Hybrid models of the cell cycle molecular machinery

Abstract

Piecewise smooth hybrid systems, involving continuous and discrete variables, are suitable models for describing the multiscale regulatory machinery of the biological cells. In hybrid models, the discrete variables can switch on and off some molecular interactions, simulating cell progression through a series of functioning modes. The advancement through the cell cycle is the archetype of such an organized sequence of events. We present an approach, inspired from tropical geometry ideas, allowing to reduce, hybridize and analyse cell cycle models consisting of polynomial or rational ordinary differential equations

New algorithms for solving tropical linear systems

“…Nevertheless, there are practical problems that would benefit from developments in this field. For instance, Noel, Grigoriev, Vakulenko, and Radulescu have recently proposed a way to use algorithms for solving tropical linear systems to study stable states of reaction networks in biology [27,26]. Thus, problems of tropical linear algebra are important from both theoretical and practical points of view.…”

Dynamical properties of a two-gene network with hysteresis

“…In [7] and [8], the authors apply hybrid systems tools to model a variety of cell biology problems. More recently, hybrid models have been used in [9] for the study of molecular interactions. It is important to note that hysteresis behavior, which is typically present in genetic regulatory networks, has not been considered in the models mentioned above.…”

“…9 For each t such that (t, j) ∈ dom(x, q), the function j : R ≥0 → N is given by j(t) = j ′ , where j ′ = max {j : (t, j) ∈ dom(x, q) }. 10 A function β is of class KL if it is continuous, r → β(r, s) is zero at zero and nondecreasing, and PSfrag replacements…”

A hybrid mammalian cell cycle model

“…Finally, as discussed in [14] monomial approximations occur naturally in "tropically-truncated" polynomial systems, ie in systems where polynomial or rational rate functions are replaced by a few dominating monomials. As compared with our previous work [14,18], in this paper we use the tropicalization only heuristically to obtain simpler reaction kinetic laws, whose parameters are then fitted. The switching of the monomial terms is not given by the max-plus rule as in [14,18], but is commanded by thresholding functions depending on parameters to be fitted.…”