In this dissertation I study the role of stochastic fluctuations in numbers of key biochemical molecules in implementing the regulation of gene expression, the processes that cells use to convert information in the gene into gene products. All cellular processes rely on chemical processes with small population sizes and nonlinear birth and death rates. A quantitative description of life in the single cell then entirely relies on mathematical descriptions of the corresponding stochastic processes.
Gene regulation is implemented primarily by proteins, who act upon communications received from their genes through messenger RNAs (mRNAs) and translate that information into appropriate biochemical responses. The response to the signal received from the gene is expressed in terms of the numbers of the relevant proteins.
Chemical reactions are inherently probabilistic events. In the context of gene expression, since the numbers of key biochemical molecules (genes, mRNAs, proteins) involved are small (for e.g., each cell typically has 1-2 copies of the gene), even if genetically identical cells in identical fixed environments are considered, the resultant cell to cell variability in protein numbers may be significant. Thus the statistics of large numbers is typically not applicable to proteins. An important biological issue is whether the activity of a particular protein is determined predominantly by just its abundance or by its cell to cell variation as well.
Most current treatments of stochastic gene expression address this question by either using computer simulations of multi-parameter models, or deriving analytical results for small white-noise added to the rate-kinetic (mean) description of the system, in steady-state. In this dissertation, I develop analytical descriptions of both transient (time-dependent) and non-linear (as in feedback models) effects that are absent in such treatments. Methods of non-equilibrium statistical mechanics are used to achieve this. Some of the results derived are used to analyze experimental data for an immunological system, generated by experimentalists at the Mount Sinai School of Medicine. Since regulated gene expression forms the basis of all life processes, this is vital to understanding how biological systems are designed, function and evolve.