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TR&D 2: Physics to Numerics (NUMERICAL TOOLS FOR MODELING IN CELL BIOLOGY)Specific Aims:1. Algorithms for modeling diffusion-advection-reaction systems in domains with moving boundaries (Moving Boundaries). Algorithms will be developed for a wide range of models formulated in domains with moving boundaries, whose velocities are either given or functions of state variables. The methods will enable a new VCell capability of modeling processes in cells that dynamically change their shape and/or migrate.2. Numerical tools for modeling cell mechanics in VCell (Cell mechanics in VCell).A sufficiently general mathematical description of key factors affecting cell mechanics will be formulated. Using this formulation as a basis, we will develop and prototype robust numerical techniques for computing cell velocities that are essential for modeling cell kinematics (Specific Aim 1). 3. Enhancing usability of VCell spatial solvers: New approaches and capabilities (Better, faster solvers). Usability of VCell spatial solvers will be markedly enhanced. Integration of the PETSc library will facilitate optimization of solvers performance through adaptive time-stepping and support for parallel execution, specifically designed for solvers enabling mesh refinement and deterministic-stochastic simulations in VCell, as well as the moving boundary code. New capabilities of the mesh refinement solver to handle advection and surface diffusion and implementation of methods of reductionof dimensionality in parts of the domain will facilitate modeling of multiple scales with VCell. The latter will be applied to electrophysiology and calcium dynamics in neuronal cells whose geometry includes quasi-1D axons and branched dendrites and 3D cell bodies. Moving BoundariesProgress reportResearch planDeveloped, prototyped, and published a novel conservative algorithm for parabolic problems in domains with moving boundaries, with order of convergence in space between 1 and 2 (based on 2D tests with exact kinematics)

Integrated the FronTier library for tracking boundaries based on membrane velocities that are either given or functions of state variables

Established that the algorithm coupled with FronTier retains its original accuracy if extrapolation of the solution near the boundary and redistribution of points on the boundary are sufficiently accurate. Error analysis was performed against accurate numerical solutions of benchmark tests obtained by alternative methods.

C++ production code for the algorithm was developed and thoroughly tested for cases with exact kinematics

Novak & Slepchenko, J Comput Phys, 2014

error metricsExpanding circle test: reference solution obtained by solving an equivalent problem in a fixed domain

The ultimate goal here is to expand capabilities of the algorithm so that the models that are currently solved by VCell in fixed geometries, can also be solved in domains with moving boundaries. We propose to achieve this goal through the following steps:

Develop a capability of handling membrane variables and arbitrary cross membrane fluxes.Develop a capability of solving reaction-transport problems on both sides of a moving boundary.Develop a prototype for solving 3D models in domains with moving boundaries.Combine the moving boundary code with VCell deterministic-stochastic hybrid solver.Cell mechanics in VCellPreliminary resultsIn DBP by Alex Mogilner (Courant Institute, NYU), minimal models of actin-driven motility have been shown to describe spontaneous cell polarization and transitions from non-motile to motile states. Elements of cell mechanics included in the models, viscous flow of actin u induced by forces generated by myosin M and slowed by focal adhesions, , are coupled with advection/diffusion of myosin and membrane kinematics influenced by actin polymerization. The models are solved by a custom code powered by the moving boundary algorithm. A manuscript is in preparation.

In DBP by Tom Pollard (Yale U.), a continuous 3D model of actin dynamics is sought to describe forces exerted by actin patches on the membrane during endocytosis in fission yeast. Actin movements near an endocytic tubule are modeled by approximating actin network as a viscoelastic medium with repulsive stress due to polymerization. Starting with fixed geometries, we used a simplified approach based on Darcys law. The resulting diffusion approximation was implemented in VCell. Moving beyond fixed geometries requires more detailed description of cell mechanics and additional numerical tools.

Simulation of actin dynamics around an endocytic tubule with two rings of nucleation-promoting factors. XZ cross-section of 3D geometry; extracellular space is white. Density of F-actin (pseudo-color) and its velocities (arrows) correspond to 20 seconds into patch formation. Research planThe objective here is to develop numerical tools for computing membrane velocities based on a sufficiently general mathematical description of key elements of cell mechanics. We plan to achieve this goal through the following steps:

Formulate deterministic equations for cytoskeletal dynamics; we will start with the viscoelastic approximation, using actin-driven motility as an example. Mathematically, models of this type commonly include a nonlinear elliptic (Stokes) equation coupled with hyperbolic conservations laws. Implement robust techniques for solving coupled nonlinear elliptic and hyperbolic equations. A goal here is to ensure numerical stability and second-order accuracy in space.Combine the code prototyped in step 2 with the moving boundary code of Aim 1.Example of a model with zero-stress boundary conditions for actin velocity, also used as a moving/ deforming cell test against Comsol Multiphysics (ALE FEM )Zero-velocity model:

and zero-velocity (Dirichlet) boundary conditions for U :

Zero-stress model:

(originally, )

and zero-stress boundary conditions for U:Driving Biological Project, Alex Mogilner (Courant Institute, NYU)Minimal models of actin-driven motility based on relatively simple approximations of actin polymerization at cell membrane and actomyosin dynamics are studied both analytically and numerically. The models account for cell self-polarization and various modes of migration, both expected and unexpected.

(M is myosin concentration and U is actin flow velocity.)Advection-diffusion of myosin:

Boundary conditions :

Membrane kinematics:Cell mechanics and adhesion:

(Vp is actin polymerization rate and Vf is membrane velocity.)MechanismsTwo types of models

Zero-velocity model

(v0, b)=(1, 2)

(v0, b)=(2.5, 1.5)instability of steady migration at large v0steady migrationDimensionless parameters:

( is average myosin concentration)

5Zero-stress model: symmetry break and transition to motility(v0, b)=(1.5, 0.25)

(v0, b)=(1.5, 0.5)

Dimensionless parameters:

( is average myosin concentration)

The goal of the project is to examine a two-ring hypothesis of actin polymerization-based force generation at endocytic sites in fission yeast by a mathematical model based on realistic 3D geometry. Using a fixed geometry as a first step, a spatial model was formulated on the basis of a previously published nonspatial model (Berro et al. 2010). The model yielded reasonable estimates and testable predictions. Moving beyond fixed geometries requires further development (TR&Ds 2,3,4). Driving Biological Project, Tom Pollard (Yale U.)Two-ring hypothesis (Arasada and Pollard 2011).where A is concentration of F-actin , v is actin velocity, and R stands for reaction terms. Continuity (mass balance) equation for F-actin:

Assumptions:Local balance of forces (viscoelasticity of actin):Diffusion approximation in fixed geometries

whereand

is the repulsive stress tensor due to actin polymerization.

1. The repulsive stress is an increasing function of F-actin concentration; as a simple approximation, linear proportionality is assumed :2. The viscous drag is described by Darcys law:

The assumptions yield effectively diffusive transport of F-actin.Model yields reasonable estimates for actin viscosity and predicts non-uniform distribution of F-actin in endocytic patches

Simulation of actin polymerization around a tubule of cell membrane with two rings of nucleation-promoting factors. Left: XZ cross-section of 3D geometry; extracellular space is white; density of F-actin (pseudo-color) and its velocities (arrows) correspond to 20 seconds into patch formation. Right: distribution of total F-actin (mM) along the vertical surface of the endocytic tubule.

Model geometryRings of NPFBetter, faster solversProgress reportResearch planUsability of VCell spatial solvers will be brought to a new level through expanding their scope of applicability and using new methods for optimizing their performance.

New capabilities:Develop new capabilities of mesh refinement solver to handle advection (directed flow) and surface diffusion; implement support for nonlinear transport.Explore and implement tools for solving PDEs in geometries allowing for partial approximation by lower dimension operators.Apply the capability of step 2 to modeling electrophysiology and calcium dynamics in neuronal cells and other multiscale problems. Explore Voronoi meshes in application to 3D electrophysiology models.Performance optimization:Integrate the PETSc library in VCell; Apply efficient time discretization schemes and adaptive time-stepping provided in PETSc for improving stability and performance of the mesh refinement solver, the hybrid deterministic-stochastic simulator, and the moving boundary code;Implement parallel versions of the solvers and develop efficient preconditioners. A mesh refinement solver using the cut-cell technology (VCell-EBChombo) was implemented and deployed in VCell Alpha. The solver has basic capabilities required by typical VCell models, including membrane variables.

A spatial deterministic-stochastic solver was developed, tested and deployed to VCell Alpha, Beta, and Release. The solver appropriately combines capabilities of VCell spatial semi-implicit fixed time step solver and those of Smoldyn, a spatial particle-based Monte Carlo simulator. A manuscript describing mathematical underpinnings of the solver and its applications has been submitted to J. Comput. Phys.

t =2

t =20

t =1000

t =2000

t =200S(r)Performance issues are partly addressed by supporting parallel runs and, most recently, by allowing for different integration time steps and different frequency of saving results at different stages of a simulation. VCell spatial deterministic-stochastic solver is applied to a hybrid model of spontaneous cell polarization: coalescence of a multi-cluster system of membrane-bound proteins into a single cluster.

Implicit formulations for spatial electrophysiology