57 research outputs found
Survival probability of a diffusing particle in the presence of Poisson-distributed mobile traps
The problem of a diffusing particle moving among diffusing traps is analyzed
in general space dimension d. We consider the case where the traps are
initially randomly distributed in space, with uniform density rho, and derive
upper and lower bounds for the probability Q(t) (averaged over all particle and
trap trajectories) that the particle survives up to time t. We show that, for
1<=d<2, the bounds converge asymptotically to give where and D
is the diffusion constant of the traps, and that for d=2. For d>2 bounds can still be derived, but they no longer
converge for large t. For 1<=d<=2, these asymptotic form are independent of the
diffusion constant of the particle. The results are compared with simulation
results obtained using a new algorithm [V. Mehra and P. Grassberger, Phys. Rev.
E v65 050101 (2002)] which is described in detail. Deviations from the
predicted asymptotic forms are found to be large even for very small values of
Q(t), indicating slowly decaying corrections whose form is consistent with the
bounds. We also present results in d=1 for the case where the trap densities on
either side of the particle are different. For this case we can still obtain
exact bounds but they no longer converge.Comment: 13 pages, RevTeX4, 6 figures. Figures and references updated;
equations corrected; discussion clarifie
Survival Probability of a Ballistic Tracer Particle in the Presence of Diffusing Traps
We calculate the survival probability P_S(t) up to time t of a tracer
particle moving along a deterministic trajectory in a continuous d-dimensional
space in the presence of diffusing but mutually noninteracting traps. In
particular, for a tracer particle moving ballistically with a constant velocity
c, we obtain an exact expression for P_S(t), valid for all t, for d<2. For d
\geq 2, we obtain the leading asymptotic behavior of P_S(t) for large t. In all
cases, P_S(t) decays exponentially for large t, P_S(t) \sim \exp(-\theta t). We
provide an explicit exact expression for the exponent \theta in dimensions d
\leq 2, and for the physically relevant case, d=3, as a function of the system
parameters.Comment: RevTeX, 4 page
Coupled differentiation and division of embryonic stem cells inferred from clonal snapshots
The deluge of single-cell data obtained by sequencing, imaging and epigenetic markers has led to an increasingly detailed description of cell state. However, it remains challenging to identify how cells transition between different states, in part because data are typically limited to snapshots in time. A prerequisite for inferring cell state transitions from such snapshots is to distinguish whether transitions are coupled to cell divisions. To address this, we present two minimal branching process models of cell division and differentiation in a well-mixed population. These models describe dynamics where differentiation and division are coupled or uncoupled. For each model, we derive analytic expressions for each subpopulation's mean and variance and for the likelihood, allowing exact Bayesian parameter inference and model selection in the idealised case of fully observed trajectories of differentiation and division events. In the case of snapshots, we present a sample path algorithm and use this to predict optimal temporal spacing of measurements for experimental design. We then apply this methodology to an \textit{in vitro} dataset assaying the clonal growth of epiblast stem cells in culture conditions promoting self-renewal or differentiation. Here, the larger number of cell states necessitates approximate Bayesian computation. For both culture conditions, our inference supports the model where cell state transitions are coupled to division. For culture conditions promoting differentiation, our analysis indicates a possible shift in dynamics, with these processes becoming more coupled over time
Mean-field analysis of the q-voter model on networks
We present a detailed investigation of the behavior of the nonlinear q-voter
model for opinion dynamics. At the mean-field level we derive analytically, for
any value of the number q of agents involved in the elementary update, the
phase diagram, the exit probability and the consensus time at the transition
point. The mean-field formalism is extended to the case that the interaction
pattern is given by generic heterogeneous networks. We finally discuss the case
of random regular networks and compare analytical results with simulations.Comment: 20 pages, 10 figure
Kinetochore assembly and heterochromatin formation occur autonomously in Schizosaccharomyces pombe
Kinetochores in multicellular eukaryotes are usually associated with heterochromatin. Whether this heterochromatin simply promotes the cohesion necessary for accurate chromosome segregation at cell division or whether it also has a role in kinetochore assembly is unclear. Schizosaccharomyces pombe is an important experimental system for investigating centromere function, but all of the previous work with this species has exploited a single strain or its derivatives. The laboratory strain and most other S. pombe strains contain three chromosomes, but one recently discovered strain, CBS 2777, contains four. We show that the genome of CBS 2777 is related to that of the laboratory strain by a complex chromosome rearrangement. As a result, two of the kinetochores in CBS 2777 contain the central core sequences present in the laboratory strain centromeres, but lack adjacent heterochromatin. The closest block of heterochromatin to these rearranged kinetochores is ∼100 kb away at new telomeres. Despite lacking large amounts of adjacent heterochromatin, the rearranged kinetochores bind CENP-ACnp1 and CENP-CCnp3 in similar quantities and with similar specificities as those of the laboratory strain. The simplest interpretation of this result is that constitutive kinetochore assembly and heterochromatin formation occur autonomously
Nonequilibrium Dynamics in Low Dimensional Systems
In these lectures we give an overview of nonequilibrium stochastic systems.
In particular we discuss in detail two models, the asymmetric exclusion process
and a ballistic reaction model, that illustrate many general features of
nonequilibrium dynamics: for example coarsening dynamics and nonequilibrium
phase transitions. As a secondary theme we shall show how a common mathematical
structure, the q-deformed harmonic oscillator algebra, serves to furnish exact
results for both systems. Thus the lectures also serve as a gentle introduction
to things q-deformed.Comment: 48 pages LaTeX2e with 9 figures and using elsart.cls (included);
Lectures at the International Summer School on Fundamental Problems in
Statistical Physics X, August-September 2001, Altenberg, Germany. v2 corrects
some errors and includes further discussion/reference
Phase diagram of a generalized ABC model on the interval
We study the equilibrium phase diagram of a generalized ABC model on an
interval of the one-dimensional lattice: each site is occupied by a
particle of type \a=A,B,C, with the average density of each particle species
N_\a/N=r_\a fixed. These particles interact via a mean field
non-reflection-symmetric pair interaction. The interaction need not be
invariant under cyclic permutation of the particle species as in the standard
ABC model studied earlier. We prove in some cases and conjecture in others that
the scaled infinite system N\rw\infty, i/N\rw x\in[0,1] has a unique
density profile \p_\a(x) except for some special values of the r_\a for
which the system undergoes a second order phase transition from a uniform to a
nonuniform periodic profile at a critical temperature .Comment: 25 pages, 6 figure
Epidemic spreading with immunization and mutations
The spreading of infectious diseases with and without immunization of
individuals can be modeled by stochastic processes that exhibit a transition
between an active phase of epidemic spreading and an absorbing phase, where the
disease dies out. In nature, however, the transmitted pathogen may also mutate,
weakening the effect of immunization. In order to study the influence of
mutations, we introduce a model that mimics epidemic spreading with
immunization and mutations. The model exhibits a line of continuous phase
transitions and includes the general epidemic process (GEP) and directed
percolation (DP) as special cases. Restricting to perfect immunization in two
spatial dimensions we analyze the phase diagram and study the scaling behavior
along the phase transition line as well as in the vicinity of the GEP point. We
show that mutations lead generically to a crossover from the GEP to DP. Using
standard scaling arguments we also predict the form of the phase transition
line close to the GEP point. It turns out that the protection gained by
immunization is vitally decreased by the occurrence of mutations.Comment: 9 pages, 13 figure
Exclusive Queueing Process with Discrete Time
In a recent study [C Arita, Phys. Rev. E 80, 051119 (2009)], an extension of
the M/M/1 queueing process with the excluded-volume effect as in the totally
asymmetric simple exclusion process (TASEP) was introduced. In this paper, we
consider its discrete-time version. The update scheme we take is the parallel
one. A stationary-state solution is obtained in a slightly arranged matrix
product form of the discrete-time open TASEP with the parallel update. We find
the phase diagram for the existence of the stationary state. The critical line
which separates the parameter space into the regions with and without the
stationary state can be written in terms of the stationary current of the open
TASEP. We calculate the average length of the system and the average number of
particles
The propagation of a cultural or biological trait by neutral genetic drift in a subdivided population
We study fixation probabilities and times as a consequence of neutral genetic
drift in subdivided populations, motivated by a model of the cultural
evolutionary process of language change that is described by the same
mathematics as the biological process. We focus on the growth of fixation times
with the number of subpopulations, and variation of fixation probabilities and
times with initial distributions of mutants. A general formula for the fixation
probability for arbitrary initial condition is derived by extending a duality
relation between forwards- and backwards-time properties of the model from a
panmictic to a subdivided population. From this we obtain new formulae,
formally exact in the limit of extremely weak migration, for the mean fixation
time from an arbitrary initial condition for Wright's island model, presenting
two cases as examples. For more general models of population subdivision,
formulae are introduced for an arbitrary number of mutants that are randomly
located, and a single mutant whose position is known. These formulae contain
parameters that typically have to be obtained numerically, a procedure we
follow for two contrasting clustered models. These data suggest that variation
of fixation time with the initial condition is slight, but depends strongly on
the nature of subdivision. In particular, we demonstrate conditions under which
the fixation time remains finite even in the limit of an infinite number of
demes. In many cases - except this last where fixation in a finite time is seen
- the time to fixation is shown to be in precise agreement with predictions
from formulae for the asymptotic effective population size.Comment: 17 pages, 8 figures, requires elsart5p.cls; substantially revised and
improved version; accepted for publication in Theoretical Population Biolog
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