# Estimating Parameters and Hidden Variables in a Non-linear State-space Model of Regulatory Networks

Understanding and identifying biological complex systems at work in the cell requires to develop models able to capture the stochastic nature of biological processes as well as their dynamics. Focusing on gene regulatory networks, we propose a new quantitative model in the form of a dynamical Bayesian network that allows to represent both genes and proteins in the same framework. We start from the nonlinear differential equations of Michaelis-Menten which are the gold-standard to represent biochemical interactions and develop a discrete-time and probabilistic model from these equations. Compared to previous works such as Nachman et al [1], our model takes into account the dependency between the regulatory proteins and the genes that code for them as well as protein-protein interactions and protein degradations. In the resulting nonlinear dynamical system, the proteins concentrations are hidden while gene expressions are observed. In order to learn the model's parameters, we first construct a discrete-time probabilistic model corresponding to our continuous-time state-space model and then derive a Kalman smoother algorithm based on the unscented transformation [2] to recursively estimate the parameters and unobserved protein activities. The generality of the learning method opens the door to various adaptations of the model if required by the biology.

Numerical results on parameter and state estimation for the repressilator [3] and other several small networks are presented and show the relevance of the model.

*Author: Minh Quach, University of Evry*