Algebraic approaches to molecular information processing

Abstract

This dissertation describes a series of applications and extensions of a framework for mathematical modeling of biochemical systems: the graph-based “linear framework” for timescale separation. In Chapter 2, we introduce “parameter geography” as a general approach for the quantitative assessment of parametric robustness in biochemical systems, and demonstrate how the geometric and topological features of a parametric region of interest may be translated into interpretations of robustness. In performing this analysis, we also integrate the linear framework—with which we can describe the steady-state behavior of an entire class of two-enzyme post-translational modification (PTM) systems without having to specify mechanistic details—with recent advances in numerical algebraic geometry, as implemented in the software packages Bertini, Paramotopy, and alphaCertified. These innovations collectively allow us to survey the steady- state of these PTM systems across parameter space at unprecedented scale, and thereby draw several novel observations regarding the robustness of bistability in PTM. In Chapter 3, we extend the linear framework into the “transient” regime of Markov processes, by providing a systematic account of first-passage times (FPTs). We explicate several surprising and beautiful connections between the original framework, which applies the Matrix-Tree theorem to describe the steady-state of any given Markov process, and the extended framework, which applies the All-Minors Matrix-Tree theorem to quantify any FPT on any given Markov process; and we illustrate how the extended framework subsumes various previously reported calculations. Finally, in Chapter 4, we employ this extended framework to undertake a rigorous analysis of the mechanism by which Streptococcus pyogenes Cas9 recognizes whether a DNA substrate matches the loaded guide RNA. By combining the extended framework with asymptotic analysis, dynamic programming, and an arsenal of other techniques, we are able to characterize how multiple disparate measures of specificity are differentially influenced by the model parameters.