An Ag-Dependent Approach Based on Adaptive Mechanisms for Investigating the Regulation of the Memory B Cell Reservoir

Immunology is the branch of biomedical sciences to study of the immune system physiology both in healthy and diseased states. Some aspects of autoimmunity draws our attention to the fact that it is not always associated with pathology. For instance, autoimmune reactions are highly useful in clearing off the excess, unwanted or aged tissues from the body. Also, generation of autoimmunity occurs after the exposure to the non-self antigen that is structurally similar to the self, aided by the stimulatory molecules like the cytokines. Thus, a narrow margin differentiates immunity from auto-immunity as already discussed. Hence, finding answers for how the physiologic immunity turns to pathologic autoimmunity always remains a question of intense interest. However, this margin could be cut down only if the physiology of the immune system is better understood. The individual chapters included in this book will cover all the possible aspects of immunology and pathologies associated with it. The authors have taken strenuous effort in elaborating the concepts that are lucid and will be of reader's interest.


Introduction
Twenty years ago, Farmer, Packard, and Perelson presented an elegant dynamical model [1] to study Idiotype Network theory [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19], in which they showed that every molecular and cellular binding site (cell receptor) can be modeled by binary bit-strings of length ℓ. In such a model, an antibody molecule can always recognize an antigen when there is complementarity between their bit-strings. The coincidence of antigens and lymphocyte receptors (lock-and-key model) is determined by considering the number of complementary bits [8,20]. For instance, if a B lymphocyte is represented by a binary string 00010101 (ℓ = 8) and an antigen is represented by the 11101010 binary string, the immune response is activated (Fig. 1). The match between bit-strings does not need to be perfect, however; some bit positions are allowed in which two strings differ. These differences between strings (mismatches) reflect the degree of affinity between the entities of the immune system in mammals and determine the quality of the response. Fig. 1. Pictorial representation of the binding site (cell antigen) by means of a bit-string frame [6].
In another work, Lagreca et al. (2001) [21] also proposed a dynamic model that was based on the recognition of shapes or patterns using bit-strings, but used the iterative solution of a coupled map system that enabled the treatment of high dimensions. In the model created by Lagreca et al. (2001), the B cell and the antibody populations are treated as a clone pools because their receptors are represented by the same bit-strings. Because a bit-string can be considered as the binary representation of an integer, the model indexes each clone to an entire σ, and the temporal evolution of the populations is described by the N (σ, t) concentration. The model also considers a source term that simulates the role played by the bone marrow, where new bit-strings are presented. The death or depletion of clones occurs in two ways: 1) by means of natural death (apoptosis), described by the parameter d; and 2) by means of a general suppression mechanism, described by a Verhulst-like factor [22]. This factor is widely used in simulations of biological systems, because it limits the maximum population that can survive in a particular environment [22]. The Lagreca et al. model (2001) considers this maximum B cell population ( max N ) to be the same for every clone, and the populations are normalized by the max (, ) (, ) / ytNtN    function.
Thus, considering a discrete temporal evolution, the following coupled map set proposed by Lagreca et al. [21] allows part of an adaptive immunological system to be simulated: ( 1 )( ,) where (1 ( , )) yt   is the Verhulst-like factor; y F (σ,t) describes the antigen population, characterized by a σ bit-string which, in this case, represents distinct antigenic determinants; σ represents the perfect complementary shape of σ; and i  are the nearest neighbors of σ in a B-dimensional hypercube. The term m represents the population of cells produced by bone marrow; the (1-d) term represents the percentage of the lymphocyte population that survives a natural cell death (apoptosis); and the other terms describe the clonal proliferation y(σ,t) that occurs because of interaction with complementary B cells and/or antigens.
The b parameter is a clonal proliferation constant (typically related to the mean number of new cells produced by the pre-existing cells), and () Tot y t is the total population, given by equation 2: The parameter h a is the connectivity factor between a specific bit-string and the specular image of its neighbors. When h a = 0.0, only a perfect coincidence of complementary shapes is valid. When h a = 0.5, a bit-string can recognize equally both its own specular image and the nearest neighbors of its specular image. The temporal evolution of the antigen pool is defined by equation 3. where k is an antigen removal parameter that represents the interactions with the clonal populations.
In fact, it is well-known that the soluble antibody population is one of the essential mechanisms of immunological response regulation [23][24][25]. However, despite the pioneering work of Lagreca et al. (2001) in developing a coupled map for studying the behavior of the mammalian immune system, their model did not consider these populations [1][2][3][4][5][6], which makes the model incomplete with respect to the regulation of the immune response by adaptive mechanisms. This omission opens up the possibility of extending their work by taking the soluble antibody populations into account. We have performed that work and present our immunological modeling and simulation findings in this paper.

Materials and methods
In this section, we briefly describe the Verhulst approach and provide details of an extension to the Lagreca et al. (2001) model, which includes an antibody variable to address the regulation of the structural mechanisms that are mediated by the immunoglobulin population. This variable was not considered in the simplified model proposed by Lagreca et al. [21].

The Verhulst approach
Since the early nineteenth century, studies on population dynamics have been developed to identify possible nonlinear behaviors. One of the first efforts aimed at predicting biological population behavior was made by Pierre François Verhulst (1804-1849), a Belgian mathematician. He proposed a nonlinear model in which the death rate was proportional to the square of the number of individuals in the population. The model can be expressed by differential equations [26][27][28][29][30], as follows: where N is the number of individuals, and A and B are constants related to the growth rate and the population growth limitation, respectively.
The Verhulst model was used again in 1976, by Robert May [27], to study insect population dynamics. In his experiments, he replaced the original differential method by what is now known as the map methodology, in which each value is obtained by its anterior value: At the limit of saturation, AN max -BN max 2 = 0, then N max = 0 or N max = A/B. x n+1 = Ax n -Bx n 2 .A/B. Defining the parameter A (birth rate) = r (control parameter), we obtain:

Solving
In equation (4), known as a logistic map, the values for [0,1] n x  and r are dimensionless and represent population fractions as a function of each of the n iterations, respectively, while r is a constant that represents the population growth rate in each new iteration. The term (1x n ) is known as the Verhulst factor [21,31,32].  In Fig. 2, the attractor is a fixed point up to the first bifurcation. For each bifurcation, there occurs a period of duplication before the system reaches the chaotic phase. However, to illustrate the dynamics of this simple model, it is important to show that, for r between 0 and 1, the population death rate is not dependent on the initial population. With r between 1 and 3, the population is prone to an attractor of a fixed point type. For r greater than 3.54, the population wiggles between values of 8, 16, 32, and so on. At approximately r=3.57, the end of the cascade duplication period occurs and chaos begins. From this value, small variations in the initial population produce very different results over time, which is the fundamental characteristic of chaos. For r greater than 4, the populations are outside the [0,1] interval.
It is possible to demonstrate that the Lagreca et al. (2001) model for clonal populations reduces to equation 4 when there is no further exposure of the system to the antigens. This reduction occurs because, under this one condition, the additive term (m) in equation 1, which represents the bone marrow contribution for the immune repertoire, is very small when compared with the clonal proliferation parameter (b) [34,35]. A detailed An Ag-Dependent Approach Based on Adaptive Mechanisms for Investigating the Regulation of the Memory B Cell Reservoir 427 demonstration of this assertion is presented in subsection 3 of the section on the model parameters.

Simulation model
As in the Lagreca et al. (2001) model [21], our extended Ag-dependent model has molecular receptors of B cells that are represented by bit-strings with 2 B of diversity, where B is the number of bits in the string. The individual components of the immune system represented in the extended model are B cells, antibodies, and antigens located at the vertices of hypercubes of size B. B cells (clones) are characterized by their surface receptors and are modeled by a binary bit-strings. The epitopes [1,8,[17][18][19], which are portions of an antigen that can be connected by the B cell receptor (BCR), are also represented by bit-strings. The antibodies have receptors (paratopes) [1,8,[17][18][19] that are represented by the same bit-string models as the BCR B cell that produced them. Thus, the new dynamic equations that describe the behavior of the adaptive immune system, taking into account the inclusion of antibody populations, are the following: for a clonal population, with complementary shapes included in the term (,) The clonal populations can range from the value generated by bone marrow (m) up to its maximum value (unity) because the Verhulst factor is a limiting factor [21,31,32].
In the model presented in this paper, the () Tot t y term represents the sum of the components that belong to an adaptive subset of the immune system, as described in the introduction to this work. Such elements, when added to antibody populations, are expressed as bit-string concentrations.
Therefore, the sum of every adaptive component considered by our model is given by equation (6).
The temporal evolution of the antigens can be defined by equation (7).
The antibody population is described by a group of 2 B variables, also defined by a Bdimensional hypercube, interacting with the antigen populations of equation (8).

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Recent Advances in Immunology to Target Cancer, Inflammation and Infections where b A is the antibody proliferation parameter; and k is the parameter related to the antibodies and antigens that will be removed.
In our model, equation 8, which considers the adaptive interactions that have been described in the specialized literature, is included. Thus, antibody proliferation is given by the recognition (, )  The role performed by the clonal regulation factor, in addition to helping with the B cell response regulation, is fundamental to the regulation of the memorization ability and clonal homeostasis [37][38][39]. The importance of the effect of the clonal regulation factor over immune system memory evolution is shown in   3 illustrates the situation in which growth capacity increases indefinitely, which is when the clonal regulation factor is suppressed in the modeling phase. This shows that clonal regulation can be fundamental to the immune system reaching clonal homeostasis.
In the proposed Ag-dependent model, each bit-string is associated with an integer that is situated in an interval , 02 1 B M   , and each represents a clonal population, antigen, or antibody located in the B-dimensional hypercube vertex. The neighbors i of a specific  or  are expressed by the Boolean functions respectively. The complementary way of obtaining  is obtained by An example of the way in which the B cell, antibody, or antigen populations are localized in 3-dimensional space is shown in Fig. 4.  Also, for better visualization, we used a 3-dimensional spatial configuration. Similar constructions for this work were made for B-dimensional spaces. Therefore, equations (5) to (8) constitute a set of maps that describes the main interactions of the immune system between the entities that interact through the lock-and-key type of connection, in other words, adaptive immune system entities that self-recognize. Such an equation set is iteratively resolved, considering various initial conditions.

Simulation dynamics
In this section, we present the dynamics of the simulations that were used to reproduce the proposed experiments in silico, and we evaluate the behavior of the proposed model. The antigen injection simulations were performed every 1,000 temporal steps (in arbitrary units -a.u.), representing the administration of a new antigen dose in a hypothetical mammal. In the first experiment, we injected 110 different antigen populations in the sample, in the second, 250, and, in the third, 350. To represent the mutation within a population of the same antigen, we used 10 different seeds for the pseudo-random number generator. In the first experiment, a seed was associated with each sample, and the same set of seeds was used to perform the other experiments. In this way, to represent the mutation, we considered that inoculated antigens in the same position belonged to the same species and underwent a mutation for each different sample. The difference between the samples is in the bit-string variation of the inoculated antigens, and the difference between the experiments is in the duration of the time steps. The design of the experiments and the antigen identification used in this work are shown in Fig. 5.
In the schematic diagram shown in Fig. 5, the antigen (i.e., a virus strain) is identified as V1E12, which is the mutation of the antigen V1E11 (belonging to an antigen population of the same species), and the antigen V2E11 is different from the V1E11 antigen (which belongs to various antigen populations). In each experiment, different lifetimes were considered for individual hypothetical mammals. The lifetime ("lifespan") for E1, E2, and E3 is 110000, 250000, and 350000 respectively.

Model parameters
The following table shows the ranges for the parameters used in our simulations, based on the literature.  Table 1. Parameters used in the proposed model.

The apoptosis clonal parameter (d)
In the extended model presented in this paper, d represents the fraction of cells that is subjected to natural death (apoptosis) or programmed death; thus, 1 sd   , where 1 sd  www.intechopen.com is the fraction of cells that avoids apoptosis. In the literature, the apoptosis of lymphocytes is typically assumed to occur in percentages not less than 95% [41.43]. For the simulations developed in this study, the natural death parameter was fixed at 0.99 (99%). To give an idea of the effect of varying this parameter, the performances of the model for two different apoptotic events and for the first inoculation antigen were compared (See Fig. 6). The virgin state of the system is the range of 0 to 1000.

The source term (m)
The source term m simulates the stochastic behavior of the bone marrow in the production of new lymphocytes [21.38].
In the model described in this work, if the pseudo-random number generator returns a value less than o r equal to 0.1 p  , the source term takes the value 7 10 m   , because m is experimentally small compared with the levels of lymphocytes produced in the immune response [34,35,38]

The clonal proliferation parameter (b)
Both the pioneering work and the recent work in the literature on theoretical immunology present results on the dynamics of the immune system and the search for attractors of the fixed point type to determine in machine clonal homeostasis (equilibrium) in the virgin state (antigen without inoculation) and in the excited state (when an antigen is recognized by some clonal population) [37,38,40].
The condition 0 FA yy   is satisfied when the virgin state of the immune system is considered, i.e., without the presentation of antigens, no antibodies are produced. Also, considering that the system only allows high-affinity connections (connections between perfectly complementary shapes), the connectivity factor 0 h a  .
In the virgin state, the sum total of the immune populations is restricted to B lymphocytes:  [17,18,19]. A more detailed explanation can be found in the results section.
Because the bone marrow term in the absence of infection (virgin state of the immune system) is much smaller than the clonal proliferation parameter (m<<b) [34,35], we have the following: In the simulations presented in this paper, the clonal proliferation parameter b was set to 2.0 .

The antibody and antigen removal parameter (k)
The parameter for the removal of antigens and antibodies k was set to 0.1 , to ensure that the populations of antigens and antibodies decay to zero before the antigen is presented.
This procedure, which is adopted for a new antigen, is applied only after the previous antigen has been completely removed [21.45].

Connectivity ( h a )
The connectivity parameter used was 7, so that 99% of the populations are coupled to their perfect complement, and only 1% of the populations are coupled to the first neighbors of their complement. The quality of the immune response is directly related to the degree of affinity among the elements of the adaptive system [8].

Bit-string length (B)
Considering the available hypercube immune populations represented by the model, the length of the bit-string B was set at 12. This value corresponds to 12 2 4,096  different antigens.

Antibody proliferation parameter ( A b )
In the model presented in this work, the initial antigenic dose () initial Ag was set to study the influence of parameter A b on the immune memory in some simulations.
In other simulations, this parameter was set to study the consequences to the memory of varying the antigen dosage. To clarify, the limit value of 0.0 A b  corresponds to the model previously proposed by Lagreca et al. [21], and the limit value of 0.0 initial F Ag y  corresponds to the virgin state of the immune system.

Results
The clonal populations that were excited after selection by an antigen (or an antigen population) are shown in Fig. 7, as follows: (a) for the first antigen inoculation; and (b) for the second antigen inoculation, with a dosage of 0.1. In this evolution, two populations were excited with the first antigen inoculation at step 1000: the clonal population that recognized the specific antigen (B1 -Burnet idiotypic cells [18.19]) and the clonal population (J1 -Jerne anti-idiotypic cells [18.19]) complementary to B1. At step 2000, the second antigen was inoculated, and four populations survived: the clonal population that was selected by the second antigen (B2), the clonal population (J2) that is complementary to B2, the clonal population that was selected by the first antigen (B1), and the clonal population (J1) that is complementary to B1.
At step 1000, clonal populations B1 and J1 are excited when they are selected by the first antigen, as shown in Fig. 7 (a). However, in step 2000, when populations B2 and J2 are excited, the clonal populations B1 and J1 are already memories of the first antigen. To maintain the homeostasis of the system, there is a decrease in the concentrations of the four remaining populations, as shown in Fig. 7 (b).

Antigen persistence
The temporal evolution (kinetics) of the Burnet cells is shown in Fig. 8 for each antigen i. This behavior occurs because the immune system has a maximum number of cells that it can support; in other words, when new antigens are memorized, others need to be forgotten (immune homeostasis turnover). At time step 7,000, when the seventh inoculation is performed, the first population begins to increase, indicating that it can be stored for a long period. In our Ag-dependent approach, this behavior indicates that an increase in the lifetime (lifespan) of memory can be generated by antigen survival (antigenic dependence).

Antigen mutation
To study the influence of antigenic mutation on memory (B cell antigen-dependent memory), simulations of inoculations of the 30 samples were also performed, with an antigenic dosage of 0.1 . . In Fig. 9 (a), for example, all of the lifetimes (lasting memories) are related to antigens of different species (V1E11...V110E11). In contrast, the first lifetime in Fig. 9 (a)-(j) refers to an antigen that has already undergone mutation (V1E11...V1E110). Similar memory developments for experiments E2 and E3 are also shown in Fig. 9. The behavior of the average durability of the memories is shown in Fig. 10 (a) -(c). The average lifespans are calculated from the memory lifespans generated by each mutated antigen, as follows: -For experiment E1 (Fig. 10 (c)), the first average lifetime is obtained by For experiment E2 ( Fig. 10 (b)), the first average lifetime is obtained by 10 12 For experiment E3 ( Fig. 10 (a)), the first average lifetime is obtained by

Discussion and conclusion
In this paper, an Ag-dependent mathematical model was used to explore how the key elements of the adaptive immune system function. The same model was also used to investigate the factors that are potentially responsible for maximum immunization capacity [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55]. Our results suggest that not only antigen type but also antigen mutation can influence the durability of immunizations, indicating that the role of antigen persistence is important for prolonging immune memory. These results were discussed with respect to recent work, and we refer to the adoption of parameter values chosen among data gathered from the literature. The model used in this study took into consideration that the immune system is a network of molecules and cells that can recognize itself [1][2][3][4][5][6]17]. The cells that recognize antigens select a complementary set of clones (anti-idiotypic antibodies) that can react with the idiotypes of other cells. Thus, the clonal expansion of complementary cells can also occur when these two types of cells interact through lock-and-key connections [8]. In the results presented here, such behavior was observed when an antigen was inoculated into the system and two B cell populations were excited: the population of cells that recognized the antigen and the population of cells that recognized its complementary shape, as shown in Fig. 7.
The results also show that an important factor in the durability of immunological memory is the mutation of antigen populations. In 2009, Tarlinton et al. [49] published a review paper, suggesting that the homeostasis of immune memory can only occur if new memory populations arise over others, i.e., to create dynamic equilibrium among memory cells, some need to disappear for others to arise, because the immune system has a maximum memory capacity [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55]. Choo et al., in a recent paper published in The Journal of Immunology [57], reported the same finding, based on the Ag-independent premise. Choo et al. (2010) have determined, by means of a quantitative analysis, that the homeostatic turnover of Agspecific CD8 memory T cells is stochastic rather than deterministic.
Then, the results we show in Fig. 8 indicate, in part, an alignment with the work of Choo et al.(2010) and with that of Tarlinton et al., because some populations were "forgotten" so that others could be "memorized", thereby complying with the principle of homeostatic turnover. However, Tarlinton et al.(2008) [49] and Choo et al. (2010) [57] suggest that the mechanism for achieving homeostasis is stochastic, contrary to earlier work of Matzinger (1995) [50] and Nayak et al. (2001) [19], who indicated that the durability of memory depends on the antigen type.
The results presented in Figs. 9 and 10 suggest that the homeostatic turnover of a memory B cell depends on the antigen type and also on their mutation(s). Thus, our model aligns best with the earlier work of Nayak et al.(2001) and Matzinger et al. (1995), and it also aligns to some extent with the work of Tarlinton et al., specifically with respect to storage capacity (homeostatic turnover). However, our results do not line up with a hypothesis of randomness (stochastically) for the kinetics of immune memory, as inferred by Choo et al.
The results presented here considered a pool of B cells, but similar conclusions can be drawn from a pool of CD4 T cells. In our simulations, memory lifespan is dependent on the antigen, and the dynamic behavior of memory is strongly deterministic. These results are especially interesting, because they may suggest a deterministic chaotic behavior for the immune memory. In chaotic behavior, there is a mix of stochasticity and determinism, i.e., there exists a well-defined mathematical function for the problem, but small changes in initial conditions can lead to unpredictable results. In conclusion, our results have shown that Choo et al.(2010) may have inferred an "apparent" stochastic behavior for homeostatic turnover in their work; however, this behavior may be linked to a deterministic-chaotic dynamic equilibrium. Nevertheless, this finding also indicates that, although the memory behavior is deterministic, just is possible to predict the durability of immunization inferred by a vaccine within a limited interval of antigenic concentration, i.e., outside chaotic region. Immunology is the branch of biomedical sciences to study of the immune system physiology both in healthy and diseased states. Some aspects of autoimmunity draws our attention to the fact that it is not always associated with pathology. For instance, autoimmune reactions are highly useful in clearing off the excess, unwanted or aged tissues from the body. Also, generation of autoimmunity occurs after the exposure to the non-self antigen that is structurally similar to the self, aided by the stimulatory molecules like the cytokines. Thus, a narrow margin differentiates immunity from auto-immunity as already discussed. Hence, finding answers for how the physiologic immunity turns to pathologic autoimmunity always remains a question of intense interest. However, this margin could be cut down only if the physiology of the immune system is better understood. The individual chapters included in this book will cover all the possible aspects of immunology and pathologies associated with it. The authors have taken strenuous effort in elaborating the concepts that are lucid and will be of reader's interest.

How to reference
In order to correctly reference this scholarly work, feel free to copy and paste the following: Alexandre de Castro (2012). An Ag-Dependent Approach Based on Adaptive Mechanisms for Investigating the Regulation of the Memory B Cell Reservoir, Recent Advances in Immunology to Target Cancer, Inflammation and Infections, Dr. Jagat Kanwar (Ed.), ISBN: 978-953-51-0592-3, InTech, Available from: http://www.intechopen.com/books/recent-advances-in-immunology-to-target-cancer-inflammation-andinfections/an-ag-dependent-approach-based-on-adaptive-mechanisms-for-investigating-the-regulation-of-thememory