11,413 research outputs found
Tree-valued Fleming-Viot dynamics with mutation and selection
The Fleming-Viot measure-valued diffusion is a Markov process describing the
evolution of (allelic) types under mutation, selection and random reproduction.
We enrich this process by genealogical relations of individuals so that the
random type distribution as well as the genealogical distances in the
population evolve stochastically. The state space of this tree-valued
enrichment of the Fleming-Viot dynamics with mutation and selection (TFVMS)
consists of marked ultrametric measure spaces, equipped with the marked
Gromov-weak topology and a suitable notion of polynomials as a separating
algebra of test functions. The construction and study of the TFVMS is based on
a well-posed martingale problem. For existence, we use approximating finite
population models, the tree-valued Moran models, while uniqueness follows from
duality to a function-valued process. Path properties of the resulting process
carry over from the neutral case due to absolute continuity, given by a new
Girsanov-type theorem on marked metric measure spaces. To study the long-time
behavior of the process, we use a duality based on ideas from Dawson and Greven
[On the effects of migration in spatial Fleming-Viot models with selection and
mutation (2011c) Unpublished manuscript] and prove ergodicity of the TFVMS if
the Fleming-Viot measure-valued diffusion is ergodic. As a further application,
we consider the case of two allelic types and additive selection. For small
selection strength, we give an expansion of the Laplace transform of
genealogical distances in equilibrium, which is a first step in showing that
distances are shorter in the selective case.Comment: Published in at http://dx.doi.org/10.1214/11-AAP831 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Evolution of the most recent common ancestor of a population with no selection
We consider the evolution of a population of fixed size with no selection.
The number of generations to reach the first common ancestor evolves in
time. This evolution can be described by a simple Markov process which allows
one to calculate several characteristics of the time dependence of . We also
study how is correlated to the genetic diversity.Comment: 21 pages, 10 figures, uses RevTex4 and feynmf.sty Corrections :
introduction and conclusion rewritten, references adde
Mathematical Models and Biological Meaning: Taking Trees Seriously
We compare three basic kinds of discrete mathematical models used to portray
phylogenetic relationships among species and higher taxa: phylogenetic trees,
Hennig trees and Nelson cladograms. All three models are trees, as that term is
commonly used in mathematics; the difference between them lies in the
biological interpretation of their vertices and edges. Phylogenetic trees and
Hennig trees carry exactly the same information, and translation between these
two kinds of trees can be accomplished by a simple algorithm. On the other
hand, evolutionary concepts such as monophyly are represented as different
mathematical substructures are represented differently in the two models. For
each phylogenetic or Hennig tree, there is a Nelson cladogram carrying the same
information, but the requirement that all taxa be represented by leaves
necessarily makes the representation less efficient. Moreover, we claim that it
is necessary to give some interpretation to the edges and internal vertices of
a Nelson cladogram in order to make it useful as a biological model. One
possibility is to interpret internal vertices as sets of characters and the
edges as statements of inclusion; however, this interpretation carries little
more than incomplete phenetic information. We assert that from the standpoint
of phylogenetics, one is forced to regard each internal vertex of a Nelson
cladogram as an actual (albeit unsampled) species simply to justify the use of
synapomorphies rather than symplesiomorphies.Comment: 15 pages including 6 figures [5 pdf, 1 jpg]. Converted from original
MS Word manuscript to PDFLaTe
Bayesian inference of sampled ancestor trees for epidemiology and fossil calibration
Phylogenetic analyses which include fossils or molecular sequences that are
sampled through time require models that allow one sample to be a direct
ancestor of another sample. As previously available phylogenetic inference
tools assume that all samples are tips, they do not allow for this possibility.
We have developed and implemented a Bayesian Markov Chain Monte Carlo (MCMC)
algorithm to infer what we call sampled ancestor trees, that is, trees in which
sampled individuals can be direct ancestors of other sampled individuals. We
use a family of birth-death models where individuals may remain in the tree
process after the sampling, in particular we extend the birth-death skyline
model [Stadler et al, 2013] to sampled ancestor trees. This method allows the
detection of sampled ancestors as well as estimation of the probability that an
individual will be removed from the process when it is sampled. We show that
sampled ancestor birth-death models where all samples come from different time
points are non-identifiable and thus require one parameter to be known in order
to infer other parameters. We apply this method to epidemiological data, where
the possibility of sampled ancestors enables us to identify individuals that
infected other individuals after being sampled and to infer fundamental
epidemiological parameters. We also apply the method to infer divergence times
and diversification rates when fossils are included among the species samples,
so that fossilisation events are modelled as a part of the tree branching
process. Such modelling has many advantages as argued in literature. The
sampler is available as an open-source BEAST2 package
(https://github.com/gavryushkina/sampled-ancestors).Comment: 34 pages (including Supporting Information), 8 figures, 1 table. Part
of the work presented at Epidemics 2013 and The 18th Annual New Zealand
Phylogenomics Meeting, 201
Genealogies of rapidly adapting populations
The genetic diversity of a species is shaped by its recent evolutionary
history and can be used to infer demographic events or selective sweeps. Most
inference methods are based on the null hypothesis that natural selection is a
weak or infrequent evolutionary force. However, many species, particularly
pathogens, are under continuous pressure to adapt in response to changing
environments. A statistical framework for inference from diversity data of such
populations is currently lacking. Toward this goal, we explore the properties
of genealogies in a model of continual adaptation in asexual populations. We
show that lineages trace back to a small pool of highly fit ancestors, in which
almost simultaneous coalescence of more than two lineages frequently occurs.
While such multiple mergers are unlikely under the neutral coalescent, they
create a unique genetic footprint in adapting populations. The site frequency
spectrum of derived neutral alleles, for example, is non-monotonic and has a
peak at high frequencies, whereas Tajima's D becomes more and more negative
with increasing sample size. Since multiple merger coalescents emerge in many
models of rapid adaptation, we argue that they should be considered as a
null-model for adapting populations.Comment: to appear in PNA
Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization
We consider a family of models describing the evolution under selection of a
population whose dynamics can be related to the propagation of noisy traveling
waves. For one particular model, that we shall call the exponential model, the
properties of the traveling wave front can be calculated exactly, as well as
the statistics of the genealogy of the population. One striking result is that,
for this particular model, the genealogical trees have the same statistics as
the trees of replicas in the Parisi mean-field theory of spin glasses. We also
find that in the exponential model, the coalescence times along these trees
grow like the logarithm of the population size. A phenomenological picture of
the propagation of wave fronts that we introduced in a previous work, as well
as our numerical data, suggest that these statistics remain valid for a larger
class of models, while the coalescence times grow like the cube of the
logarithm of the population size.Comment: 26 page
Tree-valued Feller diffusion
We consider the evolution of the genealogy of the population currently alive
in a Feller branching diffusion model. In contrast to the approach via labeled
trees in the continuum random tree world, the genealogies are modeled as
equivalence classes of ultrametric measure spaces, the elements of the space
. This space is Polish and has a rich semigroup structure for the
genealogy. We focus on the evolution of the genealogy in time and the large
time asymptotics conditioned both on survival up to present time and on
survival forever. We prove existence, uniqueness and Feller property of
solutions of the martingale problem for this genealogy valued, i.e.,
-valued Feller diffusion. We give the precise relation to the
time-inhomogeneous -valued Fleming-Viot process. The uniqueness
is shown via Feynman-Kac duality with the distance matrix augmented Kingman
coalescent. Using a semigroup operation on , called concatenation,
together with the branching property we obtain a L{\'e}vy-Khintchine formula
for -valued Feller diffusion and we determine explicitly the
L{\'e}vy measure on . From this we obtain for
the decomposition into depth- subfamilies, a representation of the process
as concatenation of a Cox point process of genealogies of single ancestor
subfamilies. Furthermore, we will identify the -valued process
conditioned to survive until a finite time . We study long time asymptotics,
such as generalized quasi-equilibrium and Kolmogorov-Yaglom limit law on the
level of ultrametric measure spaces. We also obtain various representations of
the long time limits.Comment: 93 pages, replaced by revised versio
Statistical properties of genealogical trees
We analyse the statistical properties of genealogical trees in a neutral
model of a closed population with sexual reproduction and non-overlapping
generations. By reconstructing the genealogy of an individual from the
population evolution, we measure the distribution of ancestors appearing more
than once in a given tree. After a transient time, the probability of
repetition follows, up to a rescaling, a stationary distribution which we
calculate both numerically and analytically. This distribution exhibits a
universal shape with a non-trivial power law which can be understood by an
exact, though simple, renormalization calculation. Some real data on human
genealogy illustrate the problem, which is relevant to the study of the real
degree of diversity in closed interbreeding communities.Comment: Accepted for publication in Phys. Rev. Let
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