Bifurcation analysis informs Bayesian inference in the Hes1 feedback loop

A Markov chain Monte Carlo approach can then be used to sample from the appropriate posterior probability distributions, provided that suitable prior distributions can be found for the unknown parameter values. Choosing these priors is therefore a vital first step in the inference process. We study here a negative feedback loop in gene regulation where an ODE incorporating a time delay has been proposed as a realistic model and where experimental data is available. Our aim is to show that a priori mathematical analysis can be exploited in the choice of priors.By focussing on the onset of oscillatory behaviour through a Hopf Bifurcation, we derive a range of analytical expressions and constraints that link the model parameters to the observed dynamics of the system. Computational tests on both simulated and experimental data emphasise the usefulness of this analysis.Mathematical analysis not only gives insights into the possible dynamical behaviour of gene expression models, but can also be used to inform the choice of priors when parameters are inferred from experimental data in a Bayesian setting.Mathematical models can help biologists understand the mechanisms and dynamics behind their experimental observations (Tomlin et al. [1]). The most widely used approach to modelling the dynamics of a genetic network is to employ systems of ordinary differential equations (ODEs) (Voit [2] and de Jong [3]). These models have biological parameters, some of which can be measured experimentally and some of which cannot. Parameter estimation, that is, recovering unknown parameters from experimental data, is an important step towards obtaining a good model that can not only explain observed results but can also be used for prediction and "what if" scenarios.The inference of parameters, from real biological data, within a Bayesian framework is a relatively new, albeit currently very active area. Bayesian inference and Markov chain Monte Carlo (MCMC) methods have been recently advo