lmmunochernistry, 1977, Vol. 14, pp. 733-739. Pergamon Press. Printed in Great Britain
SOME REMARKS ON THE STABILITY OF THE IDIOTYPIC NETWORK J. HIERNAUX Service de Chimie physique II, Campus Plaine, U.L.B., CP231, Boulevard du Triomphe, B-1050 Brussels, Belgium
(First received 10 March 1977; in revised form 30 May 1977) Abstract--The regulation of the immune response is a complex phenomenon whose details are not yet well known. Recently, Jeme has described the immune system as a complex idiotypic network. He has proposed that the interactions between the elements involved in the latter are responsible for the main regulatory patterns of the immune system. Some experimental results are in favour of this hypothesis. A few theoretical models have been developed to describe more precisely the nature of the interactions between the lymphocytes and the antibodies inside the network as well as their implications. In such a model, the time evolution of the concentration of those elements is described by kinetic equations whose integration over time can reproduce roughly the dynamic behaviour of the immune system. Those theoretical models should at least account for phenomena such as memory or tolerance, which implies the maintenance of a stationary state. This latter statement can be directly related to the mathematical concept of stability. We illustrate this point on various models. More precisely, we show how each mode of the immune response may be interpreted as a transition from one steady state to another one, following antigenic or any other stimulation.
I. INTRODUCTION & Bangasser, 1975; Urbain et al., 1975). Autogeneous The regulation of the various modes of the immune anti-idiotypic antibodies have also been obtained response involves complex mechanisms based on the experimentally (Rodkey, 1974; Kluskens & K/Shler, interactions of the lymphocytes and of the substances 1974; Cosenza, 1976; Urbain, 1976; Binz & Wigzell, produced by them. The most widely accepted theor- 1976). It seems likely that those anti-idiotypic antietical framework related to the immune response is bodies play a regulatory role at the level of the imthe clonal selection theory of Burnet (1959), which mune response, either suppressing it (Eichmann, 1975; describes how an (injected) antigen can be specifically Cosenza, 1976) and inducing tolerance (Binz & Wigrecognized by the receptors of pre-committed lym- zell, 1976) or stimulating it (Rajewsky & Eichmann, phocytes, thus eliciting the proliferation of clones pro1975; Trenkner & Riblet, 1975; Black et al., 1976). ducing specific antibodies. The latter react with the This idiotypic network should be functional, i.e. its antigen which is degraded. However, this reaction is regulation should be responsible for the various strongly dose dependent: below a certain threshold modes of the immune response. So, according to of antigen concentration and above another one, the Jerne (1974), the basic regulatory pattern of this animal does not eliminate the antigen if the latter network is suppression because it can guarantee the is injected continuously, the animal is said to be toler- maintenance of a steady state: for example, it can ant. Experimentalists and theoreticians have proposed keep the concentration of the antibodies at a very various models to account for different aspects of the low level in the absence of antigen. The latter is supimmune response (Bell, 1970, 1971, 1973; Bruni et al., posed to perturb the suppressed steady state of the 1975; Bretscher, 1974; Pimbley, 1974; Coutinho & immune network, thus inducing the specific synthesis M/~ller, 1975; Perelson et al., 1976). of Abl. Each immune response should correspond to Recently, Jerne (1974, 1975) has described the im- a well-defined regulatory pattern. Actually, the exact mune system as a "web of V domains" based on idio- nature of the interactions between the humoral and typic interactions: the antibodies reacting with an the cellular elements of the immune system is not exogeneous antigen (which we shall designate by Abl) well known. So, we can be interested in the following are characterized by specific idiotypes (Oudin & questions: what is the role of the T-B cooperation? Michel, 1963; Oudin, 1966, 1974)and they stimulate What is the difference between a T-dependent and the production of anti-idiotypic antibodies (which we a T-independent response (Quintans & Cosenza, shall designate by Ab2), the idiotypes of Abl being 1976)? How do the T-helper or T-suppressor cells act? antigenic with respect to Ab 2. In turn, Ab2 could elicit How are the cytotoxic cells involved inside the netthe production of an anti-idiotypic set Aba, and so work? Answers to those and other questions can on. Several experimental results are in favour of probably be provided by careful experiments. But, it Jerne's hypothesis (Jerne, 1974, 1976; Urbain, 1976). seems to us that the complexity of the regulation is The production of anti-idiotypic antibodies has been such that a theoretical approach, based on experimeninduced in serum coming from animals of other species tal results, becomes worthwhile since it allows us to (Cosenza & K6hler, 1972; Hart et al., 1972; Rowley focus on some particular aspects in order to underet al., 1973; Eichmann, 1974; K6hler, 1975; Nisonoff stand them better. 733
734
J. HIERNAUX
Up to now, two models have been proposed to describe the regulation of the immune network (Richter, 1975, 1977; Hoffmann, 1975, 1977). The model of Richter is described in more detail in the following section. Hoffmann's model considers essentially two species: the 'positive' one (which is in fact Abl) interacts with the antigen and the 'negative' one (or Ab2) recognizes the first one. The dichotomy between T and B-cells, as well as the role of the cytotoxic cells, are taken into account. The basic regulations are positive and negative interactions between the V domains of both species. Hoffmann shows bow a tolerant state as well as a normal immune response can be generated. ° Both models are based on non-linear kinetic equations describing the time evolution of the concentrations of the various populations Abe, Ab2... as well as of other elements involved in the regulation. The numerical integration over time of those equations will generally give an approximate description of some aspects of the dynamical behaviour of the immune system. In most of the cases, the system evolves to a steady state. In principle, the latter is expected to be compatible with the external as well as with the internal constraints of the organism. In simple situations, there will exist only one steady state for well-defined constraints. Moreover, slight modifications of those constraints may lead the system towards new steady states qualitatively similar to the initial state. However, in a number of situations involving non-linear kinetics, it is known that the system admits several states for the same values of the parameters. In such cases, a slight variation of the constraints is sufficient to drive the system toward qualitatively different configurations. Several biological illustrations of this behavior have been analyzed recently (Nicolis & Prigogine, 1977). A general feature accompanying these transitions is the loss of stability of the initial 'reference' state. Stability appears therefore as a necessary physical requirement to be imposed on any physiologically acceptable state. This joins the notion of homeostasis which is frequently used to characterize biological regulation. These considerations suggest the following picture for the immune system regulation. In the absence of antigenic challenge, the interactions (corresponding to the internal constraints of the immune system) between the elements of the immune network are supposed to limit the lymphocyte proliferation, and consequently, the production of antibodies. In this way, they establish an homeostatic state, which is expected to be stable with respect to small perturbations. Now, the antigenic challenge corresponds to a macroscopic perturbation which can drive the system from the original virgin state to a new one. The existence of the immune memory implies then the stability of the latter. It appears that the antigenic challenge induces a temporary modification of the external constraints, whose result is the evolution of the system towards a new homeostatic (or stationary) state. An irrevers* The same problem has been investigated for the prebiotic evolution (Eigen, 1971; Prigogine et al., 1972), for the formation of genetic nets (Kauffman, 1969) and for the ecosystems (May, t973; Allen, t975).
ible tolerant state can also be defined as mathematically stable. Let us insist on the fact that the stability is defined as long as the various parameters characterizing the regulatory pattern are maintained constant. The properties which we have described are related to the immune system of an adult animal. Another problem is the ontogeny of the adult immune system (Bruyns et al., 1976; Urbain, 1976; Augustin & Cosenza, 1976). It has been theoretically investigated in the framework of the network concept (Adam & Weiler, 1976; Adam, 1977). The model of Adam and Weiler deals more precisely with the ontogeny of antibody diversity on the basis of the interactions between the V domains of Abt and Ab2 clones. In this case, one has to understand the formation of a stable complex system, i.e. of a stable network, during the development of an organism. As we have already mentioned, the stability is a requirement of the homeostasis of the biological systems.* We assume thus that the adult immune network is the result of a maturation process which implies a dynamic cooperation over time between the external and the internal constraints. Moreover, we suppose that the latter, which are related to the regulatory processes, assure the stability by controlling the proliferation of the various clones. In this paper, we are only concerned with the stability of the mature network. We investigate the model of Richter and some slightly modified versions of the latter from this point of view. We think that a theoretical model should account on the one hand for the global dynamic behaviour of the immune system during the immune response, and on the other hand for such phenomena as tolerance and memory. As we have already mentionedl the latter are closely related to stability. Now, the naiure of the dynamic behaviour of a network also depends on its configuration: for example, the 'connectance' of the various elements must be specified. In this paper, we have limited ourselves to small cycles as well as to weakly interacting cycles.
I1. DESCRIPTION OF THE MODEL
The basic scheme of regulation of Richter's model (1975) is directly inspired from the network structure proposed by Jerne (1974). In fact, Richter has considered a linear set of V interactiofis whose starting point is the antigen. The latter elicits specifically the proliferation of lymphocytes producing the populations of antibodies Abl, which eliminate the antigen. Beyond this event, the production of Ab2, Ab3...is stimulated sequentially. As explained previously, there is a negative feedback since, broadly speaking, Ab i suppresses Abi_l which stimulates its production. Moreover, the model supposes the existence of concentration thresholds for activation and for suppression: the latter is assumed to be lower than the former in order to limit the proliferation of the lymphocytes in absence of antigenic stimulation, in agreement with the actual biological situation. On the basis of this regulatory pattern, Richter has described the normal immune response as well as the low- and the highzone tolerance phenomena. The time evolution of the concentration of the antibodies of species i, Ahi is
Immune Network Stability described by a phenomenological equation of the following form (the concentrations of the antibodies and of the lymphocytes of species i are considered as being proportional):
1974). Those aspects can easily be incorporated into the equation (1), in the following way:
dt dai_{1 dt
1 fa(ai-l, ai+l)}ai fb(ai- 1, at+ 1) - z~o
735
S+
fb(ai-1, ai+l)
(1) 1 fd(ai-1, ai+l)}ai -- kdai (2)
TD
where at is the concentration of Abe. 1/zB and 1/zD are kinetic constants corresponding respectively to the production and to the elimination of Abv The "birth and death" terms fb(ai- 1,a~+1) and f~(ai- 1,ai+1) are typical threshold curves (Richter, 1975) which represent respectively the fraction of Abl whose production is stimulated by Abi_ 1 and the fraction of Abi whose suppression results from Abe+1. According to Jerne (1974), in the absence of antigenic stimulation, the immune network is functional: it is assumed to be in a suppressed state. On the contrary, Richter has supposed that, before the antigenic challenge, the concentration of the various antibodies is lower than the threshold for suppression, which means that the network is ineffective. This seems unlikely with respect to the ontogeny of the immune network. In the framework of this hypothesis, three kinds of behavior have been obtained with Richter's equations by simulating the injection of the antigen (Richter, 1975). During the course of the normal immune response, the antigen elicits the production of Abl, which stimulates the Ab2 synthesis. The latter in turn induces the production of Ab3, but the following elements stay unstimulated. So, the dynamic behavior of the network tends towards a state where the concentrations of Abt and Ab3--which has been defined more recently as a helper level, i.e. a suppressor of the suppressor, by Richter (1977)--are high whereas the concentration of Ab 2 is very low. Now, for the same value of the parameters, a lower antigen concentration will only induce the synthesis of Abl and Abe, the latter suppressing the formation of the former. This corresponds to the establishment of lowzone tolerance. Another dynamic behavior is still obtained if the initial concentration of antigen is very high. In this case, the synthesis of Abe and Ab4 are stimulated, which suppresses the production of Abl and Ab3: this could define some kind of high-zone tolerance. We shall discuss the stability of those solutions in the next section. Let us mention that Richter (1977) has also proposed a slightly modified version of his original model, whose qualitative results remain similar to the one presented in this paragraph. In the sequel we also study the implications of models for which we have introduced the following modifications with respect to Richter's model: (i) As we have mentioned previously, the kinetic equation, considered by Richter, accounts only for the regulatory interactions inside the network, without describing other processes related to the production or the degradation of Abv For example, there is a constant production of lymphocytes of species i, resulting from the differentiation of virgin cells. On the other hand, there is a constant degradation of Ab~ (lymphocytes and antibodies), related to the high turnover of the elements of the immune system (Jerne,
where S is a constant source term and ka is the kinetic constant characterizing the turnover. It should be noted that Hoffmann has considered similar terms in his model. (ii) Richter has assumed an exponential decay of the antigen concentration, which is described by the following equation:
dAg dt
--
kaZg
(3)
where A9 is the antigen concentration and ka is a kinetic constant. The value of the latter is arbitrarily chosen according to the expected type of behavior. In fact, the degradation of the antigen results from its interaction with the antibodies Abl. For this reason, it could be more natural to describe the concentration variation of the antigen in the following self-consistent way, which should be a good approximation if the concentration of Abl is not too high:
dAg dt
-
kaAgAbl.
(4)
(iii) Richter has only considered a 'linear network', which starts from the antigen. This is obviously a very simple picture of the network concept. In the sequel, we analyze the behavior of other configurations. The simplest picture of a network is probably given by a cycle, as the one presented in Fig. 1, or by two interacting cycles having cross-reactive idiotypes, as is shown on Fig. 2. Obviously, these remain hypothetical representations of the immune network, whose real structure has not yet been clarified. Nevertheless, we may point out that the basic regulatory pattern--the interaction between the idiotype and the anti-idiotype, i.e. Abl and Ab2, as it has been proposed by Hoffmann (1975) and Kthler (1975)--constitutes by itself a simple cycle. Moreover, recent experi-
Abs
Ab2
Ab5
Ab 3
Fig. 1. Even cycle of interacting antibodies and lymphocytes of species Abe, Ab2.., The full arrows correspond to activation processes and the dashed ones to suppressive processes.
736
J. HIERNAUX
#~'" Ab1"~" "~r" ~ Abs Ab2 ',
1:
Ab~
.Ablo ~'i "--'%
SOME REMARKS ON THE STABILITY OF THE IDIOTYPIC NETWORK J. HIERNAUX Service de Chimie physique II, Campus Plaine, U.L.B., CP231, Boulevard du Triomphe, B-1050 Brussels, Belgium
(First received 10 March 1977; in revised form 30 May 1977) Abstract--The regulation of the immune response is a complex phenomenon whose details are not yet well known. Recently, Jeme has described the immune system as a complex idiotypic network. He has proposed that the interactions between the elements involved in the latter are responsible for the main regulatory patterns of the immune system. Some experimental results are in favour of this hypothesis. A few theoretical models have been developed to describe more precisely the nature of the interactions between the lymphocytes and the antibodies inside the network as well as their implications. In such a model, the time evolution of the concentration of those elements is described by kinetic equations whose integration over time can reproduce roughly the dynamic behaviour of the immune system. Those theoretical models should at least account for phenomena such as memory or tolerance, which implies the maintenance of a stationary state. This latter statement can be directly related to the mathematical concept of stability. We illustrate this point on various models. More precisely, we show how each mode of the immune response may be interpreted as a transition from one steady state to another one, following antigenic or any other stimulation.
I. INTRODUCTION & Bangasser, 1975; Urbain et al., 1975). Autogeneous The regulation of the various modes of the immune anti-idiotypic antibodies have also been obtained response involves complex mechanisms based on the experimentally (Rodkey, 1974; Kluskens & K/Shler, interactions of the lymphocytes and of the substances 1974; Cosenza, 1976; Urbain, 1976; Binz & Wigzell, produced by them. The most widely accepted theor- 1976). It seems likely that those anti-idiotypic antietical framework related to the immune response is bodies play a regulatory role at the level of the imthe clonal selection theory of Burnet (1959), which mune response, either suppressing it (Eichmann, 1975; describes how an (injected) antigen can be specifically Cosenza, 1976) and inducing tolerance (Binz & Wigrecognized by the receptors of pre-committed lym- zell, 1976) or stimulating it (Rajewsky & Eichmann, phocytes, thus eliciting the proliferation of clones pro1975; Trenkner & Riblet, 1975; Black et al., 1976). ducing specific antibodies. The latter react with the This idiotypic network should be functional, i.e. its antigen which is degraded. However, this reaction is regulation should be responsible for the various strongly dose dependent: below a certain threshold modes of the immune response. So, according to of antigen concentration and above another one, the Jerne (1974), the basic regulatory pattern of this animal does not eliminate the antigen if the latter network is suppression because it can guarantee the is injected continuously, the animal is said to be toler- maintenance of a steady state: for example, it can ant. Experimentalists and theoreticians have proposed keep the concentration of the antibodies at a very various models to account for different aspects of the low level in the absence of antigen. The latter is supimmune response (Bell, 1970, 1971, 1973; Bruni et al., posed to perturb the suppressed steady state of the 1975; Bretscher, 1974; Pimbley, 1974; Coutinho & immune network, thus inducing the specific synthesis M/~ller, 1975; Perelson et al., 1976). of Abl. Each immune response should correspond to Recently, Jerne (1974, 1975) has described the im- a well-defined regulatory pattern. Actually, the exact mune system as a "web of V domains" based on idio- nature of the interactions between the humoral and typic interactions: the antibodies reacting with an the cellular elements of the immune system is not exogeneous antigen (which we shall designate by Abl) well known. So, we can be interested in the following are characterized by specific idiotypes (Oudin & questions: what is the role of the T-B cooperation? Michel, 1963; Oudin, 1966, 1974)and they stimulate What is the difference between a T-dependent and the production of anti-idiotypic antibodies (which we a T-independent response (Quintans & Cosenza, shall designate by Ab2), the idiotypes of Abl being 1976)? How do the T-helper or T-suppressor cells act? antigenic with respect to Ab 2. In turn, Ab2 could elicit How are the cytotoxic cells involved inside the netthe production of an anti-idiotypic set Aba, and so work? Answers to those and other questions can on. Several experimental results are in favour of probably be provided by careful experiments. But, it Jerne's hypothesis (Jerne, 1974, 1976; Urbain, 1976). seems to us that the complexity of the regulation is The production of anti-idiotypic antibodies has been such that a theoretical approach, based on experimeninduced in serum coming from animals of other species tal results, becomes worthwhile since it allows us to (Cosenza & K6hler, 1972; Hart et al., 1972; Rowley focus on some particular aspects in order to underet al., 1973; Eichmann, 1974; K6hler, 1975; Nisonoff stand them better. 733
734
J. HIERNAUX
Up to now, two models have been proposed to describe the regulation of the immune network (Richter, 1975, 1977; Hoffmann, 1975, 1977). The model of Richter is described in more detail in the following section. Hoffmann's model considers essentially two species: the 'positive' one (which is in fact Abl) interacts with the antigen and the 'negative' one (or Ab2) recognizes the first one. The dichotomy between T and B-cells, as well as the role of the cytotoxic cells, are taken into account. The basic regulations are positive and negative interactions between the V domains of both species. Hoffmann shows bow a tolerant state as well as a normal immune response can be generated. ° Both models are based on non-linear kinetic equations describing the time evolution of the concentrations of the various populations Abe, Ab2... as well as of other elements involved in the regulation. The numerical integration over time of those equations will generally give an approximate description of some aspects of the dynamical behaviour of the immune system. In most of the cases, the system evolves to a steady state. In principle, the latter is expected to be compatible with the external as well as with the internal constraints of the organism. In simple situations, there will exist only one steady state for well-defined constraints. Moreover, slight modifications of those constraints may lead the system towards new steady states qualitatively similar to the initial state. However, in a number of situations involving non-linear kinetics, it is known that the system admits several states for the same values of the parameters. In such cases, a slight variation of the constraints is sufficient to drive the system toward qualitatively different configurations. Several biological illustrations of this behavior have been analyzed recently (Nicolis & Prigogine, 1977). A general feature accompanying these transitions is the loss of stability of the initial 'reference' state. Stability appears therefore as a necessary physical requirement to be imposed on any physiologically acceptable state. This joins the notion of homeostasis which is frequently used to characterize biological regulation. These considerations suggest the following picture for the immune system regulation. In the absence of antigenic challenge, the interactions (corresponding to the internal constraints of the immune system) between the elements of the immune network are supposed to limit the lymphocyte proliferation, and consequently, the production of antibodies. In this way, they establish an homeostatic state, which is expected to be stable with respect to small perturbations. Now, the antigenic challenge corresponds to a macroscopic perturbation which can drive the system from the original virgin state to a new one. The existence of the immune memory implies then the stability of the latter. It appears that the antigenic challenge induces a temporary modification of the external constraints, whose result is the evolution of the system towards a new homeostatic (or stationary) state. An irrevers* The same problem has been investigated for the prebiotic evolution (Eigen, 1971; Prigogine et al., 1972), for the formation of genetic nets (Kauffman, 1969) and for the ecosystems (May, t973; Allen, t975).
ible tolerant state can also be defined as mathematically stable. Let us insist on the fact that the stability is defined as long as the various parameters characterizing the regulatory pattern are maintained constant. The properties which we have described are related to the immune system of an adult animal. Another problem is the ontogeny of the adult immune system (Bruyns et al., 1976; Urbain, 1976; Augustin & Cosenza, 1976). It has been theoretically investigated in the framework of the network concept (Adam & Weiler, 1976; Adam, 1977). The model of Adam and Weiler deals more precisely with the ontogeny of antibody diversity on the basis of the interactions between the V domains of Abt and Ab2 clones. In this case, one has to understand the formation of a stable complex system, i.e. of a stable network, during the development of an organism. As we have already mentioned, the stability is a requirement of the homeostasis of the biological systems.* We assume thus that the adult immune network is the result of a maturation process which implies a dynamic cooperation over time between the external and the internal constraints. Moreover, we suppose that the latter, which are related to the regulatory processes, assure the stability by controlling the proliferation of the various clones. In this paper, we are only concerned with the stability of the mature network. We investigate the model of Richter and some slightly modified versions of the latter from this point of view. We think that a theoretical model should account on the one hand for the global dynamic behaviour of the immune system during the immune response, and on the other hand for such phenomena as tolerance and memory. As we have already mentionedl the latter are closely related to stability. Now, the naiure of the dynamic behaviour of a network also depends on its configuration: for example, the 'connectance' of the various elements must be specified. In this paper, we have limited ourselves to small cycles as well as to weakly interacting cycles.
I1. DESCRIPTION OF THE MODEL
The basic scheme of regulation of Richter's model (1975) is directly inspired from the network structure proposed by Jerne (1974). In fact, Richter has considered a linear set of V interactiofis whose starting point is the antigen. The latter elicits specifically the proliferation of lymphocytes producing the populations of antibodies Abl, which eliminate the antigen. Beyond this event, the production of Ab2, Ab3...is stimulated sequentially. As explained previously, there is a negative feedback since, broadly speaking, Ab i suppresses Abi_l which stimulates its production. Moreover, the model supposes the existence of concentration thresholds for activation and for suppression: the latter is assumed to be lower than the former in order to limit the proliferation of the lymphocytes in absence of antigenic stimulation, in agreement with the actual biological situation. On the basis of this regulatory pattern, Richter has described the normal immune response as well as the low- and the highzone tolerance phenomena. The time evolution of the concentration of the antibodies of species i, Ahi is
Immune Network Stability described by a phenomenological equation of the following form (the concentrations of the antibodies and of the lymphocytes of species i are considered as being proportional):
1974). Those aspects can easily be incorporated into the equation (1), in the following way:
dt dai_{1 dt
1 fa(ai-l, ai+l)}ai fb(ai- 1, at+ 1) - z~o
735
S+
fb(ai-1, ai+l)
(1) 1 fd(ai-1, ai+l)}ai -- kdai (2)
TD
where at is the concentration of Abe. 1/zB and 1/zD are kinetic constants corresponding respectively to the production and to the elimination of Abv The "birth and death" terms fb(ai- 1,a~+1) and f~(ai- 1,ai+1) are typical threshold curves (Richter, 1975) which represent respectively the fraction of Abl whose production is stimulated by Abi_ 1 and the fraction of Abi whose suppression results from Abe+1. According to Jerne (1974), in the absence of antigenic stimulation, the immune network is functional: it is assumed to be in a suppressed state. On the contrary, Richter has supposed that, before the antigenic challenge, the concentration of the various antibodies is lower than the threshold for suppression, which means that the network is ineffective. This seems unlikely with respect to the ontogeny of the immune network. In the framework of this hypothesis, three kinds of behavior have been obtained with Richter's equations by simulating the injection of the antigen (Richter, 1975). During the course of the normal immune response, the antigen elicits the production of Abl, which stimulates the Ab2 synthesis. The latter in turn induces the production of Ab3, but the following elements stay unstimulated. So, the dynamic behavior of the network tends towards a state where the concentrations of Abt and Ab3--which has been defined more recently as a helper level, i.e. a suppressor of the suppressor, by Richter (1977)--are high whereas the concentration of Ab 2 is very low. Now, for the same value of the parameters, a lower antigen concentration will only induce the synthesis of Abl and Abe, the latter suppressing the formation of the former. This corresponds to the establishment of lowzone tolerance. Another dynamic behavior is still obtained if the initial concentration of antigen is very high. In this case, the synthesis of Abe and Ab4 are stimulated, which suppresses the production of Abl and Ab3: this could define some kind of high-zone tolerance. We shall discuss the stability of those solutions in the next section. Let us mention that Richter (1977) has also proposed a slightly modified version of his original model, whose qualitative results remain similar to the one presented in this paragraph. In the sequel we also study the implications of models for which we have introduced the following modifications with respect to Richter's model: (i) As we have mentioned previously, the kinetic equation, considered by Richter, accounts only for the regulatory interactions inside the network, without describing other processes related to the production or the degradation of Abv For example, there is a constant production of lymphocytes of species i, resulting from the differentiation of virgin cells. On the other hand, there is a constant degradation of Ab~ (lymphocytes and antibodies), related to the high turnover of the elements of the immune system (Jerne,
where S is a constant source term and ka is the kinetic constant characterizing the turnover. It should be noted that Hoffmann has considered similar terms in his model. (ii) Richter has assumed an exponential decay of the antigen concentration, which is described by the following equation:
dAg dt
--
kaZg
(3)
where A9 is the antigen concentration and ka is a kinetic constant. The value of the latter is arbitrarily chosen according to the expected type of behavior. In fact, the degradation of the antigen results from its interaction with the antibodies Abl. For this reason, it could be more natural to describe the concentration variation of the antigen in the following self-consistent way, which should be a good approximation if the concentration of Abl is not too high:
dAg dt
-
kaAgAbl.
(4)
(iii) Richter has only considered a 'linear network', which starts from the antigen. This is obviously a very simple picture of the network concept. In the sequel, we analyze the behavior of other configurations. The simplest picture of a network is probably given by a cycle, as the one presented in Fig. 1, or by two interacting cycles having cross-reactive idiotypes, as is shown on Fig. 2. Obviously, these remain hypothetical representations of the immune network, whose real structure has not yet been clarified. Nevertheless, we may point out that the basic regulatory pattern--the interaction between the idiotype and the anti-idiotype, i.e. Abl and Ab2, as it has been proposed by Hoffmann (1975) and Kthler (1975)--constitutes by itself a simple cycle. Moreover, recent experi-
Abs
Ab2
Ab5
Ab 3
Fig. 1. Even cycle of interacting antibodies and lymphocytes of species Abe, Ab2.., The full arrows correspond to activation processes and the dashed ones to suppressive processes.
736
J. HIERNAUX
#~'" Ab1"~" "~r" ~ Abs Ab2 ',
1:
Ab~
.Ablo ~'i "--'%
