Monte Carlo simulation of the shape space model of immunology

Abstract: This review describes a body of work on computational immune systems that behave analogously to the natural immune system. These artificial immune systems (AIS) simulate the behavior of the natural immune system and in some cases have been used to solve practical engineering problems such as computer security. AIS have several strengths that can complement wet lab immunology. It is easier to conduct simulation experiments and to vary experimental conditions, for example, to rule out hypotheses; it is easier to isolate a single mechanism to test hypotheses about how it functions; agent-based models of the immune system can integrate data from several different experiments into a single in silico experimental system.

Abstract: The immune system is a complex system of cells and molecules that can provide us with a basic defense against pathogenic organisms. Like the nervous system, the immune system performs pattern recognition tasks, learns, and retains a memory of the antigens that it has fought. The immune system contains more than 10{sup 7} different clones of cells that communicate via cell-cell contact and the secretion of molecules. Performing complex tasks such as learning and memory involves cooperation among large numbers of components of the immune system and hence there is interest in using methods and concepts from statistical physics. Furthermore, the immune response develops in time and the description of its time evolution is an interesting problem in dynamical systems. In this paper, the authors provide a brief introduction to the biology of the immune system and discuss a number of immunological problems in which the use of physical concepts and mathematical methods has increased our understanding. {copyright} {ital 1997} {ital The American Physical Society}

Abstract: Agent-based models have been employed to describe numerous processes in immunology. Simulations based on these types of models have been used to enhance our understanding of immunology and disease pathology. We review various agent-based models relevant to host-pathogen systems and discuss their contributions to our understanding of biological processes. We then point out some limitations and challenges of agent-based models and encourage efforts towards reproducibility and model validation.

Abstract: In recent years, the study of immune response behaviour through mathematical and computational models has attracted considerable efforts. The dynamics of key cell types, and their interactions, has been a primary focus in terms of building a picture of how the immune system responds to a threat. Discrete methods, based on lattice Monte-Carlo (MC) models, with their flexibility and relative simplicity have previously been used to model the immune system behaviour. However, due to speed and memory constraints, large-scale simulations cannot be done on a single computer. Key issues in the reduction of simulation time are code optimisation and code parallelisation. In this paper, optimisation and parallelisation solutions are discussed, with reference to existing MC simulation code for dynamics of HIV infection.

Abstract: The immune system is the natural defense of an organism. It comprises a network of cells, molecules, and organs whose primary tasks are to defend the organism from pathogens and maintain its integrity. The cooperation between the components of the immune system network realizes effectively and efficiently the processes of pattern recognition, learning, and memory. Our knowledge of the immune system is still incomplete and mathematical modelling has been shown to help better understanding of its underlying principles and organization. In this chapter we provide a brief introduction to the biology of the immune system and describe several approaches used in mathematical modelling of the immune system.

Abstract: A large-scale model of the immune network is analyzed, using the shape-space formalism. In this formalism, it is assumed that the immunoglobulin receptors on B cells can be characterized by their unique portions, or idiotypes, that have shapes that can be represented in a space of a small finite dimension. Two receptors are assumed to interact to the extent that the shapes of their idiotypes are complementary. This is modeled by assuming that shapes interact maximally whenever their co-ordinates in the space-space are equal and opposite, and that the strength of interaction falls off for less complementary shapes in a manner described by a Gaussian function of the Euclidean “distance” between the pair of interacting shapes. The degree of stimulation of a cell when confronted with complementary idiotypes is modeled using a log bell-shaped interaction function. This leads to three possible equilibrium states for each clone: a virgin, an immune, and a suppressed state. The stability properties of the three possible homogeneous steady states of the network are examined. For the parameters chosen, the homogeneous virgin state is stable to both uniform and sinusoidal perturbations of small amplitude. A sufficiently large perturbation will, however, destabilize the virgin state and lead to an immune reaction. Thus, the virgin system is both stable and responsive to perturbations. The homogeneous immune state is unstable to both uniform and sinusoidal perturbations, whereas the homogeneous suppressed state is stable to uniform, but unstable to sinusoidal, perturbations. The non-uniform patterns that arise from perturbations of the homogeneous states are examined numerically. These patterns represent the actual immune repertoire of an animal, according to the present model. The effect of varying the standard deviation σ of the Gaussian is numerically analyzed in a one-dimensional model. If σ is large compared to the size of the shape-space, the system attains a fixed non-uniform equilibrium. Conversely if σ is small, the system attains one out of many possible non-uniform equilibria, with the final pattern depending on the initial conditions. This demonstrates the plasticity of the immune repertoire in this shape-space model. We describe how the repertoire organizes itself into large clusters of clones having similar behavior. These results are extended by analyzing pattern formation in a two-dimensional (2-D) shape-space. A lattice mapping is employed, whose rules are rigorously derived from a simplified version of the underlying differential equations via a logarithmic transformation of variables. A novel feature of the lattice model is that the neighborhood of cell (i, j) is centered around cell (−j, −j). Thus, interactions are non-local. The 2-D patterns that emerge are reminiscent of those found in reaction-diffusion systems, and contain many hills and valleys. (In contrast with most reaction-diffusion models, pattern formation in this model is not dependent on long-range inhibition and short-range activation.) The scale of the pattern depends on neighborhood size, with small neighborhoods generating fine scale patterns with narrow peaks, and large neighborhoods generating large scale patterns with wide peaks and valleys. Both one- and two-dimensional models support patterns in which a fraction of the clones are not stimulated by network interactions. The fraction of such “disconnected clones” increases with both dimensionality and σ.

Abstract: For the reactions among the antibodies of the immune system, fighting against a foreign antigen, the recent model of de Boer, Segel and Perelson introduced a cellular automata approximation for the interaction of different types of B cells (bone marrow derived lymphocytes) In contrast to most physics models, here each lattice site interacts mainly with its mirror image (with respect to the lattice center) in the opposite part of the lattice We simplify their model and then generalize it to include more than one or two shape-space parameters Thus instead of simulating a chain or square lattice, we simulate a d -dimensional hypercubic lattice with dimension d up to 10 In particular, we check for the concentration of B cells and the stability against localized perturbations (“damage spreading”)