5,891 research outputs found
Reactive dynamics on fractal sets: anomalous fluctuations and memory effects
We study the effect of fractal initial conditions in closed reactive systems
in the cases of both mobile and immobile reactants. For the reaction , in the absence of diffusion, the mean number of particles is shown to
decay exponentially to a steady state which depends on the details of the
initial conditions. The nature of this dependence is demonstrated both
analytically and numerically. In contrast, when diffusion is incorporated, it
is shown that the mean number of particles decays asymptotically as
, the memory of the initial conditions being now carried by the
dynamical power law exponent. The latter is fully determined by the fractal
dimension of the initial conditions.Comment: 7 pages, 2 figures, uses epl.cl
Non-equilibrium phase transitions in biomolecular signal transduction
We study a mechanism for reliable switching in biomolecular
signal-transduction cascades. Steady bistable states are created by system-size
cooperative effects in populations of proteins, in spite of the fact that the
phosphorylation-state transitions of any molecule, by means of which the switch
is implemented, are highly stochastic. The emergence of switching is a
nonequilibrium phase transition in an energetically driven, dissipative system
described by a master equation. We use operator and functional integral methods
from reaction-diffusion theory to solve for the phase structure, noise
spectrum, and escape trajectories and first-passage times of a class of minimal
models of switches, showing how all critical properties for switch behavior can
be computed within a unified framework
Numerical Methods for Parabolic Partial Differential Equations on Metric Graphs
The major motivation for this work arose from the problem of simulating diffusion type processes in the human brain network. This thesis addresses numerical methods for parabolic partial differential equations (PDEs) on network structures interpreted as metric spaces (metric graphs). Such domains frequently occur in the context of quantum graphs, where they are studied together with a differential operator and coupling conditions at the vertices of the metric graph. Quantum graphs are popular models for thin, branched structures, and there is a great interest in their studies also from the theoretical point of view. The present work aims to bridge the gap between the theoretical work and the practical usage of quantum graph models by studying arising numerical problems. The main focus is on initial boundary value problems governed by (semilinear) parabolic partial differential equations that involve a second order spatial derivative posed on the
edges of the graph. The particularity of these problems are the coupling conditions of the PDEs on their common vertices.
The two central methods studied in this thesis are a Galerkin discretization with linear finite elements and a spectral Galerkin discretization with basis functions obtained from an eigenvalue problem on the metric graph. Both approaches follow the method of lines, i.e., Galerkin’s method is applied for the spatial discretization resulting in a system of ordinary differential equations. Spectral accuracy can be obtained with the spectral discretization in space for sufficiently smooth functions that fulfill certain coupling conditions at the vertices.
In the finite element approach, the semidiscretization is solved with classical implicit-explicit time stepping methods combined with a graph specific multigrid solver for the arising systems of linear equations in each time step. In the spectral method, the stiffness matrix is diagonal such that exponential integrators can be applied efficiently to solve the semidiscretized system. The difficulty of the spectral method, by contrast, is the computation of an eigenfunction basis.
The computation of quantum graph spectra thus is the last important aspect of this work. The problem of computing eigenfunctions can be reduced to a nonlinear eigenvalue problem (NEP). In the particular case of equilateral graphs, the NEP even simplifies to a linear eigenvalue problem in the size of the number of vertices of the underlying graph. The proposed NEP solver applies equilateral approximations combined with a nested iteration approach to obtain initial guesses for a Newton-trace iteration.
Human connectomes interpreted as metric graphs are consulted to test the applicability of the methods to real world, large scale problems. Experiments on simulating distribution of tau proteins in the brain of Alzheimer’s disease patients complete this work
Mathematical approaches to differentiation and gene regulation
We consider some mathematical issues raised by the modelling of gene
networks. The expression of genes is governed by a complex set of regulations,
which is often described symbolically by interaction graphs. Once such a graph
has been established, there remains the difficult task to decide which
dynamical properties of the gene network can be inferred from it, in the
absence of precise quantitative data about their regulation. In this paper we
discuss a rule proposed by R.Thomas according to which the possibility for the
network to have several stationary states implies the existence of a positive
circuit in the corresponding interaction graph. We prove that, when properly
formulated in rigorous terms, this rule becomes a theorem valid for several
different types of formal models of gene networks. This result is already known
for models of differential or boolean type. We show here that a stronger
version of it holds in the differential setup when the decay of protein
concentrations is taken into account. This allows us to verify also the
validity of Thomas' rule in the context of piecewise-linear models and the
corresponding discrete models. We discuss open problems as well.Comment: To appear in Notes Comptes-Rendus Acad. Sc. Paris, Biologi
Persistence properties of a system of coagulating and annihilating random walkers
We study a d-dimensional system of diffusing particles that on contact either
annihilate with probability 1/(q-1) or coagulate with probability (q-2)/(q-1).
In 1-dimension, the system models the zero temperature Glauber dynamics of
domain walls in the q-state Potts model. We calculate P(m,t), the probability
that a randomly chosen lattice site contains a particle whose ancestors have
undergone exactly (m-1) coagulations. Using perturbative renormalization group
analysis for d < 2, we show that, if the number of coagulations m is much less
than the typical number M(t), then P(m,t) ~ m^(z/d) t^(-theta), with theta=d Q
+ Q(Q-1/2) epsilon + O(epsilon^2), z=(2Q-1) epsilon + (2 Q-1) (Q-1)(1/2+A Q)
epsilon^2 +O(epsilon^3), where Q=(q-1)/q, epsilon =2-d and A =-0.006. M(t) is
shown to scale as t^(d/2-delta), where delta = d (1 -Q)+(Q-1)(Q-1/2) epsilon+
O(epsilon^2). In two dimensions, we show that P(m,t) ~ ln(t)^(Q(3-2Q))
ln(m)^((2Q-1)^2) t^(-2Q) for m << t^(2 Q-1). The 1-dimensional results
corresponding to epsilon=1 are compared with results from Monte Carlo
simulations.Comment: 12 pages, revtex, 5 figure
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