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

这个项目的目标是开发新的计算方法来研究最初导致抗生素耐药性传播或治疗耐药肿瘤等生物现象的小型突变种群的增长和扩散。进化模拟中一个出了名的难题是非常大和非常小的种群共存。这是一种常见的现象，因为随机突变会产生相对较小的克隆，这可能在进化中发挥重要作用。例如，这些小克隆可能会在以后的某个时间藏匿进一步的突变，导致形成一个最终接管种群的“超级突变体”。正是这种情景的模拟带来了严重的计算问题，因为总体人口越大，计算过程就越慢。在许多现实场景中，空间限制会带来更大的复杂性。例如实体肿瘤和生物膜--具有复杂空间结构的细菌群落，与公共卫生和工业都有关联。为了说明计算技术的应用，将研究黑色素瘤(皮肤癌)治疗的优化，其中存在对该药物产生“成瘾”的耐药突变。数学可以指导如何对治疗的开/关期进行计时，以遏制耐药性。类似的考虑也适用于抗药性细菌。作为对这项研究的补充，将努力通过暑期学校向未被充分代表的少数族裔高中生介绍科学探究的过程，并有机会参与计算生物学的项目。提高计算速度的核心是开发一种通用的技术，能够有效地模拟大型异质随机总体。基于空间代理的细胞生长模型将被认为是广泛的生态和进化建模努力的共同之处。它们考虑了细胞分裂和死亡的过程，以及突变和空间相互作用。这种空间随机过程的确定性(PDE)近似通常不会产生准确的时间序列。在该项目中，首先，通过推导基于代理的模型的随机主方程并使用矩闭合技术来构造基于代理的模型的确定性表示。这一步将从根本上修正基于平均场行为的方程。然后，将开发一种空间混合随机-确定性算法。主要问题是典型进化系统的“僵硬”，这是由于小而波动的种群的存在可能对最终结果至关重要。在传统方法中，这会导致大种群的步长急剧减小。本文提出了一种解决这一问题的方法，将种群动态划分为大子种群和小子种群，假设大子种群可以用确定性定律很好地描述，并且与小子种群的影响解耦，同时小子种群的随机性保持不变(大种群仍然影响它们的动态)。这些方法将能够以可控的速度模拟大型、多组件、空间结构的进化过程。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.