Hybrid Deterministic-Stochastic Methodology for Simulating Spatial Evolution in Large Populations

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The goal of this project is to develop new computational methods to study the growth and spread of initially small mutant populations that drive biological phenomena like spread of antibiotic resistance or development of treatment resistant tumors. One notoriously difficult problem in evolutionary simulations is the coexistence of very large and very small populations. This is a common occurrence, because random mutations give rise to relatively small clones, which could play an important role in evolution. For example, these small clones could at some later time harbor further mutations that lead to the formation of a 'super-mutant,' which eventually takes over the population. It is the simulation of such scenarios that presents serious computational problems, because the larger the overall population, the slower the computational process. In many realistic scenarios, further complications arise from spatial constraints. Examples include solid tumors and biofilms--bacterial communities with complex spatial structures, implicated both in public health and in industry. To illustrate application of the computational techniques, the optimization of melanoma (skin cancer) treatment will be studied in the presence of resistant mutants that have developed 'addiction' to the drug. Mathematics can guide how the on/off periods of therapy need to be timed to contain resistance. Similar considerations apply to antibiotic-resistant bacteria. Complementing the research will be efforts to introduce under-represented minority high-school students to the process of scientific inquiry through a summer school, with the opportunity to work on projects in computational biology. At the center of enhancing computational speed is the development of a versatile technique capable of efficiently simulating large heterogeneous stochastic populations. Spatial agent-based models of cellular growth will be considered that are common to a wide variety of ecological and evolutionary modeling endeavors. They take into account the processes of cell division and death, as well as mutations and spatial interactions. Deterministic (PDE) approximations of such spatial, stochastic processes generally do not yield accurate time series. In this project, first, deterministic representations of the agent-based models will be constructed by deriving the stochastic master equation of the agent-based model and using the moment closure techniques. This step will provide a fundamental correction to equations based on mean-field behavior. Then, a spatial hybrid stochastic-deterministic algorithm will be developed. The main problem is the 'stiffness' of typical evolutionary systems, resulting from the existence of small, fluctuating populations that can be essential to the final outcome. In traditional methods, this leads to a dramatic decrease of the step size for large populations. Here, a solution to this problem is proposed, by dynamically partitioning the population into small and large subpopulations, with the assumptions that large subpopulations are well described by deterministic laws and are decoupled from the influence of small subpopulations, while the stochasticity of small subpopulations is preserved (and the large populations still affect their dynamics). These approaches will enable simulation of evolutionary processes in large, multi-component, spatially structured evolutionary processes at manageable speeds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.