Multistationarity, the basis of cell differentiation and memory. I. Structural conditions of multistationarity and other nontrivial behavior

A biological introduction serves to remind us that differentiation is an ep igenetic process, that multistationarity can account for epigenetic differe nces, including those involved in cell differentiation, and that positive f eedback circuits are a necessary condition for multistationarity and, by in ference, for differentiation. The core of the paper is comprised of a forma l description of feedback circuits and unions of disjoint circuits. We intr oduce the concepts of full-circuit (a circuit or union of disjoint circuits which involves all the variables of the system), and of ambiguous circuit (a circuit whose sign depends on the location in phase space). We describe the partition of phase space (a) according to the signs of the ambiguous ci rcuits, and (b) according to the signs of the eigenvalues or their real par t. We introduce a normalization of the system versus one of the circuits; i n two variables, this permits an entirely general description in terms of a common diagram in the "circuit space." The paper ends with general stateme nts concerning the requirements for multistationarity, stable periodicity, and deterministic chaos. (C) 2001 American Institute of Physics.