535 research outputs found
Evolutionary dynamics on degree-heterogeneous graphs
The evolution of two species with different fitness is investigated on
degree-heterogeneous graphs. The population evolves either by one individual
dying and being replaced by the offspring of a random neighbor (voter model
(VM) dynamics) or by an individual giving birth to an offspring that takes over
a random neighbor node (invasion process (IP) dynamics). The fixation
probability for one species to take over a population of N individuals depends
crucially on the dynamics and on the local environment. Starting with a single
fitter mutant at a node of degree k, the fixation probability is proportional
to k for VM dynamics and to 1/k for IP dynamics.Comment: 4 pages, 4 figures, 2 column revtex4 format. Revisions in response to
referee comments for publication in PRL. The version on arxiv.org has one
more figure than the published PR
Mutation, selection, and ancestry in branching models: a variational approach
We consider the evolution of populations under the joint action of mutation
and differential reproduction, or selection. The population is modelled as a
finite-type Markov branching process in continuous time, and the associated
genealogical tree is viewed both in the forward and the backward direction of
time. The stationary type distribution of the reversed process, the so-called
ancestral distribution, turns out as a key for the study of mutation-selection
balance. This balance can be expressed in the form of a variational principle
that quantifies the respective roles of reproduction and mutation for any
possible type distribution. It shows that the mean growth rate of the
population results from a competition for a maximal long-term growth rate, as
given by the difference between the current mean reproduction rate, and an
asymptotic decay rate related to the mutation process; this tradeoff is won by
the ancestral distribution.
Our main application is the quasispecies model of sequence evolution with
mutation coupled to reproduction but independent across sites, and a fitness
function that is invariant under permutation of sites. Here, the variational
principle is worked out in detail and yields a simple, explicit result.Comment: 45 pages,8 figure
Central limit theorem for multiplicative class functions on the symmetric group
Hambly, Keevash, O'Connell and Stark have proven a central limit theorem for
the characteristic polynomial of a permutation matrix with respect to the
uniform measure on the symmetric group. We generalize this result in several
ways. We prove here a central limit theorem for multiplicative class functions
on symmetric group with respect to the Ewens measure and compute the covariance
of the real and the imaginary part in the limit. We also estimate the rate of
convergence with the Wasserstein distance.Comment: 23 pages; the mathematics is the same as in the previous version, but
there are several improvments in the presentation, including a more intuitve
name for the considered function
Information and (co-)variances in discrete evolutionary genetics involving solely selection
The purpose of this Note is twofold: First, we introduce the general
formalism of evolutionary genetics dynamics involving fitnesses, under both the
deterministic and stochastic setups, and chiefly in discrete-time. In the
process, we particularize it to a one-parameter model where only a selection
parameter is unknown. Then and in a parallel manner, we discuss the estimation
problems of the selection parameter based on a single-generation frequency
distribution shift under both deterministic and stochastic evolutionary
dynamics. In the stochastics, we consider both the celebrated Wright-Fisher and
Moran models.Comment: a paraitre dans Journal of Statistical Mechanics: Theory and
Application
Preservation of information in a prebiotic package model
The coexistence between different informational molecules has been the
preferred mode to circumvent the limitation posed by imperfect replication on
the amount of information stored by each of these molecules. Here we reexamine
a classic package model in which distinct information carriers or templates are
forced to coexist within vesicles, which in turn can proliferate freely through
binary division. The combined dynamics of vesicles and templates is described
by a multitype branching process which allows us to write equations for the
average number of the different types of vesicles as well as for their
extinction probabilities. The threshold phenomenon associated to the extinction
of the vesicle population is studied quantitatively using finite-size scaling
techniques. We conclude that the resultant coexistence is too frail in the
presence of parasites and so confinement of templates in vesicles without an
explicit mechanism of cooperation does not resolve the information crisis of
prebiotic evolution.Comment: 9 pages, 8 figures, accepted version, to be published in PR
Stochastic slowdown in evolutionary processes
We examine birth--death processes with state dependent transition
probabilities and at least one absorbing boundary. In evolution, this describes
selection acting on two different types in a finite population where
reproductive events occur successively. If the two types have equal fitness the
system performs a random walk. If one type has a fitness advantage it is
favored by selection, which introduces a bias (asymmetry) in the transition
probabilities. How long does it take until advantageous mutants have invaded
and taken over? Surprisingly, we find that the average time of such a process
can increase, even if the mutant type always has a fitness advantage. We
discuss this finding for the Moran process and develop a simplified model which
allows a more intuitive understanding. We show that this effect can occur for
weak but non--vanishing bias (selection) in the state dependent transition
rates and infer the scaling with system size. We also address the Wright-Fisher
model commonly used in population genetics, which shows that this stochastic
slowdown is not restricted to birth-death processes.Comment: 8 pages, 3 figures, accepted for publicatio
Ordering in voter models on networks: Exact reduction to a single-coordinate diffusion
We study the voter model and related random-copying processes on arbitrarily
complex network structures. Through a representation of the dynamics as a
particle reaction process, we show that a quantity measuring the degree of
order in a finite system is, under certain conditions, exactly governed by a
universal diffusion equation. Whenever this reduction occurs, the details of
the network structure and random-copying process affect only a single parameter
in the diffusion equation. The validity of the reduction can be established
with considerably less information than one might expect: it suffices to know
just two characteristic timescales within the dynamics of a single pair of
reacting particles. We develop methods to identify these timescales, and apply
them to deterministic and random network structures. We focus in particular on
how the ordering time is affected by degree correlations, since such effects
are hard to access by existing theoretical approaches.Comment: 37 pages, 10 figures. Revised version with additional discussion and
simulation results to appear in J Phys
Landscape statistics of the low autocorrelated binary string problem
The statistical properties of the energy landscape of the low autocorrelated
binary string problem (LABSP) are studied numerically and compared with those
of several classic disordered models. Using two global measures of landscape
structure which have been introduced in the Simulated Annealing literature,
namely, depth and difficulty, we find that the landscape of LABSP, except
perhaps for a very large degeneracy of the local minima energies, is
qualitatively similar to some well-known landscapes such as that of the
mean-field 2-spin glass model. Furthermore, we consider a mean-field
approximation to the pure model proposed by Bouchaud and Mezard (1994, J.
Physique I France 4 1109) and show both analytically and numerically that it
describes extremely well the statistical properties of LABSP
Spectral Simplicity of Apparent Complexity, Part I: The Nondiagonalizable Metadynamics of Prediction
Virtually all questions that one can ask about the behavioral and structural
complexity of a stochastic process reduce to a linear algebraic framing of a
time evolution governed by an appropriate hidden-Markov process generator. Each
type of question---correlation, predictability, predictive cost, observer
synchronization, and the like---induces a distinct generator class. Answers are
then functions of the class-appropriate transition dynamic. Unfortunately,
these dynamics are generically nonnormal, nondiagonalizable, singular, and so
on. Tractably analyzing these dynamics relies on adapting the recently
introduced meromorphic functional calculus, which specifies the spectral
decomposition of functions of nondiagonalizable linear operators, even when the
function poles and zeros coincide with the operator's spectrum. Along the way,
we establish special properties of the projection operators that demonstrate
how they capture the organization of subprocesses within a complex system.
Circumventing the spurious infinities of alternative calculi, this leads in the
sequel, Part II, to the first closed-form expressions for complexity measures,
couched either in terms of the Drazin inverse (negative-one power of a singular
operator) or the eigenvalues and projection operators of the appropriate
transition dynamic.Comment: 24 pages, 3 figures, 4 tables; current version always at
http://csc.ucdavis.edu/~cmg/compmech/pubs/sdscpt1.ht
Rank Statistics in Biological Evolution
We present a statistical analysis of biological evolution processes.
Specifically, we study the stochastic replication-mutation-death model where
the population of a species may grow or shrink by birth or death, respectively,
and additionally, mutations lead to the creation of new species. We rank the
various species by the chronological order by which they originate. The average
population N_k of the kth species decays algebraically with rank, N_k ~ M^{mu}
k^{-mu}, where M is the average total population. The characteristic exponent
mu=(alpha-gamma)/(alpha+beta-gamma)$ depends on alpha, beta, and gamma, the
replication, mutation, and death rates. Furthermore, the average population P_k
of all descendants of the kth species has a universal algebraic behavior, P_k ~
M/k.Comment: 4 pages, 3 figure
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