2,334 research outputs found
Survival of interacting Brownian particles in crowded 1D environment
We investigate a diffusive motion of a system of interacting Brownian
particles in quasi-one-dimensional micropores. In particular, we consider a
semi-infinite 1D geometry with a partially absorbing boundary and the hard-core
inter-particle interaction. Due to the absorbing boundary the number of
particles in the pore gradually decreases. We present the exact analytical
solution of the problem. Our procedure merely requires the knowledge of the
corresponding single-particle problem. First, we calculate the simultaneous
probability density of having still a definite number of surviving
particles at definite coordinates. Focusing on an arbitrary tagged particle, we
derive the exact probability density of its coordinate. Secondly, we present a
complete probabilistic description of the emerging escape process. The survival
probabilities for the individual particles are calculated, the first and the
second moments of the exit times are discussed. Generally speaking, although
the original inter-particle interaction possesses a point-like character, it
induces entropic repulsive forces which, e.g., push the leftmost (rightmost)
particle towards (opposite) the absorbing boundary thereby accelerating
(decelerating) its escape. More importantly, as compared to the reference
problem for the non-interacting particles, the interaction changes the
dynamical exponents which characterize the long-time asymptotic dynamics.
Interesting new insights emerge after we interpret our model in terms of a)
diffusion of a single particle in a -dimensional space, and b) order
statistics defined on a system of independent, identically distributed
random variables
Early appraisal of the fixation probability in directed networks
In evolutionary dynamics, the probability that a mutation spreads through the
whole population, having arisen in a single individual, is known as the
fixation probability. In general, it is not possible to find the fixation
probability analytically given the mutant's fitness and the topological
constraints that govern the spread of the mutation, so one resorts to
simulations instead. Depending on the topology in use, a great number of
evolutionary steps may be needed in each of the simulation events, particularly
in those that end with the population containing mutants only. We introduce two
techniques to accelerate the determination of the fixation probability. The
first one skips all evolutionary steps in which the number of mutants does not
change and thereby reduces the number of steps per simulation event
considerably. This technique is computationally advantageous for some of the
so-called layered networks. The second technique, which is not restricted to
layered networks, consists of aborting any simulation event in which the number
of mutants has grown beyond a certain threshold value, and counting that event
as having led to a total spread of the mutation. For large populations, and
regardless of the network's topology, we demonstrate, both analytically and by
means of simulations, that using a threshold of about 100 mutants leads to an
estimate of the fixation probability that deviates in no significant way from
that obtained from the full-fledged simulations. We have observed speedups of
two orders of magnitude for layered networks with 10000 nodes
Universality of Long-Range Correlations in Expansion-Randomization Systems
We study the stochastic dynamics of sequences evolving by single site
mutations, segmental duplications, deletions, and random insertions. These
processes are relevant for the evolution of genomic DNA. They define a
universality class of non-equilibrium 1D expansion-randomization systems with
generic stationary long-range correlations in a regime of growing sequence
length. We obtain explicitly the two-point correlation function of the sequence
composition and the distribution function of the composition bias in sequences
of finite length. The characteristic exponent of these quantities is
determined by the ratio of two effective rates, which are explicitly calculated
for several specific sequence evolution dynamics of the universality class.
Depending on the value of , we find two different scaling regimes, which
are distinguished by the detectability of the initial composition bias. All
analytic results are accurately verified by numerical simulations. We also
discuss the non-stationary build-up and decay of correlations, as well as more
complex evolutionary scenarios, where the rates of the processes vary in time.
Our findings provide a possible example for the emergence of universality in
molecular biology.Comment: 23 pages, 15 figure
Transition probabilities for general birth-death processes with applications in ecology, genetics, and evolution
A birth-death process is a continuous-time Markov chain that counts the
number of particles in a system over time. In the general process with
current particles, a new particle is born with instantaneous rate
and a particle dies with instantaneous rate . Currently no robust and
efficient method exists to evaluate the finite-time transition probabilities in
a general birth-death process with arbitrary birth and death rates. In this
paper, we first revisit the theory of continued fractions to obtain expressions
for the Laplace transforms of these transition probabilities and make explicit
an important derivation connecting transition probabilities and continued
fractions. We then develop an efficient algorithm for computing these
probabilities that analyzes the error associated with approximations in the
method. We demonstrate that this error-controlled method agrees with known
solutions and outperforms previous approaches to computing these probabilities.
Finally, we apply our novel method to several important problems in ecology,
evolution, and genetics
Analytical Solution of a Stochastic Content Based Network Model
We define and completely solve a content-based directed network whose nodes
consist of random words and an adjacency rule involving perfect or approximate
matches, for an alphabet with an arbitrary number of letters. The analytic
expression for the out-degree distribution shows a crossover from a leading
power law behavior to a log-periodic regime bounded by a different power law
decay. The leading exponents in the two regions have a weak dependence on the
mean word length, and an even weaker dependence on the alphabet size. The
in-degree distribution, on the other hand, is much narrower and does not show
scaling behavior. The results might be of interest for understanding the
emergence of genomic interaction networks, which rely, to a large extent, on
mechanisms based on sequence matching, and exhibit similar global features to
those found here.Comment: 13 pages, 5 figures. Rewrote conclusions regarding the relevance to
gene regulation networks, fixed minor errors and replaced fig. 4. Main body
of paper (model and calculations) remains unchanged. Submitted for
publicatio
Langevin Trajectories between Fixed Concentrations
We consider the trajectories of particles diffusing between two infinite
baths of fixed concentrations connected by a channel, e.g. a protein channel of
a biological membrane. The steady state influx and efflux of Langevin
trajectories at the boundaries of a finite volume containing the channel and
parts of the two baths is replicated by termination of outgoing trajectories
and injection according to a residual phase space density. We present a
simulation scheme that maintains averaged fixed concentrations without creating
spurious boundary layers, consistent with the assumed physics
Minimal Absent Words in Prokaryotic and Eukaryotic Genomes
Minimal absent words have been computed in genomes of organisms from all domains of life. Here, we explore different sets of minimal absent words in the genomes of 22 organisms (one archaeota, thirteen bacteria and eight eukaryotes). We investigate if the mutational biases that may explain the deficit of the shortest absent words in vertebrates are also pervasive in other absent words, namely in minimal absent words, as well as to other organisms. We find that the compositional biases observed for the shortest absent words in vertebrates are not uniform throughout different sets of minimal absent words. We further investigate the hypothesis of the inheritance of minimal absent words through common ancestry from the similarity in dinucleotide relative abundances of different sets of minimal absent words, and find that this inheritance may be exclusive to vertebrates
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
Universal exit probabilities in the TASEP
We study the joint exit probabilities of particles in the totally asymmetric
simple exclusion process (TASEP) from space-time sets of given form. We extend
previous results on the space-time correlation functions of the TASEP, which
correspond to exits from the sets bounded by straight vertical or horizontal
lines. In particular, our approach allows us to remove ordering of time moments
used in previous studies so that only a natural space-like ordering of particle
coordinates remains. We consider sequences of general staircase-like boundaries
going from the northeast to southwest in the space-time plane. The exit
probabilities from the given sets are derived in the form of Fredholm
determinant defined on the boundaries of the sets. In the scaling limit, the
staircase-like boundaries are treated as approximations of continuous
differentiable curves. The exit probabilities with respect to points of these
curves belonging to arbitrary space-like path are shown to converge to the
universal Airy process.Comment: 46 pages, 7 figure
Economical (k,m)-threshold controlled quantum teleportation
We study a (k,m)-threshold controlling scheme for controlled quantum
teleportation. A standard polynomial coding over GF(p) with prime p > m-1 needs
to distribute a d-dimensional qudit with d >= p to each controller for this
purpose. We propose a scheme using m qubits (two-dimensional qudits) for the
controllers' portion, following a discussion on the benefit of a quantum
control in comparison to a classical control of a quantum teleportation.Comment: 11 pages, 2 figures, v2: minor revision, discussions improved, an
equation corrected in procedure (A) of section 4.3, v3: major revision,
protocols extended, citations added, v4: minor grammatical revision, v5:
minor revision, discussions extende
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