Frank L. H. Brown University of California, Santa Barbara

1. Brownian Dynamics with Hydrodynamic Interactions: Application to Lipid Bilayers and Biomembranes Frank L. H. Brown University of California, Santa Barbara

2. Journal of Chemical Physics, 69, 1352-1360 (1978). (910 citations)

3. 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 • Moore’s law: this might be possible in 46 yrs.

4. 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

5. Helfrich bending free energy: Linear response for normal modes: Ornstein-Uhlenbeck process for each mode: Linear response, curvature elasticity model L h(r) Kc: Bending modulus L: Linear dimension T: Temperature : Cytoplasm viscosity

6. Relaxation frequencies Non-inertial Navier-Stokes eq: Nonlocal Langevin equation: Oseen tensor (for infinite medium) Bending force Solve for relaxation of membrane modes coupled to a fluid in the overdamped limit: R. Granek, J. Phys. II France, 7, 1761-1788 (1997).

7. Membrane Dynamics

8. 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).

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

10. Helfrich bending free energy + additional interactions: Overdamped dynamics: (or generalized expressions) • 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). Extension to non-harmonic systems

11. Fourier Space Brownian Dynamics Evaluate F(r) in real space (use h(r) from previous time step). FFT F(r) to obtain Fk. Draw k’s 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.

12. Protein motion on the surface of red blood cells

13. S. Liu et al., J. Cell. Biol., 104, 527 (1987).

14. 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)

15. Proposed Models

16. Dynamic undulation model Dmicro Kc=2x10-13 ergs =0.06 poise L=140 nm T=37oC Dmicro=0.53 m2/s h0=6 nm

17. 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).

18. Dynamics with repulsive spectrin

19. Escape rate for protein from a corral • Macroscopic diffusion constant on cell surface • (experimentally measured) 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=23s (time to diffuse over spectrin)

20. Median experimental value Calculated Dmacro • Used experimental median value of corral size L=110 nm

21. 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).

22.   Membrane near an Impermeable wall Different viscoscities on both sides of membrane Membrane near a semi-permeable wall Dynamics in inhomogeneous fluid environments is possible And various combinations Seifert PRE 94, Safran and Gov PRE 04, Lin and Brown JCTC 06.

23. Way off! Timescales consistent with experiment Fluctuations of supported bilayers (dynamics) Impermeable wall (different boundary conditions) No wall x10-5

24. Fluctuations of active membranes (experiments) J.-B. Manneville, P. Bassereau, D. Levy and J. Prost, PRL, 4356, 1999. koff kon Off Push down koff kon Off Push up Fluctuations of “active” bilayers L. Lin, N. Gov and F.L.H. Brown, JCP, 124, 074903 (2006).

25. 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.

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