Application of Large Deviations to Genetic Evolution of Bacterial Populations

细菌或病毒的种群通过出现和固定有益的突变表现出强大的适应性。例如，这种突变可能与细菌对抗生素的耐药性或从动物转移到人类的病毒株的出现有关。 目前，存在各种描述细菌或病毒种群进化的数学模型，但仍缺乏可用于研究大型细菌或病毒种群进化轨迹的数学工具。因此，这项工作的主要目标是开发适当的数学工具，用于研究细菌或病毒中的罕见突变事件。特别是，研究人员将使用人口轨迹的大偏差方法来发展理论基础和实用的计算工具，以研究细菌种群的进化轨迹，并将其具体应用于大肠杆菌的长期实验室实验分析。大偏差技术将在人口直方图离散模型的新背景下发展。与经典的Wentzell-Freidlin方法的“小”扩散，有没有一般推导的成本函数的离散马尔可夫链的连续状态空间，使具体的分析和数值结果的离散模型是相当稀缺。然而，大的偏差提供了显着的计算优势，在进化动力学的背景下，由于这个问题的特殊结构。 特别是，进化轨迹的成本函数将以显式形式导出。这将导致开发一种新的离散差分方程的成本最小化的轨迹，从而使一个有效的数值“测地线射击”在相反的时间来计算给定的初始和终端状态之间的最有可能的人口演变。 这种新的算法将被用来估计（非常小）的概率，罕见的遗传事件，如两个连续的基因型固定之间的过渡，并计算最有可能的序列的基因型固定导致从祖先基因型到目前占主导地位的基因型。此外，在拟议的研究过程中开发的数字代码将被用来研究各种上位性的情况。因此，这项工作将产生创新的计算和概念工具，从而更好地了解细菌或病毒种群的遗传进化。

Populations of bacteria or viruses exhibit strong adaptivity through emergence and fixation of beneficial mutations. For instance, such mutations can be related to bacterial resistance to antibiotics or emergence of viral strains transferable from animal to humans. Currently, there exist various mathematical models which describe the evolution of bacterial or viral populations, but mathematical tools which can be used to study evolutionary trajectories of large bacterial or viral populations are still lacking. Therefore, the main goal of this work is to develop proper mathematical tools for studying rare mutational events in bacteria or viruses. In particular, the investigators will use the large deviation approach for population trajectories to develop theoretical foundation and practical computational tools to study the evolutionary trajectories of bacterial populations with concrete applications to the analysis of long term laboratory experiments on Escherichia Coli.The large deviations technique will be developed in a novel context of discrete models for population histograms. In contrast with the classical Wentzell-Freidlin approach for "small" diffusions, there is no general derivation of cost functions for discrete Markov chains with continuous state space, so that concrete analytical and numerical results for discrete models are rather scarce. However, large deviations offer significant computational advantages in the context of evolutionary dynamics described here due to the special structure of this problem. In particular, the cost function for evolutionary trajectories will be derived in an explicit form. This will lead to the development of a novel discretized difference equation for cost minimizing trajectories, thus enabling an efficient numerical "geodesic shooting" in reverse time to compute the most likely population evolution between given initial and terminal states. This novel algorithm will be used to estimate (very small) probabilities of rare genetic events such as the transitions between two successive genotype fixations and to compute the most likely sequence of genotype fixations leading from an ancestor genotype to a currently dominant genotype. Moreover, the numerical codes developed in the course of the proposed research will be used to study various epistasis scenarios. Thus, this work will generate innovative computational and conceptual tools leading to a better understanding of genetic evolution of bacterial or viral populations.