Dissertation

Dynamical models in gene expression
Michal Hojčka
PhD thesis advisor: Pavol Bokes

Abstract: We study the methods and approaches used in simulating systems of biochemical reactions. We present a stochastic model motivated by gene expression which includes production of protein molecules and their interactions with decoy binding sites. Then we formulate the associated Master equation. We focus on the distribution of free protein which cannot be expressed in a closed form. Therefore we present three different approaches to obtain it: employing singular perturbation reduction to obtain quasi-steady-state solution, simulating through stochastic algorithms and solving the associated system of ODEs. We also add large-system-size scaling to obtain statistical characteristics of free protein distribution like the Fano factor in a very simple form. We show that the Fano factor is greater than one for the intermediate levels of binding sites in contrast with Poissonian character (the Fano factor equals one) for no binding sites of their excess. In addition, we investigate the mRNA – microRNA system of reactions. Also here we derive quasisteady- state solution and express the formula for the Fano factor in a closed form. It yields values below one for non-extreme levels of interaction strength. All results are supported and illustrated with the help of numerical simulations.

References
[1] Michal Hojcka, Pavol Bokes: Non-monotonicity of Fano factor in a stochastic model for protein expression with sequesterisation at decoy binding sites, Biomath Vol. 6, No. 2 (2017).