Natural and Induced Tolerance in an Immune Network Model

J. theor. Biol. (1998) 193, 519–534 Article No. jt980720

Natural and Induced Tolerance in an Immune Network Model K L´*, J C†‡, R P´*, E M*  A L* *Centro de Immunologı´ a Molecular, P.O. Box 16040, Habana 11600, Cuba and †Instituto Gulbenkian de Ciencia, Oeiras, Portugal (Received on 1 December 1997, Accepted in revised form 1 April 1998)

It has been proposed that the immune system can be partitioned into central and peripheral immune systems. Recently, Carneiro et al. (1996a, b) proposed a network model incorporating B and T lymphocytes that explicitly accounts for that partition. This model, however, had some limitations that are tackled here. Two main changes were introduced: the average idiotypic connectivity is now an explicit function of time based on empirical evidence; and the activation of T lymphocytes by antigen is described by a log-bell shaped dose response curve. The new model, which also accounts for the CIS and PIS distinction, shows more reasonable results since the frequencies of tolerant, immune or autoimmune responses to an antigen are now correct. The model provides a new interpretation for tolerance induction during the neonatal period, and for the adult tolerance induction by low or high doses of antigen. It predicts that natural tolerance for antigens available during the neonatal period can be kept indefinitely upon their removal, while tolerance induced in the adult stages is rapidly lost upon transient removal of the antigen. A semiquantitative analysis of the model provides a simple explanation for the different results in terms of the frequency at which a limited set of canonical connectivity structures emerge during ontogenesis. 7 1998 Academic Press

1. Introduction The idiotypic network theory of Jerne (1974) has inspired many different research programs. In a first generation of idiotypic network models (Jerne, 1974, 1984; Richter, 1975) the main concern of theoreticians was the regulation of immune responses by local networks or regulatory circuits. In this respect the network views added little to the classic theory of clonal selection (Varela & Coutinho, 1991). In the second half of the 1980s, some experimental observations in normal non-immunized animals suggested that an autonomous idiotypic network may exist, and that may participate in the selection and in the dynamics of lymphocyte repertoires (Hooijkaas et al., 1984; Forni et al., 1988; Lundkvist et al., 1989;

‡Present address: Theoretical Biology, University of Utrecht, Utrecht, The Netherlands. 0022–5193/98/150519 + 16 $30.00/0

Holmberg et al., 1984). Direct experimental support for this hypothesis was, however, not available. In this context, mathematical modeling arose as a fundamental tool for the elaboration of the problems raised by the network theory, and as an important resource in the identification of those network properties which are amenable to experimental testing. On this premise the second generation immune network models were born (Varela & Coutinho, 1991). These were essentially one or two component models accounting for the dynamic behavior of B cell clones—the ‘‘B models’’—or of the B cell clones and the immunoglobulins they produce—the ‘‘AB models’’ (Weisbuch et al., 1990a, b; De Boer et al., 1992, 1993a, b; Detours et al., 1994; Calenbuhr et al., 1995). Based on the initial results obtained with these models, and on some indirect experimental evidence, it was proposed that the immune system is composed of two compartments: the Central Immune System (CIS) and the Peripheral Immune System (PIS) 7 1998 Academic Press

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(Coutinho, 1989). The CIS is a connected network of lymphocyte clones, which is responsible for ‘‘self assertion’’ and tolerance; and the PIS is a large set of disconnected clones, that mount classical clonal responses when specifically stimulated by antigen. Although the second generation models have contributed for the understanding of the generic properties of the idiotypic networks, they failed to provide a reasonable explanation for the putative difference between a CIS and a PIS. The difficulties inherent are two-fold: (1) the network repertoire tends to expand by the recruitment of new clones until it becomes virtually complete and there is no ‘‘room’’ left for a meaningful PIS (De Boer & Perelson, 1991; Stewart & Varela, 1991; Detours et al., 1994; Takumi & De Boer, 1996); (2) the connected network is unstable when interacting with antigens which are continuously available and in optimal stimulatory concentrations (Detours et al., 1994; Calenbuhr et al., 1995; Takumi & De Boer, 1996). Recently, Carneiro et al. (1996a, b) criticized these models along these lines, and proposed their extension to include the dynamics of T-lymphocyte clones and their cooperation with B-lymphocyte clones. The new model was shown to be able to account for the emergence of an appropriate distinction between a CIS and a PIS, under non-trivial parameter regimes. During the simulation of the model, the network develops stably in the presence of a set of founder antigens, and, by a process of self-organization, reduces its repertoire, leaving a reasonable proportion of disconnected clones. These results impinge on the classical problem of ‘‘self–non-self discrimination’’. Similar to rather classical proposals (Burnet, 1957; Cohn & Langman, 1990), the model suggests that the immune system becomes tolerant to antigens which are present during early stages of ontogenesis, and raises immune responses to antigens that are introduced in adult life. But the model is also original in many aspects: it represents an explicit, hitherto putative, mechanism by which tolerance is maintained in an active dominant way and not by the simple elimination of antigen specific lymphocytes; it suggests that in order for the immune system to be tolerant to the founder antigens, these must be diverse and presented simultaneously. Although the model was promising, its original implementation had some limitations. Hence, the lymphocyte clones that remain disconnected once the network matures, and are interpreted as representing the PIS, do not systematically give rise to immune responses when stimulated by antigen. Three patterns of response were obtained and have a biological

interpretation: a tolerant pattern, an immune response pattern and an autoimmune pattern. However, the relative frequency of the three patterns of response is essentially distorted and the model results too tolerant to a later antigen. This represents a major limitation of the model, in terms of its relation with the real immune system, and also in terms of its capacity to address ‘‘self–non-self discrimination’’. The present article tries to tackle this problem. A few modifications to the implementation proposed before (Carneiro et al., 1996b) are introduced, explored and discussed here. The main modifications are: (1) the introduction of an appropriate change of the idiotypic connectivity during the ontogenesis; and (2) a more refined description of the antigen presentation and its dependence on antigen concentration. The modified model is tested in some machine experiments designed to study some aspect of immune dynamics: the existence of a tolerance window during the neonatal period; the induction of tolerance during adult life by the classical schemes of Mitchison; and the consequences of eliminating an antigen for which the immune system is tolerant. The model is also used to simulate cancer disease.

2. The Model 2.1.

     

2.1.1. Dynamics The dynamic of the model comprises a set of differential equations describing the evolution of the main variables: the size of the T cell clones, the size of the B cells clones and the concentration of the immunoglobulins the later produce. The dynamics of any T cell clone, indexed k, is described by: dTk = −KDT ·Tk+KPT ·aT (pk ,hk ,Tk )+KST dt

(1)

where KDT is the clonal death rate, KPT is the maximal proliferation rate of the populations of activated cells, KST is the number of T cell belonging to clone k which are produced from hematopoietic organs per day, and aT (. . .) is the number of activated T cells in the clone k. It is assumed that the only negative influence on T cell activation and proliferation is its inhibition by free immunoglobulins with affinity for T cell receptor (TCR). This is the most stringent simplification in the model.

      The dynamics of any B cell clone, indexed k, is described by: dBk = −KDB ·Bk+KPB ·aB (sk ,tk ,Bk ) dt

(2)

where KDB is the clonal death rate, KPB is the maximal proliferation rate of a population of activated cells, and a(. . .) is the number of activated B cells in the clone k. The process of B cell activation was divided into two steps: the first step is the induction which accounts for the recognition of the ligand, and the second step is the cooperation between induced B cell and activated T cell which leads to full B cell activation. There are two different mechanisms of cooperation: one is the presentation of MHC-peptide on the membrane of the B cell to the TCR of the T cell; the other is the direct recognition of TCR by membrane immunoglobulins of B cell. The latter is an important postulate in the model because this mechanism has no direct experimental validation but is essential (according to the model) for the self-organization of the system. The dynamics of the free Igs, produced by the B cell clone k, is described by: dFk = −(KDF + KDC ·sk )·Fk+KSF ·aB (sk ,tk ,Bk ) (3) dt where KDF is the elimination rate of the Igs in isolated form, KDC is the elimination rate of Igs when complexed with ligand, and KSF is the rate of Ig secretion per activated B cell. 2.1.2. Metadynamics and general operation aspects The antigens are introduced in the model with a constant concentration and each one has associated one T cell clone. Once the antigen is introduced its T cell clone is switched on with initial value TK = 0. New B cell clones are supplied periodically (NSB = new B clones per cycle) to the system with initial conditions set at: Bk = 1.0, Fk = 0.0. To simplify calculations, the B cell clones are removed from the system when Bk R LB and Fk R LF . There is a set of antigens, referred to as ‘‘founder’’ antigens, that are present in the system for the whole ontogenic process. The number of such ‘‘founder’’ antigens is set through parameter NA . Other antigens can be introduced at any time in the system and therefore are called ‘‘late’’ antigens. The main concern in the present paper is with the late antigens. The affinity of interactions between different molecular species (TCRs, Antigens, Igs) are established by the generator of affinity coefficients (GAC)

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procedure. Briefly, this procedure works as follows: once a new molecular specimen is introduced to the system, its affinity interactions with the rest of the molecular species, already in it, are set. Each pairwise affinity coefficient (m) is generated by a two step procedure: the first step answers the question: Is there any interaction between this two molecules? The probability of affirmative answers is adjusted to be a theoretical parameter PR , which accounts for the probability of connectivity in the system. The second step assigns a value to the mAB coefficient following the rules: (1) If the answer of first step is no, then m = 0. (2) If the answer is yes, then m is generated according to the following probability distribution: P(m) =

8

if log(m) E aM P = 0

$ 0

if log(m) q aM P = exp −

am − log(m) dm

1 %9 2

(4) where aM and dM are parameters to be adjusted. It is to be noted that in the original model, the probability that the interaction between any two molecules could be positive, was calculated as a binary function of a parameter associated to each of the individual molecules; this parameter was understood as measurement of the multireactivity or specificity of individual molecules (Carneiro et al., 1996b). In so doing, some constraints were imposed on the affinity matrices produced by the original GAC—a multireactive antibody was always a multireactive antibody; these constraints are no longer present in the new GAC, where PB is a supraclonal parameter defining the probability of interaction between the immunoglobulin expressed by any new lymphocyte coming out of the bone marrow, and the molecules that can be found in the periphery (other Igs, TCRs or antigens). It may be useful to pinpoint the different practical implications of the two GAC. Consider two molecules i and j whose interactions with a given sample of N molecules are scored, by a procedure such as those described by Kearney et al. (1987) or Holmberg et al. (1984). Let us assume that molecule i scores more positive interactions with this set of molecules than molecule j, i.e. that molecule i scored as more multireactive than molecule j. The rationale underlying the previous GAC was that, on average, molecule i would always score as more multireactive than molecule j independently of the particular sample (assuming, of

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course, that the size N of the initial sample was high enough to produce a good estimate of parameter associated to molecules i and j). The rationale underlying the new GAC is that new specificities are selected (recursively) according to its degree of multireactivity in the context of the actual repertoire components (either of the CIS or PIS compartments), so any new VH region with a high degree of connectivity to either the elements of the CIS or the elements of the PIS are negatively selected. We think that up to date no experimental data exists allowing us to decisively exclude one of these hypothesis. Anyway we have found that the results presented in Section 3, can be obtained with both GACs, but using a different parameters range.* Since the new GAC is less expensive in computational terms it has been adopted here, leading us to work in a parameter range that enlightens the results. 2.2.   In this section the major modifications made to the model as reported by Carneiro et al. (1996b) are presented. Their implementation and immunological explanation are provided. Only the changes to the model or to its implementation are mentioned. Any other aspects remain the same as before (Carneiro et al., 1996a, b).

2.2.1. Idiotypic connectivity as an explicit function of time We implemented a seminal idea proposed in Carneiro et al. (1996b). Nowadays the variation of idiotypic connectivity during the ontogenesis of immune system is an experimental fact. The works of Holmberg (Holmberg et al., 1984; Kearney et al., 1987) demonstrated the existence of high idiotypic connectivity amongst antibodies isolated from newborn mice, in contrast with a low idiotypic connectivity amongst the antibodies isolated from adult animals. These observations are one of the corner stones of immune network theory. What is not yet established is the mechanism underlying this developmental switch and whether or not it has any physiological relevance. Several experimental observations seem to indicate that this switch may result *We semiempirically found that a system implemented with the new GAC procedure and parameters values P1 = P1o P2 = P2o and SB = SBo (see Section 2.4 for a parameter description), gives similar result to a system using the old GAC implementation with parameter values P1 = P1o P2 = (P2o )2/P 1o and SB = SBo P2o/P 2 (the other parameters remain the same). It is easy to note from those parameter relations that the old GAC implementation quickly becomes computationally prohibitive because of the high values of SB .

from some genetic encoded mechanism operating at the level of single cells: the finding of a biased VH gene expression during the neonatal period in contrast with a practical homogeneous usage during adult period (Yancopoulos et al., 1984; Holmberg et al., 1984), the absence or low frequency of N-additions during the neonatal period which is closely related to the low expression of TdT (Carlson et al., 1990, 1992; Gu et al., 1990). Alternatively, it was proposed that the changes in the VH gene may result from selection processes operating on cell survival in the bone marrow and/or in the periphery (Sundblad et al., 1991; Coutinho et al., 1992; Freitas et al., 1995). This second view is more consistent with the idiotypic network concept as suggested by the results of mathematical models (De Boer & Perelson, 1991; Stewart & Varela, 1991; Grandien et al., 1992). The available experimental evidence does not allow to exclude any of these possibilities, which may very well be both operative. It is very important to note that, in the present model, we make no explicit account for the mechanism underlying the time dependence of the connectivity structure, which is introduced as a basic postulate, and implemented in its simplest way. As in the original model, the idiotypic connectivity of clones is determined by the quantity (PR ), which is now defined as the following function of time: PR (t) =

8

P1 if t E t1

$

P2 + (P1 − P2)·exp −0.5·

0 1% t−t1 t2

2

9

if t1 E t

(5) where P1 and P2 are the values in early and late ontogenic stages, respectively; t1 and t2 are parameters to be adjusted. As will be presented later, the implementation of this developmental switch of the value of PR is sufficient to obtain an appropriate frequency distribution of three patterns of response to a late antigen, i.e. to ensure that an immune response is almost systematically raised against the late antigens. 2.2.2. The activation function of T cells is modified to take in account some effects of antigen presentation A considerable amount of experimental evidence indicates that antigen presentation plays an important role in the way the immune system couples with an immunizing antigen. We try to account for this aspect imposing to the efficiency in antigen presentation a dependence of antigen concentration. Of course this is a radical oversimplification of reality, and many

      other variables related to the context under which presentation takes place may also be relevant. However, as we will see later, this simple implementation is enough to increase the plasticity of the model, and entails a whole set of biologically reasonable results. In the original model the number of T cells activated was a function of the total number of cells in clone l (Tl ), a stimulatory factor (pl ) representing the efficiency of antigen presentation, and an inhibitory signal (hl ) mediated by free anti-TCR Igs. At that moment the stimulatory factor (pl ) was assumed to be constant and unitary for reasons of simplicity. We introduce a log bell-shaped dependence of this factor on antigen concentration:

$ 0

pl (A) = exp −

log(A) − Ap1 Ap2

1% 2

(6)

where A is the antigen concentration; Ap1 and Ap2 are parameters to be adjusted. This function represents a critical feature of the new model. It allows us to study the effect of the introduction of antigens in non-optimal concentrations, and to carry out simulations that mimic in vivo antigen dose response curves. Without this feature, the studies on the classical protocols of tolerance induction (Section 3.2) could not be carried out. 2.3.

 

2.3.1. Activated and resting B cells have different life spans The equation describing B cell dynamics [eqn (2)] was modified to take into account the different life spans of activated and resting B cells. The influence of the activation state on the life span of B lymphocytes is supported by many experimental reports (Freitas et al., 1995): activated cells have an average life span of about 24 hours and resting lymphocytes have a life span of several days (around 7 days). The equation describing B cell dynamics now looks as follows: dBl R A = −KDB ·BlR−KDB ·aB (l,t)+KPB ·aB (l,t) dt

(7)

where Bl is the total number of cells in clone l, BlR is the number of resting cells, aB (l,t) is the number of B activated cells, KDB is the death rate of resting cells, A KDB is the death rate of activated cells, and j(l,t) is the metadynamic production of new cells belonging to clone l.

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2.3.2. Somatic hypermutation and affinity maturation Another mechanism introduced in the model is the somatic hypermutation and affinity maturation. Somatic hypermutation is known to take place in germinal center, in advanced stages of an immune response when responding cell populations have undergone significant expansion, and to result in affinity maturation (Wagner et al., 1996). To account for these phenomena, we introduced the following procedure in simulations: (1) once the field over an antigen reaches a value greater than FL a new B cell clone is generated every Nd days (Nd = 5); (2) this clone is supposed to have the property of recognizing the antigen, and its affinity coefficient m is set using a probability distribution P(m) [eqn (4)] with a different set parameters (Am = 0.0, Dm = 1.0), i.e. the average affinities are higher and the distribution is wider; (3) the other possible connections of this clone are established by the standard procedure.   The parameters of the model dynamics are changed from that originally proposed by Carneiro et al. (1996a, b). A first group was set following Zuckier et al. (1989). The death rates of lymphocytes in different R A activation states (KDB = 0.14, KDB = 1.0, KDT = 1.0, −1 dimensions day ) and the elimination rates of free immunoglobulins (KDF = 5.0e − 2, dimensions day − 1; KDC = 8.0e − 5 day − 1 molecules − 1) belong to this group. A second group, composed by the proliferation and secretion rates was adjusted to obtain an appropriate behavior (KPB = 2.0, KPT = 2.0, dimensions day − 1, KST = 0.1, dimensions cells.day − 1, KSF = 10.0, dimensions molecules.cell − 1.day − 1) The original model established an implicit difference between free immunoglobulins and antigens. Specific free immunoglobulins were assumed not to circulate in concentrations and frequencies which were able to stimulate T cell clones. In practice, this semiquantitative argument was implemented such that while antigens were systematically associated with a specific T cell clone, Igs could not stimulate T cell clones at any concentration. Note that by this assumption, the model departed from the traditional immune network ideas that assume that Ig and antigen are operationally equivalent. Besides, with original parameters, the average Ig concentration in simulations was actually greater than the concentration of the antigens, in a violation of the assumption above. Based on these considerations, we modified the parameters defining the activation 2.4.

. ´ ET AL .

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functions of B cell and T cells (b1 = log(1000), b2 = log(2.1), aT1 = log(80), aT2 = log(2.1), s = 0.0005, AP1 = log(1000), AP2 = 2.3) in order to obtain average Ig concentrations which are at least one order of magnitude lower than antigen concentrations. The parameters related to general operation and metadynamics are changed, but this time the values are set to obtain a behavior of the system which is appropriate: NA = 100, t1 = 50 days, t2 = 200 days, FL = 1.0e5 molecules, LB = 0.1 cells, and LF = 0.5 cells. The remaining parameters are used in either of the two combinations: Set (1: NSB = 300 cells.day − 1, aM = −0.3, dM = 0.5, P1 = 5.0e − 2 and P2 = 5.0e − 4 Set (2: NSB = 200 cells.day − 1, aM = −0.5, dM = 1.0, P1 = 1.0e − 1, and P2 = 1.0e − 3 The major differences between these two parameter sets are the variation of average connectivity on time and the range of affinity values from where the coefficients in the connectivity matrix are generated (illustrated in Fig. 1). In computational terms the first set is more expensive than the second one, and it was only used in the calculations illustrated in Fig. 3. A better parameter study will be necessary to understand all the mechanism operating in the model. 3. Results and Discussion     The two prototypical modes by which the model immune system can couple with an antigen were identified before (Carneiro et al., 1996a). They are: a 3.1.

1.0

p(m)

0.8 0.6 0.4 0.2 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

m F. 1. Probability distributions of the affinity coefficients [(m) generated by the GAC procedure under the two parameter settings (see main text for details)]. Parameter set (1 (– –) generates affinity coefficients on a narrower range than set (2 (——), which is also closer to the value mAB = 1.0.

stable mode in which the sizes of antigen specific clones remain bounded, and an unstable mode in which antigen specific clones undergo unbounded exponential growth. The first mode is interpreted as tolerance to the antigen, while the second mode is interpreted as (auto)immunity to the antigen. In this section, we study how these modes of coupling arise as a function of the time point when the antigen becomes available during the ontogenesis of the system. The simulation protocols are similar to those reported in Carneiro et al. (1996b). Simulations always start with a set of NA ( = 100) antigens at concentrations which remain constant and are optimal in terms of the efficiency of antigen presentation (Ai = 1000; i = 1, . . ., 100). This set of antigens were called founder because they entail the development of a connected network and sustain its dynamics. It is appropriate to first consider the situation in which no antigens other than the founder are made available. This is the modeling equivalent to an antigen free mouse (Hooijkas et al., 1984; Pereira et al., 1986). As discussed before (Carneiro et al., 1996a) the system can attain a (meta)dynamical steady state, such that the total number of species is constant in time and the fields over the founder antigens remain bounded [Fig. 2(a)]. In the simulations reported here, the parameters have been optimized such that this global tolerance happens systematically. This is appropriate because these results are interpreted as the development and maintenance of natural tolerance to somatic antigens. At the steady state—when we say the system is mature—the network is structured around the founder antigens and has a restricted repertoire. The potential lymphocyte repertoire is partitioned into a network and a set of disconnected clones, and this result is interpreted as the emergence of the CIS and the PIS compartments (Carneiro et al., 1996b). As in Carneiro et al. (1996b), a typical immunization (i.e. the injection of an antigen) is simulated here by introducing, into the mature system, a single antigen with constant optimal concentration. Because these antigens were not introduced together with the founder antigens (that gave raise to the network), they are called late antigens. The three patterns of response to the perturbation by such late antigen are also depicted in Fig. 2. The dashed lines represent the field over late antigens and the solid lines represent the field over founder antigens. Figure 2(b) shows the immune response pattern where the system couples with the late antigen in an unstable mode, but remains stable regarding all the founder antigens. Figure 2(c) is the tolerance pattern where the system couples

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F. 2. Prototypical modes in which the model immune system couples with antigen. Coupling with founder antigens in tolerant mode is illustrated (top left). The remaining graphs illustrate coupling with a late antigen introduced at time t = 1010 day, which can occur in immune response mode (top right), tolerant mode (bottom left) or autoimmune mode (bottom right). The graphs are timeplots of the sum of antibodies against the antigen during simulations. Solid lines and dashed lines represent founder and late antigens, respectively.

stably with the late antigen. Figure 2(d) is the autoimmune pattern where the stable coupling with the founder antigens is lost upon the immunization with the late antigen. The relative frequency of the three types of response to late antigens are listed in Table 1. As noted before, the same patterns of response were obtained with the original model but their frequency

distribution was distorted regarding what is expected by the immunological common sense. The mature system was mostly tolerant to late antigens, under the parameter regimes that would systematically lead to stable coupling with the founder antigens. The relative frequencies of the patterns of response were rendered biologically reasonable by implementing PR as an explicit function of time.

T 1 The relative frequencies of the patterns of response to late antigens Pattern of response Immunity Tolerance Autoimmunity

New model New model set (1 set (2 85 10 5

77 12 11

Original model*

Expected for adult animals

5 85 10

High Low Very low

*The parameters used in the original model are those of Fig. 8(a) Caneiro et al., 1996b.

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core [Fig. 4(a)]. During the tolerant mode the connectivity is characterized by the existence of a B cell clone that recognizes some antigen in the system (founder or late) and the TCR of the T cell clone driven by the late antigen [Fig. 4(b,c,d)]. The autoimmune mode of coupling with the late antigen is characterized by connectivity structures in which a B cell clone driven by the late antigen recognizes also a founder antigen [Fig. 4(e,f)]. These results are easy

Percentage of simulation runs

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Time of antigen challenge F. 3. The percentage of three modes of coupling with a late antigen as a function of the time at which the antigen is introduced: immune response (E), tolerance (R) and autoimmunity (W). Each point in the curves represents the result of 200 simulation runs.

(b) A

A second type of simulation protocol was designed to investigate how the time elapsed between the ‘‘introduction’’ of the founder antigens and the ‘‘introduction’’ of the late antigen can change the relative frequencies of tolerant, immune or autoimmune patterns of response to a late antigen. The results are depicted in Fig. 3. Each time point is the result of 200 runs. The results show that the system has an ontogenic period of time where it systematically couples with newly introduced antigens in a stable mode. This is the period when the network is being organized and its repertoire is essentially complete, both because of the initial diversification of the repertoire (Carneiro et al., 1996b) and because any new B clone is very multireactive. Any antigen appropriately presented to the system at this state will be coupled in a tolerant way, integrating the set of founder antigens. In the classical immunological jargon this amounts to say that the antigen is ‘‘treated as self’’. After this period, while the network ‘‘focuses’’ and restricts its repertoire (Carneiro et al., 1996a, b) and the average probability of interactions of the new B cell clones decreases, the system, as a whole, becomes predominantly immune to newly presented antigens. In order to understand the proximal causes of the three modes of coupling with late antigens in simulations, we examined the actual connectivity structures. Each mode of coupling was always associated with one or two canonical connectivity structures (Fig. 4). The structures found during the immune response mode are characterized by the existence of at least a B cell clone that recognizes the late antigen, and is disconnected from the network

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F. 4. Connectivity structures typically associated with three modes of coupling with a late antigen: immune response (a), tolerance (b, c and d) and autoimmunity (e and f). The elements in the diagram are: the immunoglobulin molecules, the TCR molecules, the MHC–peptide complex, and the lymphocytes. The shaded elements belong to the central immune system and are coupled with the founder antigens; the white elements are triggered by late antigen.

     

3.2.     :       

In this section we study the effect of antigen concentration on the way the mature system, partitioned as a network and disconnected compartments, couples with a late antigen. This is an important aspect of the model that bears a direct relationship with the two classical protocols for the induction of antibody tolerance in adult animals, that were reported by Mitchison (1964). The first protocol consists in immunizing the animal several times with a low dose of the antigen, and the second one consists in a single injection of a very high dose of the antigen. An animal that undergoes either of these treatments becomes unable to raise an antibody response to an immunogenic dose of the same antigen. A machine experiment was designed to test the effects of these protocols in our model of immune system. The experiment consists in the introduction of a late antigen at constant concentration (A = a0) during a time period (TP ) and then set its concentration to the optimum value (A = 1000). The results are shown on Fig. 5, where the frequency of tolerant patterns is plotted as a function of a0 for two values of TP (30 and 40 days). Each time point represents 100 runs. Five concentration ranges result in three types of response which are biologically relevant. Very low or very high concentrations have no effect on the system, relatively low or relatively high concentrations lead predominantly to tolerance, and concentrations around the optimal stimulatory value lead predominantly to immune response (Fig. 5). In the concentration ranges a0 Q 40 or a0 q 30 000 an antigen is unable to stimulate the growth of its specific T-cell clone. In these conditions, there is no effective coupling of the system with the antigen, such that when the antigen concentration is raised to the optimum value the system will respond as if the antigen was introduced by the very first time. At such very low or very high concentrations, the antigen does not perturb the system. This result invokes the concept of immunological ignorance that has been proposed notably by Mamula et al. (1993), who related it to an inability of the APC to present the antigen. In the present model, which does not account for qualitative differences in the processing and presentation of antigens by APCs, immunological ignorance is explained alone by the log bell-shaped

response curve of T-cells to antigen, and its impact on clonal dynamics. A late antigen which is introduced at a concentration within the ranges 40 E a0 Q 500 or 2000 e a0 q 30 000 perturbs the system, but it is predominantly coupled in the tolerant mode. As the concentration approaches the optimal value, the frequency of tolerant responses decreases slowly while the frequency of immune responses increases until the proportions at the optimal concentration are reached. It is to be noted that although the antigen presentation efficiency p(A) is symmetric in log scale, low and high doses are not formally equivalent in the model, because B–T cell cooperation mediated by MHC + peptide presentation is assumed to be a linear function of antigen concentration. This non-equivalence between low and high doses explains the asymmetry of the frequency distribution in Fig. 4. Figure 5 also illustrates another result obtained with the model which is the fact that the longer the time delay TP the greater the frequency of tolerant responses at relatively low or relatively high doses of antigen. The result for low concentration is in close agreement with the low dose tolerance scheme obtained by Mitchison (1964). There are two important aspects of the induction of tolerance using this protocol. First, the antigen concentration used for tolerance induction must be sufficiently high to effectively perturb the system, but must be sufficiently 100 Percentage of tolerant runs

to understand upon a consideration of the dynamical consequences of the prototypical modes of B–T cell cooperation, that were characterized before (Carneiro et al., 1996a).

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ao F. 5. Simulation of classical low and high dose tolerance. The simulations consisted in introducing a late antigen at t = 1010 day, at the concentrations indicated in the graph and setting its concentration to the optimum stimulatory value, at time t = 1010 + Tp. The graph represents the percentage of tolerant responses in the second challenge as the function of the concentration A in the first challenge, for two delays, Tp. Each point is the result of 100 simulation runs. (Q) Tp = 30; (R) Tp = 40.

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low not to trigger an immune response. Second, the longer the time delay Tp the greater the probability that the system becomes tolerant. At higher concentrations, the relation between our machine experiment and Mitchison high dose tolerance scheme is not so straightforward and it requires a further assumption that when an animal is immunized with very high doses of antigen the antigen remains on the system at a constant concentration (at least during the period Tp). A more realistic way to implement Mitchison’s high dose tolerance is to make antigen concentration decrease exponentially with time. As long as the decay rate is low, the results of this type of simulation is quite similar to the ones obtained with the fix concentration (data not shown). In general terms, the results presented in this section illustrate how antigen presentation protocol can change the way by which the immune network reorganizes and couples with a specific late antigen.     : ‘‘ ’’ Memory to an immunizing antigen is currently supposed to be derived from the existence of some long-lived cell (whose specific phenotype has not been identified yet), to be due to the persistence of the antigen (Gray & Matzinger, 1991), or to be ensured by the sustained dynamics of cooperating subpopulations of lymphocytes (Weisbuch et al., 1990b). Tolerance as one type of activity of the immune system, may also reveal some kind of memory, and in our view, this memory can only be understood as a systemic property emerging from a dynamic process. We show in this section the result of simple machine experiments that illustrate the conditions under which the immune network can sustain tolerance towards antigens that become transiently absent, i.e. tolerance memory. The simulations involve the following steps: (1) the system is allowed to mature in the presence of the founder antigens; (2) optionally, tolerance is induced to a late antigen (see Section 3.2); (3) an antigen for which the system is tolerant, is ‘‘removed’’ by setting its concentration to a negligible value; (4) the system is left unperturbed for a period (Tp), and then concentration is raised back to the optimal value; (5) the class of response by the system is scored. The same experiment is applied to either a founder or a late antigen. The result of this type of experiment can be used in the prediction and interpretation of the observations in vivo related to the establishment of neonatal tolerance or to tolerance induction in adult life. The results are shown on Fig. 6, where the frequency of tolerant runs is plotted as a function of 3.3.

Tp, for the two classes of antigens. While a single founder antigen can be removed for any period of time without loss of specific tolerance, the situation can be very different for a late antigen. In only 20% of the cases the system will remain tolerant to the reintroduction of a late antigen as it happens with founder antigens; in the majority of the cases the system remains tolerant to the reintroduction of the antigen only during a relatively short period of time and following this period the probability of getting a tolerant response drops suddenly to the values found when a new late antigen is introduced. In the model, tolerance requires that the antigen driven T cell clones are controlled by an anti-idiotypic B cell clone (Carneiro et al., 1996a). The maintenance of tolerance in the absence of the antigen, requires that this (or these) anti-idiotypic B clone(s) is (are) somehow maintained in the system (Fig. 7). For this to happen it is necessary that they get help from the other T cell clones that remain in the system. The difference between founder and late antigens in terms of tolerance memory is directly mapped to a difference in connectivity. In the simulations, the central B cell clones that control the T cell clones driven by the founder antigens are multireactive and typically connected to more than 7 T cell clones; since they get help from multiple sources, they do not go extinct during the absence of antigen (Fig. 7). Concerning tolerance to late antigens two different situations are observed, corresponding to the cases of presence or absence of memory for tolerance. The first one [Fig. 4(b)] is characterized by the existence of a B cell clone of the central network that directly recognize the T cell

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Tp F. 6. The graph illustrates how the tolerance to a founder antigen is retained on the system for an indefinite period of time in the absence of the antigen, while the induced tolerance to a late antigen (80% of total cases of tolerance to late antigens) is rapidly lost. (W) Founder antigen; (E) late antigen induced tolerance).

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of the antigen (Fig. 7). We propose to label these subclasses of tolerance to late antigen as natural and induced. It is interesting to point out here, that the increased frequency of tolerance obtained by manipulation of the protocols of antigen presentation (Section 3.1) is directly correlated with an increase in the frequency of induced tolerance, and not of the natural tolerance. The practical result is that tolerance induced by these protocols has no memory. Since the founder antigens are presented simultaneously to the system, this may have important consequences for the way the system organizes, and the way tolerance is maintained. To try to address this possibility, we investigated the behavior of the mature system following the removal of different numbers of founder antigens. The results are depicted in Fig. 8, where each point represents a statistic over 100 simulation runs. The behavior of the system can be classified into three categories: (1) sustained memory to tolerance, corresponding to the situations in which the system remains tolerant to each and every antigen removed; (2) loss of memory to tolerance, corresponding to the situations which the system loses tolerance to at least one of the antigens transiently removed; and, finally, (3) system crash, corresponding to those situations in which, upon removal of several founder antigens, tolerance is lost to the remaining antigens because a critical fraction of the network collapses. These results suggested that memory to ‘‘founder antigens’’ may be collective—or as proposed by Carneiro & Stewart (1995) the founder antigens form a coherent Gestalt. It is an intriguing possibility that

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Number of removed antigens F. 8. The impact of removing several antigens simultaneously on tolerance memory. The graph represents the percentage of the three types of behavior of the mature system—sustained memory (R), no memory (Q) or system collapse (W)—following the removal of the number of antigens indicated in the x axis. Each point represents 100 simulation runs.

this modelling result bares some relation to the finding that patients suffering from organ specific autoimmunity tend to display a higher susceptibility to other autoimmune diseases. A particularly cogent example, is Hashimoto’s thyroiditis which is frequently associated with chronic active hepatitis, dermatitis herpetiforme and lupus erythematosus (Baker, 1991). The set of results presented in this section reinforce the notions that the system couples differently with later and founder antigens, and that this may be due to the developmental switch from the preferential expression of multireactive antibodies during the perinatal period to the expression of more specific antibodies in adulthood (Holmberg, 1984). 3.4.

     

A simple simulation of the immunology of tumors is presented in this section. It is our first attempt to apply the model to some practical purposes. We study the kinetics of cancer growth as a possible mechanism for tumor-induced tolerance. The tumor is simulated as a late antigen, which grows exponentially from very low concentration (A = 10, typically on ignorance range) to the optimal concentration value (A = 1000). It is important to acknowledge from the outset that this is an oversimplification that is probably unrealistic: although tumor growth follows an exponential curve with exponential coefficient decreasing in time, it would be a major coincidence that it would precisely arrest at the optimal effective concentration to stimulate lymphocytes. The statistics on the classes of response are studied as a function of the tumor growth rate (r). The results are depicted in Fig. 8. When the growth rate is very small (r Q 0.05) the tumor is predominantly tolerized. As the growth rate increases the frequency of tolerant responses decreases until it reaches the typical value obtained with an antigen at optimal immunizing concentration (the limit as r : infinity). The result establishes a direct relation between the tumor growth rate and the tumor rejection capacity by the immune system, in the sense that slow growing tumors are more frequently tolerated than the fast growing ones. It should be noted, that according to classical clonal selection models, a tumor that escapes rejection is typically understood as the result of an inability of the immune system to recognize the tumors most typically due to a non-appropriate presentation (Fucks & Matzinger, 1996), or because it grows so fast that the immune system cannot keep up with its pace (Prehn, 1996) The present model offers a radically different interpretation: a tumor would

      escape rejection as long as it engages the immune system but its slow growth rate entails a process of active tolerance. This interpretation is an explanatory hypopthesis for the finding that the immunogenicity of tumors induced by MCA is inversely proportional to the latency period of the tumor (the time required for the formation of an appreciable tumor) (Bartlett, 1972; Old, 1962). 3.5.

       

3.5.1. Tolerance to late antigen—adult tolerance The different canonical connectivity structures associated to each type of response to late antigen were illustrated in Fig. 4. The emergence of one or another structure in simulations is a very complex process, involving the stochastic process by which the clones with the necessary specificities are generated, and the sustained dynamics of those clones that engage in positive and negative interactions amongst them and with the remaining clones in the system. A major feature of the dynamic rules and parameters is the dominant character of the tolerant connectivity structures [fig. 4(b,c), and (d)], meaning that when they emerge they overcome other pre-existing structures and force a tolerant coupling with antigen. This prompted us to try to estimate the frequency at which those tolerant connectivity structures could emerge by the stochastic GAC process, neglecting the detailed dynamics. This radical simplification can hopefully provide some insight into what is going on in simulations. Let us define some quantities that will be used during the following analysis: NA = total number of antigens (founders + late antigens) NCB = number of B cells clones integrating the central system NSB = number of B cell clones newly generated per unit of time PBT = probability that a randomly peaked B cell clone idiotypically recognizes a randomly peaked T cell clone by direct TCR–BCR interactions PBA = probability that a randomly peaked B cell clone recognizes a randomly peaked antigen. PBB = probability that there is a productive interaction between two randomly peaked B cell clones. According to the GAC procedure used here: PBB = PR and PBT = PBA = PR /z3

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Natural tolerance: the connectivity structure associated to natural tolerance is illustrated in Fig. 4(b). The frequency of this form of structure can be estimated by the probability that the T cell clone driven by a late antigen is idiotypically recognized by at least one of the NCB clones in central system. This probability denoted PNT is: PNT = 1 − (1 − pBT )NCB

(8)

It is important to note that the values of both PR and NCB change in time. PR is externally defined by eqn (5), such that in the late stage we have PR = P2. NCB is an emergent result of the organization of the network, that changes in time in a characteristic way described before by Carneiro et al. (1996b). During an initial period the value of NCB rises but after this period of diversification, its value decreases until it reaches a constant value. In our simulations and using both parameters settings (Section 2.4) at this late stage NCB = 70 clones. Induced tolerance: the connectivity structure typically associated to induced tolerance is illustrated in Fig. 4(c). This is the type of structure that results in no memory as discussed in Section 3.3. The frequency of this type of behaviour in simulations maps to the probability that at least one of the B cell clones generated after the introduction of the late antigen fulfils two conditions: it recognizes idiotypically the T cell clone driven by the late antigen and it recognizes any of the NA antigens in the system. This probability denoted PIT is: PIT = 1 − (1 − pBT ·pBA )SB ·Dt·NA

(9)

where Dt is lapse of time the system takes to explode in a typical immune response to an optimal stimulatory antigen dose. In the simulation we have 20 Q Dt Q 40 day. In the following calculation we will use the average Dt = 30 day. The total frequency of tolerant coupling with late antigens is approximately: PT = PTI + PNT

(10)

under the reasonable assumption that clones that could intervene in both eqns (8) and (9) can be neglected. Table 2 compared the expected frequencies of these two modes of tolerance with the actual results obtained by simulations for the two set of parameters. The expected frequencies roughly agree with the actual data. The deviation observed, which is particularly evident in the last column of Table 2, may be partially attributed to the character of the estimation (many values have been numerically

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T 2 Expected vs. actual values of incidence probability of tolerance patterns Tolerance pattern PNT PIT Total (PT ) PNT /PT

Expected value (set (1)

Actual value (set (1)

2% 7.5% 9.5% 21%

3% 7% 10% 30%

estimated: NBC and Dt) and the existence of a small number of machine experiments (from a statistical point of view) in the determination of the actual values. 3.5.2. Transition from a neonatal tolerant system to an adult competent one As noted on the last section frequency of tolerance to late antigens, PT , is a function of the ontogenic time since it depends on the time dependent quantities PR and NBC . The transition point from mostly tolerance to mostly immune responses can be made explicit by the instant t1/2 when PT = 0.5. Let us try to use our simplified probabilistic analysis to generate an expected value of t1/2 and compare it to the actual value 600 day found in simulations (Fig. 3). By the results of simulations we know that at t1/2 the reduction of clones in the central system NBC has already taken place such that we have NBC = 70 clones. In these special conditions we can use eqn (10) to obtain the PR value such that PT = 0.5, and use it in eqn (5) to obtain the expected value of t1/2. The value obtained by this procedure is t1/2 = 640 day, which is in close agreement with the actual value obtained in numeric simulations. 3.5.3. Tolerance induction of low and high doses of antigen The effect of antigen concentration on the probability that the mature system couples with a late antigen in the tolerant mode, can be understood immediately from eqn (9). Low or high doses of antigen make the antigen driven clones grow more slowly as compared with the optimal intermediate doses [eqns (1) and (6)] such that the period Dt that *It is important to note that the new model also keeps the most stringent simplifications of the old one: there is only one T cell clone for each antigen and the only limiting factor for the T cell growth is the presence of free Igs. Despite all these simplifications the new model allows us to address some important questions, but some other problems, as those exposed by Carneiro et al. (1996b, Section 4.2), remain unsolved.

Expected value (set (2)

Actual value (set (2)

4% 20% 24% 16.7%

2.5% 9.5% 12% 20.8%

the system takes to explode increases accordingly. At the limit in which antigen driven clones do not grow at all the period Dt tends to infinitely big and the quantitity PIT tends to unity. Because the dose of the antigen does not affect the probability of natural tolerance PNT [eqn (8)], our semiquantitative scheme also accounts for the finding that tolerance obtained using high or low doses of antigen has no memory (Section 3.3).

4. Final Remarks The model of immune system presented here extends and complements the CIS and PIS model proposed before by Carneiro et al. (1996a, b). In the original work, the main concern was the self-organization of the immune system in the absence of immunizing antigens, and how this process brought about the CIS and PIS distinction and the behavior that is classically known as ‘‘self–non-self discrimination’’. We now complement this approach by concentrating on the way the immune system actually couples with the immunizing antigen, paying special attention to mechanisms underlying the acquisition of tolerance in the mature stages. The core postulates and the main results* of the original model still hold, even if a considerable number of changes have been introduced to it. Tolerance is essentially maintained by circulating anti-TCR antibodies that under certain conditions define regulatory feedback loops. Natural tolerance is established to founder antigens that are present during an early period of ontogenesis when the repertoire of the network is complete, such that any antigen specific TCR cannot escape regulation; the immune response to late antigens is only made possible if the repertoire of the mature system is no longer complete, and antigen specific TCR can escape regulation. However, under the parameter settings now explored, the new model achieves these results in a way which is different from the one originally described. Hence, at variance with Carneiro et al.

      (1996b)†, the differences now observed during early and late stages of the system are directly mapped to an ‘‘external’’ developmental switch in the connectivity structure‡ (predefined as a function of time), and only marginally related to the number of founder antigens. The main result in most second generation immune networks is the functional correlation between connectivity and tolerance. In the same line we found that high connectivity is associated to tolerance to antigen, and a low connectivity is associated to the immune response to antigen. However, the analysis of our model suggests that there is more to be said about the relation between tolerance and the local connectivity structures involving antigen specific clones. Some canonical structures can be identified that correlate with other properties of the tolerance states. Hence, some particular structures characterized by a high degree of connectivity, are typically associated to founder antigens, and can be extended to late antigens under situations of natural tolerance; these structures allow for a collective tolerance to founder antigens that shows memory. Other connectivity structures, typically obtained when tolerance is induced to a single late antigen, are characterized by a lower degree of connectivity. They are not robust to the elimination of the antigen, and do not display memory. The introduction of some aspects of antigen presentation process demonstrates that ontogenic time—i.e. the founder vs. late distinction—although a determinant fact, may not be the only relevant aspect for coupling with antigen. Antigen availability is also a determinant factor, meaning that an antigen will only be recognized as founder, if it is available in the system during the correct period of time, and if it is efficently presented to the T lymphocytes. Reciprocally, a late antigen can be tolerated if it is not efficiently presented, although the tolerance is maintained under conditions which are different from those by which tolerance is maintained to founder antigens. †Carneiro et al. (1996b) mainly studied a version of the model where the average connectivity was constant in time. Under the parameters studied, it was found that the frequency distribution of the three classes of responses was abnormally biased towards tolerance. The authors raised the hypothesis that a more appropriate distribution of responses should be expected at higher number of founder antigens; arguing that this would be formally equivalent to the situation, explored here, in which the connectivity is high when coupling with founder antigens takes place. This hypothesis was not addressed here and remains to be explored. ‡The possibility of integrating the time dependence of connectivity probability as a consequence of system self-organization throughout a detailed modelling of the recursive selection process remains to be explored.

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An interesting feature of the model is that it brings together two different views in immunology: the notion that the process of antigen presentation plays a key role in the immune response (Matzinger, 1994), and the notion that a self-organized network determines when and how an immune response will take place (Varela & Coutinho, 1991; Cohen, 1992). Although, it is easy to conceive events as dominated by either antigen presentation or network regulation, there is an extensive region where both mechanisms are operative. By changing the efficiency of antigen presentation we can make use of the dynamic plasticity of the immune network in order to achieve different modes of coupling with antigen. This cooperation region is the most interesting from the point of view of system dynamics, and is what made it possible to interpret some core experimental observations such as Mitchison’s low and high dose tolerance. Probably more relevant, this region where both antigen presentation and network organization cooperate represents an open field for future studies, where different presentation protocols could be designed in order to get different behaviors of the network. Hitherto the predictive capacity of immune network models has been practically null. Most of the times modeling strategies try to address general issues raised by formal networks, or, alternatively, try to provide some interpretation to some generic observations in immunology. Both these approaches are reasonable in their own right, and provide some indirect support for network theories. However, it is our conviction that raising the predictive capacity of the models could help to establish a more efficient relation between theory and experimentation. On this very first analysis of our extension of the model of Carneiro et al. (1996a, b), we have tried to identify and classify its behavior in terms of coupling with immunizing antigen, with the aim of enlightening lines of research that could bring the model to an effective empirical test. To establish an initial basis, we simulated the response of the immune system to classical protocols for the induction of tolerance, with results that reasonably match the experimental observations. The simulation of new experimental protocols in machina, under the same parameter settings, allowed us to predict the existence of two different classes of tolerance—natural and induced tolerance. The existence of these forms of tolerance in the natural system could in fact be tested by comparable experimental protocols in vivo. The simple simulation of tumor growth represented a first attempt to give the model some clinical application. We hope that further works on this model could lead

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to more quantitative and more accurate predictions, and therefore guide the precise design of experimental tests.

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