Introduction

The quantity of biological information available for analysis is expanding at a tremendous rate, especially information obtained from molecular genetics techniques, but the methods of analysis are lagging behind. One of the most interesting challenges from the theoretical and practical points of view is to unravel the networks of regulation, involving for example metabolism or cell differentiation. These networks are expected to exhibit stability (since loss or change of cell differentiation are usually observed under unfrequent conditions, respectively anaplasia and metaplasia) and multistationarity (as there are many different cell types in metazoans), and we thus investigate both aspects. Our approach is to find conditions that are fulfilled by biological systems; our goal is to derive both new theoretical concepts, and practical constraints on interactions networks, to make more tractable the task of reverse-engineering such networks from partial, sometimes unreliable, experimental data. To become viable, regulation models are in need of quantification, as pointed out by Koshland (1998). Boolean networks do not meet this need entirely, and biological systems have not been proven to exhibit behaviours easily modelled in a boolean fashion (Kringstein et al., 1998, have even shown in detail that eukaryotic transcription can give a graded response to concentrations of molecules); we thus consider continuous systems, which are much more difficult to deal with, but which provide much more accurate modelling of real systems. There have been attempts to study properties of generic regulation systems of dimension 1 or 2; for example, a beautiful characterisation of multistability has been achieved by Cherry & Adler (2000). But higher-dimensional systems display much greater complexity, and probably are the rule rather than the exception in real-world systems (it is becoming clearer and clearer that signalling pathways are extremely intricate, see for example Jordan et al., 2000). It seems difficult to scale up the results in 2 dimensions to higher dimensions, which is why we try not to make assumptions about the dimensions of the systems we study, even if it is at the cost of sometimes less powerful results. In the first part of this article, we study steady states of a generic system, and derive qualitative and quantitative properties of the system according to its behaviour at these steady states. In the second part, we build on these results to tackle "Delbrück's conjecture"; Max Delbrück proposed in 1948 that cell differentiation could be established by a unique regulating system having distinct attractors (Delbrück, 1949). In 1980, R. Thomas made the conjecture that "the presence of a positive circuit in the logical structure of an autonomous differential system is a necessary, although not sufficient, condition for multistationarity" (Thomas, 1981). The conjecture has been proven in the case of boolean networks (see Aracena et al., 2000, and Demongeot et al., 2000b), and partial demonstrations in the case of continuous systems have been given (Plahte et al., 1995, as well as Snoussi, 1998, and Gouze, 1998, introduced by Demongeot, 1998), but their usefulness is deeply undermined by restrictive hypotheses. We propose another demonstration which introduces softer constraints. Finally, we study a multistable switch to illustrate our results.