Comparing different ODE modelling approaches for gene regulatory networks

Abstract

A fundamental step in synthetic biology and systems biology is to derive appropriate mathematical models for the purposes of analysis and design. For example, to synthesize a gene regulatory network, the derivation of a mathematical model is important in order to carry out in silico investigations of the network dynamics and to investigate parameter variations and robustness issues. Different mathematical frameworks have been proposed to derive such models. In particular, the use of sets of nonlinear ordinary differential equations (ODEs) has been proposed to model the dynamics of the concentrations of mRNAs and proteins. These models are usually characterized by the presence of highly nonlinear Hill function terms. A typical simplification is to reduce the number of equations by means of a quasi-steady-state assumption on the mRNA concentrations. This yields a class of simplified ODE models. 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. We will show that different models often lead to conflicting conclusions concerning the existence and stability of equilibria and stable oscillatory behaviours. Moreover, we shall discuss, where possible, the viability of making certain modelling approximations (e.g. quasi-steady-state mRNA dynamics or piecewise-linear approximations of Hill functions) and their effects on the overall system dynamics.

Polynomial representations of piecewise-linear differential equations arising from gene regulatory networks

“…However, models with Boolean interactions represent a level of resolution which may be too coarse to describe sliding trajectories, while incorporating sigmoid-type responses helps to solve this problem. Minding this, we stress that unlike the paper [9] we derive sigmoid-based models from Boolean-based models, and not vice versa.…”

“…A possible biological motivation is more controversial, as the multilinearity assumption on regulatory functions is widely accepted in the literature (see e.g. [1], [2], [4], [9] and references therein), although our overall impression is that, in fact, not much is really known about the precise biological and mathematical mechanisms of gene regulations ( [1], [4]). …”

“…The mathematical and biological reasons for that can e.g. be found in [9] and are not discussed here. However, models with Boolean interactions represent a level of resolution which may be too coarse to describe sliding trajectories, while incorporating sigmoid-type responses helps to solve this problem.…”

Analysis of Deterministic Cyclic Gene Regulatory Network Models with Delays

“…Therefore, deterministic models should be studied along with the stochastic models to have a good idea of the system under consideration. Different ODE modeling approaches for gene regulatory networks have been compared in [25]. Dynamic patterns of gene regulation have been investigated in [26] for the two-gene system.…”

Mathematical Modelling of Gene Regulatory Networks

“…Using ordinary differential equations for representing gene regulatory networks concentrations of proteins, mRNAs and other molecules are presented as continuous time variables (Polynikis et al, 2009). Flexibility of ordinary deferential equations allows the description of complex relations between components of the net.…”