2026/07/17 by Armaan Ahmed, Jasmine Foo, Einar Gunnarsson +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #math.PR #q-bio.PE
We consider a supercritical two-type continuous-time linear birth-death process with mutation and selection, in which wild-type individuals give rise to mutant offspring with a larger net growth rate. In this setting, we investigate the site frequency spectrum (SFS) of driver mutations, describing the number of driver mutations present at any given frequency in the population. First, we derive exact moments for the SFS and establish asymptotic power laws at large times and frequencies. Then, strong laws of large numbers for the driver SFS are proven by constructing suitable L2-approximations. These results apply both to the case when all driver clones have the same selective advantage and when the selective advantage is random. Finally, we allow the frequency to vary with time to examine the number of "intermediate" and "large" driver clones, identifying a cutoff frequency at which there are order 1 number of mutant clones. Overall, our results provide quantitative insights into how selection shapes the site frequency spectrum both at small and large frequencies, which can in principle be leveraged to construct estimators of relevant evolutionary parameters, including the selective advantage of driver mutations.