Probability Problems from Genetics and Ecology

我们建议研究由遗传学和生态学引起的概率论中的一些问题。我们将扩展我们对有益突变固定影响的分析，开发分析结果以帮助量化Hill-Robertson干扰的影响，当第二个有利突变在第一个突变达到固定之前发生时，并研究密码子使用偏差相关问题的扩散近似。我们将获得最近作为重组谱系近似方法引入的序列马尔可夫聚结的严格结果，研究果蝇和其他具有大有效种群规模的生物的调控序列进化的本质；研究基因重复持久性的亚功能化解释，并考虑随机布尔网络，以深入了解基因网络。受生态学的启发，我们将使用空间模型来理解个体之间出现和维持合作所需的条件，我们将研究当控制捕食者-猎物和流行病模型中相互作用的参数演变时发生的变化。在过去的十年里，我们看到了大量DNA序列数据的产生。然而，掌握控制生命的说明书只是一个开始，我们需要了解基因的表达是如何被控制的，以及基因如何在生物网络中相互作用。这些相互作用不能直接观察到，所以需要建立数学模型来检验假设，并推断遗传系统是如何进化的。特别是，我们的研究将调查(i)由于调控序列进化而发生的变化有多快，这是人类和黑猩猩之间许多差异的常见解释，（ii）亚功能化假说是否为基因重复保存的悖论提供了令人满意的解释，以及（iii）提高我们对适应性进化引起的变异模式的理解。这仅仅是生物学和数学之间众多交叉领域的三个例子。我们建议的一个重要方面是培养研究生在这个令人兴奋的领域工作。

We propose to investigate a number of problems in probability theory that arise from genetics and ecology. We will extend our analysis of the impact of the fixation of beneficial mutations, develop analytical results to help quantify the effects of Hill-Robertson interference, which occurs when a second advantageous mutation occurs before the first one reaches fixation and study diffusion approximations for the related problem of codon usage bias. We will obtain rigorous results for the sequentially Markovian coalescent, which has recently been introduced as an approximation for genealogies with recombination, investigate the nature of regulatory sequence evolution in Drosophila and other organisms with large effective population size; study the subfunctionalization explanation of the persistence of gene duplicates, and consider random Boolean networks, in order to gain insights into gene networks. Inspired by ecology, we will use spatial models to understand conditions needed for the emergence and maintenance of cooperation between individuals, and we will investigate changes that occur when the parameters governing the interactions in predator-prey and epidemic models evolve.The last decade has seen the generation of incredible amounts of DNA sequence data. However, having the text of the instruction manual that controls life is just the beginning, we need to understand how the expression of genes is controlled and how genes interact in biological networks. These interactions cannot be directly observed so one needs to develop mathematical models to test hypotheses, and to infer how genetic systems evolved to operate as they do. In particular, our research will investigate (i) how rapidly changes can occur due to regulatory sequence evolution, which is commonly quoted explanation for many of the differences between humans and chimpanzees, (ii) whether or not the subfunctionalization hypothesis provides a satisfactory explanation of the paradox of the preservation of gene duplicates, and (iii) to improve our understanding the patterns of variations caused by adaptive evolution. These are just three examples of many at the interface between biology and mathematics. An important aspect of our proposal is to train graduate students to work in this exciting area.