Simple models for biomembrane
dynamics and structure
Frank L. H. Brown
Department of Chemistry and Biochemistry, UCSB
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 simulations for membrane dynamics
– Protein motion on the surface of the red blood cell
– Supported bilayers and “active” membranes
• Molecular membrane model
– Flexible lipid, solvent-free
• Correspondence with experimental physical properties and
stress profile
• Protein induced deformation of the bilayer
• Elastic model for peristaltic bilayer deformations
– Fluctuation spectra
– Protein deformations
Linear response, curvature elasticity model
L
Helfrich bending free energy:
2
Kc
Kc
2
2
4
E
 h(r ) dxdy  2  k hk


2
2L k
h(r)
Linear response for normal modes:
d
hk t     k hk   (t)
dt
K ck 3
k 
4
Kc: Bending modulus
L: Linear dimension
T: Temperature
: Cytoplasm viscosity
Ornstein-Uhlenbeck process for each mode:
P(hk ) 
 Kc k 4
Kc k 4
2 
exp

h
2
2
k 

2 kB TL
 2k B TL

 K k 4 h (t)  h (0)e  k t 2 
Kc k 4
k
k
c
Phk (t ) | hk (0) 
exp

2 t

2 t

2 k BTL2 (1  e k )
2k BTL2
(1  e k )


Relaxation frequencies
Solve for relaxation of membrane modes coupled to a fluid in
the overdamped limit:
Non-inertial Navier-Stokes eq:
p  2 u  fext
 u  0
Nonlocal Langevin equation:
K ck 3
k 
4
hÝ(r,t)   dr' H(r  r' ) f (r ' ,t)   (r ,t)
H(r ) 
1
8r
f (r )  
E
h(r )
Oseen tensor
(for infinite medium)
Bending force
R. Granek, J. Phys. II France, 7, 1761-1788 (1997).
1 
4 
ˆ( k,t)
Ý
hk (t )   Kck 
hk (t)  
4k 
Generalization to other fluid
environments is possible


Different viscoscities
on both sides of
membrane
Membrane near an
Impermeable wall
And various combinations
Membrane near a
semi-permeable wall
Seifert PRE 94,
Safran and Gov PRE 04,
Lin and Brown (In press)
Membrane Dynamics
QuickTime™ and a
decompressor
are needed to see this picture.
Extension to non-harmonic systems
Helfrich bending free energy + additional interactions:
2
Kc
Kc
2
2
4
E
 h(r ) dxdy  V[h(r)]  2  k hk  Vk [h(r)]


2
2L k
Overdamped dynamics:
d
 dr ' H(r  r ') f (r ',t)   (r,t) dt hk t    k Fk [h(r,t)]   k (t)
E
f (r )  
1
(or generalized
h(r )
k 
4k expressions)
Solve via Brownian dynamics
hÝ(r ,t) 
•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).
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
Why membrane undulations?
• The mechanism has not been
considered before.
• “Thermal” membrane fluctuations are
known to exist in red blood cells (flicker
effect).
• Physical constants necessary for a
simple model are available from
unrelated experiments.
Dynamic undulation model
Dmicro
Kc=2x10-13 ergs
=0.06 poise
L=140 nm
T=37oC
Dmicro=0.53 m2/s
h0=6 nm
F. Brown, Biophys. J., 84, 842 (2003).
Explicit Cytoskeletal Interactions
• Harmonic anchoring of spectrin
cytoskeleton to the bilayer
Kc

E  A d r 
 2
  h(r ) 
2
2
1
 V(r)h 2 (r) 


2
V(r)     (r  Ri )
i
• Additional repulsive interaction along
the edges of the corral to mimic spectrin
Frep    dr e  h (r ) / 
A
  (ax  by
i
i
L. Lin and F. Brown, Biophys. J., 86, 764 (2004).
L. Lin and F. Brown, Phys. Rev. Lett., 93, 256001 (2004).
i
 ci )
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=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
Simulation
Geometry
Dmacro (cm2s-1 )
L
Free Membrane
Square
L
Pinned
Square

Pinned
Square
9  1012
5 10 12
7  1010
Pinned &
Repulsive
Square
Pinned &
Repulsive
Triangular


• Median experimental value
Dmacro  6.6 10-11 cm2s-1
3.4 1010
2 1010
Fluctuations of supported bilayers
Y. Kaizuka and J. T. Groves, Biophys. J., 86, 905 (2004).
Lin and Brown (submitted).
Fluctuations of supported
bilayers (dynamics)
Impermeable wall (different boundary conditions)
No wall
x10-5
Timescales consistent with
experiment
Way off!
Fluctuations of “active” bilayers
Fluctuations of active membranes (experiments)
J.-B. Manneville, P. Bassereau, D. Levy and J. Prost, PRL, 4356, 1999.
kon
Off
Off
koff
kon
koff
Push down
Push up
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.
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
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YUV420 codec decompressor
are needed to see this picture.
(128 lipids)
Phase Behavior
Fluid (kBT = 0.9)
“Gel” (kBT = 0.5)
Flexible (and tunable)
kc: 1 - 8 * 10-20 J
kA: 40-220 mJ/m2
Stress Profile
(-surface tension vs. height)
R. Goetz and R.Lipowsky, JCP 108 p.
7397, (1998)
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
fit to elastic

r
h(r)  h0
h0

H. Aranda-Espinoza et. al., Biophys. J., 71, 648 (1996).
G. Brannigan and FLH Brown, Biophys. J., 90, 1501 (2006).
The fit is meaningful
Deformation profile has three independent
constants formed from
combinations of elastic properties
h0
0
kc
kA
c0
c0
’
monolayer
thickness
area/lipid
bending
modulus
Stretching
modulus
Spontaneous
curvature
(monolayer)
area derivative
of above
Using physical
constant inferred from
homogeneous bilayer
simulations we can
predict the “fit” values.
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
k c 2  2
2
 2
 z   k     int    2 intz   
 dxdy

2
A
kA  2
z 2  kc 2  2
2 
'
 2 z   2k c c 0 z  2kc c 0  c 0 0   z   z 
t0
2
2t 0
k     int    2 intz  
2
2
Undulations (bending & protrusion) k

Peristaltic
bending
Peristaltic protrusions
(and coupling to bending)
c

: Monolayer bending modulus
2
k : Protrusion binding constant
 int: Interfacial (water/hydrocarbon) tension
kA
: Monolayer compressibility modulus
2
c 0 : Monolayer spontaneous curvature
A consistent elastic model
•Predicts thermal fluctuation amplitudes for
both undulation and peristaltic modes
–Continuous as function of q
–Spontaneous curvature included
–Elastic constants are interrelated
–5 fit constants over two data sets
•Predicts response to hydrophobic mismatch
–Same elastic constants as fluctuations
–Protrusion modes explicitly considered
Fits to fluctuation spectra
Lindahl &Edholm
Marrink & Mark
Scott & coworkers
hq
tq
2
2

kB T
kB T

k c q 4 2k    intq 2 
kB T

kcq4  4
kc
2
c 0  c '0 0 q 2  k A /t 0

t0

kB T
2k   intq 2 
Effect of spontaneous curvature
Positive spontaneous curvature & conservation
of volume favors modulated thickness
These molecules are
unhappy, but, there are
relatively few of them.
These molecules are
happy because they’re
in their preferred
curvature.
Fluctuation spectra: spontaneous
curvature
Of the available simulation data, this effect
is most pronounced for DPPC:
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Lindahl, E. and O. Edholm.
Biophysical Journal 79:426(2000)
gramicidin A channel lifetimes
Rate vs. monolayer
thickness
) e  Fela s (TS )Fela s (D )1
   (Fela s (TS )Fela s (D ))
) e
2
(1)
0
(2)
0
kd (z
kd (z
In progress…
Free energy calculations to establish
potential of mean force.
Summary
•Implicit solvent models can reproduce an array
of bilayer properties and behaviors.
•Computation is significantly reduced relative to
explicit solvent models, enabling otherwise
difficult simulations.
•A single elastic model can explain peristaltic
fluctuations, height fluctuations and response to
a protein deformations (if you include
spontaneous curvature and protrusions)
Acknowledgements
Lawrence Lin
Grace Brannigan
Adele Tamboli
Peter Phillips
Jay Groves,
Larry Scott, Jay Mashl
Siewert-Jan Marrink
Olle Edholm
NSF, ACS-PRF, Sloan Foundation,UCSB