J. theor. Biol. (2000) 203, 000}000 doi:10.1006/jtbi.2000.2001, available online at http://www.idealibrary.com on Evolution of Speci5city in an Immune Network KOUJI HARADA* AND TAKASHI IKEGAMI Institute of Physics, ¹he Graduate School of Arts and Sciences, ; niversity of ¹okyo, 3-8-1, Komaba, Meguro-ku, ¹okyo 153, Japan (Received on 17 December 1998, Accepted in revised form on 18 January 2000) A dynamic antigen response of the immune network is discussed, based on shape-space modelling. The present model extends the shape-space modelling by introducing the evolution of speci"city of idiotypes. When the amount of external antigen increases, a measure of stability of the immune network is lost and thus the network can respond to the antigen. It is shown that speci"c and non-speci"c responses emerge as a function of antigen amounts. A speci"c response is observed with a "xed-point attractor, and a non-speci"c response is observed with a chaotic attractor for the lymphocyte population dynamics. The network topology also changes between "xed-point and chaotic attractors. For some antigen amounts, chaotic attractors will vanish or become long-lived super-transient states. A dynamic bell- shaped response function will thus emerge. The relevance of long-lived chaotic transient states embedded in "xed-point attractors is discussed with respect to immune functions.  2000 Academic Press 1. Introduction Complementarity between molecular shapes is a basic and commonly used mechanism of com- munication in many biochemical systems. In the case of immune systems, it is used for self-/non- self-discrimination. Thus, non-self-antigens are "rst detected by complementary shape matching (binding a$nity) between the antibody surface ligands and lymphocyte receptors. The degree of complementarity or &&speci"city' is not "xed in time and/or in speci"c types of lymphocyte clones. Speci"city should be #exible. For example, Wedemayer et al. (1997) showed that the structure of the antigen-binding site of germ- line antibodies can change slightly, according to the reaction opponent. The mechanism of #exible * Author to whom correspondence should be addressed. E-mail: harada@sacral.c.u-tokyo.ac.jp speci"city relates to how some kinds of chemical bonds are formed within the antigen-binding site. In immune systems, no single antibody type determines self-/non-self-discrimination. It is, rather, a manifestation of the immune system as a whole (Varela et al., 1988). Therefore, it is rea- sonable to assume that immune responses are controlled globally by a network. To sustain such global control, speci"c components of the im- mune network, namely di!erent lymphocytes, must be functionally coupled in certain ways. One possibility for such a coupling was "rst pro- posed by Jerne (1974), who suggested that lymphocytes form the so-called && idiotype net- work'' (see also Bona & Kohler 1983). Each lymphocyte has its own molecular shape, called its idiotype, which is recognized by idiotypes of other lymphocytes by complementarity between their respective shapes. SRB BALA yamuna JTB 20002001 DATE 11/2/00 DISC USED YES 0022}5193/00/000000#00 $35.00/0  2000 Academic Press