Modelling periodic oscillation of biological systems with multiple timescale networks

Abstract

In this paper, we aim to develop a new methodology to model and design periodic oscillators of biological networks, in particular gene regulatory networks with multiple genes, proteins and time delays, by using multiple timescale networks (MTN). Fast reactions constitute a positive feedback-loop network (PFN), while slow reactions consist of a cyclic feedback-loop network (CFN), in MTN. Multiple timescales are exploited to simplify models according to singular perturbation theory. We show that a MTN has no stable equilibrium but stable periodic orbits when certain conditions are satisfied. Specifically, we first prove the basic properties of MTNs with only one PFN, and then generalise the result to MTNs with multiple PFNs. Finally, we design a biologically plausible gene regulatory network by the cI and Lac genes, to demonstrate the theoretical results. Since there is less restriction on the network structure of a MTN, it can be expected to apply to a wide variety of areas on the modelling, analysing and designing of biological systems.

An Integrated Approach to Global Synchronization and State Estimation for Nonlinear Singularly Perturbed Complex Networks

“…On the other hand, in practice, many dynamical systems possess two-time-scale characteristics, namely, an interaction of 'fast' and 'slow' dynamics such as aircraft and racket systems [34], electric power systems [1], [29] and biological systems [37]. Such kind of systems is governed by both fast and slow dynamics, and customarily referred to as the singularly perturbed systems (SPSs).…”

Bounded synchronisation of singularly perturbed complex network with an application to power systems

“…In other words, there exist small parasitic parameters multiplying the time derivatives of part of the system states. Hence, the dynamics of all nodes can be presented in singularly perturbed system in power networks [11,12] and biological networks [14,15]. Thus, it is important to study the synchronisation of singularly perturbed complex networks.…”

Mean square exponential stability of stochastic genetic regulatory networks with time-varying delays

“…On the contrary, extrinsic noise originates from the random variation of one or more of the externally set control parameters, e.g., the rate constants associated with a given set of reactions. In recent years, there has been a growing interest in the development of stochastic models for describing the behavior of intrinsic and extrinsic noise [2,14,18,29,31,33]. Basically, there are two types of genetic regulatory networks models, i.e., master equation and reactive-rate equation.…”