This proposal concentrates on both stochastic and deterministic models of intracellular reaction networks. If the abundances of the constituent molecules of a reaction network are sufficiently high then the reaction network is typically modeled as a coupled set of ordinary differential equations. If, however, the abundances are low then the standard deterministic models do not provide a good representation of the behavior of the system and stochastic models are used. The simplest stochastic model treats the system as a continuous time Markov chain. More complicated models tend to be hybrid in nature with some components modeled discretely, some diffusively, and some absolutely continuously (that is, solutions of ordinary differential equations with stochastic inputs from the other components). This project has two related goals. The first is to develop rigorous results relating network structure---typically the simplest information to obtain experimentally about networks---with qualitative properties of the dynamics of reaction networks, and to do so in both the deterministic and stochastic settings. The second goal is to develop and analyze numerical simulation methods for stochastically modeled systems that account for both the natural scalings and basic mathematical structure of reaction networks. Stochastically modeled reaction networks often involve multiple natural scales and it is crucial that these natural scales be accounted for when developing and analyzing numerical simulation methods. Also, all of the equations that describe reaction networks are of a very specific mathematical structure that can be utilized to develop efficient algorithms in both the discrete and continuous stochastic contexts.
All of the questions being addressed in this proposal have motivation from the biosciences. Recent advances in experimental methods concerning the dynamics of the cell (such as green fluorescent protein) have increased interest in how biological molecules interact at the cellular level over the course of time. Because Brownian forces are significant at this level, stochastic models for these intracellular reaction networks, combined with analytical and computational tools, are essential if they are to be well understood. This project will also provide a fertile training ground for graduate students. In particular, there is a high demand for well-trained mathematical scientists with the interest and expertise necessary to contribute to the solution of problems arising in biology.
Intracellular reaction networks may be classified as gene regulatory networks, protein interaction networks, or metabolic networks. Together these networks determine the full set of reactions or molecular interactions between a genome and its environment, ultimately defining the growth and development of an organism. If the abundances of the constituent molecules are sufficiently high then the reaction network is typically modeled as a coupled set of ordinary differential equations. If, however, the abundances are low then the standard deterministic models do not provide a good representation of the behavior of the system and stochastic models are used. There were two major goals of this project. Goal 1. It is generally the case that key system parameters of reaction networks are unknown. However, it is often possible to prove theorems that relate the qualitative behavior of the system to network structure alone. Providing such theorems, in both the stochastic and deterministic settings, was the first major goal of this proposal. Goal 2. The second goal of the project was the development of new computational methods for the numerical study of reaction networks with stochastic dynamics. The primary objectives were new Monte Carlo methods for the efficient approximation of (i) expectations and (ii) parametric sensitivities (derivatives of expectations with respect to the key parameter values). The stated goals of the project were largely met. The results obtained under the support of this project will play a role in increasing our understanding of cellular processes, both through analytical results and by providing general purpose computational tools for biologists. For Goal 1, the major results were the following: We proved the so called "Global Attractor Conjecture" for a broad class of networks (those with a single "linkage class"). The Global Attractor Conjecture is the most well known theorem in the field of chemical reaction network theory and has been open for over 40 years. The mathematical methods developed in A Proof of the Global Attractor Conjecture in the Single Linkage Class Case were later used to prove the "Boundedness Conjecture", and have been utilized by other researchers in the study of "endotactic systems." In Stochastic analysis of biochemical reaction networks with absolute concentration robustness, we elucidated the long-term behavior of a large and interesting class of stochastically modeled biochemical processes. Importantly, we proved that systems with a certain network structure and stochastic dynamics will necessarily undergo an extinction event, whereas systems with the same network structure and deterministic dynamics will exhibit a very special form of robustness. Such differences in long-term behavior had been observed previously in isolated examples. The novelty of the our research was in categorizing a whole class of relevant models exhibiting this behavior. For Goal 2, the major results were the following: The development of multi-level Monte Carlo methods for the biochemical setting. We showed how to extend a recently proposed multi-level Monte Carlo approach to the continuous time Markov chain setting, thereby greatly lowering the computational complexity needed to compute expected values of functions of the state of the system to a specified accuracy. The extension was non-trivial, exploiting a coupling of the requisite processes that is easy to simulate while providing a small variance for the estimator. Further, and in a stark departure from other implementations of multi-level Monte Carlo, we showed how to produce an unbiased estimator that is significantly less computationally expensive than the usual unbiased estimator arising from exact algorithms in conjunction with crude Monte Carlo. We thereby dramatically improved, in a quantifiable manner, the basic computational complexity of current approaches. The development of new methods for the computation of parametric sensitivities. We developed multiple methods for the computation of sensitivities. Such computations often for the bottleneck for optimization problems, and so will make many previously impossible computational experiments possible. One highlight of the research was the development of a new finite difference scheme which is easy to implement, and provides estimates in significantly less time than what was the previous state of the art.
