Liénard systems and potential-Hamiltonian decomposition: applications in biology

In separated notes, we described the mathematical aspects of the potential-Hamiltonian (PH) decomposition, in particular, for n-switches and Liénard systems [J. Demongeot, N. Glade, L. Forest, Liénard systems and potential-Hamiltonian decomposition - I. Methodology, II. Algorithm and III. Applications, C. R. Acad. Sci., Paris, Ser. I, in press]. In the present note, we give some examples of biological regulatory systems susceptible to be decomposed. We show that they can be modelled in terms of 2D ordinary differential equations belonging to n-switches and Liénard system families [O. Cinquin, J. Demongeot, High-dimensional switches and the modeling of cellular differentiation, J. Theor. Biol. 233 (2005) 391-411]. Although simplified, these models can be decomposed into a set of equations combining a potential and a Hamiltonian part. We discuss about the advantage of such a PH-decomposition for understanding the mechanisms involved in their regulatory abilities. We suggest a generalized algorithm to deal with differential systems having a second part of rational-fraction type (frequently used in metabolic systems). Finally, we comment what can be interpreted as a precise signification in biological systems from the dynamical behaviours of both the potential and Hamiltonian parts.