Most biological regulatory systems involve complex networks of interactions. Theoretical modelling, together with simulations and computational approaches, provides a useful framework for integrating data and gaining insights into the dynamical and functional properties of such networks. In this perspective, a major aim of the research is to contribute to the understanding of how regulatory mechanisms at various scales (e.g. molecular, cellular and intercellular) act synergistically or competitively to achieve degrees of regulation not attainable by one mechanism alone. Key issues are the variety of attractors possible for a network, the nature of transition states and transition dynamics, and the role of the network in emergent behaviour. These issues are examined in terms of systems of differential equations, automata networks and probabilistic models. Specific research themes include:

Generic properties of regulatory structures
Considering the importance of the notion of feedback circuits in biological systems, we address the general question of the relation between the circuit structure or "logical structure" of a network and its main qualitative dynamical properties. Using the techniques of dynamical systems theory, we study the generic properties of typical regulatory modules, namely the properties that rely on the combination of feedback circuits rather than on the precise nature of the individual interactions. This approach leads to formulate general laws concerning the relation between structure, dynamics and function in regulatory networks and provides ways to identify key control points in a network circuitry.

Immune and genetic regulatory networks
Through the elaboration of models on various scales the aim is to generate a coherent picture of the regulation and dynamics of some specific immune and genetic processes, including the interplay between genetic and immunological aspects. The following problems are being investigated:

• T cell signalling
• Cellular versus humoral immunity (Th1/Th2 decision)
• Regulation in cytokine networks
• T cell development
• Neural development in Drosophila melanogaster
• Molecular basis of sex determination in Drosophila melanogaster

Selected publications:

• M. Kaufman, F. Andris and O. Leo, "A Model for Antigen-Induced T-Cell Unresponsiveness Based on Autophosphorylative Protein Tyrosine Kinase Activity", Intl. Immunol. 8, 613-624 (1996).
• R. Thomas, "Laws for the dynamics of regulatory networks", Int. Dev. Biol. 42, 479-485 (1996).
• M. Kaufman, F. Andris and O. Leo, "A Logical Analysis of T Cell Activation and Anergy", Proc. Natl. Acad. Sci. USA 96, 3894-3899 (1999).
• R. Thomas, (1999), "Deterministic chaos seen in terms of feedback circuits: analysis, synthesis, labyrinth chaos". Intl. J. Bifurcations & Chaos 9, 1889-1905 (1999).
• R. Thomas and M. Kaufman, "Multistationarity, the Basis of Cell Differentiation and Memory. I. Structural Conditions of Multistationarity and Other Non-Trivial Behaviour, and II. Logical Analysis of Regulatory Networks in Terms of Feedback Circuits", Chaos 11, 170-195 (2001).

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(a) Cytokine network involved in the decision between cellular (Th1) and humoral (Th2) immunity.
(b) Two coexisting chaotic attractors for a system with a simple circuit structure described by the differential equations: dx/dt = - ax + y - y³; dy/dt = - ay + z - z³; dz/dt = - az +x - x³, with a= 0.28.