Positive feedback loops and multistationarity

Abstract

This paper aims to draw the attention of mathematicians to a class of problems whose resolution has recently become possible from the practical viewpoint but still badly lacks a firm mathematical basis. These problems arise in biological systems which involve multiple interacting feedback loops. Such systems may be described in terms of differential equations. However, the non-linear character of most of these equations precludes an analytic treatment, and it is often very difficult to obtain a global idea of their complex dynamics. Logical methods (briefly summarised in this paper) have been developed (Thomas [13–15]). In spite of the somewhat caricatural character inherent to logical descriptions, these methods grasp the essential qualitative features of the dynamics of the systems, and, for complex systems, they greatly help in clarifying the continuous description. Graphs are involved at two distinct levels; first, the logical structure of a system can be described by a graph of interactions, and at a later stage of the analysis one obtains a graph of the logical sequences of states. Although algorithms permit the second type of graph to be derived from the first, there is so far no general mathematical analysis of their relation. The paper gives, as an illustration of this type of problem, the relation between positive loops (in the graph of interactions) and multiple steady states (which are found as the final states in the graph of the sequences of states). In short, a system comprising n positive loops may have up to 3 n steady states (2 n attractors); interactions between the loops reduce these numbers in a predictable way.