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 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

4. 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:

5. 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, (1997).

6. Membrane Dynamics

7. 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, (2004).
L. Lin and F. Brown, Phys. Rev. E, 72, (2005).

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. 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)

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, (2004).

15. Dynamics with repulsive spectrin

16. 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)

17. 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

18. 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).

19. 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

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

21. 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, (2006).

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

23. 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.

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

26. Curved proteins diffuse more slowly than flat proteins

27. 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).

28. Molecular membrane models with implicit solvent

29. 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.

30. 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)

31. Motivation
D. Marsh, Biochim. Biophys. Acta, 1286, (1996).

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

33. Self-Assembly
(128 lipids)

34. Flexible (and tunable)
kc: * J
kA: mJ/m2

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

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

37. 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).

38. 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

39. 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)

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

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

42. 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.

43. 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)

44. 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).

45. 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.

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