The Monte Carlo EM method for the parameter estimation of biological models

It is often the case in modeling biological phenomena that the structure and the effect of the involved interactions are known but the rates of the interactions are neither known nor can easily be determined by experiments. This paper deals with the estimation of the rate parameters of reaction networks in a general and abstract context. In particular, we consider the case in which the phenomenon under study is stochastic and a continuous-time Markov chain (CTMC) is appropriate for its modeling. Further, we assume that the evolution of the system under study cannot be observed continuously but only at discrete sampling points between which a large amount of reactions can occur. The parameter estimation of stochastic reaction networks is often performed by applying the principle of maximum likelihood. In this paper we describe how the Expectation-Maximisation (EM) method, which is a technique for maximum likelihood estimation in case of incomplete data, can be adopted to estimate kinetic rates of reaction networks. In particular, because of the huge state space of the underlying CTMC, it is convenient to use such a variant of the EM approach, namely the Monte Carlo EM (MCEM) method, which makes use of simulation for the analysis of the model. We show that in case of mass action kinetics the application of the MCEM method results in an efficient and surprisingly simple estimation procedure. We provide examples to illustrate the characteristics of the approach and show that it is applicable in case of systems of reactions involving several species.