Application of Large Deviations to Genetic Evolution of Bacterial Populations

细菌或病毒群体通过出现和固定有益的突变表现出很强的适应性。例如，这种突变可能与细菌对抗生素的抗药性或出现可从动物传染给人类的病毒株有关。目前，已有各种描述细菌或病毒种群进化的数学模型，但可用于研究大规模细菌或病毒种群进化轨迹的数学工具仍然缺乏。因此，这项工作的主要目标是开发适当的数学工具来研究细菌或病毒中罕见的突变事件。特别是，研究人员将使用种群轨迹的大偏差方法来开发研究细菌种群进化轨迹的理论基础和实用计算工具，并将具体应用于对大肠杆菌的长期实验室实验分析。大偏差技术将在种群直方图离散模型的新背景下开发。与经典的“小”扩散的Wentzell-Freidlin方法相比，离散状态空间的离散马氏链的代价函数没有得到普遍的推导，因此离散模型的具体分析和数值结果相当匮乏。然而，由于这个问题的特殊结构，在这里描述的进化动力学的背景下，大偏差提供了显著的计算优势。特别是，进化轨迹的成本函数将以显式形式导出。这将导致一个新的离散化的差分方程式的发展，以最大限度地减少成本的轨迹，从而使有效的数值“测地线射击”在反向时间，以计算最可能的种群演化之间的给定的初始和结束状态。这种新的算法将被用来估计罕见遗传事件的(非常小的)概率，例如两个连续的基因定位之间的转变，并计算从祖先基因到当前主导基因的最可能的基因固定序列。此外，在拟议的研究过程中开发的数字代码将用于研究各种上位情景。因此，这项工作将产生创新的计算和概念工具，从而更好地理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.