as wehave modelled it has the self-organizing property thatits repertoire is restricted essentially to the set of

‘‘founder’’ antigens that are present during the earlyontogeny of the system. The structure of the matureCIS constitutes a mapping of the antigenic compo-sition in terms of both B- and T-cell determinants. Itrepresents an ontogenical record of the antigeniccomposition which relates across primary structure(peptide sequence) to the tertiary structures of thenative antigens. Antigens which engage the CIS bydual recognition, both as whole molecules and asMHC-presented peptides, will couple in a tolerantmode; antigens which do not so engage the CIS willbe recognized by the residual PIS and will provoke animmune response.

The notion that the immune system establishes aprivileged relationship with those antigens that areavailable during a given time-window when thesystem is building itself up is not of course new. Theprivileged status of a ‘‘founder’’ antigenic compo-

541

sition has been made evident in multiple spontaneoussituations and experimental settings, where themature system responds differently to antigens thatwere available in early phases of its developmentcompared to antigens that were not so available(Billingham et al., 1953). At a purely descriptive level,this observation is independent of the nature of theantigen itself; it is very consistent, and is not in doubt.In classical immunology, however, this observationhas been interpreted as a consequence of clonaldeletion, with the corollary that tolerance is viewedessentially as a passive, recessive phenomenon. Theshortcomings of that view have been extensivelyreviewed (notably by Coutinho, 1989; Coutinho et al.,1992) and we will not elaborate on that point here. Itis nevertheless worth recalling here the consistentfinding that fully functional lymphocytes bearingreceptors directed against body molecules are easilydetectable in the neonatal period and become muchless frequent in the adult stage (Sakaguchi et al., 1982;Holmberg et al., 1986). From the point of view of theclonal deletion theory this observation can only beregarded as aberrant. However, in the context of thepresent model this corresponds to the spontaneousshift from early antigen driven lymphocyte activationto a mature B-T network, that is systematicallyobserved in simulations of viable ontogenesis. Theoriginality of our proposed model is that it providesan actual mechanism whereby tolerance to founderantigens can be interpreted as the result of an active,dominant process; in an inversion of the classicalview, it is rather the immune response mode (and thePIS) that is residual and recessive.

4.2.

One of the major innovative features of the presentmodel (compared with previous ones) is that it placesthe core of immune regulation on the interfacebetween B and T cells, and in particular ascribes acrucial role to soluble Ig molecules. The existence ofsuch interactions is based in the experimental datareferred to in the companion article (Carneiro et al.,1996). The behaviour emerging from these inter-actions is also in general agreement with experimentalobservations; it provides, for example, an explanationto the classical finding that the production of naturalserum idiotypes requires co-operation with T-lym-phocytes through a pathway which is not MHCrestricted (Bottomly & Mosier, 1981). However, themodel goes further than the direct evidence, andattributes a central role to these interactions: itpredicts that a fully functional immune system, withcoexistent CIS and PIS compartments, would onlyappear following the ontogenic establishment of an

appropriately structured interface between B and Tcells. In this section we wish to discuss the refutablepredictions which may make it possible to test thisnew model.

At first sight, the best test of our model would bethe analysis of gene-targeted animals. In TCRb-deficient mice (Mombaerts et al., 1992a), which lackmature ab T-lymphocytes, the performance of B cellsshould be strongly perturbed; this prediction isfulfilled to some extent, since these mice are indeedunable to mount normal immune responses. How-ever, these mice appear to have near-normal levels ofserum IgM (as do nude mice), and this indicates thatour basic postulate, according to which all B-cellactivation is T-cell dependent, is an oversimplifica-tion. Our model also predicts that mMT mice(Kitamura et al., 1991), which lack mature B-lympho-cytes, should have abnormally high frequencies ofthose T-cell clones that are driven by peripheralantigens. Here also, the observation is that theremaining lymphoid compartment as a whole seemsto show a remarkably normal size.

These initial and global observations, however, donot constitute a decisive refutation of the presentmodel. The general reason is that biological systemsare rather typically organized with a substantialdegree of fail-safe redundancy. Thus, faced with theknock-out of a major component, previouslyirrelevant or subordinate mechanisms can take overand compensate at least superficially. A good exampleis the recently reported finding that mice lacking bothMHC class I and II are nevertheless able to rejectallogeneic grafts with normal kinetics (Dierich et al.,1993). This reveals that alternative mechanisms areoperative in these mice; but this observation does notrefute the very well established fact that normally themajor histocompatibility complex molecules play amajor role in graft rejection (as their name implies).Similarly, it could be that the ‘‘normality’’ of theremaining lymphoid compartments in m and ab

deficient mice is only apparent. Such special pleadingcan obviously not in itself validate our model; theonus must be on us to identify the alternativemechanisms responsible for the apparent normality ofthe remaining lymphoid compartments in m and ab

deficient mice, and to show that in normal mice theyare indeed subsidiary to the mechanisms we havepostulated. In addition, we require experimentsinvestigating whether or not, despite appearances, theremaining lymphoid compartments in gene-targetedmice are in fact abnormal. In particular, we lackinformation concerning the clonal sizes and diversityin the naturally activated T cell compartment in mMTmice as compared to normal mice.

. .542

The sort of additional experiment we have in mindwould consist in making strong perturbations to the‘‘partially defective’’ immune system that would tendto push the system far away from its steady stateequilibrium. The compensation of such perturbationsis likely to engage all the regulatory mechanisms, sothat by comparing the compensatory capacity of thepartially defective lymphoid system and a normalfully-fledged immune system, it may be possible todetect the consequences of an abnormal B-T interfacepredicted by our model.

Some very suggestive experiments of this kind havebeen recently reported in adult nude mice reconsti-tuted with syngeneic T cells, which were shown to bemuch more susceptible to superantigen induced toxicshock (a major perturbation) than normal syngeneicmice (Williams et al., 1994). This result would in factbe predicted by our model, since the T cellreconstitution in adult stages can be expected not togenerate an ‘‘appropriate’’ co-selected interfacebetween B and T cell compartments. We wouldactually predict that similar results should be expectedin m deficient animals, where ‘‘superantigen’’ pertur-bations to the remaining T cell compartment shouldreveal the absence of the regulatory mechanisms thatour model predicts.

4.3. ‘‘’’

‘‘-’’

We wish finally to comment on the consequences ofour proposed mechanism for the concepts ofimmunological ‘‘self’’ and ‘‘non-self’’. The ‘‘self/no-self’’ distinction is indeed a cornerstone of classicalimmunology; it seems quite obvious that if theimmune system destroys ‘‘non-self’’ molecules, then itmust be capable of distinguishing these moleculesfrom the ‘‘self’’ molecules of the body, on pain ofproducing catastrophically dysteleological autoim-munity. The distinction between ‘‘non-self’’(notablypathogenic bacteria and viruses) and ‘‘self’’(somaticantigens) is quite clear and unproblematic forimmunologists; however, it is anything but clear howthe immune system itself makes a distinction of thisorder, particularly since the molecules to be sodistinguished may be very similar in terms of organicchemistry—and different for each genetically distinctindividual. The conceptual pitfalls in this difficult areahave been discussed by Vaz & Varela (1978) and,more recently, by Chernyak & Tauber (1991).Molecules which are radically ‘‘non-self’’ in terms ofthe cognitive repertoire of the immune system itselfwould fail to engage it at all, so that the label‘‘non-sense’’ would be less misleading than ‘‘non-self’’. It should be stated clearly that a molecular

entity can only be promoted to the category of an‘‘antigen’’ if it can, at least potentially, couple with theimmune system. Necessarily, the nature, availabilityand concentration of such a molecule as well as thatof its receptors will determine its promotion to thestatus of an ‘‘antigen’’. However, although useful andindeed essential, such conceptual considerations donot in themselves provide a positive answer to thequestion of the mechanisms actually involved.

In terms of the model proposed here, the proximatecause of the putative ‘‘self-nonself’’ distinction is thestatus of the T-cells that recognize the MHC-pre-sented peptides of the antigen in question. If the TCRinvolved is recognized by immunoglobulins present inthe idiotypic network, then the coupling with theantigen will occur in a ‘‘tolerant’’ mode; if the TCRis not so recognized, the coupling will take the formof an immune response. It may be noted that in thecase of tolerance, the idiotype concerned can beconsidered as an ‘‘internal image’’ of the(MHC+peptide). It is important to emphasize thatsuch an ‘‘internal image’’, reflected in the ‘‘mirror’’ ofthe TCR, is purely functional and does not implyany general similarity between the idiotype andthe MHC-presented peptide. On this proviso, theoperational distinction actually effected by theimmune system could be described as one betweenantigens which have ‘‘internal images’’ and antigenswhich do not. Overall, however, the best descriptionof this distinction is probably ‘‘CIS vs. PISengagement’’. This acknowledges that the actualmechanisms involved are multiple and complex,involving dual recognition of native epitopes andMHC-presented peptides in the context of a B-T cellinterface.

When this primary distinction is placed in thecontext of the ontogeny of a fullyfledged system withdecreasing PR, the distinction becomes one between‘‘founder antigens’’ collectively present throughoutearly ontogeny, versus ‘‘late isolated antigens’’. Thetotal biological situation is such that, in the end, this‘‘founder vs, late’’ distinction correlates ratherreliably with the immunologist’s ‘‘self vs. non-self’’distinction; but it is important to recognize that thiscorrespondence is essentially contingent. If we wish todescribe what the system itself is actually doing(rather than what immunologists, as externalobservers, can heuristically project onto the system),then the terminology ‘‘CIS vs. PIS engagement’’would seem to be preferable.

We would like to thank all our colleagues at the PasteurInstitut for providing us with the unique interdisciplinaryenvironment that made this work possible. We also thank

543

J. Faro, H. Bersini, V. Calenbuhr and V. Detours for manyfruitful discussions on the simulation of ontogenesis of‘second generation’ idiotypic network models. J. Carneiroacknowledges the financial support of ‘‘Junta Nacional deInvestigacao Cientıfica e Tecnologica-Programa Ciencia’’,Lisbon (grant BD/2319/92-ID), for which he is mostgrateful. This work was supported by grants from ANRS(France) and the European Union.

REFERENCES

A, J. D., C, C. & S, R. H. (1986). Highfrequency of non-random distribution of alloreactivity in T cellclones selected for recognition of foreign antigen in associationwith self class II molecules. J. Immunol. 136, 389–395.

B, R. E., B, L. & M, P. B. (1953). ‘‘Activelyacquired tolerance’’ of foreign cells. Nature 172, 603–606.

B, K. & M, D. E. (1981). Antigen specific helper Tcells required for dominant idiotype expression are not H-2restricted. J. Exp. Med. 154, 411–421.

B, J. H., J, T. S., G, J. C., S, L. J., U,R. G., S, J. L. & W, D. C. (1993). Three-dimen-sional structure of the human class II histocompatibility antigenHLA-DR1. Nature 364, 33–39.

C, V., B, H., S, J. & V, J. (1995).Natural tolerance in a simple immune network. J. theor. Biol.177, 199–213.

C, J., C, A., F, J. & S, J. (1996). Amodel of the immune network with B-T cell co-operation.I—Basic structure-dynamics correlates. J. theor. Biol. 182,

513–529.C, J. & S, J. (1994). Rethinking ‘‘shape space’’:

evidence from simulated docking suggests that steric shapecomplementarity is not limiting for antibody-antigen recognitionand idiotypic interactions. J. theor. Biol. 169, 391–402.

C, L. & T, A. I. (1991). The dialectical self:immunology’s contribution. In: Organism and the Origin of Self.(A. I. Tauber, ed.) pp. 109–156. Dordrecht: Kluwer AcademicPublishers.

C, C. & L, A. M. (1987). Canonical structures for thehypervariable regions of immunglobulins. J. Mol. Biol. 196,

901–917.C, P. M. (1988). Structure of antibody-antigen complexes:

implications for immune recognition. Adv. Immun. 43, 99–132.C, A. (1989). Beyond clonal selection and network.

Immunol. Rev. 110, 63–87.C, A., C, G., G, A., M, M. A. R. &

B, A. (1992). Some reasons why deletion and anergy donot satisfactorily account for natural tolerance. Res. Immunol.143, 345–354.

C, A., F, L., H, D. & I, F. (1983). Is thenetwork theory tautologic? In: Nobel Symposium 55: Genetics ofthe Immune Response. (Moller, ed.) pp. 273–297. New York:Plenum Press.

D B, R. J. & P, A. S. (1991). Size and connectivity asemergent properties of a developing network. J. theor. Biol. 149,

381–424.D, V., B, H., S, J. & V, F. (1994).

Development of an idiotypic network in shape space. J. theor.Biol. 170, 401–414.

D, A., C, S. H., B, C. & M, D. (1993). Graftrejection by T cells not restricted by conventional majorhistocompatibility complex molecules. Eur. J. Immunol. 23,

2725–2728.E, K., F, I., M, I. & S, M. M. (1980).

Quantitative studies on T cell diversity. I. Determination of theprecursor frequencies for two types of Streptococcus A-specifichelper cells in nonimmune, polyclonally activated splenic T cells.J. Exp. Med. 152, 477–492.

E, V. H. (1994). Structure of peptides associated withclass I and class II MHC molecules. Ann. Rev. Immunol. 12,

181–207.F, A. A., G, B., H, D., W, G.,

C, A. & A, S. (1986). Analysis of autoantibodyreactivities in hybridoma collections derived from normal adultBALB/c mice. Ann. Inst. Pasteur. 137 D, 33–45.

H, D., F, S., I, F. & C, A. (1984).Reactions amongst IgM antibodies derived from normal,neonatal mice. Eur. J. Immunol. 14, 435–441.

H, D., W, G., A, L. & C, A.(1986). The high idiotypic connectivity of ‘‘natural’’ newbornantibodies is not found in adult, mitogen-reactive B cellrepertoires. Eur. J. Immunol. 16, 82–87.

H, F., J, F., P-R, C., V, F. &C, A. (1988a). Autoimmunity: The moving boundariesbetween physiology and pathology. J. Autoimmunity. 1, 507–518.

H, F., L-S, E.-L., P, P., Pı, D. &C, A. (1988b). T cell dependence of ‘‘natural’’auto-reactive B cell activation in the spleen of normal mice. Eur.J. Immunol. 18, 1615–1622.

K, D., R, J., K, R. & R, K. (1991). A B-celldeficient mouse by targeted disruption of the membrane exon ofthe immunoglobulin mu chain gene. Nature 350, 423–426.

K, M., S, G., H, L. E. & S, N. (1986). Themolecular genetics of the T-cell antigen receptor and T-cellantigen recogition. Ann. Rev. Immunol. 4, 529–591.

M, S. C., A, O., H, R. E., H, J. C.,F, K. A., S, S. F. & R, E. L. (1983).Evidence for the T3-associated 90K heterodimer as the T-cellreceptor. Nature 303, 808–810.

M, P., C, A. R., R, M. A., I, J.,I, S., L, J. J., W, L., I, Y., J,R., H, M. L. & T, S. (1992a). Mutations in theT-cell antigen receptor alpha and beta block thymocytedevelopment at different stages. Nature 360, 225–231.

M, P., I, J., J, R. S., H, K.,T, S. & P, V. E. (1992b). RAG-1 deficientmice have no mature B and T lymphocytes. Cell 68, 869–877.

P, A. S. (1989). Immune network theory. Immunol. Rev.110, 5–36.

P, A. S. & O, G. F. (1979). Theoretical studies of clonalselection: minimal antibody repertoire and reliability ofself-non-self discrimination. J. theor. Biol. 81, 645–670.

Pı, D., F, A., H, D., B, A. &C, A. (1986). Immunocompetent autoreactive B lym-phocytes are activated cycling cells in normal mice. J. Exp. Med.164, 25–35.

P, W. H., T, S. A., V, W. T. & F,B. P. (1992). Numerical Recipes in C. The art of scientificcomputing. Cambridge: Cambridge University Press.

R, A. Y., R, S., P-H, P., M, D.B. & J, J. C. A. (1991). On the complexity of self. Nature353, 660–662.

S, S., T, T. & N, Y. (1982). Study oncellular events in post-thymectomy autoimmune oophoryties inmice. II. Requirement of Lyt-1 cells in normal female micefor the prevention of oophoritis. J. Exp. Med. 156,

1577–1586.S, J. & V, F. J. (1991). Morphogenesis in shape-space.

Elementary meta-dynamics in a model of the immune network.Theoret. Biol. 153, 477–498.

S, B., V H, J. L. & B, U. (1994). Central immunesystem, the self and autoimmunity. Bull. Math. Biol. 56,

1009–1040.T, J. A., G, E. D., P, Y., O, A. J. &

L, R. A. (1985). The atomic mobility component of proteinantigenecity. Ann. Rev. Immunol. 3, 501–535.

T, S. (1983). Somatic generation of antibody diversity.Nature 302, 575–581.

V, F., C, A., D, B. & V, N. (1988). Cognitivenetworks: immune, neural and otherwise. In: In Theoretical

. .544

Immunology. (Perelson, ed.) Vol. III. pp. 359–374. RedwoodCity: Addison-Wesley Publ. Co., Inc.

V, N. M. & V, F. G. (1978). Self and nonsense: anorganism-centered approach to immunology. Med. Hypothesis 4,

231–267.V, A. C., C, A. & F, A. A. (1992). Differential

expression of VH gene families in peripheral B cell repertoires ofnewborn or adult immunoglobulin H chain congenic mice. J.Exp. Med. 175, 1449–1456.

W, E., A, D., M, D., B, A. C.,M, A., K, A. & V R, V. (1984).Correlation between segmental mobility and the location ofantigenic determinants in proteins. Nature, 311, 123–126.

W, O., A, L. S. & Mı, A. C. (1994). Absenceof peripheral clonal deletion and anergy in immune responses ofT cell-reconstituted athymic mice. Eur. J. Immunol. 24, 579–584.

APPENDIX A

The Equations Governing the Dynamics of the Model

Immune System

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 .

Antigens are introduced with a fixed concentrationcorresponding to the stimulatory peak in B-lympho-cyte induction (see definition of b1 below):

Ak =0 or Ak = b1 (A.1)

Once antigen k is introduced it drives acorresponding T-lymphocyte clone k, whose concen-tration Tk is from then on described by:

dTk

dt= − kDT ·Tk + kPT ·aT (pk , hk , Tk )+ kST (A.2)

with kPT =0.2, kDT =0.15, kST =0.1.The size of the B-lymphocyte clone, Bi, and the

concentration of Ig-molecules it produces, Fi, aredescribed by:

dBi

dt= − kDB ·Bi + kPB ·aB (si , ti , Bi ) (A.3)

dFi

dt= −(kDF + kDC ·si )·Fi + kSF ·aB (si , ti , Bi ) (A.4)

The activation state of T-lymphocyte clone k isinfluenced by both stimulatory (pk) and inhibitory (hk)signals, such that the number of activated cells in themodel is described by:

aT (pl , hl , Tl )= pl ·Tl ; if log(hl )Q aT1 (A.5a)

aT (pl , hl , Tl )= exp$−0log(hl )− aT1

aT2 12

%×pl ·2Tl ; if log(hl )e aT1 (A.5b)

with aT1 = log(80.0), aT2 = log(2.1); pk =1 (if theclone is driven by antigen and pk =0 otherwise), and:

hk = sNB

j=1

MTFkj ·Fj (A.6)

The number of activated cells in the B-clone i isdescribed by:

aB (si , ti , Bi )=ti ·bi

ti + bi(A.7)

where bi is the number of cells induced byligand-mediated crosslinking:

bi =exp$−0log(si )− b1

b2 12

%·Bi (A.8)

with b1 = log(80.), b2 = log(2.1), and:

si = sNB

j=1

MBFij ·Fj + s

NA

k=1

MBAik ·Ak (A.9)

and ti is the effective number of T-helper cellsavailable for clone i:

ti= sNT

l=1

$BTil

m·

$BTil ·bi

sNB

j=1

$BTjl ·bj

·n ·aT (pl , hl , Tl ) (A.10)

with n=1.0, m=1.0.The coefficient of B-T co-operation $BT is either:

$BTil =MBT

il (A.11a)

or

$BTil = s · s

NA

k=1

MTAlk ·MBA

ik ·Ak (A.11b)

whether the cooperation is mediated by directmembrane Ig-TCR interactions or antigenic-peptidepresentation.

545

APPENDIX B

Detailed Procedures for the Generation of Affinity

Coefficients (GAC), and Biological Justification for the

Simplifying Postulates Involved

The completeness of the Ig repertoire implies thatIg-molecules can bind to the specific determinants inboth TCRs and MHC+peptide complexes (Cout-inho et al., 1983). A crucial question now concerns thepossible diversity of TCRs and MHC+peptidecomplexes as Ig-determinants: are all the determi-nants in these molecules recognized by a small subsetof Ig-clonotypes; or alternatively, can they be sodifferent from each other that they are effectivelydistributed over the whole repertoire of Ig-clono-types? The second view is supported by twoconvergent arguments. The first (strong) argument isthat monoclonal Igs have been derived which arespecific (in the sense of discriminatory) for particularTCRs (Meuer et al., 1983), or for particularMHC+peptide complexes (Rudensky et al., 1991).The second (weak) argument relates to the potentialdiversity of the involved molecules. The mechanismsof somatic DNA rearrangement are essentially thesame for TCR genes and Ig genes, and the variabilityof TCRs is comparable to that of Igs (Tonegawa,1983; Kronenberg et al., 1986; Mombaerts et al.,1992b). As for MHC II molecules, they are chargedwith peptides with variable sequences and lengths(Brown et al., 1993; Engelhard, 1994) comparable tothose of the CDR3 region of Ig-molecules which isbelieved to account for much of the latter’suniqueness (Chothia & Lesk, 1987).

The issue of TCR-antigen interactions involves anumber of questions, starting with those of peptideMHC complex formation. Unfortunately, however,there is a dearth of the empirical observations wewould need to answer them in detail. For example,there are some indications that the probability that agiven TCR specifically recognizes a random antigen iscomparable to that of Ig molecules (Eichmann et al.,1980; Ashwell et al., 1986), but the truth of the matteris that experimentally measured matrices for TCRantigen interactions are entirely missing. Accuratequantitative data on peptide processing, MHCloading, etc. are also lacking. Hence, we can onlyproceed on ad hoc suppositions, and we have decidedto simplify to a maximum: in the present model wesuppose that each antigen gives rise to one and onlyone MHC+peptide complex, which can be recog-nized by one and only one TCR molecule. Thissimplification of the T-lymphocyte realm allows us tofocus our attention on the B-T interface and itsinvolvement with the idiotypic network.

Initial work on the generation of simulated affinitycoefficients for modelling the ontogeny of immunenetworks, both by ourselves and others (De Boer &Perelson, 1991; Stewart & Varela, 1991; Detourset al., 1994), has employed the shape-space concept(Perelson & Oster, 1979; Perelson, 1989). However, ina recent study (Carneiro & Stewart, 1994), we havebeen led to question the validity of this concept: thereare reasons to believe that the interaction betweentwo proteins (including idiotypic interactions) isessentially a relational phenomenon. It follows thatthe affinity may be virtually unpredictable from thecharacteristics of the two isolated molecules, since itis determined by the nature of the specific assemblyof the molecules. In order to capture this unpre-dictability, in this paper, affinity matrices have beengenerated by a procedure based on randomassignment (which implies no intrinsic topology in theresulting matrices).

We will first describe the very simple procedure forTCR-ligand interactions. For each antigen Ak

available in the system, we attribute a single specificT-cell clone, Tk. An optimal interaction strengthbetween the pair is assumed by setting pk =1.0, thusmaximizing the fraction of cell in the clone Tk whichare driven by the antigen Ak.

The Ig-ligand interactions (with other Ig, nativeantigen or TCR) are calculated as follows:

First, we define the overall probability pA thatmolecule A interacts significantly with any particularpotential ligand; this parameter pA is a measure of thegeneral multireactivity (or conversely the specificity)of the individual molecule A. In the population of allpossible A molecules there will be an average value ofpA, but also a range since individual molecules will bemore or less multireactive. Determining the distri-butions of the values of pA for the different molecularcategories composing our model (Ig, TCR,MHC+peptide complexes or antigen) is in principlean experimental issue; but in the absence of preciseinformation we have assumed that pA follows aGaussian distribution:

0E pA E 1 with P(pA )= e−$

aP −log(pA )

dP%

2

. (B.1)

On this basis, the probability PAB that a particularpair of molecules A and B have a significantinteraction strength is given by:

PAB =zpA ·pB . (B.2)

If a pair of molecules A and B do interact, theiractual affinity MAB is defined according to aprototypical distribution: we suppose that the values

. .546

of MAB follow a one-armed log-normal distribution[eqn (7)]:

MAB e aM with P(MAB )= e−$

aM −log(MAB )

dM%

2

(B.3)

In combination, the probability distributions ofeqns (5–7) mimic the general shape of experimentaldistribution of Ig-ligand interaction strength asmesured by ELISA (Freitas et al., 1986; Fig. 1). Thethreshold aM that we are imposing for any log(MAB)to be relevant corresponds to the threshold (abovebackground noise) that experimentalists require toconsider that a signal (ELISA titre or optical density)is significant. The generators with adequate distri-butions were implemented using the ‘‘ran 1’’ and‘‘gaussdev’’ routines from’’Numerical Recipes in C0(Press et al., 1992).

We have explored the parameter ranges0.001E aP E 0.1 and 0.0 E dP E 0.5. All the resultsthat illustrate this article were obtained using dP 1 0.0(no deviation from the value aP) and setting the aP

value for T-lymphocytes and antigens to 1/3 of thecorresponding value for the immunoglobulins. Theinteraction strength distribution was essentially fixedto aM =0.1 and dM =1.0 (Fig. 1).

APPENDIX C

A Simplified Probabilistic Model for the Parameter

Dependence of the Transition Between Unstable and

Stable Development of the Model Immune System

Let SB be the source term of recruitable B-clones,i.e., the number of resting clones, freshly produced bythe bone-marrow, which can be integrated into thesystem at any given time if they happen to recognizeits components. NA is the number of availableantigens and NT is the number of T-clones theyspecifically activate. PR is the average probability thatan Ig-molecule recognizes a random ligand. In thepresent formulation of the model all these quantitiesare constant in time; the only variable is the numberof B-cell clones at time t, denoted NB(t).

Let us first establish an adequate expression for thecondition for stable development, which is onlyachieved if and only if all NT activated T-cells arespecifically recognized by the Ig-molecules producedby the NB(t) B-clones in the system. Mathematicallythis implies that the probability that a randomlypicked T-clone escapes recognition by the NB(t)clones must be practically negligible, i.e.:

(1−PR )NB (t) 3 0 (C.1)

As a corollary of condition C.1, the repertoire of theactivated B-clones in the system must becomepractically ‘‘complete’’.

The parameter dependence of this condition can bebetter understood if we re-write eqn (C.1) in adifferent way. Since PR is much smaller than 1.0, wecan reasonably approximate eqn (C.1) by:

e−PR ·NB (t) 3 0 (C.2)

Thus, the crucial quantity that will determinewhether the system develops stably or continues togrow exponentially is the product PR.NB(t). Tounderstand how the parameters will influence thisquantity amounts to understanding how theyinfluence the dynamics of variable NB(t), which mustgrow and reach a critical value (of the order of I/PR)within a time-window Dt (of the order of 50time-steps, before the exponentially growing T-cellclones reach uncontrollable levels).

By reducing the dynamics of individual clones toinclusion or not in the system, we can describe theearly expansion phase by the following differentialequation:

dNB (t)dt

SB ·r(t); with NB1(0)=0 (C.3)

where r(t) is the probability that a randomly pickedclone amongst the SB candidates will be recruited intothe system at instant t.

Since in this early phase the antigen driven T-cellsare not regulated, they are growing exponentially. Wecan assume that they are not limiting for the B-clonesand thus the only restriction for recruitment of newB-clones is their ability to recognize a componentalready present in the system. Thus, the recruitmentprobability r(t) can be estimated as:

r(t)= [1− (1−PR )NA ]+ [1− (1−PR )NB (t)]

×[1− (1−PR )NT ]− {. . .} (C.4)

where the first term is the probability that the clonerecognizes at least one of the NA available antigens (asufficient condition for its induction and fullactivation). The second term is the probability that acandidate clone recognizes simultaneously both oneof the NB(t) B-clones (induction) and one of the NT

T-clones (help); it corresponds to the quantitativecontribution of the idiotypic recruitment cascade. Thethird term (in brackets, not explicit) corresponds tothe intersection of two previous probabilities that weneglect for the simplicity of the argument.

Once again we can approximate eqn (C.4) aboveby:

r(t)= [1− e−PR NA ]+ [1− e−PR ·NB (t)]·[1− e−PR ·NT ]

(C.5)

547

Note that the first term is constant while the secondone is the product of a constant (the probability ofrecognizing a T-cell) by a function which is negligiblewhile the product PR.NB(t) is small and tendsasymptotically to 1.0 as NB(t) increases. This impliesthat NB will grow linearly with a rate which will startas: SB ·[1− e−PR ·NA ], and will tend asymptotically to:SB ·[(1− e−PR ·NT )]+ (1− e−PR ·NT )]. Note that in thepresent formulation of the model, since there is a oneto one correspondence of antigens and T-cell clones,the asymptotic rate is twice the initial one.

Since the growth is linear it could be expected thatconditions (C.1) and (C.2) will sooner or later beattained; however, in simulations this is not the case.The reasons for this are easy to understand. Thesimplifications underlying eqns (C.4) and (C.5) areonly valid during a given time-window, Dt, corre-sponding to the time previously recruited B-clones

take to reach threshold sizes: on the one hand, theIg-molecules being produced reach non-stimulatoryconcentrations for the complementary clones (thesecond term is no longer valid); on the other hand, therelative sizes of already existing clones vs. newcandidates is such that the latter are competed out intheir attempt to get help. Thus, the expression for r(t)above is only valid while t QDt; once t reaches Dt theeffective recruitment rate decreases dramatically[eqns (C.4) or (C.5) now represent unreasonableover-estimations].

The necessary condition for the system to developstably is thus that the value of NB must satisfycondition (C.1) within time window Dt; if this is notverified the system will inevitably explode. The crucialimportance of SB and the product PR.NA, observedempirically in our simulations, can now be easilyunderstood.