Stochastic modeling and simulation of biological systems

The high complexity of biological systems creates numerous challenges in their modeling and simulation. The dissertation is concerned with some fundamental problems in stochastic modeling and control of genetic regulatory network systems, primarily we explore three issues: firstly, the feasibility of using average behavior of stochastic master equation models to generate control policies for altering the behavior of biological systems; the second issue is design of approaches to reduce the complexity of stochastic master equation model simulation and the third topic considered is the design of control approaches when the stochastic model is unknown. Further explanation of the three topics is presented in the following paragraphs.

Stochastic master equation (SME) models can provide detailed representation of genetic regulatory system but their use is restricted by the large data requirements for parameter inference and the inherent computational complexity involved in its simulation. To mitigate the complexity issue, we approximate the expected value of the output distribution of the SME by the output of a deterministic Differential Equation (DE) model. The mapping provides a technique to simulate the average behavior of the system in a computationally inexpensive manner and enables us to use existing tools for DE models to control the system. The effectiveness of the mapping and the subsequent intervention policy design was evaluated through a biological example.

In the next chapter, we attempt to tackle the inherently high computational complexity of the stochastic master equation by designing efficient approaches for its simulation. We present a new approach to stochastic model simulation based on Kronecker product analysis and approximation of Zassenhaus formula for matrix exponentials. Simulation results on model biological systems illustrate the comparative performance of our modeling approach to stochastic master equations with significantly lower computational complexity. We also provide a stochastic upper bound on the deviation of the steady state distribution of our model from the steady state distribution of the stochastic master equation.

The third issue explored in the dissertation is concerned with the scenario when the stochastic model is unknown and we are interested in design of drug combinations that can produce a desired system response. We consider an exploration approach where we keep learning about the system based on response to different combination inputs.

The enormous number of possible drug combinations constrains exhaustive experimentation approaches and personal variations in genetic diseases restrict the use of prior knowledge in optimization. We present a stochastic search algorithm consisting of a parallel experimentation phase followed by a combination of focused and diversified sequential search. We evaluated our approach on seven synthetic examples four of them evaluated twice with different parameters and two biological examples on bacterial and lung cancer cell inhibition response to combination drugs. The performance of our approach as compared to recently proposed Adaptive Reference Update approach was superior for all the examples considered, achieving an average of 45% reduction in the number of experimental iterations.