Overview of Current Projects

Computing Invariant Manifolds in Coupled Chemical Reactions

Chemical reactions often unfold over short (fast) and long (slow) timescales. When the kinetics of a particular reaction is subject to a disparity in timescales, the majority of the reaction occurs on the slow timescale(s) and, in the slow regime, the differential equation (DE) model that describes the reaction (either deterministic or stochastic) can be reduced via slow manifold projection. Slow manifolds are invariant manifolds that exist in the phase-space of the DE model, and they describe the quasi-steady-state (QSS) evolution of the reaction. While the manifold structure of two-dimensional (i.e., two chemical species) is fairly well understood, many coupled reactions (which are equipped with more than two chemical species) often contains multiple separated fast and slow timescales (i.e., 3 or 4 timescales). Moreover, the phase-space description of coupled reactions often contain a hierarchy of invariant manifolds. Current research seeks to develop algorithms capable of computing the manifold structure present in the phase-space of coupled reactions.

The Kinetics of Phase-Separated Droplets in Cell Cytoplasm

Membraneless organelles have been shown to emerge in the cytoplasm of cells as a result of liquid-liquid phase-separation (LLPS), and the specific role of these organelles is currently (and intensely) being studied by both experimentalists and theorists. In a nutshell, these organelles consist of locally concentrated cellular material such as (and not necessarily limited to) reactants (substrates) and catalysts (enzymes). It has been suggested that the general function of these phase-separated droplets (organelles) is to adjust and regulate specific biochemical reactions within the cell. However, the dynamics and kinetics of reactions occurring within a phase-separated environment is poorly understood. From a mathematical point of view this is fascinating: first, even without diffusion, chemical reactions often occur over a fast and slow timescales; second, the presence of diffusion introduces another timescale (i.e., the diffusion timescale). In a phase-separated environment, the diffusion timescales of the reactants and catalysts will naturally be widely separated. Hence, multiple timescales exist not only due to the rate kinetics but also due to disparate diffusion rates among chemical specifies. Third, in certain environments, the length scale of the droplets may be small in comparison to the natural length scale of the reaction domain. Thus, disparate spatial scales can coexist with disparate timescales. Current research seeks to analyze the compartmental chemistry and kinetics of phase-separated environments within cells through asymptotic analysis, geometric singular perturbation theory, and computational methods and numerical simulation.

Inverse Problems and Parameter Estimation

The in vitro quantification of enzyme activity is a particularly important component of drug targeting and drug development. Specifically, enzyme activity is usually quantified through means of an assay, in which the product that forms of a result of an enzyme catalyzed reaction is recored in the form of a progress curve. In turn, the progress curve can be employed to estimate the kinetic parameters (i.e., rate constants) of the  particular mathematical model that describes the reaction. However, not all enzyme reactions can be observed experimentally; thus, non-observable primary reactions must be coupled to a secondary reaction that is experimentally observable. Consequently, the enzyme activity of the non-observable reaction must be extracted from the progress curve of the observable reaction. Current research aims to develop protocols and algorithms through which enzyme activity can be accurately estimated and quantified from coupled enzyme assays.