1,059 research outputs found
From individual to collective behaviour of coupled velocity jump processes: a locust example
A class of stochastic individual-based models, written in terms of coupled velocity jump processes, is presented and analysed. This modelling approach incorporates recent experimental findings on behaviour of locusts. It exhibits nontrivial dynamics with a “phase change” behaviour and recovers the observed group directional switching. Estimates of the expected switching times, in terms of number of individuals and values of the model coefficients, are obtained using the corresponding Fokker-Planck equation. In the limit of large populations, a system of two kinetic equations with nonlocal and nonlinear right hand side is derived and analyzed. The existence of its solutions is proven and the system’s long-time behaviour is investigated. Finally, a first step towards the mean field limit of topological interactions is made by studying the effect of shrinking the interaction radius in the individual-based model when the number of individuals grows
On chemisorption of polymers to solid surfaces
The irreversible adsorption of polymers to a two-dimensional solid surface is
studied. An operator formalism is introduced for chemisorption from a
polydisperse solution of polymers which transforms the analysis of the
adsorption process to a set of combinatorial problems on a two-dimensional
lattice. The time evolution of the number of polymers attached and the surface
area covered are calculated via a series expansion. The dependence of the final
coverage on the parameters of the model (i.e. the parameters of the
distribution of polymer lengths in the solution) is studied. Various methods
for accelerating the convergence of the resulting infinite series are
considered. To demonstrate the accuracy of the truncated series approach, the
series expansion results are compared with the results of stochastic
simulation.Comment: 20 pages, submitted to Journal of Statistical Physic
Hybrid modelling of individual movement and collective behaviour
Mathematical models of dispersal in biological systems are often written in terms of partial differential equations (PDEs) which describe the time evolution of population-level variables (concentrations, densities). A more detailed modelling approach is given by individual-based (agent-based) models which describe the behaviour of each organism. In recent years, an intermediate modelling methodology – hybrid modelling – has been applied to a number of biological systems. These hybrid models couple an individual-based description of cells/animals with a PDEmodel of their environment. In this chapter, we overview hybrid models in the literature with the focus on the mathematical challenges of this modelling approach. The detailed analysis is presented using the example of chemotaxis, where cells move according to extracellular chemicals that can be altered by the cells themselves. In this case, individual-based models of cells are coupled with PDEs for extracellular chemical signals. Travelling waves in these hybrid models are investigated. In particular, we show that in contrary to the PDEs, hybrid chemotaxis models only develop a transient travelling wave
Multi-resolution polymer Brownian dynamics with hydrodynamic interactions
A polymer model given in terms of beads, interacting through Hookean springs
and hydrodynamic forces, is studied. Brownian dynamics description of this
bead-spring polymer model is extended to multiple resolutions. Using this
multiscale approach, a modeller can efficiently look at different regions of
the polymer in different spatial and temporal resolutions with scalings given
for the number of beads, statistical segment length and bead radius in order to
maintain macro-scale properties of the polymer filament. The Boltzmann
distribution of a Gaussian chain for differing statistical segment lengths
gives a Langevin equation for the multi-resolution model with a mobility tensor
for different bead sizes. Using the pre-averaging approximation, the
translational diffusion coefficient is obtained as a function of the inverse of
a matrix and then in closed form in the long-chain limit. This is then
confirmed with numerical experiments.Comment: Submitted to Journal of Chemical Physic
Time scale of random sequential adsorption
A simple multiscale approach to the diffusion-driven adsorption from a solution to a solid surface is presented. The model combines two important features of the adsorption process: (i) the kinetics of the chemical reaction between adsorbing molecules and the surface; and (ii) geometrical constraints on the surface made by molecules which are already adsorbed. The process (i) is modelled in a diffusion-driven context, i.e. the conditional probability of adsorbing a molecule provided that the molecule hits the surface is related to the macroscopic surface reaction rate. The geometrical constraint (ii) is modelled using random sequential adsorption (RSA), which is the sequential addition of molecules at random positions on a surface; one attempt to attach a molecule is made per one RSA simulation time step. By coupling RSA with the diffusion of molecules in the solution above the surface the RSA simulation time step is related to the real physical time. The method is illustrated on a model of chemisorption of reactive polymers to a virus surface
Stochastic modelling of reaction-diffusion processes:\ud algorithms for bimolecular reactions
Several stochastic simulation algorithms (SSAs) have been recently proposed for modelling reaction-diffusion processes in cellular and molecular biology. In this paper, two commonly used SSAs are studied. The first SSA is an on-lattice model described by the reaction-diffusion master equation. The second SSA is an off-lattice model based on the simulation of Brownian motion of individual molecules and their reactive collisions. In both cases, it is shown that the commonly used implementation of bimolecular reactions (i.e. the reactions of the form A+B → C, or A+A → C) might lead to incorrect results. Improvements of both SSAs are suggested which overcome the difficulties highlighted. In particular, a formula is presented for the smallest possible compartment size (lattice spacing) which can be correctly implemented in the first model. This implementation uses a new formula for the rate of bimolecular reactions per compartment (lattice site)
Taxis Equations for Amoeboid Cells
The classical macroscopic chemotaxis equations have previously been derived
from an individual-based description of the tactic response of cells that use a
"run-and-tumble" strategy in response to environmental cues. Here we derive
macroscopic equations for the more complex type of behavioral response
characteristic of crawling cells, which detect a signal, extract directional
information from a scalar concentration field, and change their motile behavior
accordingly. We present several models of increasing complexity for which the
derivation of population-level equations is possible, and we show how
experimentally-measured statistics can be obtained from the transport equation
formalism. We also show that amoeboid cells that do not adapt to constant
signals can still aggregate in steady gradients, but not in response to
periodic waves. This is in contrast to the case of cells that use a
"run-and-tumble" strategy, where adaptation is essential.Comment: 35 pages, submitted to the Journal of Mathematical Biolog
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