Frank L' H' Brown

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Brownian Dynamics with Hydrodynamic Interactions
Application to Lipid Bilayers and Biomembranes
Frank L. H. Brown University of California, Santa
Barbara
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Journal of Chemical Physics, 69, 1352-1360
(1978). (910 citations)
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Limitations of Fully Atomic Molecular Dynamics
Simulation

• A recent large membrane simulation
• (Pitman et. al., JACS, 127, 4576 (2005))
• 1 rhodopsin, 99 lipids, 24 cholesterols, 7400
  waters
• (43,222 atoms total)
• 5.5 x 7.7 x 10.3 nm periodic box for 118 ns
  duration

• Length/time scales relevant
• to cellular biology
• ms, ?m (and longer)
• A 1.0 x 1.0 x 0.1 ?m simulation for 1 ms
• would be approximately 2 x 109 more
• expensive than our abilities in 2005

• Moores law this might be possible in 46 yrs.

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Outline

• Elastic membrane model (Energetics)
• Elastic membrane model (Dynamics)
• Brownian dynamics of Fourier modes
• Protein motion on the surface of the red blood
  cell
• Fluctuations in intermembrane junctions and
  active membranes

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Linear response, curvature elasticity model
L
h(r)
Kc Bending modulus L Linear dimension T
Temperature ? Cytoplasm viscosity
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Relaxation frequencies
Solve for relaxation of membrane modes coupled to
a fluid in the overdamped limit
R. Granek, J. Phys. II France, 7, 1761-1788
(1997).
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Membrane Dynamics
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Harmonic Interactions

• Membrane is pinned to the cytoskeleton at
  discrete points
• Add interaction term to Helfrich free energy

• When g is large, interaction mimics localized
  pinning

L. Lin and F. Brown, Biophys. J., 86, 764 (2004).
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Pinned Membranes

• Can diagonalize the free energy with interactions
  and find eigenmodes
• Eigenmodes are described by Ornstein-Uhlenbeck
  processes

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Extension to non-harmonic systems
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Fourier Space Brownian Dynamics

• Evaluate F(r) in real space (use h(r) from
  previous time step).
• FFT F(r) to obtain Fk.
• Draw ?ks from Gaussian distributions.
• Compute hk(t?t) using above e.o.m..
• Inverse FFT hk(t?t) to obtain h(r) for the next
  iteration.

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Protein motion on the surface of red blood cells
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S. Liu et al., J. Cell. Biol., 104, 527 (1987).
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S. Liu et al., J. Cell. Biol., 104, 527 (1987).

• Spectrin corrals protein diffusion
• Dmicro 5x10-9 cm2/s (motion inside corral)

• Dmacro 7x10-11 cm2/s (hops between corrals)

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Proposed Models
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Dynamic undulation model
Dmicro
Kc2x10-13 ergs ?0.06 poise L140
nm T37oC Dmicro0.53 ?m2/s h06 nm
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Explicit Cytoskeletal Interactions

• Harmonic anchoring of spectrin cytoskeleton to
  the bilayer

• Additional repulsive interaction along the edges
  of the corral to mimic spectrin

L. Lin and F. Brown, Biophys. J., 86, 764 (2004).
L. Lin and F. Brown, Phys. Rev. Lett., 93, 256001
(2004).
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Dynamics with repulsive spectrin
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Information extracted from the simulation

• Probability that thermal bilayer fluctuation
  exceeds h06nm at equilibrium (intracellular
  domain size)
• Probability that such a fluctuation persists
  longer than t023?s (time to diffuse over
  spectrin)

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Calculated Dmacro

• Used experimental median value of corral size
  L110 nm

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Fluctuations of supported bilayers
Y. Kaizuka and J. Groves, Biophys. J., 86, 905
(2004). L. Lin, J. Groves and F. Brown, Biophys
J., 91,3600 (2006).
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Dynamics in inhomogeneous fluid environments is
possible
And various combinations Seifert PRE 94, Safran
and Gov PRE 04, Lin and Brown JCTC 06.
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Fluctuations of supported bilayers (dynamics)
Impermeable wall (different boundary conditions)
No wall
x10-5
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Fluctuations of active bilayers
L. Lin, N. Gov and F.L.H. Brown, JCP, 124, 074903
(2006).
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Summary (elastic modeling)

• Elastic models for membrane undulations can be
  extended to complex geometries and potentials via
  Brownian dynamics simulation.
• Thermal undulations appear to be able to
  promote protein mobility on the RBC.
• Other biophysical and biochemical systems are
  well suited to this approach.

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Acknowledgements
Lawrence Lin Ali Naji NSF, ACS-PRF, Sloan
Foundation,UCSB