Euler-Maruyama approximation and maximum likelihood estimator for a stochastic differential equation model of the signal transduction process

Abstract

The conversion of an external signal by the cell into internal molecules is called the signal transduction process. In this paper, the role of the G-protein coupled receptors (GPCRs) is considered because GPCRs constitute the largest family of protein on eukaryotic cell membrane. Furthermore, GPCRs can detect the external signals and transduce them into the cell leading to the production of the secondary hormone or massager such as cAMP (cyclic adenosine monophosphate). The abnormality of the signal transduction process can cause many serious diseases. Better understanding of GPCRs and the signal transduction process should be greatly beneficial for pharmacological research. Here, a stochastic differential equation (SDE) model of the signal transduction in the cell has been proposed and investigated. An SDE model has been modified from the deterministic model proposed by Rattanakul et al. (2009) to take into account the observation that experimental data on cAMP measurements often show random fluctuations (Ueda and Shibata, 2007). The model parameters are then estimated by using the Euler-Maruyama approximation and maximum likelihood estimators. With the estimated parameters, the stochastic model simulations are found to provide a better dynamic representation of the transduction system with noise, in comparison to the deterministic model which does not take into account the random fluctuations in the production of the secondary signaling hormone, cAMP, which could significantly impact the amplification effect that it has on the primary signaling hormone. Such stochastic behavior can significantly influence the outcome of the process which controls the proper function of the human body. We discuss the simulation results of the SDE model with estimated parametric values in comparison with those obtained from the deterministic model proposed by Ratanakul et al. [80], with parameter values estimated by a genetic algorithm.