Brewster David A, Svoboda Jakub, Roscow Dylan, Chatterjee Krishnendu, Tkadlec Josef, Nowak Martin A
John A. Paulson School of Engineering and Applied Sciences, Harvard University, Boston, MA 02134, USA.
Department of Molecular and Cellular Biology, Harvard University, Cambridge, MA 02138, USA.
PNAS Nexus. 2025 Aug 8;4(8):pgaf252. doi: 10.1093/pnasnexus/pgaf252. eCollection 2025 Aug.
We examine population structures for their ability to maintain diversity in neutral evolution. We use the general framework of evolutionary graph theory and consider birth-death (bd) and death-birth (db) updating. The population is of size . Initially all individuals represent different types. The basic question is: what is the time until one type takes over the population? This time is known as consensus time in computer science and as total coalescent time in evolutionary biology. For the complete graph, it is known that is quadratic in for db and bd. For the cycle, we prove that is cubic in for db and bd. For the star, we prove that is cubic for bd and quasilinear ( ) for db. For the double star, we show that is quartic for bd. We derive upper and lower bounds for all undirected graphs for bd and db. We also show the Pareto front of graphs (of size ) that maintain diversity the longest for bd and db. Further, we show that some graphs that quickly homogenize can maintain high levels of diversity longer than graphs that slowly homogenize. For directed graphs, we give simple contracting star-like structures that have superexponential time scales for maintaining diversity.
我们研究种群结构维持中性进化中多样性的能力。我们使用进化图论的一般框架，并考虑生死（bd）和死-生（db）更新。种群规模为 。最初，所有个体代表不同类型。基本问题是：直到一种类型占据种群的时间 是多少？这个时间在计算机科学中称为共识时间，在进化生物学中称为总合并时间。对于完全图，已知对于db和bd， 是 的二次函数。对于循环图，我们证明对于db和bd， 是 的三次函数。对于星型图，我们证明对于bd， 是三次函数，对于db是拟线性（ ）。对于双星图，我们表明对于bd， 是四次函数。我们推导了所有无向图对于bd和db的上下界。我们还展示了对于bd和db维持多样性时间最长的图（规模为 ）的帕累托前沿。此外，我们表明一些快速同质化的图比缓慢同质化的图能更长时间地维持高水平的多样性。对于有向图，我们给出了具有超指数时间尺度以维持多样性的简单收缩星状结构。
