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 deterministicstochastic 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 reduction of 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.
Progress report
Moving Boundaries
Novak & Slepchenko, J Comput Phys, 2014
error metrics
-
Developed, 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
Research plan
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:
Expanding circle test: reference solution
obtained by solving an equivalent problem
in a fixed domain
1. Develop a capability of handling membrane variables and arbitrary cross
membrane fluxes.
2. Develop a capability of solving reaction-transport problems on both sides of
a moving boundary.
3. Develop a prototype for solving 3D models in domains with moving
boundaries.
4. Combine the moving boundary code with VCell deterministic-stochastic
hybrid solver.
Cell mechanics in VCell
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 )
Preliminary results
-
In 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,  T u  Δu    ( MI )  u , 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 Darcy’s law. The resulting ‘diffusion’ approximation was
implemented in VCell. Moving beyond fixed geometries requires more detailed
description of cell mechanics and additional numerical tools.
Research plan
The 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:
Simulation of actin dynamics around an endocytic tubule
with two rings of nucleation-promoting factors. XZ crosssection of 3D geometry; extracellular space is white.
Density of F-actin (pseudo-color) and its velocities
(arrows) correspond to 20 seconds into patch formation.
1. 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.
2. 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.
3. Combine the code prototyped in step 2 with the moving boundary code of Aim 1.
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.
Mechanisms
Advection-diffusion of myosin:
 t M  DΔM    (UM )
(M is myosin concentration and U is actin flow velocity.)
Boundary conditions : n  ( DM  (U  Vf ) M ) |  0
Cell mechanics and adhesion:
 t U  ΔU    ( MI )  U
Membrane kinematics: n  (Vf  U) |  Vp
(Vp is actin polymerization rate and Vf is membrane velocity.)
Two types of models
‘Zero-velocity’ model: Vp 
V0
K | |
M 0  M |
and zero-velocity (Dirichlet)
boundary conditions for U :
n  U |  0
‘Zero-stress’ model: Vp 
V0
K | |
M0
(originally, Vp 
and zero-stress boundary
conditions for U:
V0
| |
n  ( U  M  I ) |  0.
)
‘Zero-velocity’ model
Dimensionless parameters:
b  M , v0 
V0
M
( M is average myosin
concentration)
steady migration
(v0, b)=(1, 2)
(v0, b)=(2.5, 1.5)
instability of steady migration at large v0
‘Zero-stress’ model: symmetry
break and transition to motility
Dimensionless parameters:
b  M , v0 
V0
M
( M is average myosin
concentration)
(v0, b)=(1.5, 0.25)
(v0, b)=(1.5, 0.5)
Driving Biological Project, Tom Pollard (Yale U.)
‘Two-ring’ hypothesis (Arasada and Pollard 2011).
The goal of the project is to examine a ‘tworing’ 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).
‘Diffusion” approximation in fixed geometries
Continuity (mass balance) equation for F-actin:
 t A    (vA)  R
where A is concentration of F-actin , v is actin velocity,
and R stands for reaction terms.
Local balance of forces (viscoelasticity of actin):
Factin polymerization  Fviscous drag  0,
where
Factin polymerization    ˆ
and ˆ is the repulsive stress tensor due to actin polymerization.
Assumptions:
1. The repulsive stress is an increasing function of Factin concentration; as a simple approximation,
linear proportionality is assumed :
ˆ  AIˆ.
2. The viscous drag is described by Darcy’s law:
Fviscous drag  v.
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
Model geometry
Rings of NPF
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 (M) along the vertical surface of the endocytic
tubule.
Better, faster solvers
Progress report
-
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.
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.
S(r)
t =2
t =20
- 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 particlebased Monte Carlo simulator. A manuscript describing mathematical underpinnings of
the solver and its applications has been submitted to J. Comput. Phys.
Research plan
t =200
t =1000
t =2000
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.
Usability 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:
1. Develop new capabilities of mesh refinement solver to handle advection
(directed flow) and surface diffusion; implement support for nonlinear transport.
2. Explore and implement tools for solving PDEs in geometries allowing for partial
approximation by lower dimension operators.
3. 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:
1. Integrate the PETSc library in VCell;
2. 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;
3. Implement parallel versions of the solvers and develop efficient preconditioners.
Implicit formulations for spatial electrophysiology