On Stochastic Differential Equations: Theory and Biochemical Applications

Abstract

The time evolution of chemical systems is traditionally modeled using deterministic ordinary differential equations. The deterministic approach, in general, describes the average time series behavior of the system, but is incapable of capturing the random nature of chemical reactions. Thus, in order to be able to construct physically realistic models, stochastic methods have to be used. This is especially the case in many biochemical applications in which the reaction volume is small, the molecular concentrations are low, and the impact of random fluctuations is significant.

Biochemical reactions can be modeled stochastically using numerous different approaches. For instance, stochastic differential equations have been proposed as a promising method to model biochemical reactions stochastically. The main goal of this Master of Science Thesis is to study the theory behind stochastic differential equations and to apply the theory to the modeling of biochemical systems. First, probability theory and stochastic processes are studied from the measure-theoretic point of view. Second, fundamental definitions and results from the field of stochastic calculus are presented. Applications based on the theory will then follow.

In the simulation part, two biochemical systems, the chemical degradation and the Lotka reactions, are modeled by means of stochastic differential equations. The simulation of Lotka reactions proves the excellence of this stochastic approach. The stochastic differential equation model takes the natural fluctuations into account, and is thus capable of describing the dynamical properties of the system, even in the case, in which the traditional deterministic model fails to capture the temporal behavior.

Asiasanat: stochastic differential equations, Ito calculus, modeling biochemical systems