||nology has not described the behavior of single cells quantitatively. There are no equivalents in immunology of the Hodgkin-Huxley (1952) equations, which describe how a nerve impulse travels down an axon (see Cronin, 1987). Yet the "macroscopic behavior" of the immune system, as probed in a specific experiment, can be well characterized. The problem then arises of selecting a simple representation for the elementary interactions that would give rise to the organized behavior observed in the immune system. The adventure of statistical physics is full of equivalent endeavors, starting from the description of thermal properties of gases and solids based, respectively, on the assumption of independent particles composing a perfect gas and the coupling of harmonic oscillators, to the more recent description of neural nets. This kind of approach is especially suited to theoretical immunology because of our ignorance about the detailed mechanisms responsible for the observed behaviors of the immune system. To be more specific, we shall look for generic properties among models of the immune system. As in the case of phase transitions in condensed-matter physics, we are interested in semiquantitative laws, such as scaling laws, which depend only on the general features of the model, and not on its details.
In Sec. II, we shall describe the theory of clonal selection and provide quantitative estimates of how the ability of the immune system to recognize foreign cells and molecules scales with the size of the system. We shall show, with parameters estimated for the mammalian immune system, that an immune system needs at least 105 different elements to function. The basic elements of the immune system are a class of white blood cells known as lymphocytes. The size of an organism determines the maximum number of its lymphocytes. Thus mice have of the order of 108 lymphocytes, while humans have of the order of 1012. Thus system sizes are large but not as large as Avogadro's number.
Cells are already macroscopic systems far from equilibrium from the point of view of thermodynamics, and there is little hope of starting from a simple Hamiltonian as is often done in statistical mechanics. But basically, the simple system of differential equations described in Sec. IV plays the role of the dynamics of an Ising Hamiltonian with respect to diluted magnetic systems or of the discrete time logistic equation for chaos and turbulence. Although it represents a strong simplification of the interactions present in the system, it is expected to belong to the same class of universality as a "true" model of the immune system and to exhibit the same generic properties. In the case of a dynamic system, the generic properties concern the attractors of the dynamics. Some of the questions that we shall address are: Are the attractors limit points, limit cycles, or chaotic? What is their number? What are their basins of attraction? How do these properties relate to the parameters of the differential system? How can one force transitions among the different dynamical regimes? If our hypotheses about the universality of a model are true, the generic properties, qualitative classification of the attractors, and scal-
ing laws should be evident in the phenomenology of the mammalian immune system.
Besides issues of dynamics, we shall address questions of a probabilistic nature. For example, in Sec. II, we ask how well the immune system performs the task of distinguishing self components from foreign or nonself components. We also address a design question: in order to perform efficient self-nonself discrimination, how large a region on a molecule should the immune system examine? If the immune system looks at very small regions, say one or two amino acids, then with a rather limited set of receptors all foreign molecules could be recognized. However, since self molecules are made from the same building blocks, such a recognition system would also recognize all self molecules. If the immune system looks at a very large region of molecules, then a receptor may need to be so specific that it might be able to recognize only one particular molecule, and with finite resources many foreign molecules may escape detection.
Section III begins our foray into dynamics. We first discuss models based on the physical chemistry of receptor-ligand interactions, which underlie the ability of lymphocytes to detect antigen. We then abstract from the chemistry more phenomenological laws that govern the growth and differentiation (or maturation) of lymphocytes into cells with specialized functions, such as plasma cells that secrete antibody at very high rates. We also look at a phenomenon, known as affinity maturation, by which the immune system can improve the average equilibrium binding constant of antibody for antigen.
The immune system is more than a collection of independently operating lymphocytes. While many chemical signals are involved in setting up communication between these cells, here we focus on one class of models, called idiotypic network models, in which signals are propagated via specific interactions between cell surface receptors and antibody molecules. Thus a lymphocyte that detects a foreign antigen can begin secreting antibody A]_, which is a molecule that has a very specific structure and that can bind the antigen. Molecules At are proteins, and their novel or "idiosyncratic" parts may be detected by other lymphocytes in the immune system. These "second-level lymphocytes" may respond to seeing At by secreting a complementary or "anti-idiotypic" antibody A2. Some molecules in class A2 may resemble the antigen and hence help encode memory of the encounter, while others may be distinct and excite a third level of response. Section IV deals with different approaches to modeling idiotypic networks and understanding their dynamics.
Quantitative information that can be used to evaluate differential equation models is often lacking in immunology. Given this state of affairs, it is frequently desirable to formulate models in which the variables have a limited number of states, e.g., 1 and 0, cells are activated or not. Automata models of this type are summarized in Sec. V.
Rev. Mod. Phvs., Vol. 69, No. 4, October 1997