58 research outputs found
Adaptation dynamics of the quasispecies model
We study the adaptation dynamics of an initially maladapted population
evolving via the elementary processes of mutation and selection. The evolution
occurs on rugged fitness landscapes which are defined on the multi-dimensional
genotypic space and have many local peaks separated by low fitness valleys. We
mainly focus on the Eigen's model that describes the deterministic dynamics of
an infinite number of self-replicating molecules. In the stationary state, for
small mutation rates such a population forms a {\it quasispecies} which
consists of the fittest genotype and its closely related mutants. The
quasispecies dynamics on rugged fitness landscape follow a punctuated (or
step-like) pattern in which a population jumps from a low fitness peak to a
higher one, stays there for a considerable time before shifting the peak again
and eventually reaches the global maximum of the fitness landscape. We
calculate exactly several properties of this dynamical process within a
simplified version of the quasispecies model.Comment: Proceedings of Statphys conference at IIT Guwahati, to be published
in Praman
Escalation of error catastrophe for enzymatic self-replicators
It is a long-standing question in origin-of-life research whether the
information content of replicating molecules can be maintained in the presence
of replication errors. Extending standard quasispecies models of non-enzymatic
replication, we analyze highly specific enzymatic self-replication mediated
through an otherwise neutral recognition region, which leads to
frequency-dependent replication rates. We find a significant reduction of the
maximally tolerable error rate, because the replication rate of the fittest
molecules decreases with the fraction of functional enzymes. Our analysis is
extended to hypercyclic couplings as an example for catalytic networks.Comment: 6 pages, 4 figures; accepted at Europhys. Let
Records in a changing world
In the context of this paper, a record is an entry in a sequence of random
variables (RV's) that is larger or smaller than all previous entries. After a
brief review of the classic theory of records, which is largely restricted to
sequences of independent and identically distributed (i.i.d.) RV's, new results
for sequences of independent RV's with distributions that broaden or sharpen
with time are presented. In particular, we show that when the width of the
distribution grows as a power law in time , the mean number of records is
asymptotically of order for distributions with a power law tail (the
\textit{Fr\'echet class} of extremal value statistics), of order
for distributions of exponential type (\textit{Gumbel class}), and of order
for distributions of bounded support (\textit{Weibull class}),
where the exponent describes the behaviour of the distribution at the
upper (or lower) boundary. Simulations are presented which indicate that, in
contrast to the i.i.d. case, the sequence of record breaking events is
correlated in such a way that the variance of the number of records is
asymptotically smaller than the mean.Comment: 12 pages, 2 figure
Locating the minimum : Approach to equilibrium in a disordered, symmetric zero range process
We consider the dynamics of the disordered, one-dimensional, symmetric zero
range process in which a particle from an occupied site hops to its nearest
neighbour with a quenched rate . These rates are chosen randomly from the
probability distribution , where is the lower cutoff.
For , this model is known to exhibit a phase transition in the steady
state from a low density phase with a finite number of particles at each site
to a high density aggregate phase in which the site with the lowest hopping
rate supports an infinite number of particles. In the latter case, it is
interesting to ask how the system locates the site with globally minimum rate.
We use an argument based on local equilibrium, supported by Monte Carlo
simulations, to describe the approach to the steady state. We find that at
large enough time, the mass transport in the regions with a smooth density
profile is described by a diffusion equation with site-dependent rates, while
the isolated points where the mass distribution is singular act as the
boundaries of these regions. Our argument implies that the relaxation time
scales with the system size as with for and
suggests a different behaviour for .Comment: Revtex, 7 pages including 3 figures. Submitted to Pramana -- special
issue on mesoscopic and disordered system
Evolutionary trajectories in rugged fitness landscapes
We consider the evolutionary trajectories traced out by an infinite
population undergoing mutation-selection dynamics in static, uncorrelated
random fitness landscapes. Starting from the population that consists of a
single genotype, the most populated genotype \textit{jumps} from a local
fitness maximum to another and eventually reaches the global maximum. We use a
strong selection limit, which reduces the dynamics beyond the first time step
to the competition between independent mutant subpopulations, to study the
dynamics of this model and of a simpler one-dimensional model which ignores the
geometry of the sequence space. We find that the fit genotypes that appear
along a trajectory are a subset of suitably defined fitness \textit{records},
and exploit several results from the record theory for non-identically
distributed random variables. The genotypes that contribute to the trajectory
are those records that are not \textit{bypassed} by superior records arising
further away from the initial population. Several conjectures concerning the
statistics of bypassing are extracted from numerical simulations. In
particular, for the one-dimensional model, we propose a simple relation between
the bypassing probability and the dynamic exponent which describes the scaling
of the typical evolution time with genome size. The latter can be determined
exactly in terms of the extremal properties of the fitness distribution.Comment: Figures in color; minor revisions in tex
Evolution in random fitness landscapes: the infinite sites model
We consider the evolution of an asexually reproducing population in an
uncorrelated random fitness landscape in the limit of infinite genome size,
which implies that each mutation generates a new fitness value drawn from a
probability distribution . This is the finite population version of
Kingman's house of cards model [J.F.C. Kingman, \textit{J. Appl. Probab.}
\textbf{15}, 1 (1978)]. In contrast to Kingman's work, the focus here is on
unbounded distributions which lead to an indefinite growth of the
population fitness. The model is solved analytically in the limit of infinite
population size and simulated numerically for finite . When
the genome-wide mutation probability is small, the long time behavior of
the model reduces to a point process of fixation events, which is referred to
as a \textit{diluted record process} (DRP). The DRP is similar to the standard
record process except that a new record candidate (a number that exceeds all
previous entries in the sequence) is accepted only with a certain probability
that depends on the values of the current record and the candidate. We develop
a systematic analytic approximation scheme for the DRP. At finite the
fitness frequency distribution of the population decomposes into a stationary
part due to mutations and a traveling wave component due to selection, which is
shown to imply a reduction of the mean fitness by a factor of compared to
the limit.Comment: Dedicated to Thomas Nattermann on the occasion of his 60th birthday.
Submitted to JSTAT. Error in Section 3.2 was correcte
Persistence in Cluster--Cluster Aggregation
Persistence is considered in diffusion--limited cluster--cluster aggregation,
in one dimension and when the diffusion coefficient of a cluster depends on its
size as . The empty and filled site persistences are
defined as the probabilities, that a site has been either empty or covered by a
cluster all the time whereas the cluster persistence gives the probability of a
cluster to remain intact. The filled site one is nonuniversal. The empty site
and cluster persistences are found to be universal, as supported by analytical
arguments and simulations. The empty site case decays algebraically with the
exponent . The cluster persistence is related to the
small behavior of the cluster size distribution and behaves also
algebraically for while for the behavior is
stretched exponential. In the scaling limit and with fixed the distribution of intervals of size between
persistent regions scales as , where is the average interval size and . For finite the
scaling is poor for , due to the insufficient separation of the two
length scales: the distances between clusters, , and that between
persistent regions, . For the size distribution of persistent regions
the time and size dependences separate, the latter being independent of the
diffusion exponent but depending on the initial cluster size
distribution.Comment: 14 pages, 12 figures, RevTeX, submitted to Phys. Rev.
Nonequilibrium Statistical Mechanics of the Zero-Range Process and Related Models
We review recent progress on the zero-range process, a model of interacting
particles which hop between the sites of a lattice with rates that depend on
the occupancy of the departure site. We discuss several applications which have
stimulated interest in the model such as shaken granular gases and network
dynamics, also we discuss how the model may be used as a coarse-grained
description of driven phase-separating systems. A useful property of the
zero-range process is that the steady state has a factorised form. We show how
this form enables one to analyse in detail condensation transitions, wherein a
finite fraction of particles accumulate at a single site. We review
condensation transitions in homogeneous and heterogeneous systems and also
summarise recent progress in understanding the dynamics of condensation. We
then turn to several generalisations which also, under certain specified
conditions, share the property of a factorised steady state. These include
several species of particles; hop rates which depend on both the departure and
the destination sites; continuous masses; parallel discrete-time updating;
non-conservation of particles and sites.Comment: 54 pages, 9 figures, review articl
Phases of a conserved mass model of aggregation with fragmentation at fixed sites
To study the effect of quenched disorder in a class of reaction-diffusion
systems, we introduce a conserved mass model of diffusion and aggregation in
which the mass moves as a whole to a nearest neighbour on most sites while it
fragments off as a single monomer (i.e. chips off) from certain fixed sites.
Once the mass leaves any site, it coalesces with the mass present on its
neighbour. We study in detail the effect of a \emph{single} chipping site on
the steady state in arbitrary dimensions, with and without bias. In the
thermodynamic limit, the system can exist in one of the following phases -- (a)
Pinned Aggregate (PA) phase in which an infinite aggregate (with mass
proportional to the volume of the system) appears with probability one at the
chipping site but not in the bulk. (b) Unpinned Aggregate (UA) phase in which
\emph{both} the chipping site and the bulk can support an infinite aggregate
simultaneously. (c) Non Aggregate (NA) phase in which there is no infinite
cluster. Our analytical and numerical studies show that the system exists in
the UA phase in all cases except in 1d with bias. In the latter case, there is
a phase transition from the NA phase to the PA phase as density is increased. A
variant of the above aggregation model is also considered in which total
particle number is conserved and chipping occurs at a fixed site, but the
particles do not interact with each other at other sites. This model is solved
exactly by mapping it to a Zero Range Process. With increasing density, it
exhibits a phase transition from the NA phase to the PA phase in all
dimensions, irrespective of bias. Finally, we discuss the likely behaviour of
the system in the presence of extensive disorder.Comment: RevTex, 19 pages including 11 figures, submitted to Phys. Rev.
Activated Random Walkers: Facts, Conjectures and Challenges
We study a particle system with hopping (random walk) dynamics on the integer
lattice . The particles can exist in two states, active or
inactive (sleeping); only the former can hop. The dynamics conserves the number
of particles; there is no limit on the number of particles at a given site.
Isolated active particles fall asleep at rate , and then remain
asleep until joined by another particle at the same site. The state in which
all particles are inactive is absorbing. Whether activity continues at long
times depends on the relation between the particle density and the
sleeping rate . We discuss the general case, and then, for the
one-dimensional totally asymmetric case, study the phase transition between an
active phase (for sufficiently large particle densities and/or small )
and an absorbing one. We also present arguments regarding the asymptotic mean
hopping velocity in the active phase, the rate of fixation in the absorbing
phase, and survival of the infinite system at criticality. Using mean-field
theory and Monte Carlo simulation, we locate the phase boundary. The phase
transition appears to be continuous in both the symmetric and asymmetric
versions of the process, but the critical behavior is very different. The
former case is characterized by simple integer or rational values for critical
exponents (, for example), and the phase diagram is in accord with
the prediction of mean-field theory. We present evidence that the symmetric
version belongs to the universality class of conserved stochastic sandpiles,
also known as conserved directed percolation. Simulations also reveal an
interesting transient phenomenon of damped oscillations in the activity
density
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