A Comparison of Simulation Accuracy with URDME, MesoRD and STEPS

This appendix shows simulation results using URDME and MesoRD for a simple di↵usion problem. This problem illustrates the advantage of unstructured meshes: the ability to resolve processes on curved surfaces without causing a unnecessarily fine discretization. We also evaluate the accuracy of the software on a non-trivial reaction-di↵usion system by simulating the example of Min oscillations presented in the main text of URDME, MesoRD and STEPS. The output metric used to study the results of the simulations is the mean oscillation period. Simple di↵usion to a target on the surface of a sphere A major strength of using a tetrahedral and triangular mesh over a Cartesian mesh is the possibility to better resolve curved boundaries, and the ability to model 2D processes on membranes embedded in a 3D volume by di↵usion on the surface mesh. To illustrate this, we used URDME and MesoRD to simulate di↵usion on the surface of a sphere. When molecules hit a small circular patch on one of the poles of the sphere, they become absorbed. For a sphere with radius 1μm, a di↵usion constant = 10 12 m2 and a patch with a small radius r, the exact solution is well approximated by [59, 60] ⌧ = 2 ln 2 (1 cos r) (1 + cos r) 1 We used r = 0.1, giving ⌧ ⇡ 5.01 s. Figure 1 shows simulation results for URDME for varying number of voxels. Di↵usion in URDME is modeled using Comsol’s capability to discretize di↵usion on a curved surface. For details, consult the model file in Additional File 7. Figure 1 shows the results from simulations with URDME for varying mesh resolutions. As can be seen, even for the coarsest mesh with only 500 voxels, the error is ⇡ 4%. For the finest mesh with 4343 voxels the error is as small as 0.2%. In order to simulate this model in MesoRD, we created a corresponding 3D geometry consisting of the di↵erence between two spheres, one with radius 1.05μm and one with radius 0.95μm for a membrane width of 100nm. To model the target, we let a sphere with radius 0.1μm centered 1 0 50