Analytical and Computational Methods for Biological Interaction Systems
Abstract
This research concentrates on the stochastic (i.e. random) mathematical models that arise in cell biology, and resides at the interface of biology, chemical engineering, stochastic analysis, and computational mathematics. The work includes: (i) theoretical/foundational research aimed at elucidating the general principles of biological interaction networks and (ii) applied/computational research designed to help researchers outside of mathematics greatly increase their research capabilities. Biological interaction systems (which include intracellular processes, viral infections, population interaction networks, etc.) are typically modeled in one of three ways. If the counts of the constituent species are high, then the evolution of their concentrations are often modeled via a system of ordinary differential equations. If the counts are moderate (perhaps order 1,000 or 10,000), then they may be approximated by some form of continuous diffusion process. If the counts are low, then the system is typically modeled stochastically as a discrete-space continuous-time Markov chain. Due in part to the appearance of new technologies -- most notably fluorescent proteins -- there is now a large literature demonstrating that the fluctuations arising from the effective randomness of individual interactions can have significant consequences on the emergent behavior of the system. In such cases, stochastic models combined with both analytical and computational tools are essential if these systems are to be well understood. The models of interest can be depicted graphically via a reaction graph, which is a graphical representation of the interactions between the constituents of the model. These reaction graphs can be extraordinarily complex; for example, there are over 20,000 genes in the human genome and the proteins they encode may be modified in myriad ways. Further, cellular systems often have different sub-systems that operate on multiple different scales (both temporally and in terms of copy numbers), with the species operating at one scale greatly influencing those at a different scale. However, hidden within the complexity there are often underlying structures that, if properly quantified, give great insight into the dynamical or stationary behavior of the system. As mathematics is particularly adept at finding patterns in complexity, this is one arena, in determining how general network structure relates to system properties, that mathematics has a significant role to play in biology. This topic constitutes the first major focus of this proposal. Specifically, we will relate properties of the reaction graph, which are easy to check, with dynamical properties of the underlying mathematical model. The ultimate goal is to discover the hidden principles of biology and, in particular, to understand how the complex networks found in biology produce their emergent behaviors. While analytical techniques give great insight into classes of models, it is often numerical techniques that are utilized to understand specific models. The second major focus of this proposal therefore pertains to computational methods. It is rarely the case that generating a single realization of a given process is challenging. Instead, the challenge often comes from requiring ensembles of trajectories in order to approximate distributions, expectations, and/or to perform optimization. The projects therefore focus on two well-defined problems that form the bottleneck for many computational experiments in systems biology, (i) Monte Carlo for expectations, and (ii) Monte Carlo for parametric sensitivities (derivatives of expectations with respect to parameters, which are critical for optimization problems). The research will make feasible many realistic modeling and simulation scenarios that are beyond the range of existing techniques.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Feb 19, 2019
- Source ID
- W911NF1810324
Entities
People
- D. B. Anderson
Organizations
- Army Contracting Command
- United States Army
- University of Wisconsin–Madison