Abstract
BackgroundThe linear noise approximation (LNA) is commonly used to predict how noise is regulated and exploited at the cellular level. These predictions are exact for reaction networks composed exclusively of first order reactions or for networks involving bimolecular reactions and large numbers of molecules. It is however well known that gene regulation involves bimolecular interactions with molecule numbers as small as a single copy of a particular gene. It is therefore questionable how reliable are the LNA predictions for these systems.ResultsWe implement in the software package intrinsic Noise Analyzer (iNA), a system size expansion based method which calculates the mean concentrations and the variances of the fluctuations to an order of accuracy higher than the LNA. We then use iNA to explore the parametric dependence of the Fano factors and of the coefficients of variation of the mRNA and protein fluctuations in models of genetic networks involving nonlinear protein degradation, post-transcriptional, post-translational and negative feedback regulation. We find that the LNA can significantly underestimate the amplitude and period of noise-induced oscillations in genetic oscillators. We also identify cases where the LNA predicts that noise levels can be optimized by tuning a bimolecular rate constant whereas our method shows that no such regulation is possible. All our results are confirmed by stochastic simulations.ConclusionThe software iNA allows the investigation of parameter regimes where the LNA fares well and where it does not. We have shown that the parametric dependence of the coefficients of variation and Fano factors for common gene regulatory networks is better described by including terms of higher order than LNA in the system size expansion. This analysis is considerably faster than stochastic simulations due to the extensive ensemble averaging needed to obtain statistically meaningful results. Hence iNA is well suited for performing computationally efficient and quantitative studies of intrinsic noise in gene regulatory networks.
Highlights
The linear noise approximation (LNA) is commonly used to predict how noise is regulated and exploited at the cellular level
The main analytical tool to address this issue has since been the linear noise approximation (LNA) of the Chemical Master Equation (CME) [15,16,17] which allows one to approximate the dynamics of the latter by a set of linear stochastic differential equations from which all moments can be computed in closed form
We use the Effective Mesoscopic Rate Equation (EMRE) and Inverse Omega Square (IOS) methods to study the parametric dependence of the coefficient of variation (CV) and Fano factors of mRNA and protein fluctuations in two examples of stochastic gene regulation involving nonlinear protein degradation, post-transcriptional, post-translational and negative feedback regulation. We show that these results agree with stochastic simulations but in many instances disagree with the LNA results
Summary
The linear noise approximation (LNA) is commonly used to predict how noise is regulated and exploited at the cellular level. The main analytical tool to address this issue has since been the linear noise approximation (LNA) of the Chemical Master Equation (CME) [15,16,17] which allows one to approximate the dynamics of the latter by a set of linear stochastic differential equations from which all moments can be computed in closed form. In this approximation, the mean concentrations of the CME are approximated by the solution of the deterministic rate equations (REs) and the probability distribution of the fluctuations is approximated by a Gaussian. To-date, it is unclear how important these corrections are for many gene regulatory networks of interest
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