Available online at www.sciencedirect.com BioSystems 93 (2008) 133–140 Stochastic simulations on a model of circadian rhythm generation Shigehiro Miura a , Tetsuya Shimokawa b , Taishin Nomura a,c,∗ a Division of Bioengineering, Graduate School of Engineering Science, Osaka University, Osaka 560-8531, Japan b Graduate School of Frontier Biosciences, Osaka University, Japan c The Center for Advanced Medical Engineering and Informatics, Osaka University, Japan Received 18 April 2008; received in revised form 1 May 2008; accepted 5 May 2008 Abstract Biological phenomena are often modeled by differential equations, where states of a model system are described by continuous real values. When we consider concentrations of molecules as dynamical variables for a set of biochemical reactions, we implicitly assume that numbers of the molecules are large enough so that their changes can be regarded as continuous and they are described deterministically. However, for a system with small numbers of molecules, changes in their numbers are apparently discrete and molecular noises become significant. In such cases, models with deterministic differential equations may be inappropriate, and the reactions must be described by stochastic equations. In this study, we focus a clock gene expression for a circadian rhythm generation, which is known as a system involving small numbers of molecules. Thus it is appropriate for the system to be modeled by stochastic equations and analyzed by methodologies of stochastic simulations. The interlocked feedback model proposed by Ueda et al. as a set of deterministic ordinary differential equations provides a basis of our analyses. We apply two stochastic simulation methods, namely Gillespie’s direct method and the stochastic differential equation method also by Gillespie, to the interlocked feedback model. To this end, we first reformulated the original differential equations back to elementary chemical reactions. With those reactions, we simulate and analyze the dynamics of the model using two methods in order to compare them with the dynamics obtained from the original deterministic model and to characterize dynamics how they depend on the simulation methodologies. © 2008 Elsevier Ireland Ltd. All rights reserved. Keywords: Circadian rhythm; Stochastic simulation; Interlocked feedback model 1. Introduction Biological phenomena are often modeled by differential equations. States of a given system are usually described by continuous real values, and in many cases the equations form a set of ordinary differential equations. Biochemical reactions with large numbers of molecules can be well described by the mass action law, and changes in concentrations of molecules are described by deterministic ordinary differential equations, although each reaction occurs stochastically. However, if the numbers of molecules involved in the reactions are small, stochastic nature of individual reactions may dominate the dynamics and changes of the numbers of molecules should be treated as discrete. In such cases, the corresponding mathemat- ∗ Corresponding author at: Division of Bioengineering, Graduate School of Engineering Science, Osaka University, Osaka 560-8531, Japan. Tel.: +81 6 6850 6532. E-mail address: taishin@bpe.es.osaka-u.ac.jp (T. Nomura). ical models describing the reactions must include stochastic effects. Circadian rhythms of living organisms are generated, in many cases, by periodic gene expression, which is a sequence of chemical reactions. Several mathematical models for circadian rhythms have been proposed (Ueda et al., 2001; Gillespie, 1976, 1977). The numbers of molecules concerned with such clock gene expressions are usually small, and the concentration of each molecule involved is low despite that the volume of the system is also small. In such cases, it is difficult to consider the concen- tration as a continuous variable. The amount of changes in the number of molecules should be regarded as discrete. Therefore, it is expected that the deterministic ordinary differential equa- tions may not be appropriate for modeling the gene expression for the circadian rhythm generation. There are several different ways to model and simulate such chemical reactions with small numbers of molecules. One may expect a standard Langevin equation with an additive Gaussian white noise to the corresponding deterministic ordinary differen- tial equation representing the chemical reactions with the mass 0303-2647/$ – see front matter © 2008 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.biosystems.2008.05.002