#### Date of Award

Spring 2012

#### Document Type

Thesis

#### Degree Name

Bachelor of Science

#### Department

Mathematics

#### First Advisor

Dr. Ovidiu Lipan

#### Abstract

This thesis originated from a specific problem from biology. Namely we need to study probabilistic models that represent molecular interactions that take place inside living cells, such as the number of molecular heat-shock proteins present in a cell. Because of the intrinsic discrete nature of the number of molecules present in cells, the fundamental mathematical models are based on Markov processes. For such processes a transition probability matrix describes the evolution of the state of the cell, whereas the state itself, i.e. the number of molecules present at a specific time, is described by a vector. The components of this vector represent the probabilities for finding specific molecule numbers. For example, consider a cell that contains between zero and ten heat-shock protein molecules. The number of heat-shock proteins that help repair a cell undergoing heat shock follows a random process. The components of our vector would represent the probabilities for having between zero and ten protein molecules. The transition probability matrix would include the probabilities of transitions between the number of molecules, in other words the probability that given the cell had five protein molecules that it would increase to six, remain at five, or decrease to four. Thus the next state of the cell is dependent upon the previous state of the cell.

#### Recommended Citation

Cates, Jordan Emile, "Mapping of Stochastic matrices into Polynomial form in the complex plane" (2012). *Honors Theses*. 63.

https://scholarship.richmond.edu/honors-theses/63