Conclusion

We have shown that the existence of positive feedback is a necessary condition for multistationarity in a wide class of systems. This is an interesting property per se, but it is also a finding of practical importance. One can never be assured that the factors in a regulation network have all been identified, but it is also difficult to be certain that factors are missing. However, if one comes across a system exhibiting multistationarity, and the system has no positive feedback, then our result shows that the interaction graph is incomplete. For example, von Dassow et al. (2000) studied a system involved in cell differentiation, for which there was no experimental data asserting the presence of a positive feedback circuit. The system displayed a very poor behaviour, which led the authors to assume two supplementary interactions in their model system. Most interestingly, these two assumed interactions each introduced a positive feedback circuit in the system, a fact which had not been explicitly pointed out by the authors. Cellular differentiation is a process of major biological and medical interest. As illustrated by our multistable switch, mathematical modelling could prove to be useful in its investigation. Molecular information is becoming to be sufficiently available for the study of specific models; for example, dosed perturbation of the proteins involved in the transdifferentiation of pancreas to liver (Shen et al., 2000) could allow to derive parameters for a possible switch.