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

1. Simple Models for Biomembrane Structure and Dynamics Frank L. H. Brown University of California, Santa Barbara

2. 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.

3. Outline • Elastic membrane model • Background and simulation methods • Protein motion on the surface of the red blood cell • Supported bilayer fluctuations and diffusion of curved proteins • Molecular membrane model • Correspondence with experimental physical properties and stress profile • Protein induced deformation of the bilayer and detailed elastic models

4. 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

5. 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).

6. Membrane Dynamics

7. 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

8. 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.

9. Protein motion on the surface of red blood cells

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

11. M. Sheetz, Semin. Hematol., 20, 175 (1983). M. Tomishige et al., J. Cell. Biol., 142, 989 (1998). • Spectrin “corrals” protein diffusion • Dmicro= 5x10-9 cm2/s (motion inside corral) • Dmacro= 7x10-11 cm2/s (hops between corrals)

12. Proposed Models

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

14. 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).

15. Dynamics with repulsive spectrin

16. 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)

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

18. Explicit diffusion of membrane“proteins” A. Naji and F.L.H. Brown, JCP, 126, 235103 (2007); E. Reister et. al., PRE, 75, 011908 (2007).

19. C B A Explicit diffusion of membraneproteins Numerically, the membrane shape is defined on a discrete grid. Protein position is continuously variable. Numerically, proteins A,B&C are not equivalent.

20. Blood P. D., Voth G. A. PNAS 2006;103:15068-15072 This problem can be solved… P. Atzberger et. al., J Comp Phys, 224, 1255 (2007).

22. Curved proteins diffuse more slowly than flat proteins

23. 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. • Curved proteins diffuse more slowly than flat proteins. • Other biophysical and biochemical systems are well suited to this approach. Annual Review of Physical Chemistry, 59, 685 (2008).

24. Molecular membrane models with implicit solvent

25. Why? • Study the basic (minimal) requirements for bilayer stability and elasticity. • A particle based method is more versatile than elastic models. • Incorporate proteins, cytoskeleton, etc. • Non “planar” geometries • Computational efficiency.

26. A range of resolutions... W. Helfrich, Z. Natuforsch, 28c, 693 (1973) J.M. Drouffe, A.C. Maggs, and S. Leibler, Science 254, 1353 (1991) Helmut Heller, Michael Schaefer, and Klaus Schulten, J. Phys. Chem 1993, 8343 (1993) Y. Kantor, M. Kardar, and D.R. Nelson, Phys. Rev. A 35, 3056 (1987) R. Goetz and R. Lipowsky, J. Chem. Phys. 108, 7397 (1998)

27. Motivation D. Marsh, Biochim. Biophys. Acta, 1286, 183-223 (1996).

28. A solvent free model G. Brannigan, P. F. Philips and F. L. H. Brown, Phys. Rev. E, 72, 011915 (2005).

29. Self-Assembly (128 lipids)

30. Flexible (and tunable) kc: 1 - 8 * 10-20 J kA: 40-220 mJ/m2

31. Stress Profile(-surface tension vs. height) Atomistic DPPC bilayer Lindahl and Edholm, J. Chem. Phys., 113, 3882 (2000). Explicit Solvent Implicit Solvent

32. Protein Inclusions Single protein inclusion in the bilayer sheet Inclusion is composed of rigid “lipid molecules”

33. Deformation of the bilayer data elastic theory r H. Aranda-Espinoza et. al., Biophys. J., 71, 648 (1996). G. Brannigan and FLH Brown, Biophys. J., 90, 1501 (2006).

34. Direct from simulation Thermal fluctuations Infer from stress Profile & fluctuations The correspondence is meaningful Using physical constants inferred from homogeneous bilayer simulations we can predict the “fit” values. Deformation profile has three independent constants formed from combinations of elastic properties

35. A consistent elastic model protrusion bending • Begin with standard treatment for surfactant monolayers (e.g. Safran) • Couple monolayers by requiring volume conservation of hydrophobic tails and equal local area/lipid between leaflets • Include microscopic protrusions (harmonically bound to z fields & include interfacial tension)

36. Undulations (bending & protrusion) Peristaltic bending Peristaltic protrusions (and coupling to bending) A consistent elastic model

37. Fits to fluctuation spectra Lindahl &Edholm Marrink & Mark Scott & coworkers

38. These molecules are happy because they’re in their preferred curvature. These molecules are unhappy, but, there are relatively few of them. Effect of spontaneous curvature Positive spontaneous curvature & conservation of volume favors modulated thickness

39. Fluctuation spectra: spontaneous curvature Of the available simulation data, this effect is most pronounced for DPPC: Lindahl, E. and O. Edholm. Biophysical Journal79:426(2000)

40. Rate vs. monolayer thickness H. Huang, Biophys. J., 50, 1061 (1986). H. Kolb and E. Bamberg, Biochim. Biophys. Acta, 464, 127 (1977). J. Elliott et. al., Biochim. Biophys. Acta, 735, 95 (1983). G. Brannigan and F. Brown, Biophys. J., 90, 1501 (2006). gramicidin A channel lifetimes

41. Summary(molecular models) • Implicit solvent models can reproduce an array of bilayer properties and behaviors. • Computation is significantly reduced relative to explicit solvent models, enabling otherwise difficult (impossible) simulations. • Coarse-grained models are especially well suited for validating and suggesting analytical theory. • A single elastic model can explain peristaltic fluctuations, height fluctuations and response to a protein deformations.

42. Lawrence Lin Grace Brannigan Ali Naji Evgeni Penev Paul Atzberger NSF, ACS-PRF, Sloan Foundation, Dreyfus Foundation, UCSB Acknowledgements