Mechanics of a Blebbing Cell

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Mechanics of a Blebbing Cell

• A Fluid-Structure Interaction Model
• Jennifer Young

Cha Cha Days 2008 UNC Chapel Hill
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Outline of Presentation

• Problem Description
• Mathematical Model
• Numerical Methods
• Results
• Conclusions and Future Work

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Problem Description
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Animal Cell
Image from www.cellsalive.com
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What is a Bleb?

• Balloon-like Protrusion
• Membrane separates from cortical filament network
• Cytosol inflates membrane
• Blebbing occurs during
• Cell Motility
• Apoptosis

Charras et al. Nature, May 2005
Charras et al. Nature, May 2005
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SH4-domain-induced plasma membrane dynamization
promotes bleb-associated cell motilityJ Cell Sci
Tournaviti et al. 120 3820
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Why model blebs?

• Blebs are a part of everyday cell life, yet the
  mechanics behind their formation is not yet
  clearly understood
• There is debate regarding the characterization of
  the cytoskeleton as individual actin filaments or
  as a porous media in blebbing cells
• It would be helpful to have a versatile model in
  order to test the various hypotheses and
  parameters

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The Cellular Components

• Cytoskeleton mesh of cross-linked actin
  filaments, which can be contracted by myosin II
• Cytosol Fluid within the cell, with Internal
  overpressure 20-200 Pa
• Membrane Lipid bilayer covering the cell
  cortex it has folds and ruffles to allow for
  shape changes

Alberts, Molecular Biology of the Cell
http//www.dreamingintechnicolor.com/IdeaLab/
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General Hypothesis
Charras et al. JCB, May 2006

• Actin network contracts
• Contraction force causes detachment of the
  membrane from the cytoskeleton
• Loose membrane is inflated by pressure-driven
  cytosol
• New actin network forms within bleb and pulls the
  membrane back to the cytoskeleton

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Modeling Approach

• Two-dimensional model of cell cross-section
• Moving Boundary Problem
• Fluid-Structure Interaction
• Model cellular components with constitutive laws
• Cytosol Newtonian fluid
• Membrane A closed elastic string
• Actin filaments Elastic strings
• Interactions occur through boundary conditions
  and external force exchanges
• Volume conservation

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Mathematical Model
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Component Equations

• Filament Equation
• Membrane Equation
• Fluid Equation
• Low Reynolds
• Stokes

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Numerical Methods
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Numerical Methods for Component Eqs.

• Membrane Equation
• Hyperbolic PDE with source terms
• Wave Propagation Method (Leveque JCP 1997)
• Finite Volume Scheme
• Fluid Equation
• Stokes equation is split into three Poisson
  equations
• Elliptic PDEs solved with Multigrid Iterations
• Each numerical scheme was checked against
  analytical solutions
• All computational work was done using the
  software package BEARCLAW
  (Mitran, http//coanda.amath.unc.edu/bearclaw)

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Orthogonal Grid Mapping

• Do computations on rectangular grid and have a
  mapping to the changing physical grid
• Via the Euler Variational Principle,
  minimizing this integral is
  equivalent
  to solving

Computational
Physical
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Volume Constraint

• Cellular volume remains constant during blebbing
  (experimental work of Charras, Nature 2005)
• Due to splitting procedure, numerical errors
  arise which do not keep the volume constant
• To correct error, impose volume conservation via
  a constrained energy minimization problem
• Minimize
• E Elastic Energy Pressure Work Filament
  Work
• Solve using Quasi-Newton method
• (BFGS method was used here)

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Volume Constraint Test
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Fluid-Structure InteractionOne Time Step

• New fluid grid is orthogonalized
• Fluid motion solved, producing new velocities and
  pressure info
• Filament endpoint forces are calculated
• Boundary pressure and filament end forces are
  used as source terms to solve for the membrane
  motion
• Membrane position is adjusted by volume
  constraint
• New membrane position resets the boundary of the
  fluid

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Results
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Set-up for Simulation

• Higher pressure inside the cell vs. out
• Filaments in extended state
• Membrane wrinkled
• Break a section of filament-membrane bonds

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Simulation Results
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Simulation Results Filaments
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Simulation Results Multiple Blebs
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Conclusions and Future Work
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Conclusions

• A fluid-structure interaction model of bleb
  formation has been created
• The individual parts of the model have been
  validated including
• The numerical schemes for the component equations
• Orthogonal grid mapping
• The volume conservation algorithm
• Several runs have been done to test the
  capabilities of the model

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Future Work

• Comparison of model to experimental work
• Exploration into the micromechanics of the actin
  cytoskeleton and cross-links
• Addition of biological factors such as levels of
  actin, myosin, and other proteins that can
  trigger bleb formation
• Thank you to my advisor Dr. Mitran, and to Dr.
  Ken Jacobson, Dr. Tim Elston, and Gabriel Weinreb
  for the research idea.

Acknowledgements
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Thank You!!!