Genealogy of branching populations, new descriptions of large random trees, and a mathematical framework for the evolution of senescence

大多数复杂的有机体不仅会随着时间的推移而衰老，而且往往会按照类似的模式退化。生物学家基于突变-选择平衡的概念提出了对这一事实的定性解释：自然选择并不反对具有有害影响的突变，如果这些影响主要是在携带者能够繁殖之后感受到的。然而，要为这一概念提供一个严格的定量版本一直是一个挑战，因为它涉及大量具有复杂相互作用的基因，而由此产生的衰老模式应该对潜在的细节相当不敏感。该项目的一部分旨在利用无限维、非线性动力系统的思想来提高对进化生物学和生物人口学中这一重要问题的理解。进化生物学中的另一个常见问题是，进化过程是否在现代物种的系统发育树中留下了某种特征。同样，随着大量数据的可用性越来越高，计算机科学家想知道，随着时间的推移，一个大型网络的当前结构是否揭示了这种增长的动力学。出于这些问题的动机，人们寻求“统计”，以某种方式总结树的形状和更一般的图形，并确定这些量的行为时，图形是由特定的机制。在数学上，这种描述符的自然候选者是所谓的图的特征值。该项目的两个进一步的目标是，首先，使用相对简单的技术，从线性代数和概率来理解大类随机图的特征值，其次，利用马丁边界理论的思想，充分描绘出当树木变大时，某些树木生长模型可以表现出的所有可能的方式。该提案的最后一个目标是研究最近的树木的年龄是如何变化的。一个种群的共同祖先（MRCA）会随着时间的推移而变化。任何无性繁殖的种群都有一个独特的MRCA，整个种群都是从它传下来的。在有性繁殖的物种中，每个非重组DNA片段也是如此。例如，我们的“线粒体夏娃”，所有现代人类都继承了他们的线粒体DNA，估计生活在大约18万年前。随着时间的推移，最终除了目前线粒体夏娃的一个女儿之外，所有的线粒体血统都会灭绝，此时新的线粒体夏娃将生活在时间的稍晚一些。拟议的研究将调查MRCA年龄变化的动力学。

Not only do most complex organisms age, in the sense that they degenerateas time passes, but they often do so according to similar patterns. Biologistshave proposed qualitative explanations of this fact based on the concept ofmutation-selection balance: that natural selection does not oppose mutationswith deleterious effects if those effects are felt primarily after the bearer has been able to reproduce. However, it has been a challenge to provide a rigorous quantitative version of this notion because it involves large numbers of genes with complex interactions, and yet the resulting patterns of aging should be fairly insensitive to the underlying specifics. One part of the project aims to improve understanding of this important problem in evolutionary biology and biodemography using ideas from infinite-dimensional, non-linear dynamical systems.Another common question in evolutionary biology is whether evolutionaryprocesses leave some sort of signature in the shape of the phylogenetic tree ofa collection of present day species. Similarly, with the increasing availability of large amounts of data, computer scientists wonder if the current structure of a large network that has grown over time reveals something about the dynamics of that growth. Motivated by such questions, one seeks "statistics" that somehow summarize the shape of trees and more general graphs, and to determine the behavior of these quantities when the graphs are generated by specific mechanisms. Mathematically natural candidates for such descriptors are the so-calledeigenvalues of the graph. Two further goals of the project are, firstly, to userelatively simple techniques from linear algebra and probability to understandthe eigenvalues of broad classes of large random graphs, and, secondly, to useideas from the theory of Martin boundaries to delineate fully all the possibleways that certain models for growing trees can behave when the tree becomes large.The last objective of the proposal is to study how the age of the most-recent-common ancestor (MRCA) of a population changes over time. Any asexually reproducing population has a unique MRCA, from whom the entire population is descended. In sexually reproducing species, the same is true for each non-recombining piece of DNA. For instance, our "mitochondrial Eve" from whom all modern-day humans inherited their mitochondrial DNA is estimated to have lived around 180,000 years ago. As time progresses into the future, eventually the mitochondrial lineages of all but one of the daughters of the current mitochondrial Eve will die out, at which point the new mitochondrial Eve will have lived somewhat later in time. The proposed research will investigate the dynamics by which the age of the MRCA varies.