45 research outputs found
Power series method for solving TASEP-based models of mRNA translation
We develop a method for solving mathematical models of messenger RNA (mRNA)
translation based on the totally asymmetric simple exclusion process (TASEP).
Our main goal is to demonstrate that the method is versatile and applicable to
realistic models of translation. To this end we consider the TASEP with
codon-dependent elongation rates, premature termination due to ribosome
drop-off and translation reinitiation due to circularisation of the mRNA. We
apply the method to the model organism {\it Saccharomyces cerevisiae} under
physiological conditions and find excellent agreements with the results of
stochastic simulations. Our findings suggest that the common view on
translation as being rate-limited by initiation is oversimplistic. Instead we
find theoretical evidence for ribosome interference and also theoretical
support for the ramp hypothesis which argues that codons at the beginning of
genes have slower elongation rates in order to reduce ribosome density and
jamming.Comment: 13 pages, 10 figure
Totally asymmetric exclusion process with long-range hopping
Generalization of the one-dimensional totally asymmetric exclusion process
(TASEP) with open boundary conditions in which particles are allowed to jump
sites ahead with the probability is studied by
Monte Carlo simulations and the domain-wall approach. For the
standard TASEP phase diagram is recovered, but the density profiles near the
transition lines display new features when . At the first-order
transition line, the domain-wall is localized and phase separation is observed.
In the maximum-current phase the profile has an algebraic decay with a
-dependent exponent. Within the regime, where the
transitions are found to be absent, analytical results in the continuum
mean-field approximation are derived in the limit .Comment: 10 pages, 9 figure
Absence of phase coexistence in disordered exclusion processes with bypassing
Adding quenched disorder to the one-dimensional asymmetric exclusion process
is known to always induce phase separation. To test the robustness of this
result, we introduce two modifications of the process that allow particles to
bypass defect sites. In the first case, particles are allowed to jump l sites
ahead with the probability p_l ~ l^-(1+sigma), where sigma>1. By using Monte
Carlo simulations and the mean-field approach, we show that phase coexistence
may be absent up to enormously large system sizes, e.g. lnL~50, but is present
in the thermodynamic limit, as in the short-range case. In the second case, we
consider the exclusion process on a quadratic lattice with symmetric and
totally asymmetric hopping perpendicular to and along the direction of driving,
respectively. We show that in an anisotropic limit of this model a regime may
be found where phase coexistence is absent.Comment: 18 pages, 10 figures, to appear in JSTA
Conditioned random walks and interaction-driven condensation
We consider a discrete-time continuous-space random walk under the
constraints that the number of returns to the origin (local time) and the total
area under the walk are fixed. We first compute the joint probability of an
excursion having area and returning to the origin for the first time after
time . We then show how condensation occurs when the total area
constraint is increased: an excursion containing a finite fraction of the area
emerges. Finally we show how the phenomena generalises previously studied cases
of condensation induced by several constraints and how it is related to
interaction-driven condensation which allows us to explain the phenomenon in
the framework of large deviation theory.Comment: 28 pages, 6 figures, invited paper for Special Issue of J. Phys. A
"Emerging talents
Inequivalence of nonequilibrium path ensembles: the example of stochastic bridges
We study stochastic processes in which the trajectories are constrained so
that the process realises a large deviation of the unconstrained process. In
particular we consider stochastic bridges and the question of inequivalence of
path ensembles between the microcanonical ensemble, in which the end points of
the trajectory are constrained, and the canonical or s ensemble in which a bias
or tilt is introduced into the process. We show how ensemble inequivalence can
be manifested by the phenomenon of temporal condensation in which the large
deviation is realised in a vanishing fraction of the duration (for long
durations). For diffusion processes we find that condensation happens whenever
the process is subject to a confining potential, such as for the
Ornstein-Uhlenbeck process, but not in the borderline case of dry friction in
which there is partial ensemble equivalence. We also discuss continuous-space,
discrete-time random walks for which in the case of a heavy tailed step-size
distribution it is known that the large deviation may be achieved in a single
step of the walk. Finally we consider possible effects of several constraints
on the process and in particular give an alternative explanation of the
interaction-driven condensation in terms of constrained Brownian excursions.Comment: 22 pages, 7 figures, minor revisio
Inherent Variability in the Kinetics of Autocatalytic Protein Self-Assembly
In small volumes, the kinetics of filamentous protein self-assembly is
expected to show significant variability, arising from intrinsic molecular
noise. This is not accounted for in existing deterministic models. We introduce
a simple stochastic model including nucleation and autocatalytic growth via
elongation and fragmentation, which allows us to predict the effects of
molecular noise on the kinetics of autocatalytic self-assembly. We derive an
analytic expression for the lag-time distribution, which agrees well with
experimental results for the fibrillation of bovine insulin. Our expression
decomposes the lag time variability into contributions from primary nucleation
and autocatalytic growth and reveals how each of these scales with the key
kinetic parameters. Our analysis shows that significant lag-time variability
can arise from both primary nucleation and from autocatalytic growth, and
should provide a way to extract mechanistic information on early-stage
aggregation from small-volume experiments.Comment: 5pp, 3 fig. + Supp. Mat. (7pp, 4 fig.), accepted for publication in
PR
Disordered exclusion process revisited:some exact results in the low-current regime
We study steady state of the totally asymmetric simple exclusion process with
inhomogeneous hopping rates associated with sites (site-wise disorder). Using
the fact that the non-normalized steady-state weights which solve the master
equation are polynomials in all the hopping rates, we propose a general method
for calculating their first few lowest coefficients exactly. In case of binary
disorder where all slow sites share the same hopping rate r<1, we apply this
method to calculate steady-state current up to the quadratic term in r for some
particular disorder configurations. For the most general (non-binary) disorder,
we show that in the low-current regime the current is determined solely by the
current-minimizing subset of equal hopping rates, regardless of other hopping
rates. Our approach can be readily applied to any other driven diffusive system
with unidirectional hopping if one can identify a hopping rate such that the
current vanishes when this rate is set to zero.Comment: 26 pages, 7 figures, iopart class, submitted to J. Phys.
Conditioned stochastic particle systems and integrable quantum spin systems
We consider from a microscopic perspective large deviation properties of
several stochastic interacting particle systems, using their mapping to
integrable quantum spin systems. A brief review of recent work is given and
several new results are presented: (i) For the general disordered symmectric
exclusion process (SEP) on some finite lattice conditioned on no jumps into
some absorbing sublattice and with initial Bernoulli product measure with
density we prove that the probability of no absorption event
up to microscopic time can be expressed in terms of the generating function
for the particle number of a SEP with particle injection and empty initial
lattice. Specifically, for the symmetric simple exclusion process on conditioned on no jumps into the origin we obtain the explicit first and
second order expansion in of and also to first order in
the optimal microscopic density profile under this conditioning. For the
disordered ASEP on the finite torus conditioned on a very large current we show
that the effective dynamics that optimally realizes this rare event does not
depend on the disorder, except for the time scale. For annihilating and
coalescing random walkers we obtain the generating function of the number of
annihilated particles up to time , which turns out to exhibit some universal
features.Comment: 25 page