Introduction

Bistable switches have been studied extensively in the theoretical biology literature, and such switches have even been artificially built into biological organisms (see Gardner et al., 2000, for a prokaryotic switch, and Becskei et al., 2001, for a eukaryotic switch), displaying the properties that were expected from theoretical studies. However, to our knowledge, no study of n-stable switches has been proposed for $n>2$ (a structure leading to such systems was mentioned by Kling & Székely, 1968, in a different context). Such switches are much more resistant to mathematical analysis than bistable switches, but one would expect that they can appear in biological systems where a decision more complicated than a binary decision has to be made, especially in the case of cellular differentiation. Demongeot et al. (2000a) have for example proposed a 3-stable switch between the apex and cotyledonary bud growth variables, in order to explain storage and recall functions observed in Bidens pilosa L. While it has long been clear that cellular differentiation is a dynamic phenomenon in developing organisms, it has only recently been widely recognised that differentiated cells are not always in a definitively fixed state, recorded for example by chromatin remodelling or DNA methylation. Indeed, there are many examples in which cells can be observed to undergo transdifferentiation or metaplasia in non-pathological situations (Slack & Tosh, 2001), regeneration in urodele amphibians is thought to involve dedifferentiation (Brockes, 1997), and in many cases stem cells have shown wider differentiation potential than expected (see for example Ferrari et al., 1998, Bjornson et al., 1999, and Clarke et al., 2000). This points to autonomous differential systems, exhibiting switch-like behaviour, as a relevant type of mathematical model to study cellular differentiation. Biologically, such switches, while stable on their own, could be made to change states if they underwent the right perturbation, for example by external signalling factors. One can think of cellular differentiation as following two different kinds of pathways; one extreme would be a hierarchical, sequential series of binary decisions, as illustrated in Figure 1a (such a pathway could correspond to the progressively more specialised progeny of stem cells; it would involve a low number of interactions between all components). Another extreme would be an "instantaneous" decision between all the possible outcomes (depending for example on external stimulations), as illustrated in Figure 1b (such a pathway structure makes it necessary that every single component has an inhibitory effect on all other components). These two proposed pathways are compatible with combinatorial choice of cell fate by different factors; we do not discuss this possibility because the way in which the cell-type is "read-out" does not influence the number of steady-states of the system. Biological studies haven't yet given a precise answer to how differentiation occurs, but it seems quite possible that real-world differentiation systems come half-way between these two extremes, and in particular that there are steps that involve "decisions" between at least 3 possible outcomes. It is thus very interesting to study n-stable switches with $n>2$, since the study of hierarchical, sequential systems comprising such switches can easily be reduced to the study of n-stable switches, because the decision process can be split up between the different decision steps. What's more, the study of such switches will give us the opportunity to illustrate remarks we made previously about the weight of positive and negative feedback circuits.
Figure 1: two basic models for cell differentiation. Empty squares correspond to repression, and arrows to activation. 1a: hierarchic decision model: genes $A_0$ and $B_0$, $A$ and $B$, and $C$ and $D$ form mutually repressive circuits, providing positive feedback. Differentiation would involve setting a first bistable switch, formed by $A_0$ and $B_0$, and then a second one, formed by either $A$ and $B$, or $C$ and $D$. 1b: simultaneous decision model: genes $A$, $B$, $C$ and $D$ form a 4-stable switch, and require mutual inhibition. Differentiation would involve setting the switch to the right state in one step.