Random behaviors in the process of immunological memory

Simulation Modelling Practice and Theory 15 (2007) 831–846

www.elsevier.com/locate/simpat

Random behaviors in the process of immunological memory

Alexandre de Castro *

Empresa Brasileira de Pesquisa Agropecuaria – EMBRAPA, Centro de Pesquisa de Pecuaria dos Campos Sulbrasileiros –

CPPSUL, Embrapa Pecuaria Sul – BR 153 Km 595, 96401-970 Bage, RS, Brazil

Universidade Estadual do Rio Grande do Sul, Centro Regional VI – UERGS 90010-030, Porto Alegre, RS, Brazil

Universidade Federal do Rio Grande do Sul, Instituto de Pesquisas Hidraulicas – IPH/UFRGS 91501-970, Porto Alegre, Brazil

Received 16 August 2004; received in revised form 22 November 2005; accepted 17 April 2007Available online 24 April 2007

Abstract

We have used a system of coupled maps to study random variations in the immune memory process and the possiblenetwork immune memory that can remain for long time in the absence of antigenic stimulation through the idiotypic–anti-idiotypic interactions. Our approach describes the behavior of the immune network proposed by Jerne and considers thatthe cell–cell interactions routine results in maintenance of memory in a dynamic equilibrium. The critical values of the con-centrations of antigens for the validity of the model and a phases diagram with three different phases are also obtained.� 2007 Elsevier B.V. All rights reserved.

Keywords: Immune memory; Antibodies evolution; Memory-regenerating

1. Introduction

In the clonal selection theory proposed by Burnet [1–3], when a living body is exposed to an antigen (inva-der), some subpopulations of its bone marrow (B lymphocytes) respond through antibodies production. Eachcell secretes only one kind of antibody, which is relatively specific for the antigen. By binding to these anti-bodies (receptors) and with a second signal, such as the T-helper cells [4–7], the antigen stimulates the prolif-eration of B-cells and antibodies secretion. While B-cells secrete antibodies, the T-cells play a central role inthe regulation of the B-cells response and are fundamental in cell mediated immune responses. Lymphocytes,in addition of proliferating and differentiating into plasma cells, also differentiate into long-lived memory pop-ulations. Memory populations circulate through blood, lymph and tissues and, when exposed to a second anti-genic stimulus differentiate into large lymphocytes capable of producing high affinity antibodies, pre-selectedby the specific antigen.

Basically, there are two theories to explain the immune memory. The first considers that after the expansionof the B-cells, there occurs the formation of plasma cell and memory cells. The second hypothesis, due Jerne[8,9], considers that the immune system presents its memory and response capacity to the second invasion of

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doi:10.1016/j.simpat.2007.04.003

* Corresponding author. Tel./fax: +55 53 32428499.E-mail address: [email protected]

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832 A. de Castro / Simulation Modelling Practice and Theory 15 (2007) 831–846

antigens as an self-organization of the system. This allows the growth of cells populations that survive for along time not a specific type cell that has a longer life than the other cells of the organism.

Different models based on clone dynamics have been proposed to describe cell proliferation and death,interaction between different cells, antibody secretion and interaction with antigens. The Celada–Seiden model[10–13], for example, is one of the most detailed lattice gas automata for the immune system response. Its com-plexity derives from the fact that in addition to the different cellular populations considered, also a molecularrepresentation of the cell and molecular binding site is given in term of specific recognition between bit strings.The match between antigen and lymphocyte receptors is given considering the number of complementary bitsin the bit-strings. For example, if the lymphocyte B is equipped with the binary string 00010101 (l = 8) and theantigen is represented by the string 11101010, then the probability to trigger a response is very high. In thismodel, the bit-string match is not required to be perfect, i.e., some mismatches are allowed like, for example,one bit from the eight bit string in the above mentioned example.

In this paper, we used an approach for immune system considering structural mechanisms of regulationthat were not contemplated in the simplified model of bit-string proposed by Lagreca et al. [14]. In ourapproach, not only we treat the antibodies bound to the surface of the B-cells (surface receptors), but alsothe populations of soluble antibodies in the blood (antibodies secreted by mature B-cells), thus making theproposed model more similar to a real immune system.

In our model, we defined clones as an ensemble of B-cells, and the populations of antibodies are treatedseparately, to more appropriately study the evolution of the components of the immune system. Consequently,the results for the time evolution of the clones reported in [14] present only the time evolution of B-cells andnot the evolution of the secreted antibodies populations. This is due to the fact that the set of coupled maps oftheir model considers only the antibodies bound to the surface of B-cells and not the dispersed antibodies inthe serum. Moreover, the regulation of the immune response in the previous model is just done by apoptosis(programmed cell death) and by the Verhulst-like factor. It is not considered the fundamental role of the anti-bodies in the mediation of the global control of the differentiation of the B-cells. The Verhulst-like factor pro-duces a local control of the populations of clones (B-cells) considering the several mechanisms of regulation.However, globally the immunological memory of the system [15] is strongly affected by the populations of sol-uble antibodies in the blood, what was contemplated in this work.

Our approach allows to model, in a more complete form the generation, maintenance and mechanisms ofregulation of the immune memory through a network memory combining the characteristics of Burnet’s clo-nal selection theory and Jerne’s network hypothesis, considering the interaction idiotypic-anti-idiotypic pro-posed for Jerne. We also studied the behavior of the model for concentrations of antigens from 0.0001 to 1.5and we obtained a diagram for three different phases of the immune system.

In the model presented here, we also considered a modified version of exact enumeration techniques,together with multispin coding [16–19] to allow the control of time evolution of populations over all high-dimensional shape space.

2. The computational model and biological entities

In the model discussed here, the B-cell molecular receptors (BCR) are represented by bit-strings with diver-sity of 2B, where B is the number of bits in the strings. The entities present in the model are B-cells, antibodiesand antigens. The B-cells are represented by clones, each clone being characterized by its surface receptorwhich is modeled by a binary string of B bits.

The epitopes portion of an antigen that can be bonded by the B-cell receptor (BCR) are also represented bybit-strings [20,21]. The antibodies have a receptor (paratope) that is represented by the same string as the BCRof the parent B-cell which produced them [4].

Each bit-string (shape) is associated to an integer r (0 6 r 6M = 2B � 1), corresponding to each clone,antigen or antibody; the neighbors to a given r are obtained by the Boolean function ri = (2i x or r). Thecomplementary shape of r is obtained as ri ¼ M � r and, through direct interaction, the time evolution ofthe concentrations of several populations are obtained as functions of integer variables r and t.

The equations that describe the behavior of the populations of B-cell clones y(r,t) are iterates, for differentparameters and initial conditions:

A. de Castro / Simulation Modelling Practice and Theory 15 (2007) 831–846 833

yðr; t þ 1Þ ¼ ð1� yðr; tÞÞ mþ ð1� dÞyðr; tÞ þ byðr; tÞyTOTðtÞ

fahðr; tÞ

� �ð1Þ

with the complementary shapes included in the term fahðr; tÞ

fahðr; tÞ ¼ ð1� ahÞðyðr; tÞ þ yFðr; tÞ þ yAðr; tÞÞ þ ah

XB

i¼1

ðyðri; tÞ þ yFðri; tÞ þ yAðri; tÞÞ;

where yA(r,t) and yF(r,t) are the antibody and antigen population, respectively. Here, b is the proliferationrate of B-cells, r and ri are the complementary shape to r and to the B nearest neighbors on the hypercube(with the ith bit flipped). The first term in the curled bracket (m) represents the bone marrow production, andis a stochastic variable. This term is small, but non-zero. The second term describes the population that sur-vives to the natural death (d), and the third term represents the clonal proliferation due to the interaction withcomplementary forms (other clones, antigens or antibodies). The parameter ah is the relative connectivityamong a given bit-string and the neighborhood of its mirror image. When ah = 0, only perfect match is al-lowed. When ah = 0.5, a string may recognize equally well its mirror image and first neighbors.

The factor yTOT (t) is given by

yTOTðtÞ ¼X

r

½yðr; tÞ þ yFðr; tÞ þ yAðr; tÞ�: ð2Þ

The time evolution of the antigens is given by:

yFðr; t þ 1Þ ¼ yFðr; tÞ � kyFðr; tÞyTOTðtÞ

� fð1� ahÞ½yðr; tÞ þ yAðr; tÞ� þ ah

XB

i¼1

½yðri; tÞ þ yAðri; tÞ�g; ð3Þ

where k is the speed with which populations of antigens or antibodies decay to zero. Here, it measures theremoval rate of the antigen due to interaction with the population of clones and antibodies.

The population of antibodies is described by a group of 2B variables, defined on the B-dimensional hyper-cube, interacting with the populations of antigens:

yAðr; t þ 1Þ ¼ yAðr; tÞ þ bA

yðr; tÞyTOTðtÞ

� ð1� ahÞyFðr; tÞ þ ah

XB

i¼1

yFðri; tÞ" #

� kyAðr; tÞyTOTðtÞ

fahðr; tÞ; ð4Þ

where the complementay shapes contribution fahðr; tÞ is again included in the last term. Here, bA is the pro-

liferation rate of antibodies; k is the antibodies removal rate that measures its interactions with otherpopulations.

We solved the Eqs. (1)–(4) considering different set of initial conditions and concentrations of antigens.

3. The dynamics

To show the capabilities of the model we present the results of some simulations, where we reproduceimmunization experiments in which the several antigens with fixed concentration are injected in each 1000time steps, to stimulate the immune response. When a new antigen is introduced, its interactions (connections)with all other components in the system are obtained according to the random generator of shape chosen.

3.1. Parameters

The length of the bit-string B, has been set equal to 12, corresponding to the potential repertoire of 4096distinct receptors and molecules. We set 110 injections of different antigens for a range of 0–110,000 time steps.First we considered the value 0.08 for the concentration of antigens populations, in the second simulation itwas set as 0.10.

We consider the apoptosis rate or natural cell death d = 0.99. The proliferation rate for clones equals to 2.0and the proliferation rate for antibodies chosen as 100. The connectivity parameter ah was 0.01. Both the valueof the antigens removal rate and antibodies removal rate have been set equal to 0.1, so that in an interval of


1000 time steps the previous populations of antigens and antibodies vanish before the next antigen ispresented.

3.2. Numerical results

The generalized shape space is a hypercube in a 12-dimensional space with non-negative integer coordi-nates. Then, there are 4096 discrete points in the shape space, what is considered as the potential immunerepertoire.

3.2.1. Antibodies and antigens evolution

First, let us consider a typical simulation with 110 types of different antigens or several infections with sev-eral different pathogens. The immune system quickly begin to reply after the start of the simulation.

We used the same seed for the random number (integer) generator for each inoculation doing with that dif-ferent antigens were inoculated in the same order in all the simulations. Several simulations were accomplishedwhere we altered the concentrations from 0.0001 to 1.5. The most significant results were obtained for twoconcentrations, as follows.

Figs. 1a and 2a indicate that the behavior of the populations of antibodies obtained by our approach, forthe concentrations of antigens 0.08 and 0.10 differs from the results obtained by Lagreca et al. [14]. When thepopulations of antibodies secreted in an immune response are separately treated, the result of the time evolu-tion is a sequence of peaks, in agreement with the expected since for a healthy individual the populations ofantibodies should decrease when the infection finishes.

Fig. 1a shows that after many injections of antigens with concentration 0.08, the amount of antibodies fluc-tuates slightly with time. Fig. 2a shows that for concentration 0.10, except for the injection 82 (in the 82,000

Fig. 1. Simulation of the elimination of antigens (pathogens) by the immune system model, for a concentration of antigens equal to 0.08.(a) Antibodies populations yA versus time; (b) foreign antigen populations yF versus time. A new antigen population is presented each timeinterval of 1000 time steps and we set 110 injections of different antigen populations.

Fig. 2. Simulation of the elimination of antigens (pathogens) by the immune system model, for a concentration of antigens equal to 0.010.(a) Antibodies populations yA versus time; (b) foreign antigens populations yF versus time. A new antigens population is presented eachtime interval of 1000 time steps and we set 110 injections of different antigens populations.


time step), the concentration of antibodies is approximately stable. This result was expected because in adynamic system with different Attractors, the response (antibodies) is not necessarily the same in each pointof the shape space.

Figs. 1b and 2b show a stable behavior for the populations of antigens. Figs. 3a–d show in detail the evo-lution of the antigens and antibodies populations for a range of 0–6000 time steps. The populations of anti-gens decrease quickly when antibodies are secreted, and the response of antibodies is proportional to theinoculated antigens dosage as would be expected for the behavior of a normal immune system.

Our approach shows that the time evolution of clones previously obtained [14] cannot be understood asbeing the evolution of an ensemble of B-cells and antibodies, but only as corresponding to the evolution ofB-cells. The soluble antibodies in the blood should be treated separately because their behavior differs fromthat of the evolution of B-cells [1,2]. In a normal immune system, the populations of antibodies are not main-tained in a fixed level after the end of the infection. When the populations of antigens decrease, the popula-tions of antibodies also decrease.

3.2.2. Regulation of cell proliferation and memory capacityFigs. 4a and 5a indicate that due to the formation of memory populations of clones, the average of the total

concentration of clones (B-cells) increase after some injections of different antigens. The increase is continuousuntil a certain point, after which the evolution saturates due to capacity of memorization (homeostasis)because the body has a maximum limit of cells it can support.

Fig. 4b and 5b show the maximum number of excited populations of clones after the 110 injections, con-sidering the two different concentrations of antigens. In this evolution the excited populations in each 1000time steps are the clonal population that recognized a specific antigen (Burnet cells), and the clonal populationwith complementary shapes (Jerne cells). In Fig. 6 it is shows the behavior of the first population of clones that

Fig. 3. Antigens yF and antibodies yA populations evolutions for a range of time steps 0–6000. Model performance for a concentration ofantigens equal to 0.08 in (a) and (c) and equal to 0.10 in (b) and (d).

Fig. 4. Model performance for a concentration of antigens equal to 0.08. (a) Average clonal population hy(r)i and (b) number of excitedpopulations Ne.


Fig. 5. Model performance for a concentration of antigens equal to 0.10. (a) Average clonal population hy(r)i and (b) number of excitedpopulations Ne.


recognizes the first inoculated antigen. The behavior is the same for both concentrations 0.08 or 0.10. Fig. 6ashows the behavior of the population of clones with the addition of populations of antibodies, while Fig. 6brepresent the system without antibodies in the set of coupled maps. In both situations, the behavior is almostthe same, indicating that the addition of antibodies in the system does not give rise to a considerable localdisturbance. However, the addition of antibodies influences in the global memory capacity, as is shown nest.

Fig. 6. Behavior of the first population of clones that recognizes the first inoculated antigen with the addition of populations of antibodieswith (a) and without populations of antibodies (b). The behavior is not altered for the concentrations 0.08 or 0.10.


Figs. 7 and 8 show the memory capacity until 16,000 time steps for the concentration of antigens equal to0.08 and concentration equal to 0.10. In Fig. 7a and Fig. 8a represent The memory capacity until 16,000 timesteps, for concentrations of antigens equal to 0.08 and 0.10. In Figs. 7 and 8, where the antibodies populationsare token into account, The network capacity is smalles than in the absence of populations of antibodies, asshown in Figs. 7b and 8b. For an antibodies dose of 0.08, 16 populations of clones are alive in the 8000 timesteps, while for concentration of 0.10, 14 populations are alive in the same period. This overall behavior indi-cates the role of the antibodies in the mechanism of regulation of the proliferation of B-cells and in the main-tenance of the immune memory.

The populations of antibodies soluble in the blood participate not only in the immune response as they helpin the regulation of the differentiation of B-cells. The presence of soluble antibodies alters the global propertiesof the network – this behavior can only be observed when the populations are treated separately. In theapproach by Lagreca et al. [14] the antibodies secreted by mature B-cell are not contemplated only the anti-bodies bound to the surface of B-cells were considered. The extension of the previous approach allows to sim-ulate the behavior of the populations of antibodies and to visualize its influence in the mechanisms of responseregulation and in the memory capacity.

Figs. 9 and 10 show the populations of clones that specifically recognize the populations of antigens for aconcentration of antigens equal to 0.08. Considering a typical simulation until the tenth injection, the fourthand octave populations of clones remain live until 11,000 time steps. All the other 8 remaining populationshave shorter life.

In Figs. 11 and 12, concentration of antigens equal to 0.10, is shown that populations surviving for a longerperiod are the first and seventh, the other 8 remaining populations have shorter life. This behavior is random,therefore, it is impossible to foresee which population of clones stays excited by a long time.

Figs. 13 and 14 show the evolution of the specific populations with more detail for each time step. In thisfigures, it can be observed that while a population is forgotten, another one is learned. Fig. 14 also shown thatthe population that recognized the first type of antigen proliferates to a certain point and with the secondinjection, it begins to decrease. This behavior happens because the immune system has a maximum capacity

Fig. 7. Memory capacity as a function of time (until 16,000 time steps) for the antigen concentration equal to 0.08. The network capacityin (a), considering the antibodies, is smaller than in (b), the absence of populations of antibodies.

Fig. 8. Memory capacity as a function of time (until 16,000 time steps) for the antigen concentration equal to 0.10. The network capacityin (a), considering the antibodies, is smaller than in (b), the absence of populations of antibodies.

Fig. 9. Performance of the model for each population of clones that recognizes a specific population of antigen (concentration equal to0.08). Figs. (a)–(e) show that all the populations (1–5) are excited after the inoculation. However, just the fourth population of clonesremain non-zero for a long time after the suppression of the antigens.


Fig. 10. Performance of the model for each population of clones that recognizes a specific population of antigens (concentration equal to0.08). Figs. (a)–(e) show that all the populations (6–10) are excited after the inoculation. However, just the eighth population of clonesremain non-zero for a long time after the suppression of the antigen.

Fig. 11. Performance of the model for each population of clones that recognizes a specific population of antigens (concentration equal to0.10). Figs. (a)–(e) show that all the populations (1–5) are excited after the inoculation. However, just the first population of clones remainnon-zero for a long time after the suppression of the antigen.


Fig. 12. Performance of the model for each population of clones that recognizes a specific population of antigens (concentration equal to0.10). Figs. (a)–(e) show that all the populations (6–10) are excited after the inoculation. However, just the seventh population of clonesremain non-zero for a long time after the suppression of the antigen.


of cells it can support. When new types of antigens are learned, others need to be forgotten. However, in 7000time step (the seventh injection) the first population begins to increase. The evolution shows that the first pop-ulation is a possible long time memory.

Fig. 13. The evolution of the populations of clones that survive until 11,000 time steps, for the concentration of antigens equal to 0.08. Thefourth and the eighth population remain excited, in agreement with Figs. 6 and 7. The other clone populations die during the simulations.Obviously, the last and the one before last population are still alive.

Fig. 14. The evolution of the populations of clones that survive until 11,000 time steps, for the concentration of antigens equal to 0.10. Thefirst and the seventh population remain excited, in agreement with Figs. 8 and 9. The other clone populations die during the simulations.Obviously, the last and the one before last population are still alive.

Fig. 15. Lifetime for the populations of clones that recognize each specific antigen. The concentration of antigens is equal to 0.10 in (a)and 0.08 in (b).



The increase of concentration of the first population after the 7000 time step shows that our modeldescribes in a very adequate way the behavior of the immune network proposed by Jerne [8,9], where theimmune memory is formed by populations of cells that last for a long time through interactions cell–cell –and not formed by a specific type of cell that has longer life than the other cells of the body.

Our approach simulates the hypothesis of relay of Nayak et al. [15] combining characteristics of the Bur-net’s clonal selection theory and Jerne’s network hypothesis. In the model described here the presence of long-living memory cells or of persisting antigen is not required, however, the individual behavior of each popula-tion is completely random. Figs. 15a and b show the global behavior of the lifetime for the populations ofclones that recognize the antigens for concentrations between 0.08 and 0.10. The results are altered whenthe concentration of antigens Varies, showing that it is impossible to foresee the population that will survivefor a long period. This results are reasonable because it is impossible to predict the duration of a vaccine in ahealthy individual.

Fig. 16. Lifetime average over the 10 samples, for the 5 first populations of clones and concentration of antigens equal to 0.0001.

Fig. 17. Average, over the 10 samples, of B-cells production, until 2000 time steps.


4. Limits of the model and phases diagram

To obtain a phases diagram we simulated several injections of antigens with several many different doses.Fig. 16 shows the average lifetime on the 10 samples, for 5 populations that recognize antigens. When we set aconcentration of antigens equal to 0.0001, the populations of clones are not excited indicating that there is aminimum concentration so that immune response is present. When the concentrations of antigens are verylow, B-cells do not respond to invasion and there is no antibodies production or this production is verylow [1]. The small concentrations do not cause damages to the individual and the antigens stay in the bodyuntil its natural death. Fig. 17 shows that until 2000 time steps, the average of production of B-cells is verylow for a concentration of 0.0001.

For a concentration equal to 1.5, the average lifetime of the clones is near 1800, as is shown in Fig. 18.However, the fifth population corresponding to time steps equal to 5000 in the range from 0 to 6000, hasshorter lifetime of about 1000 time steps. Fig. 19 shows that for the concentration of antigens equal to 1.5,

Fig. 18. Lifetime average over the 10 samples, for the 5 first populations of clones and concentration of antigens equal to 1.5.

Fig. 19. Average number of total clones, for concentration of antigens equal to 1.5.

Fig. 20. Average behavior of the populations of clones for concentration of antigens equal to 1.5. When a new population is excited theprevious population vanishes.


the average number of total clones remains the same, indicating that there is not immune memory in this case.This situation is shown more clearly in Fig. 20, where we observed that for 5 different types of antigens, when anew population is learned or excited the previous one is forgotten. This behavior indicates that a limit for thecapacity of the network model is reached. This limit value is 1.5, for the set of parameters we used.

Fig. 21. Phases diagram for the concentration of antibodies versus concentration of antigens. Two different phases (plus a third, notdistinguishable in this scale) are obtained by interaction of the system of coupled maps and variation in the concentration of antigens.


Varying the antigen concentration from 0.0 to 4.0 we obtained the phases diagram, relating the concentra-tion of antibodies with the concentration of antigens, as shown in Fig. 21. The antigen concentration 1.5 is acritical value, originating to a transition of phase. The memory phase for the system correspond to concentra-tion values below this threshold.

For values above 1.5, the system enters in the no memory phase and it acquires the capacity of the network.For concentrations of antigens near zero (below 10�4) there are not excited states and the production of anti-bodies and the concentration of clones are approximately zero, as shown in Fig. 17.

5. Discussion and conclusions

In this paper, we have extended the model proposed by Lagreca et al. [14], to include antibodies responseand to study the dynamics of an infectious disease, considering structural mechanisms of regulation that werenot previously contemplated. As a consequence we obtained that the time evolution of the clones is differentfrom the time evolution of the populations of secreted antibodies, in agreement with what is expected for anormal immune system. We considered the fundamental role of the antibodies in the mediation of the globalcontrol of the differentiation of the B-cells, showing that they affect considerably the immunological memory.Our model considers that the cell–cell interactions routine [20,21] results in maintenance of memory in adynamic equilibrium.

The three different phases – no excitation, memory and no Memory – obtained in our simulations charac-terize the behavior of an immune system in vivo. Very small concentrations of antigens do not provoke pro-duction of antibodies because the immune system does not recognize antigens in small doses or very smallsizes. Very high doses of antigens do not induce the appearance of the immune memory, and produce toler-ance of the immune system.

In this model there is no the need of persistent antigen or the existence of long-living memory lymphocytes[22–24]. The presence of Burnet cell and complementary Jerne cell stabilize the memory-regenerating systemthrough idiotypic-anti-idiotypic interactions of their surface immunoglobulins, what implies a self-perpetuat-ing [20,21,15].

References

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Random behaviors in the process of immunological memory

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