Stochastic modelling of the growth of C. Acetobutylicum with missing data

Abstract

Stochastic influences play an important role in various areas especially in the area of biological process. Stochastic differential equation is the differential equation in which the terms of their characteristic involve stochastic process or ‘white noise’. In this study, we used the stochastic differential equation to describe the population dynamics of the cell growth of C. Acetobutylicum in fermentation process. Stochasticity incorporated into the model via its growth coefficient- Umax-ymax. We used the model of stochastic logistic to model the growth of cell against time at different initial pH. The range of initial pH level is from 4.0 until 7.0. The missing data were estimated using expectation maximization (EM) and regression approach. The estimated parameters were obtained using simulated maximum likelihood. The estimated ^ max and s values of stochastic differential equation at five different initial pH level (4.0, 4.5, 5.0, 6.0, and 7.0) are (0.1098, 0.09), (0.154, 0.04), (0.41, 0.01), (2.92, 0.113) and (0.341, 0.09) respectively. Five different trajectories for different initial pH were formed based on EM and regression approximation. It was found that all trajectories based on EM show a lower mean square error as compared to those approximated using regression. Thus, EM estimate is a better estimator for missing data and the model is adequate. It was also found that the means square error for stochastic are lower than deterministic model at five different initial pH. This implies that stochastic logistic model is better in describing the growth of cell C.Acetobutylicum in fermentation process compared to deterministic model.