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Positive feedback loops and multistationarity. (English) Zbl 0639.92003

The aim of the paper is to draw the attention of mathematicians to complex systems, frequent in biology and related fields, which comprise multiple feedback loops. Such systems may be described in terms of ordinary differential equations. Since most of these differential equations are non-linear, these systems can not be treated analytically. Namely, it is often very difficult to grasp a global view of the dynamics of these systems. The authors describe some methods which permit description and treatment of such systems in an adequate way from the practical view point. However, a firm mathematical basis for these methods is lacking so far.
The authors show that crucial questions in the field can be posed in terms of graph theory. The paper discusses the following open problem. A formal model is described by a graph of interactions, from which one derives a set of logical equations. From these equations, one derives a new graph which describes the sequences of logical states compatible with the model. Although one knows how to proceed in individual cases and begins to have some global views, the general problem of the relation between the graph of interactions and the graph of the sequences of states remains open.
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). Namely, a system comprising n positive loops may have up to \(3^ n\) steady states; interactions between the loops reduce these numbers in a predictable way.
Reviewer: V.Alad’ev

MSC:

92B05 General biology and biomathematics
93C15 Control/observation systems governed by ordinary differential equations
05C99 Graph theory
92D10 Genetics and epigenetics
93D15 Stabilization of systems by feedback
93C10 Nonlinear systems in control theory
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