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