Mathematical Models of the Immune System

The immune system has some unique features, which render it appealing for mathematical modelling:

It is a highly distributed system, which carries out a complex recognition and classification task

It evolves and matures using combinatorial, evolutionary and adaptation mechanisms

Immune system models can generally be classified into continuous models, describing the immune process by sets of differential equations, and discrete models, describing the immune process as a series of interactions in discrete time steps, or utilising combinatorial methods to predict immune properties.

Traditionally, the approach to modelling the immune system involved ODE (ordinary differential equations) or PDE (partial differential equation) [30]. However, in the last two decades discrete mathematical models, and most notably, the CA approach, have become increasingly popular in the theoretical immunology community. These new trends were largely due to the wide-range use of CA in modelling complex phenomena in physics, biology, finance, and, more recently, sociology [31]. Below we briefly overview models belonging to each modelling group. A more detailed review can be found in [30]. Continuous Models

Most mathematical models in immunology employ systems of differential equations to describe the dynamic interactions of immune cells and pathogens. The system's description may include equations and parameters for proliferation and death rates of pathogens and lymphocytes, for the transitions between resting and activated states of immune cells, or between naive and memory phenotypes, transitions of the response between humoral and cellular activity, etc. Among the issues addressed using this approach are the maturation of the humoral immune response, exhibited by B cell proliferation and differentiation using clonal selection and somatic hypermu-tations [32,33], the effect of feedback in monitoring, balancing, and improving the immune response [34], the role of cross-reactive stimulation in maintaining immune memory [35], the threshold ratio between Th memory cells and antigen dose needed to establish T cell memory [36], antiviral immune response in infections, such as hepatitis B, influenza [37,38], HIV [39-42], etc. Discrete Models

One subclass of immune system models uses methods of discrete mathematics to evaluate characteristics of the immune system and to predict its behaviour. Perel-son et al. [43] have employed a "shape space" model to study aspects of the immune repertoire: how large should this repertoire be in order to be complete, and what is the probability of recognising foreign vs. self antigens. The shape space model geometrically describes the immunological receptors as points in a multi-dimensional space, each dimension representing a binding parameter such as length, width, charge, etc., and each receptor can bind epitopes within a small "recognition ball" surrounding its complement in the shape space.

A different approach was introduced by Agur et al. [44,45], who analysed the strategy of the humoral immune response as an optimisation problem. Agur et al.

employed dynamic programming methods for investigating the optimal mutation rate function in B cells, which maximises the probability that the required structure of the antigen-binding antibody will be efficiently generated during any immune response. Analytical results have pinpointed a step-function mutation rate as the globally optimal strategy, transition from minimum mutation rate to the maximum biologically possible mutation rate occurring when the size of the best performing B cell clone exceeds a well-defined threshold.

A second subclass of discrete models is that of CA. Discrete in both space and time, these models describe the immune system dynamics by deterministic rules of cells, molecules and their local interactions. In [46] the concept of "evolutionary" experiments in-machina (i.e., within computer) was introduced. Thus, computer simulation experiments were performed, where each B cell was represented by a two dimensional cellular automata with variable processing rules. Results of this work suggest that efficient immune response to antigenically homogenous pathogen favours strong contraction in phase space in antibody generation (one B-cell clone -one antibody), whereas efficient response to antigenically varying pathogen should favour weak contraction in phase-space in antibody generation (one B-cell clone -many antibodies).

These type of experiments can be used prior to any in vitro or in vivo experiments for qualitatively examining problems in immunology by fast, reproducible, and cheap means. Indeed, Celada and Seiden put forward such a CA simulation model, which attempts to capture "all" the different constituents of the immune system in one comprehensive framework (to be denoted CS-model, [15,47,48]). This model has been used to study various phenomena, including the optimal number of human leukocyte antigens (HLA) [48], the autoimmunity and T lymphocytes selection in the thymus [49], antibody selection and hyper-mutation [47], and the dynamics of various lymphocyte populations in the presence of viruses, which are characterised by infectivity, reproduction efficiency, etc., [50]. Formally, the CS-model belongs to a subclass called "stochastic CA." Stochastic Cellular Automata

Most simulators of the immune response are deterministic, assuming that a given set of initial conditions leads to only one end-state. Typically, deterministic models, constituted by a set of differential equations which represent the interactions among immune cells and molecules, are solved iteratively by numerical integration. However, the assumptions underlying the deterministic modelling method cannot represent many intra- and inter-cellular processes, which, typically, are sensitive to the behaviour of a relatively small number of cells and molecules. Under such circumstances any given set of initial conditions can lead to a plurality of end-states. Stochastic CA are models designed to represent the latter systems. In these models the caveats of the deterministic approach are avoided, since they allow for randomness in the activity of the system's operators.

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