Discrete time piecewise affine models of genetic regulatory networks

Abstract

We introduce simple models of genetic regulatory networks and we proceed to the mathematical analysis of their dynamics. The models are discrete time dynamical systems generated by piecewise affine contracting mappings whose variables represent gene expression levels. These models reduce to boolean networks in one limiting case of a parameter, and their asymptotic dynamics approaches that of a differential equation in another limiting case of this parameter. For intermediate values, the model present an original phenomenology which is argued to be due to delay effects. This phenomenology is not limited to piecewise affine model but extends to smooth nonlinear discrete time models of regulatory networks. In a first step, our analysis concerns general properties of networks on arbitrary graphs (characterisation of the attractor, symbolic dynamics, Lyapunov stability, structural stability, symmetries, etc). In a second step, focus is made on simple circuits for which the attractor and its changes with parameters are described. In the negative circuit of 2 genes, a thorough study is presented which concern stable (quasi-)periodic oscillations governed by rotations on the unit circle - with a rotation number depending continuously and monotonically on threshold parameters. These regular oscillations exist in negative circuits with arbitrary number of genes where they are most likely to be observed in genetic systems with non-negligible delay effects.

Comparing different ODE modelling approaches for gene regulatory networks

“…It was proposed that the resulting equations can be simplified by considering piecewise-linear approximations of these Hill functions [5]. Another possibility [7] is to discretize the continuous-time ODEs to obtain a discrete-time system. In what follows, we briefly outline the main features of each of these modelling approaches.…”

“…As originally proposed by [23] and more recently in [7], discrete-time models can be used to study gene regulatory networks. The idea is to derive a difference-equation model describing the change of the gene product concentrations at discrete time intervals.…”

“…The idea is to derive a difference-equation model describing the change of the gene product concentrations at discrete time intervals. It is suggested that this may be appropriate to (coarsegrain) model gene regulation where local complex chemical reactions have to be integrated over short time scales in order to produce interactions affecting expression levels on larger time scales [7]. The discrete-time model in [7], is based on the quasi-steady-state mRNA assumption.…”

“…It is suggested that this may be appropriate to (coarsegrain) model gene regulation where local complex chemical reactions have to be integrated over short time scales in order to produce interactions affecting expression levels on larger time scales [7]. The discrete-time model in [7], is based on the quasi-steady-state mRNA assumption. Hence the model has a single state variable (either mRNA or protein) for each gene.…”

“…A radically different approach is to replace the Hill functions by piecewise-linear approximations [5]. A further modelling approach is the use of discretetime maps [7] where the evolution of the system is modelled in discrete, rather than continuous, time. The aim of this paper is to discuss and compare these different modelling approaches, using a representative gene regulatory network.…”