Fixed Points of Majority Rule Cellular Automata with Application to Plasticity and Precision of the Immune System

Signal processing in the immune syst em is st udied in a t heor etical mult ilayered network. The elements in th e network have regular conn ect iviti es, with t he level of connect ivity reflecti ng t he level of signal (cyto kine) multifunctionalit y. Each element opera t es by th e same nonlinear majority rule. An exact formul a for t he/ {;umber of fixed points is derived as a fun ction of t he network's size n (n odd) and connect ivity r, as well as a genera l upper-bound estimate for t he numb er of fixed point s for all n and r. Results show th at increas­ ing connect ivity enha nces resilience by st rengthening t he globa l error dam ping capacity of t he system. It s cost is diminished general memory st orage space, reflectin g diminished ph enotypic plast icity. Laboratory experiments are suggeste d for verifying t he implications of the result s for pathogenesis and immunosuppression. The durat ion a nd am plit ude of t he immune resp onse is regu lated by a group of p eptid es called cyt okin es t hat bi nd to the cell sur face and ult im ately lea d t o a change in it s phen otypic state , t hat is , it s pro lifer ati ve or fun ct ion al b eh avior. In t he past, ea ch step in t he immune process was t ho ught to b e contro lled by a unique sing le-p urpose molecul e. Tod ay it is r ealized t hat cytokine activity is charact erized by pleiotropism a nd r edundancy: indi vid­ ua l cytokines are pleiotropic in having mult ipl e overl apping cell r egula t ory actions (de no te d a one/ m any property) , while differ en t cytokin es can have a sim ila r a ct ion (many/one proper ty ). Mo reove r , t he ult im a t e resp on se of a cell usually dep ends on two or more differen t messages received concurrent ly a t t he cell surface; t hese are selecte d fr om a larger rep ertoire of cytokines (1-3) . It thus ap pe ars that t he pr o cessing rules in t his system , who se state t rans it ions are det ermined by a combination of elements each of whi ch is insufficient a nd unnecessary on it s own , are of t he majority vote ty pe.