New in VCell 4.1 beta

2D and 3D Membrane Diffusion

Modeling Overview

Users may now introduce diffusion for membrane-localized molecular species for 2D and 3D simulations. The diffusion rates and boundary conditions (when needed) are specified the same way as in bulk diffusion (initial conditions tab of application window).

Boundary conditions types (either surface density or flux) are taken from the enclosed compartment (e.g. plasma membrane takes it’s boundary condition type from that specified for the cytosol). Note that boundary conditions are only applicable for membranes that intersect the box defining the simulation domain (e.g. when modeling only part of a cell).

[This image is of a simulated FRAP experiment where the fluorescent marker was diffusing within the membrane. Click to see movie (quicktime 1.6Mb)]

Numerical Method Testing

Both the 2D and 3D numerical methods for calculating the reaction/diffusion equation on arbitrary surfaces (including coupling to volumetric processes) has been tested extensively against exact solutions.

• First, it has been shown that the surface normals and other metrics of the mesh converge to those of the exact geometries as the element size is reduced.
• Secondly, it has been shown that numerical solutions to the resulting partial differential equations (PDE) converge to known solutions (for spheres, planes, and surfaces of revolution).

Testing surfaces of revolution

Arbitrarily accurate solutions to axisymmetric reaction-diffusion equations have been computed by recasting the problem into a one-dimensional PDE that can be solved with great accuracy. Thus, solutions of equations on surfaces of revolution that have a wide range of curvatures can be evaluated. Here we tested both for overall convergence and max error (to characterize mesh related artifacts).

Below are simulation results for meshes consisting of 21x21x21 and 81x81x81 volume elements.

Testing spheres

There exist analytic solutions for reaction/diffusion equations defined on the surface of spheres. Thus testing for convergence is very straightforward.

Below is a series of numerical solutions with an asymmetric initial condition (left column) and the log of the relative error (right column) for mesh sizes of 6x6x6, 21x21x21, 41x41x41, and 161x161x161.

numerical solution concentration at t=0.8 For asymmetric initial conditions	relative error logarithmically scaled blue = error of 10^-6 red = error of 10^0 (yellow is 0.05 or 5% relative error).