Chandrasekhar Venkataraman

Implicit--Explicit Timestepping with Finite Element Approximation of Reaction--Diffusion Systems on Evolving Domains

We present and analyze an implicit--explicit timestepping procedure with finite element spatial approximation for semilinear reaction--diffusion systems on evolving domains arising from biological models, such as Schnakenbergs (1979). We employ a Lagrangian formulation of the model equations which permits the error analysis for parabolic equations on a fixed domain but introduces technical difficulties, foremost the space-time dependent conductivity and diffusion. We prove optimal-order error estimates in the

Global existence for semilinear reaction–diffusion systems on evolving domains

{L}_{\infty}(0,T;{L}_{2}(\varOmega))

Stability analysis and simulations of coupled bulk-surface reaction–diffusion systems

and

Parameter identification problems in the modelling of cell motility

{L}_{2}(0,T;{H}^{1}(\varOmega))

A Model for Selection of Eyespots on Butterfly Wings.

norms, and a pointwise stability result. We remark that these apply to Eulerian solutions. Details on the implementation of the Lagrangian and the Eulerian scheme are provided. We also report on a numerical experiment for an application to pattern formation on an evolving domain.

Coupled bulk-surface free boundary problems arising from a mathematical model of receptor-ligand dynamics

We present global existence results for solutions of reaction–diffusion systems on evolving domains. Global existence results for a class of reaction–diffusion systems on fixed domains are extended to the same systems posed on spatially linear isotropically evolving domains. The results hold without any assumptions on the sign of the growth rate. The analysis is valid for many systems that commonly arise in the theory of pattern formation. We present numerical results illustrating our theoretical findings.

A robust and efficient adaptive multigrid solver for the optimal control of phase field formulations of geometric evolution laws

In this article, we formulate new models for coupled systems of bulk-surface reaction–diffusion equations on stationary volumes. The bulk reaction–diffusion equations are coupled to the surface reaction–diffusion equations through linear Robin-type boundary conditions. We then state and prove the necessary conditions for diffusion-driven instability for the coupled system. Owing to the nature of the coupling between bulk and surface dynamics, we are able to decouple the stability analysis of the bulk and surface dynamics. Under a suitable choice of model parameter values, the bulk reaction–diffusion system can induce patterning on the surface independent of whether the surface reaction–diffusion system produces or not, patterning. On the other hand, the surface reaction–diffusion system cannot generate patterns everywhere in the bulk in the absence of patterning from the bulk reaction–diffusion system. For this case, patterns can be induced only in regions close to the surface membrane. Various numerical experiments are presented to support our theoretical findings. Our most revealing numerical result is that, Robin-type boundary conditions seem to introduce a boundary layer coupling the bulk and surface dynamics.

Adaptive Finite Elements for Semilinear Reaction-Diffusion Systems on Growing Domains

We present a novel parameter identification algorithm for the estimation of parameters in models of cell motility using imaging data of migrating cells. Two alternative formulations of the objective functional that measures the difference between the computed and observed data are proposed and the parameter identification problem is formulated as a minimisation problem of nonlinear least squares type. A Levenberg–Marquardt based optimisation method is applied to the solution of the minimisation problem and the details of the implementation are discussed. A number of numerical experiments are presented which illustrate the robustness of the algorithm to parameter identification in the presence of large deformations and noisy data and parameter identification in three dimensional models of cell motility. An application to experimental data is also presented in which we seek to identify parameters in a model for the monopolar growth of fission yeast cells using experimental imaging data. Our numerical tests allow us to compare the method with the two different formulations of the objective functional and we conclude that the results with both objective functionals seem to agree.

Preserving invariance properties of reaction–diffusion systems on stationary surfaces

Unsolved Problem The development of eyespots on the wing surface of butterflies of the family Nympalidae is one of the most studied examples of biological pattern formation.However, little is known about the mechanism that determines the number and precise locations of eyespots on the wing. Eyespots develop around signaling centers, called foci, that are located equidistant from wing veins along the midline of a wing cell (an area bounded by veins). A fundamental question that remains unsolved is, why a certain wing cell develops an eyespot, while other wing cells do not. Key Idea and Model We illustrate that the key to understanding focus point selection may be in the venation system of the wing disc. Our main hypothesis is that changes in morphogen concentration along the proximal boundary veins of wing cells govern focus point selection. Based on previous studies, we focus on a spatially two-dimensional reaction-diffusion system model posed in the interior of each wing cell that describes the formation of focus points. Using finite element based numerical simulations, we demonstrate that variation in the proximal boundary condition is sufficient to robustly select whether an eyespot focus point forms in otherwise identical wing cells. We also illustrate that this behavior is robust to small perturbations in the parameters and geometry and moderate levels of noise. Hence, we suggest that an anterior-posterior pattern of morphogen concentration along the proximal vein may be the main determinant of the distribution of focus points on the wing surface. In order to complete our model, we propose a two stage reaction-diffusion system model, in which an one-dimensional surface reaction-diffusion system, posed on the proximal vein, generates the morphogen concentrations that act as non-homogeneous Dirichlet (i.e., fixed) boundary conditions for the two-dimensional reaction-diffusion model posed in the wing cells. The two-stage model appears capable of generating focus point distributions observed in nature. Result We therefore conclude that changes in the proximal boundary conditions are sufficient to explain the empirically observed distribution of eyespot focus points on the entire wing surface. The model predicts, subject to experimental verification, that the source strength of the activator at the proximal boundary should be lower in wing cells in which focus points form than in those that lack focus points. The model suggests that the number and locations of eyespot foci on the wing disc could be largely controlled by two kinds of gradients along two different directions, that is, the first one is the gradient in spatially varying parameters such as the reaction rate along the anterior-posterior direction on the proximal boundary of the wing cells, and the second one is the gradient in source values of the activator along the veins in the proximal-distal direction of the wing cell.

Lumped finite elements for reaction-cross-diffusion systems on stationary surfaces

We consider a coupled bulk-surface system of partial differential equations with nonlinear coupling modelling receptor-ligand dynamics. The model arises as a simplification of a mathematical model for the reaction between cell surface resident receptors and ligands present in the extra-cellular medium. We prove the existence and uniqueness of solutions. We also consider a number of biologically relevant asymptotic limits of the model. We prove convergence to limiting problems which take the form of free boundary problems posed on the cell surface. We also report on numerical simulations illustrating convergence to one of the limiting problems as well as the spatio-temporal distributions of the receptors and ligands in a realistic geometry.