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Brownian Dynamics with Hydrodynamic Interactions: Application to Lipid Bilayers and Biomembranes. Frank L. H. Brown University of California, Santa Barbara. Journal of Chemical Physics, 69 , 1352-1360 (1978). (910 citations). Limitations of Fully Atomic Molecular Dynamics Simulation. - PowerPoint PPT Presentation
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Frank L. H. Brown
University of California, Santa Barbara
Brownian Dynamics with Hydrodynamic Interactions: Application to Lipid Bilayers
and Biomembranes
Journal of Chemical Physics, 69, 1352-1360 (1978).(910 citations)
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 relevantto 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
• Moore’s law: this might be possible in 46 yrs.
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
Linear response, curvature elasticity model
h(r)
LHelfrich bending free energy:
E =
Kc
2∇2h(
r r )( )
2
∫ dxdy=Kc
2L2 k4 hk
2
k∑
Linear response for normal modes:ddt
hk t( )=−ωkhk +ζ(t)
ωk =Kck
3
4η
Ornstein-Uhlenbeck process for each mode:
P(hk)=Kck
4
2πkBTL2 exp−Kck
4
2kBTL2 hk
2⎡
⎣ ⎢ ⎤
⎦ ⎥
P hk(t) |hk(0)( ) =Kck
4
2πkBTL2(1−e−2ωkt)exp−
Kck4
2kBTL2
hk(t) −hk(0)e−ωkt 2
(1−e−2ωkt)
⎡
⎣ ⎢
⎤
⎦ ⎥
Kc: Bending modulusL: Linear dimensionT: Temperature: Cytoplasm viscosity
Relaxation frequencies
R. Granek, J. Phys. II France, 7, 1761-1788 (1997).
Solve for relaxation of membrane modes coupled to a fluid inthe overdamped limit:
Non-inertial Navier-Stokes eq:
∇p−η∇2r u = fext
∇ ⋅r u =0
Nonlocal Langevin equation:
˙ h (r r ,t) = d
r r ' H(
r r −
r r ' ) f(
r r ',t)∫ +ζ (
r r ,t)
H(r r )=
18πηr
f (r r ) =−
δEδh(
r r )
Oseen tensor(for infinite medium)
Bending force
˙ h k(t)=− Kck4
( )1
4ηk
⎛ ⎝ ⎜ ⎞
⎠ hk(t)+ˆ ζ (k,t)
ωk =Kck
3
4η
Membrane Dynamics
QuickTime™ and a decompressor
are needed to see this picture.
Harmonic Interactions• Membrane is pinned to the cytoskeleton at discrete
points• Add interaction term to Helfrich free energy
E = dA∫ r
Kc
2 ∇2h(r) [ ]
2+
12
V(r)h2(r)⎧ ⎨ ⎩
⎫ ⎬ ⎭
V(r) =γ δ(r−Ri)i∑
• When is large, interaction mimics localized pinning
L. Lin and F. Brown, Biophys. J., 86, 764 (2004).
Pinned Membranes• Can diagonalize the free energy with interactions and find
eigenmodes• Eigenmodes are described by Ornstein-Uhlenbeck processes
QuickTime™ and a decompressor
are needed to see this picture.
Extension to non-harmonic systems
Helfrich bending free energy + additional interactions:
€
E =Kc
2∇ 2h(
r r )( )
2
∫ dxdy + V[h(r)] =Kc
2L2k 4 hk
k
∑2
+ V[hk ]
Overdamped dynamics:
€
˙ h (r r , t) = d
r r 'H(
r r −
r r ') f (
r r ', t)∫ + ζ (
r r , t)
f (r r ) = −
δE
δh(r r )
Solve via Brownian dynamics•Handle bulk of calculation in Fourier Space (FSBD)
•Efficient handling of hydrodynamics•Natural way to coarse grain over short length scales
L. Lin and F. Brown, Phys. Rev. Lett., 93, 256001 (2004).L. Lin and F. Brown, Phys. Rev. E, 72, 011910 (2005).
€
d
dthk t( ) = Λk Fk[h(r, t)] + ζ k (t){ }
Λk =1
4ηk(or generalizedexpressions)
Fourier Space Brownian Dynamics
€
d
dthk t( ) = Λk Fk[h(r, t)] + ζ k (t){ } → hk t + Δt( ) = hk t( ) + ΛkFk[h(r, t)] + Γk (Δt)
€
Γk (Δt) = Λk dτξ k (τ )t
t +Δt
∫
1. Evaluate F(r) in real space (use h(r) from previous time step).
2. FFT F(r) to obtain Fk.
3. Draw Γk’s from Gaussian distributions.4. Compute hk(t+t) using above e.o.m..
5. Inverse FFT hk(t+t) to obtain h(r) for the next iteration.
Protein motion on the surface of red blood cells
S. Liu et al., J. Cell. Biol., 104, 527 (1987).
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)
Proposed Models
Dynamic undulation model
Kc=2x10-13 ergs=0.06 poiseL=140 nmT=37oCDmicro=0.53 m2/sh0=6 nm
Dmicro
Explicit Cytoskeletal Interactions
• Additional repulsive interaction along the edges of the corral to mimic spectrin
∑∫ ++= −
i
/)( )( iiih
Arep cbyaxedF δε λrr
L. Lin and F. Brown, Phys. Rev. Lett., 93, 256001 (2004).
• Harmonic anchoring of spectrin cytoskeleton to the bilayer
E = dA∫ r
Kc
2 ∇2h(r) [ ]
2+
12
V(r)h2(r)⎧ ⎨ ⎩
⎫ ⎬ ⎭
V(r) =γ δ(r−Ri)i∑
L. Lin and F. Brown, Biophys. J., 86, 764 (2004).
Dynamics with repulsive spectrin
QuickTime™ and a decompressor
are needed to see this picture.
Information extracted from the simulation
•Probability that thermal bilayer fluctuation exceeds h0=6nm at equilibrium (intracellular domain size)•Probability that such a fluctuation persists longer than t0=23s (time to diffuse over spectrin)
•Escape rate for protein from a corral•Macroscopic diffusion constant on cell surface
(experimentally measured)
Calculated Dmacro• Used experimental median value of corral size L=110 nm
∞
9×10−12
5×10−12
7×10−10
System Size Simulation
Type
Simulation
Geometry
L Free Membrane Square
L Pinned Square
Pinned Square
Pinned & Repulsive
Square
Pinned & Repulsive
Triangular
Dmacro (cm2s-1)
∞∞
€
2 ×10−10
€
3.4 ×10−10
Dmacro=6.6×10-11 cm2s-1• Median experimental value
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).
Dynamics in inhomogeneous fluid environments is possible
Different viscoscitieson both sides of membrane
Membrane near an Impermeable wall
Membrane near a semi-permeable wall
And various combinations
Seifert PRE 94, Safran and Gov PRE 04,
Lin and Brown JCTC 06.
Fluctuations of supported bilayers (dynamics)
x10-5
Impermeable wall (different boundary conditions) No wall
Timescales consistent withexperiment
Way off!
Fluctuations of “active” bilayers
koff
J.-B. Manneville, P. Bassereau, D. Levy and J. Prost, PRL, 4356, 1999.
Off Push down
Off Push up
kon
koff
kon
Fluctuations of active membranes (experiments)
L. Lin, N. Gov and F.L.H. Brown, JCP, 124, 074903 (2006).
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.
Lawrence LinAli Naji
NSF, ACS-PRF, Sloan Foundation,UCSB
Acknowledgements