The construction and the estimation of quantitative models of gene regulatory networks and metabolic networks is one of the task of Systems Biology. Such models are useful because they provide tools for simulating and predicting biological systems. Various approaches have been proposed, such as graphical models , Bayesian dynamical models or Ordinary Differential Equations (ODE's) . For the latter, one can also expect to derive parameters that often have a meaningful biological sense. We focus on the estimation of a parameter theta indexing a (vector) ODE, from an observed time series (concentration profiles) which may be nonlinear (e.g. due to the use of Michaelis-Menten dynamics or mass action law). Even when the likelihood is simple (in the case of Gaussian error noise), the computation of the Maximum Likelihood Estimator remains hard because of the burden of the optimization step. Indeed, the implicit definition of the model necessitates the integration of the ODE for each evaluation of the likelihood. Moreover, the likelihood may have numerous local maxima we need to avoid, hence the exploration of the parameter space may be computer-intensive. We propose then an alternative (frequentist) estimator of theta based on a preliminary spline estimator of the solution of the ODE. We use a simple characterization of theta that enables to derive a learning algorithm avoiding the integration of the ODE, and that can split the estimation of a vector differential equation in several estimations of scalar differential equations. We illustrate this algorithm with different models used in Systems Biology and we sketch how it can be adapted to various settings encountered by the practitioner.

Joint work with Chris Klaassen and Florence d'Alché-Buc.