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

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.

Outline Elastic membrane model Molecular 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

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

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

Membrane Dynamics

Extension to non-harmonic systems 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).

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.

Protein motion on the surface of red blood cells

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

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

Proposed Models

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

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).

Dynamics with repulsive spectrin

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) 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 System Size Simulation Type Geometry L Free Membrane Square Pinned Pinned & Repulsive Triangular 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 viscoscities on both sides of membrane Membrane near an Impermeable wall And various combinations Seifert PRE 94, Safran and Gov PRE 04, Lin and Brown JCTC 06. Membrane near a semi-permeable wall

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

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 Fluctuations of active membranes (experiments) L. Lin, N. Gov and F.L.H. Brown, JCP, 124, 074903 (2006).

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

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

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

Curved proteins diffuse more slowly than flat proteins

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).

Molecular membrane models with implicit solvent

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.

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)

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

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

Self-Assembly (128 lipids)

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

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

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

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).

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 h0 monolayer thickness 0 area/lipid kc bending modulus kA Stretching modulus c0 Spontaneous curvature (monolayer) c0’ area derivative of above Direct from simulation Thermal fluctuations Infer from stress Profile & fluctuations

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)

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

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

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

Fluctuation spectra: spontaneous curvature Of the available simulation data, this effect is most pronounced for DPPC: Lindahl, E. and O. Edholm. Biophysical Journal 79:426(2000)

gramicidin A channel lifetimes H. Huang, Biophys. J., 50, 1061 (1986). Rate vs. monolayer thickness 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).

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.

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