Probability Problems from Genetics and Ecology

我们建议调查一些问题的概率论，产生于遗传学和生态学。我们将扩展我们的有益突变的固定的影响的分析，开发分析结果，以帮助量化的Hill-Robertson干扰的影响，这发生在第二个有利的突变发生在第一个达到固定和研究扩散近似密码子使用偏置的相关问题。我们将获得严格的结果，连续马尔可夫合并，最近被引入作为一个近似的系谱重组，研究调节序列进化的性质在果蝇和其他生物体具有大的有效人口规模;研究亚功能化解释的持续性基因重复，并考虑随机布尔网络，以获得深入了解基因网络。受生态学的启发，我们将使用空间模型来理解个体之间合作的出现和维持所需的条件，并且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发生变化时发生的变化。然而，掌握控制生命的说明书文本只是一个开始，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。这些相互作用无法直接观察到，因此需要开发数学模型来测试假设，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。这些只是生物学和数学之间许多接口的三个例子。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.