On moments and timing: stochastic analysis of biochemical systems

Date
2018
Journal Title
Journal ISSN
Volume Title
Publisher
University of Delaware
Abstract
At the level of individual living cells, key species such as genes, mRNAs, and proteins are typically present in small numbers. Consequently, the biochemical reactions involving these species are inherently noisy and result in considerable cell-to-cell variability. This thesis outlines two mathematical tools to quantify stochasticity in these biochemical reaction systems: (i) a novel computational method that provides exact lower and upper bounds on statistical moments of population counts of important species, and (ii) a first-passage time framework to study noise in the timing of a cellular event that occurs when population count of an underlying regulatory protein attains a critical threshold. ☐ The method to compute bounds on moments builds upon the well-known linear dynamical system that describes the time evolution of statistical moments. However, except for some ideal cases, this dynamical system is not closed in the sense that lower-order moments depend upon some higher-order moments. To overcome this issue, our method exploits the fact that statistical moments of a random variable must satisfy constraints that are compactly represented through the positive semidefiniteness of moment matrices. We find lower and upper bounds on a moment of interest via a semidefinite program that includes linear constraints obtained from moment dynamics, along with semidefinite constraints on moment matrices. We further show that these bounds improve as the size of the semidefinite program is increased by including dynamics of more moments as well as constraints involving them. We also extend the scope of this method for stochastic hybrid systems, which are a more general class of stochastic systems that integrate discrete and continuous dynamics. ☐ The second tool proposed in this thesis - a first-passage time framework to study event timing - is based on the premise that several cellular events in living cells occur upon attainment of critical levels by corresponding regulatory proteins. Two particular examples that we study here are the lysis of a bacterial cell infected by the virus bacteriophage lambda and the cell-division in exponentially growing bacterial cells. We provide analytical calculations for the first-passage time distribution and its moments for both these examples. We show that the first-passage time statistics can be used to explain several experimentally observed behaviors in both these systems. Finally, the thesis discusses potential directions for future research.
Description
Keywords
Biological sciences, Applied sciences, First-passage time, Gene expression, Moment approximation
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