Accurate discretization of advection-diffusion equations

Outcomes of multifarious selection on the evolution of visual signals

Multifarious sources of selection shape visual signals and can produce phenotypic divergence. Theory predicts that variance in warning signals should be minimal due to purifying selection, yet polymorphism is abundant. While in some instances divergent signals can evolve into discrete morphs, continuously variable phenotypes are also encountered in natural populations. Notwithstanding, we currently have an incomplete understanding of how combinations of selection shape fitness landscapes, particularly those which produce polymorphism. We modelled how combinations of natural and sexual selection act on aposematic traits within a single population to gain insights into what combinations of selection favours the evolution and maintenance of phenotypic variation. With a rich foundation of studies on selection and phenotypic divergence, we reference the poison frog genus Oophaga to model signal evolution. Multifarious selection on aposematic traits created the topology of our model's fitness landscape by approximating different scenarios found in natural populations. Combined, the model produced all types of phenotypic variation found in frog populations, namely monomorphism, continuous variation and discrete polymorphism. Our results afford advances into how multifarious selection shapes phenotypic divergence, which, along with additional modelling enhancements, will allow us to further our understanding of visual signal evolution.

Precooling Strategy Allows Exponentially Faster Heating

What is the fastest way to heat a system which is coupled to a temperature controlled oven? The intuitive answer is to use only the hottest temperature available. However, we show that often it is possible to achieve an exponentially faster heating protocol. Surprisingly, this protocol can have a precooling stage-cooling the system before heating it shortens the heating time significantly. To demonstrate such improvements in many-body systems, we developed a projection-based method with which such protocols can be found in large systems, as we demonstrate on the 2D antiferromagnet Ising model.

A stochastic model of corneal epithelium maintenance and recovery following perturbation

Various biological studies suggest that the corneal epithelium is maintained by active stem cells located in the limbus, the so-called limbal epithelial stem cell hypothesis. While numerous mathematical models have been developed to describe corneal epithelium wound healing, only a few have explored the process of corneal epithelium homeostasis. In this paper we present a purposefully simple stochastic mathematical model based on a chemical master equation approach, with the aim of clarifying the main factors involved in the maintenance process. Model analysis provides a set of constraints on the numbers of stem cells, division rates, and the number of division cycles required to maintain a healthy corneal epithelium. In addition, our stochastic analysis reveals noise reduction as the epithelium approaches its homeostatic state, indicating robustness to noise. Finally, recovery is analysed in the context of perturbation scenarios.

Capturing Brownian dynamics with an on-lattice model of hard-sphere diffusion

Conventional master equation approaches approximate the diffusion of molecules in continuum space by the process of particles hopping on a spatial lattice. The hopping probability from one voxel (spatial lattice point) to its neighbor is usually considered to be constant throughout space. Such an assumption is only consistent with pointlike molecules and thus neglects volume-exclusion effects due to finite particle size. A few studies have attempted to introduce volume-exclusion effects by choosing the hopping probability from one voxel to a neighboring one to be a linear function of the number density. Here, we formulate an alternative master equation in which the hopping probability is equal to the fraction of available space in the neighboring voxel as estimated using scaled particle theory. This leads to the hopping probability being a nonlinear function of the number density. A mean-field approximation (mfa) leads to a partial differential equation of the advection-diffusion type. We show that the time evolution of the particle number density sampled using the stochastic simulation algorithm associated with the new master equation and the number density obtained by numerical integration of the mfa are in good agreement with each other. They are also distinctly different than the time evolution predicted by the conventional master equation and those with hopping probabilities which are linear functions of the number density. The results from the new lattice description are also shown to be in very good agreement with the lattice-free method of Brownian dynamics, even for highly crowded scenarios.

A continuum approximation to an off-lattice individual-cell based model of cell migration and adhesion

Cell-cell adhesion plays a key role in the collective migration of cells and in determining correlations in the relative cell positions and velocities. Recently, it was demonstrated that off-lattice individual cell based models (IBMs) can accurately capture the correlations observed experimentally in a migrating cell population. However, IBMs are often computationally expensive and difficult to analyse mathematically. Traditional continuum-based models, in contrast, are amenable to mathematical analysis and are computationally less demanding, but typically correspond to a mean-field approximation of cell migration and so ignore cell-cell correlations. In this work, we address this problem by using an off-lattice IBM to derive a continuum approximation which does take into account correlations. We furthermore show that a mean-field approximation of the off-lattice IBM leads to a single partial integro-differential equation of the same form as proposed by Sherratt and co-workers to model cell adhesion. The latter is found to be only effective at approximating the ensemble averaged cell number density when mechanical interactions between cells are weak. In contrast, the predictions of our novel continuum model for the time-evolution of the ensemble cell number density distribution and of the density-density correlation function are in close agreement with those obtained from the IBM for a wide range of mechanical interaction strengths. In particular, we observe 'front-like' propagation of cells in simulations using both our IBM and our continuum model, but not in the continuum model simulations obtained using the mean-field approximation.

Aggregation of chemotactic organisms in a differential flow

We study the effect of advection on the aggregation and pattern formation in chemotactic systems described by Keller-Segel-type models. The evolution of small perturbations is studied analytically in the linear regime complemented by numerical simulations. We show that a uniform differential flow can significantly alter the spatial structure and dynamics of the chemotactic system. The flow leads to the formation of anisotropic aggregates that move following the direction of the flow, even when the chemotactic organisms are not directly advected by the flow. Sufficiently strong advection can stop the aggregation and coarsening process that is then restricted to the direction perpendicular to the flow.

Steady-state solutions to the advection-diffusion equation and ghost coordinates for a chaotic flow

Steady-state solutions to the advection-diffusion equation for a passive scalar, with a chaotic divergence-free flow, are determined using a discrete-time, finite-difference model. The physical system studied is a density of particles diffusing across a chaotic layer. The impact of the advective structures on the solutions is illustrated, with special attention given to the cantori. It is argued that cantori play an important role in restricting transport and that coordinates adapted to cantori, called ghost coordinates, provide a natural framework about which the dynamics may be organized; for example, the averaged density profile becomes a smoothed devil's staircase in ghost coordinates.

Monte Carlo results for the Ising model with shear

We study the kinetics of domain growth in the Ising model with nonconserved dynamics under the action of a stochastic driving field that mimics the action of a shear flow. At late times, we found multistriped configurations with constant transversal size and linear growth in the direction of the flow. In cases with weak shear, a regime characterized by the decreasing of the transversal size is found that could correspond to previous theoretical investigations. This behavior is confirmed by the analysis of the structure factor patterns.

Many-body theory of chemotactic cell-cell interactions

We consider an individual-based stochastic model of cell movement mediated by chemical signaling fields. This model is formulated using Langevin dynamics, which allows an analytic study using methods from statistical and many-body physics. In particular we construct a diagrammatic framework within which to study cell-cell interactions. In the mean-field limit, where statistical correlations between cells are neglected, we recover the deterministic Keller-Segel equations. Within exact perturbation theory in the chemotactic coupling epsilon , statistical correlations are non-negligible at large times and lead to a renormalization of the cell diffusion coefficient D(R)--an effect that is absent at mean-field level. An alternative closure scheme, based on the necklace approximation, probes the strong coupling behavior of the system and predicts that D(R) is renormalized to zero at a critical value of epsilon, indicating self-localization of the cell. Stochastic simulations of the model give very satisfactory agreement with the perturbative result. At higher values of the coupling simulations indicate that D(R) approximately epsilon(-2) , a result at odds with the necklace approximation. We briefly discuss an extension of our model, which incorporates the effects of short-range interactions such as cell-cell adhesion.