Partial Differential Equation Models and Numerical Simulations of RNA Interactions and Gene Expression.

摘要

Our genetic information is stored in the nucleus of our cells via a double helical chain of nucleotides called deoxyribonucleic acid (DNA). DNA is transcribed into a single chain of nucleotides called ribonucleic acid (RNA) which is then translated into proteins. New discoveries of other non-coding macromolecules and their functions along with a new understanding of post-transcriptional protein regulation have influenced the study of these processes. For example, small, non-coding RNAs (sRNA) such as microRNA (miRNA) or small interfering RNA (siRNA) regulate developmental events through certain interactions with messenger RNA (mRNA). By binding to specific sites on a strand of mRNA, sRNA may cause a gene to be activated or suppressed which may turn a gene “on” or “off” To understand these interactions, we developed a mathematical model which consists of N+1 coupled partial differential equations that describe mRNA and sRNA interactions across cells and tissue. These equations illustrate how one small, non-coding RNA segment and N target mRNA segments interact with each other depending on transcription rates, independent and dependent degradation rates, and the rate of intercellular mobility of each species. By varying diffusion coefficients (mobility of each species) and time dependence (creating a steady state), the system of N+1 coupled PDEs can be studied as three separate well-posed systems of equations: a single, nonlinear diffusion equation; coupled diffusion equations at steady state; and coupled diffusion equations with time dependence. This dissertation analyzes the mathematical models created and shows the implementation of consistent, efficient numerical methods such as modified finite difference methods and a form of alternating iteration to solve these equations. The numerical simulations show that when sRNA has mobility across tissue, the concentration profiles of mRNA display a sharp interface between tissue with high mRNA concentration and tissue with low mRNA concentration. If mRNA mobility across tissue is added, the concentration profile of mRNA is smoothed across the tissue. These simulations suggest that the mobilities of sRNA and mRNA contribute to the behavior of mRNA concentrations across tissue. In addition, this model may be utilized to illustrate similar types of interactions between multiple chemical species.