2,177 research outputs found
An "All Possible Steps" Approach to the Accelerated Use of Gillespie's Algorithm
Many physical and biological processes are stochastic in nature.
Computational models and simulations of such processes are a mathematical and
computational challenge. The basic stochastic simulation algorithm was
published by D. Gillespie about three decades ago [D.T. Gillespie, J. Phys.
Chem. {\bf 81}, 2340, (1977)]. Since then, intensive work has been done to make
the algorithm more efficient in terms of running time. All accelerated versions
of the algorithm are aimed at minimizing the running time required to produce a
stochastic trajectory in state space. In these simulations, a necessary
condition for reliable statistics is averaging over a large number of
simulations. In this study I present a new accelerating approach which does not
alter the stochastic algorithm, but reduces the number of required runs. By
analysis of collected data I demonstrate high precision levels with fewer
simulations. Moreover, the suggested approach provides a good estimation of
statistical error, which may serve as a tool for determining the number of
required runs.Comment: Accepted for publication at the Journal of Chemical Physics. 19
pages, including 2 Tables and 4 Figure
Exit times in non-Markovian drifting continuous-time random walk processes
By appealing to renewal theory we determine the equations that the mean exit
time of a continuous-time random walk with drift satisfies both when the
present coincides with a jump instant or when it does not. Particular attention
is paid to the corrections ensuing from the non-Markovian nature of the
process. We show that when drift and jumps have the same sign the relevant
integral equations can be solved in closed form. The case when holding times
have the classical Erlang distribution is considered in detail.Comment: 9 pages, 3 color plots, two-column revtex 4; new Appendix and
references adde
Stochastic Chemical Reactions in Micro-domains
Traditional chemical kinetics may be inappropriate to describe chemical
reactions in micro-domains involving only a small number of substrate and
reactant molecules. Starting with the stochastic dynamics of the molecules, we
derive a master-diffusion equation for the joint probability density of a
mobile reactant and the number of bound substrate in a confined domain. We use
the equation to calculate the fluctuations in the number of bound substrate
molecules as a function of initial reactant distribution. A second model is
presented based on a Markov description of the binding and unbinding and on the
mean first passage time of a molecule to a small portion of the boundary. These
models can be used for the description of noise due to gating of ionic channels
by random binding and unbinding of ligands in biological sensor cells, such as
olfactory cilia, photo-receptors, hair cells in the cochlea.Comment: 33 pages, Journal Chemical Physic
Supersymmetry solution for finitely extensible dumbbell model
Exact relaxation times and eigenfunctions for a simple mechanical model of
polymer dynamics are obtained using supersymmetry methods of quantum mechanics.
The model includes the finite extensibility of the molecule and does not make
use of the self-consistently averaging approximation. The finite extensibility
reduces the relaxation times when compared to a linear force. The linear
viscoelastic behaviour is obtained in the form of the ``generalized Maxwell
model''. Using these results, a numerical integration scheme is proposed in the
presence of a given flow kinematics.Comment: 5 pages, 2 figure
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
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
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
Efficient Stochastic Simulations of Complex Reaction Networks on Surfaces
Surfaces serve as highly efficient catalysts for a vast variety of chemical
reactions. Typically, such surface reactions involve billions of molecules
which diffuse and react over macroscopic areas. Therefore, stochastic
fluctuations are negligible and the reaction rates can be evaluated using rate
equations, which are based on the mean-field approximation. However, in case
that the surface is partitioned into a large number of disconnected microscopic
domains, the number of reactants in each domain becomes small and it strongly
fluctuates. This is, in fact, the situation in the interstellar medium, where
some crucial reactions take place on the surfaces of microscopic dust grains.
In this case rate equations fail and the simulation of surface reactions
requires stochastic methods such as the master equation. However, in the case
of complex reaction networks, the master equation becomes infeasible because
the number of equations proliferates exponentially. To solve this problem, we
introduce a stochastic method based on moment equations. In this method the
number of equations is dramatically reduced to just one equation for each
reactive species and one equation for each reaction. Moreover, the equations
can be easily constructed using a diagrammatic approach. We demonstrate the
method for a set of astrophysically relevant networks of increasing complexity.
It is expected to be applicable in many other contexts in which problems that
exhibit analogous structure appear, such as surface catalysis in nanoscale
systems, aerosol chemistry in stratospheric clouds and genetic networks in
cells
Generalized Rayleigh and Jacobi processes and exceptional orthogonal polynomials
We present four types of infinitely many exactly solvable Fokker-Planck
equations, which are related to the newly discovered exceptional orthogonal
polynomials. They represent the deformed versions of the Rayleigh process and
the Jacobi process.Comment: 17 pages, 4 figure
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