Analytic solutions for some reaction-diffusion scenarios

RBL-2H3 Mast Cell Receptor Dynamics in the Immunological Synapse

The RBL-2H3 mast cell immunological synapse dynamics is often simulated with reaction-diffusion and Fokker-Planck equations. The equations focus on how the cell synapse captures receptors following an immune response, where the receptor capture at the immunological site appears to be a delayed process. This article investigates the physical nature and mathematics behind such time-dependent delays. Using signal processing methods, convolution and cross-correlation-type delay capture simulations give a χ -squared range of 22 to 60, in good agreement with experimental results. The cell polarization event is offered as a possible explanation for these capture delays, where polarizing rates measure how fast the cell polarization event occurs. In the case of RBL-2H3 mast cells, polarization appears to be associated with cytoskeletal rearrangement; thus, both cytoskeletal and diffusional components are considered. From these simulations, a maximum polarizing rate ranging from 0.0057 s-2 to 0.031 s-2 is obtained. These results indicate that RBL-2H3 mast cells possess both temporal and spatial memory, and cell polarization is possibly linked to a Turing-type pattern formation.

A theory of coalescence of signaling receptor clusters in immune cells

A theory of coalescence of signal receptor clusters in mast cells is developed in close connection with experiments. It is based on general considerations involving a feedback procedure and a time-dependent capture as part of a reaction-diffusion process. Characteristic features of observations that need to be explained are indicated and it is shown why calculations available in the literature are not satisfactory. While the latter involves static centers at which the reaction part of the phenomenon occurs, by its very nature, coalescence involves dynamically evolving centers. This is so because the process continuously modifies the size of the cluster aggregate which then proceeds to capture more material. We develop a procedure that consists of first solving a static reaction-diffusion problem and then imbuing the center with changing size. The consequence is a dependence of the size of the signal receptor cluster aggregate on time. A preliminary comparison with experiment is shown to reveal a sharp difference between theory and data. The observation indicates that the reaction occurs slowly at first and then picks up rapidly as time proceeds. Parameter modification to fit the observations cannot solve the problem. We use this observation to build into the theory an accumulation rate that is itself dependent on time. A memory representation and its physical basis are explained. The consequence is a theory that can be fit to observations successfully.

Binding kinetics of harmonically confined random walkers

Diffusion-mediated binding of molecules under the influence of discrete spatially confining potentials is a commonly encountered scenario in systems subjected to explicit fields or implicit fields arising from tethering restraints. Here, we derive analytical expressions for the mean binding time of two random walkers geometrically confined by means of two harmonic potentials in one- and two-dimensional systems, which show excellent agreement with Brownian dynamics simulations. As a demonstration of its utility, we use this theory to maximize the communication speed in existing DNA walkers, obtaining quantitative agreement with previously reported experimental findings. The analytical expressions derived in this paper are broadly applicable to diverse systems, providing ways to characterize communication processes and optimize the rate of signal propagation for sensing and computing applications at the nanoscale.

Analysis of Transmission of Infection in Epidemics: Confined Random Walkers in Dimensions Higher Than One

The process of transmission of infection in epidemics is analyzed by studying a pair of random walkers, the motion of each of which in two dimensions is confined spatially by the action of a quadratic potential centered at different locations for the two walks. The walkers are animals such as rodents in considerations of the Hantavirus epidemic, infected or susceptible. In this reaction-diffusion study, the reaction is the transmission of infection, and the confining potential represents the tendency of the animals to stay in the neighborhood of their home range centers. Calculations are based on a recently developed formalism (Kenkre and Sugaya in Bull Math Biol 76:3016-3027, 2014) structured around analytic solutions of a Smoluchowski equation and one of its aims is the resolution of peculiar but well-known problems of reaction-diffusion theory in two dimensions. The resolution is essential to a realistic application to field observations because the terrain over which the rodents move is best represented as a 2-d landscape. In the present analysis, reaction occurs not at points but in spatial regions of dimensions larger than 0. The analysis uncovers interesting nonintuitive phenomena one of which is similar to that encountered in the one-dimensional analysis given in the quoted article, and another specific to the fact that the reaction region is spatially extended. The analysis additionally provides a realistic description of observations on animals transmitting infection while moving on what is effectively a two-dimensional landscape. Along with the general formalism and explicit one-dimensional analysis given in Kenkre and Sugaya (2014), the present work forms a model calculational tool for the analysis for the transmission of infection in dilute systems.

Minimal model for anomalous diffusion

A random walk model with a local probability of removal is solved exactly and shown to exhibit subdiffusive behavior with a mean square displacement the evolves as 〈x^{2}(t)〉∼t^{1/2} at late times. This model is shown to be well described by a diffusion equation with a sink term, which also describes the evolution of a pressure or temperature field in a leaky environment. For this reason a number of physical processes are shown to exhibit anomalous diffusion. The presence of the sink term is shown to change the late time behavior of the field from 1/t^{1/2} to 1/t^{3/2}.

Random walks with fractally correlated traps: Stretched exponential and power-law survival kinetics

We consider the survival probability f(t) of a random walk with a constant hopping rate w on a host lattice of fractal dimension d and spectral dimension d_{s}≤2, with spatially correlated traps. The traps form a sublattice with fractal dimension d_{a}<d and are characterized by the absorption rate w_{a} which may be finite (imperfect traps) or infinite (perfect traps). Initial coordinates are chosen randomly at or within a fixed distance of a trap. For weakly absorbing traps (w_{a}≪w), we find that f(t) can be closely approximated by a stretched exponential function over the initial stage of relaxation, with stretching exponent α=1-(d-d_{a})/d_{w}, where d_{w} is the random walk dimension of the host lattice. At the end of this initial stage there occurs a crossover to power-law kinetics f(t)∼t^{-α} with the same exponent α as for the stretched exponential regime. For strong absorption w_{a}≳w, including the limit of perfect traps w_{a}→∞, the stretched exponential regime is absent and the decay of f(t) follows, after a short transient, the aforementioned power law for all times.

Theory of the transmission of infection in the spread of epidemics: interacting random walkers with and without confinement

A theory of the spread of epidemics is formulated on the basis of pairwise interactions in a dilute system of random walkers (infected and susceptible animals) moving in [Formula: see text] dimensions. The motion of an animal pair is taken to obey a Smoluchowski equation in [Formula: see text]-dimensional space that combines diffusion with confinement of each animal to its particular home range. An additional (reaction) term that comes into play when the animals are in close proximity describes the process of infection. Analytic solutions are obtained, confirmed by numerical procedures, and shown to predict a surprising effect of confinement. The effect is that infection spread has a non-monotonic dependence on the diffusion constant and/or the extent of the attachment of the animals to the home ranges. Optimum values of these parameters exist for any given distance between the attractive centers. Any change from those values, involving faster/slower diffusion or shallower/steeper confinement, hinders the transmission of infection. A physical explanation is provided by the theory. Reduction to the simpler case of no home ranges is demonstrated. Effective infection rates are calculated, and it is shown how to use them in complex systems consisting of dense populations.

Reaction-diffusion theory in the presence of an attractive harmonic potential

Problems involving the capture of a moving entity by a trap occur in a variety of physical situations, the moving entity being an electron, an excitation, an atom, a molecule, a biological object such as a receptor cluster, a cell, or even an animal such as a mouse carrying an epidemic. Theoretical considerations have almost always assumed that the particle motion is translationally invariant. We study here the case when that assumption is relaxed, in that the particle is additionally subjected to a harmonic potential. This tethering to a center modifies the reaction-diffusion phenomenon. Using a Smoluchowski equation to describe the system, we carry out a study which is explicit in one dimension but can be easily extended for arbitrary dimensions. Interesting features emerge depending on the relative location of the trap, the attractive center, and the initial placement of the diffusing particle.