TR&D 2: NUMERICAL TOOLS FOR MODELING IN CELL BIOLOGY

1. TR&D 2: NUMERICAL TOOLS FOR MODELING IN CELL BIOLOGY Software development: Jim Schaff Fei Gao Frank Morgan Math & Physics: Boris Slepchenko Diana Resasco Igor Novak Yung-Sze Choi (UConn, Storrs) Highlights of the year : 1. Implementation of stiff PDE solvers in VCell. Fully-implicit solver has been deployed after extensive testing. Tests: threshold conditions in reactions with respect to time; diffusion and reaction parameters as functions of variables; large systems with multiple coupled variables. The solver was used during the 2009 VCell summer workshop. 2. Implementation of elliptic solvers in VCell. Using VCell Poisson in the study of diffusion in crowded spaces (published in Biophys. J. (2009)). Also prototyped extension to include linear reactions. Interesting application of VCell (Nat Cell Biol. (2009)) points to future development of nonlinear solvers in VCell. 3. Mesh refinement (MR) and parallelization. Major effort on adapting capabilities of EBChombo for MR in VCell. Meeting with Chombo team in Nov 2009; preliminary performance tests using representative geometry. Need funding for productive collaboration.

2. Mesh Refinement & Parallel Solvers Why mesh refinement? Irregular cell geometries often exhibit small spatial features, such as thin flat lamellipodia or finger-like protrusions (filopodia); (other examples include neuronal axons, dendrites, and dendritic spines). VCell places them in a rectangular box and applies uniform orthogonal meshing. For geometries with disparate spatial scales, this creates a large number of grid points, and in some applications, necessary resolution cannot be achieved due to memory limitations. Adaptive mesh refinement is a good way to address the issue. “keratocyte” Purkinje cell dendrite and spines

3. Previous Work & New Progress Previously: Idea: to combine MR capability of Chombo (LBNL) with VCell geometry handling and solvers. Installed Chombo, wrote interface; performed 2D tests using matrix-free PETSc solvers. Problems: - ad-hoc treatment of the operator near membranes could be unsafe; - slow performance; - steep learning curve. • New developments: • - Official release of EBChombo in Spring 2009 combines: treatment of irregular boundaries using embedded boundary (EB) method, adaptive mesh refinement (AMR), and parallel solvers. PDE solvers maintain second order accuracy. • Attempted to incorporate EBChombo, encountered problems. Contacted Chombo team; Chombo team’s visit to UCHC in Nov 2009; since then some preliminary work has been done together on representative test problems.

4. Accuracy test: EBChombo achieves good accuracy with coarser grids. Exact Solution: Geometry: unit sphere Convergence (uniform grids) EBChombo extends finite volume formulation to irregular boundaries using cut-cell approach. PDE solvers maintain second order accuracy at the cost of extending the computational stencil for points near the boundary. VCell is ~ first order accurate (less accurate near membrane).

5. Mesh refinement with EBChombo Test example: distribution of G-actin in moving keratocyte source: cell body; sink: 0.05- mm band near leading edge. Time point t=50 VCell, Dx=0.1 mm, EBChombo, 4-level MR, Dxmin=0.025 mm Preliminary testing on two examples with irregular geometry, performed after Chombo team’s visit to UCHC, show that EBChombo has the potential to become a very useful tool for cases of complex geometry, as the solvers provide: (1) good accuracy near membranes; (2) capability to refine small features in cases when extremely fine uniform grid is no longer practical. Preliminary performance report from Chombo team shows that EB solvers can be slow (team is looking into performance improvements). Overall, interacting with Chombo team was a great boost to the project. Line scan along symmetry axis

6. Issues and limitations • Limited scope of applicability • Currently:solving only on one side of the surface. • Needed: solving on both sides of surface (generally, in multiple compartments), membrane • variables and reactions (ideally, membrane diffusion coupled to diffusion in the bulk); surface • fluxes as functions of solution. Does multigrid solver (MG) make some features harder to • develop? • Performance issues • Multiple-level refinement in 3D (most interesting application) is slow. Related issues: • geometry generation, MG convergence is slow when EB and MR are combined. • Promising possibility for VCell purposes: coupling with Sundials (some preliminary code was • already written and tested – no preconditioner). Automatic mesh refinement? Looking ahead… Most pressing issue: funding!