Estimating parameters and hidden states in biological networks with particle filters
Minh Quach, Nicolas Brunel and Florence d'Alché-Buc
In: Probabilistic Modelling of Networks and Pathways, 04 - 05 September 2007, Sheffield, UK.
Identifying biological networks requires to develop models able to capture their dynamics and statistical learning methods to estimate their parameters from time-series measurements.
In particular, Ordinary Differential Equations (ODE's) are a rich family of quantitative models, but their estimation remains a bottleneck for reverse engineering, especially when the
biological processes are nonlinear and partially observed. In a recent work , we have proposed a state-space model, derived from ODE's used in Systems Biology, that can encompass regulatory networks, metabolic networks or signaling pathways. For this model, we have derived a Bayesian estimation procedure for both parameters and hidden variables based on nonlinear filtering and a particular approximation scheme: the Unscented Kalman Filter (UKF).
Despite satisfactory results, the UKF approximation possesses some limitations such as few theoretical results and a limited range of applications. We propose then to use Sequential Monte
Carlo methods (SMC), also known as particle filters , which are now standard methods for filtering nonlinear and non Gaussian processes. SMC methods provide a nonparametric
approximation of the filtering probability discretely supported by the so-called particles whose convergence properties have been intensively studied [1, 2].
In this work, we develop a SMC approach for the Bayesian estimation of the (kinetic) parameters and hidden states, by considering the parameters as additional hidden states with no evolution. Despite the generality of SMC methods, the deterministic evolution of the hidden variables implies a fast degenerescence of standard algorithms, e.g. bootstrap filter. To overcome this problem, we use a solution proposed by Liu and West , which relies on an adapted kernel smoothing of the particle approximation. The method is illustrated on the Repressilator, an ODE proposed for a gene regulatory network, and on an ODE for the JAK-STAT signaling pathway . Experimental results show that particle filters provide similar results to UKF for the parameter estimation and lower Mean Square Error for the state estimation, while offering a greater versatility.