Benjamin AllenHarvard University, Emmanuel CollegeChristine SampleEmmanuel CollegeYulia DementievaEmmanuel CollegeRuben C. MedeirosEmmanuel CollegeChristopher PaolettiEmmanuel CollegeMartin A. NowakHarvard University
Publications of CMSA of Harvardmathscidoc:1702.38002
Over time, a population acquires neutral genetic substitutions as a consequence of random
drift. A famous result in population genetics asserts that the rate, K, at which these substitutions
accumulate in the population coincides with the mutation rate, u, at which they arise in
individuals: K = u. This identity enables genetic sequence data to be used as a “molecular
clock” to estimate the timing of evolutionary events. While the molecular clock is known to
be perturbed by selection, it is thought that K = u holds very generally for neutral evolution.
Here we show that asymmetric spatial population structure can alter the molecular clock
rate for neutral mutations, leading to either K<u or K>u. Our results apply to a general class
of haploid, asexually reproducing, spatially structured populations. Deviations from K = u
occur because mutations arise unequally at different sites and have different probabilities of
fixation depending on where they arise. If birth rates are uniform across sites, then K u. In
general, K can take any value between 0 and Nu. Our model can be applied to a variety of
population structures. In one example, we investigate the accumulation of genetic mutations
in the small intestine. In another application, we analyze over 900 Twitter networks to
study the effect of network topology on the fixation of neutral innovations in social evolution.
For a self-similar measure in d-dimensional Euclidean space with overlaps but satisfies the so-called bounded measure type condition introduced by Tang and the authors, we set up a framework for deriving a closed formula for the Lq-spectrum of the measure for nonnegative q. The framework allows us to include iterated function systems that have different contraction ratios and those in higher dimension. For self-similar measures with overlaps, closed formulas for the Lq-spectrum have only been obtained earlier for measures satisfying Strichartz second-order identities. We illustrate how to use our results to prove the differentiability of the Lq-spectrum, obtain the multifractal dimension spectrum, and compute the Hausdorff dimension of the measure.
We observe that some self-similar measures defined by finite or infinite iterated function systems with overlaps satisfy certain "bounded measure type condition", which allows us to extract useful measure-theoretic properties of iterates of the measure. We develop a technique to obtain a closed formula for the spectral dimension of the Laplacian defined by self-similar measure satisfying this condition. For Laplacians defined by fractal measures with overlaps, spectral dimension has been obtained earlier only for a small class of one-dimensional self-similar measures satisfying Strichartz second-order self-similar identities. The main technique we use relies on the vector-valued renewal theorem proved by Lau, Wang and Chu.
We obtain two-sided sub-Gaussian estimates of heat kernels for strongly
local Dirichlet forms on intervals, equipped with self-similar measures
generated by iterated function systems (IFS's) that do not satisfy the open
set condition (OSC) and have overlaps. We first give a framework for heat
kernel estimates on intervals, and then consider examples of self-similar
measures to illustrate this phenomenon. These examples include the infinite
Bernoulli convolution associated with the golden ratio, and a family of
convolutions of Cantor-type measures. We make use of Strichartz second-order
identities defined by auxiliary IFS's to compute measures of cells on
different levels. These auxiliary IFS's do satisfy the OSC and are used to
define new metrics. The walk dimensions obtained under these new metrics are
strictly greater than 2 and are closely related to the spectral dimension
of fractal Laplacians.