Biomechanics of a Blebbing Cell A Fluid-Structure Interaction Problem Jennifer Joyce1, Dr. Sorin Mitran1 1University of North Carolina at Chapel Hill, Department of Mathematics

1
Biomechanics of a Blebbing CellA Fluid-Structure
Interaction Problem Jennifer Joyce1, Dr. Sorin
Mitran11University of North Carolina at Chapel
Hill, Department of Mathematics
What is a Bleb?
Testing the Model
Weighted Residual Galerkin Method

• Connect the PDE to an integral formulation to
  find solution which minimizes the error
• Want to solve where is a
  differential operator
• Define an inner product, using test
• functions, then integrate by parts
• Stokes Example
  After IBP, 1st-order problem for FreeFEM

Cells are enclosed in plasma membranes, and
these membranes are elastic bodies that can
deform due to forces and pressures imposed by
many biological factors. A bleb is a bud or
balloon-like protrusion in the plasma membrane,
which forms when the outer membrane separates
from an underlying network of actin filaments.
This mesh of actin filaments provides structure
and shape to the cell. Blebs are one of a number
of cell motility mechanisms and they also play a
key role in the break-up of a cell during
apoptosis (cell death).
Initial Configuration This is the initial
position of the membrane, and filaments. Each
filament is attached at both ends to the membrane.
Confocal microscopy image (Charras et al. Nature,
May 2005)
Volume Test This is a test of the volume
preservation constraint, when the cell initially
has a higher pressure inside vs. outside. Green
Initial Position Red Position after 100 time
steps, with the volume preservation
constraint Blue Position after 100 time steps
without the volume preservation
constraint
The Hypothesis
How exactly a bleb forms, is unknown. The
goal for this model is to simulate the creation
of a bleb in the membrane of a cell. Beginning
with several hundred filaments tethered to the
membrane under tension, and an overpressure on
the interior of the cell, the hypothesis is that
if several connections between the filament
network and the membrane are broken in one area,
then a bleb should form at that spot.
Fluid-Structure Interaction

• A splitting procedure is used to decouple the
  fluid and elasticity problems at each time step
• Fluid motion problem is solved for given
  membrane position, velocity. The resulting fluid
  pressure is applied as an external force on the
  membrane interior when solving the elasticity
  problem (4).
• 2. Filament positions are updated in accordance
  with local fluid velocity (zero-mass hypothesis)
  using a second-order Runge-Kutta method.
• 3. Filament deformation at new positions is
  computed. Filament endpoint stresses are imposed
  on the membrane interior.
• 4. Elasticity equations within the membrane are
  solved. External forces on interior membrane are
  given by cellular fluid pressure and filament
  tensions. Exterior of membrane is subject to a
  uniform ambient pressure.
• 5. Displacement of the membrane is used to update
  the position of the fluid mesh in the cell.
• 6. The process begins all over again!

• Bleb Test
• The filaments begin in an extended state wanting
  to pull the membrane in, while the internal
  pressure of the cell wants to push the membrane
  out
• 1/9 of the total number of filament connections
  have been broken with the membrane, (concentrated
  in an upper left section of the cell)
• Below is the original, and broken connection
  membrane at 3 points in time
• Notice the formation of a bleb!

Methods of Modeling
Green Normal Membrane Red Broken
Connection Membrane

• Filaments Spring/Force Equations with no mass
• Cytoplasmic Fluid and Elastic Membrane
• Finite Element Discretization (2D Triangular
  Meshes)
• Stokes Equation for incompressible fluid
• Elasticity Equation (with Lamé coefficients)
• Boundary Stresses
• Implementation FreeFEM, a finite element PDE
  solver
• (F. Hecht et al.
  www.freefem.org)

2 sec
2.5 sec
3 sec

• Stress Test
• This is a display of the membrane stresses in
  the broken connection model.
• The blue indicates an area of lower stress,
  which is expected since the filament forces are
  not being felt there.

Volume Conservation

• The splitting procedure decouples the elasticity
  and fluid problems during a time step. The
  elasticity problem must be solved under the
  constraint of a constant enclosed volume of
  incompressible cellular fluid.
• This is implemented through a penalty method in
  the finite element formulation

Future Plans
Future plans include the addition of
biological factors such as levels of calcium,
myosin, and actin in the system, which are
thought to be the chemicals that cause the
breakage and reattachment of filaments to the
membrane.
The University of North Carolina at Chapel
Hill