2,704 research outputs found
Population genetics of neutral mutations in exponentially growing cancer cell populations
In order to analyze data from cancer genome sequencing projects, we need to
be able to distinguish causative, or "driver," mutations from "passenger"
mutations that have no selective effect. Toward this end, we prove results
concerning the frequency of neutural mutations in exponentially growing
multitype branching processes that have been widely used in cancer modeling.
Our results yield a simple new population genetics result for the site
frequency spectrum of a sample from an exponentially growing population.Comment: Published in at http://dx.doi.org/10.1214/11-AAP824 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Coexistence in stochastic spatial models
In this paper I will review twenty years of work on the question: When is
there coexistence in stochastic spatial models? The answer, announced in
Durrett and Levin [Theor. Pop. Biol. 46 (1994) 363--394], and that we explain
in this paper is that this can be determined by examining the mean-field ODE.
There are a number of rigorous results in support of this picture, but we will
state nine challenging and important open problems, most of which date from the
1990's.Comment: Published in at http://dx.doi.org/10.1214/08-AAP590 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Coexistence of grass, saplings and trees in the Staver-Levin forest model
In this paper, we consider two attractive stochastic spatial models in which
each site can be in state 0, 1 or 2: Krone's model in which 0vacant,
1juvenile and 2a mature individual capable of giving birth, and
the Staver-Levin forest model in which 0grass, 1sapling and
2tree. Our first result shows that if is an unstable fixed point
of the mean-field ODE for densities of 1's and 2's then when the range of
interaction is large, there is positive probability of survival starting from a
finite set and a stationary distribution in which all three types are present.
The result we obtain in this way is asymptotically sharp for Krone's model.
However, in the Staver-Levin forest model, if is attracting then there
may also be another stable fixed point for the ODE, and in some of these cases
there is a nontrivial stationary distribution.Comment: Published at http://dx.doi.org/10.1214/14-AAP1079 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Contact processes on random graphs with power law degree distributions have critical value 0
If we consider the contact process with infection rate on a random
graph on vertices with power law degree distributions, mean field
calculations suggest that the critical value of the infection rate
is positive if the power . Physicists seem to regard this as an
established fact, since the result has recently been generalized to bipartite
graphs by G\'{o}mez-Garde\~{n}es et al. [Proc. Natl. Acad. Sci. USA 105 (2008)
1399--1404]. Here, we show that the critical value is zero for any
value of , and the contact process starting from all vertices
infected, with a probability tending to 1 as , maintains a positive
density of infected sites for time at least for any
. Using the last result, together with the contact process duality,
we can establish the existence of a quasi-stationary distribution in which a
randomly chosen vertex is occupied with probability . It is
expected that as . Here we
show that , and so for . Thus
even though the graph is locally tree-like, does not take the mean
field critical value .Comment: Published in at http://dx.doi.org/10.1214/09-AOP471 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Waiting for regulatory sequences to appear
One possible explanation for the substantial organismal differences between
humans and chimpanzees is that there have been changes in gene regulation.
Given what is known about transcription factor binding sites, this motivates
the following probability question: given a 1000 nucleotide region in our
genome, how long does it take for a specified six to nine letter word to appear
in that region in some individual? Stone and Wray [Mol. Biol. Evol. 18 (2001)
1764--1770] computed 5,950 years as the answer for six letter words. Here, we
will show that for words of length 6, the average waiting time is 100,000
years, while for words of length 8, the waiting time has mean 375,000 years
when there is a 7 out of 8 letter match in the population consensus sequence
(an event of probability roughly 5/16) and has mean 650 million years when
there is not. Fortunately, in biological reality, the match to the target word
does not have to be perfect for binding to occur. If we model this by saying
that a 7 out of 8 letter match is good enough, the mean reduces to about 60,000
years.Comment: Published at http://dx.doi.org/10.1214/105051606000000619 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
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