Frank L. H. Brown

Department of Chemistry and Biochemistry, UCSB

Simple models for biomembranedynamics and structure

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 relevantto 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

h(r)

LHelfrich bending free energy:

E =

Kc

2∇2h(

r r )( )

2

∫ dxdy=Kc

2L2 k4 hk

2

k∑

Linear response for normal modes:ddt

hk t( )=−ωkhk +ζ(t)

ωk =Kck

3

4η

Ornstein-Uhlenbeck process for each mode:

P(hk)=Kck

4

2πkBTL2 exp−Kck

4

2kBTL2 hk

2⎡

⎣ ⎢ ⎤

⎦ ⎥

P hk(t) |hk(0)( ) =Kck

4

2πkBTL2(1−e−2ωkt)exp−

Kck4

2kBTL2

hk(t) −hk(0)e−ωkt 2

(1−e−2ωkt)

⎡

⎣ ⎢

⎤

⎦ ⎥

Kc: Bending modulusL: Linear dimensionT: Temperature: Cytoplasm viscosity

Relaxation frequencies

R. Granek, J. Phys. II France, 7, 1761-1788 (1997).

Solve for relaxation of membrane modes coupled to a fluid inthe overdamped limit:

Non-inertial Navier-Stokes eq:

∇p−η∇2r u = fext

∇ ⋅r u =0

Nonlocal Langevin equation:

˙ h (r r ,t) = d

r r ' H(

r r −

r r ' ) f(

r r ',t)∫ +ζ (

r r ,t)

H(r r )=

18πηr

f (r r ) =−

δEδh(

r r )

Oseen tensor(for infinite medium)

Bending force

˙ h k(t)=− Kck4

( )1

4ηk

⎛ ⎝ ⎜ ⎞

⎠ hk(t)+ˆ ζ (k,t)

ωk =Kck

3

4η

Generalization to other fluid environments is possible

Different viscoscitieson both sides of membrane

Membrane near an Impermeable wall

Membrane near a semi-permeable wall

And various combinations

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:

€

E =Kc

2∇ 2h(

r r )( )

2

∫ dxdy + V[h(r)] =Kc

2L2k 4 hk

2

k

∑ + Vk[h(r)]

Overdamped dynamics:

€

˙ h (r r , t) = d

r r 'H(

r r −

r r ') f (

r r ', t)∫ + ζ (

r r , t)

f (r r ) = −

δE

δh(r r )

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).

€

d

dthk t( ) = Λk Fk[h(r, t)] + ζ k (t){ }

Λk =1

4ηk(or generalizedexpressions)

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.

F. Brown, Biophys. J., 84, 842 (2003).

Dynamic undulation model

Kc=2x10-13 ergs=0.06 poiseL=140 nmT=37oCDmicro=0.53 m2/sh0=6 nm

Dmicro

Explicit Cytoskeletal Interactions

• Additional repulsive interaction along the edges of the corral to mimic spectrin

∑∫ ++= −

i

/)( )( iiih

Arep cbyaxedF δε λrr

L. Lin and F. Brown, Phys. Rev. Lett., 93, 256001 (2004).

• Harmonic anchoring of spectrin cytoskeleton to the bilayer

E = dA∫ r

Kc

2 ∇2h(r) [ ]

2+

12

V(r)h2(r)⎧ ⎨ ⎩

⎫ ⎬ ⎭

V(r) =γ δ(r−Ri)i∑

L. Lin and F. Brown, Biophys. J., 86, 764 (2004).

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=23s (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

∞

9×10−12

5×10−12

7×10−10

System Size Simulation

Type

Simulation

Geometry

L Free Membrane Square

L Pinned Square

Pinned Square

Pinned & Repulsive

Square

Pinned & Repulsive

Triangular

Dmacro (cm2s-1)

∞∞

€

2 ×10−10

€

3.4 ×10−10

Dmacro=6.6×10-11 cm2s-1• Median experimental value

Fluctuations of supported bilayers

Y. Kaizuka and J. T. Groves, Biophys. J., 86, 905 (2004).Lin and Brown (submitted).

Fluctuations of supported bilayers (dynamics)

x10-5

Impermeable wall (different boundary conditions) No wall

Timescales consistent withexperiment

Way off!

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

koff

kon

Fluctuations of active membranes (experiments)

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...

Y. Kantor, M. Kardar, Y. Kantor, M. Kardar, and D.R. Nelson, Phys. and D.R. Nelson, Phys. Rev. A Rev. A 35, 35, 3056 (1987)3056 (1987)

J.M. Drouffe, A.C. J.M. Drouffe, A.C. Maggs, and S. Leibler, Maggs, and S. Leibler, Science Science 254254, 1353 , 1353 (1991)(1991) Helmut Heller, Helmut Heller,

Michael Schaefer, Michael Schaefer, and Klaus Schulten, and Klaus Schulten, J. Phys. Chem J. Phys. Chem 1993, 1993, 8343 (1993)8343 (1993)

R. Goetz and R. R. Goetz and R. Lipowsky, J. Chem. Lipowsky, J. Chem. Phys. Phys. 108, 108, 7397 7397 (1998)(1998)

W. Helfrich, Z. W. Helfrich, Z. Natuforsch, Natuforsch, 28c28c, 693 , 693 (1973)(1973)

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

QuickTime™ and aYUV420 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 JkA: 40-220 mJ/m2

Stress Profile(-surface tension vs. height)

R. Goetz and R.Lipowsky, JCP 108 p. 7397, (1998)

ExplicitSolvent

ImplicitSolvent

Protein Inclusions

Inclusion is composed of rigid “lipid molecules”

Single protein inclusion in the bilayer sheet

Deformation of the bilayer

r

€

δ =h(r) − h0

h0

data

fit to elastic

H. Aranda-Espinoza et. al., Biophys. J., 71, 648 (1996). G. Brannigan and FLH Brown, Biophys. J., 90, 1501 (2006).

The fit is meaningfulDeformation profile has three independent

constants formed from

combinations of elastic properties

h0monolayer thickness

0area/lipid

kcbending modulus

kAStretching modulus

c0Spontaneous curvature (monolayer)

c0’ area

derivative of above

Direct from simulation

Thermal fluctuations

Infer from stressProfile & fluctuations

Using physical constant inferred from homogeneous bilayer simulations we can predict the “fit” values.

A consistent elastic model

•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)

bendingprotrusion

A consistent elastic model

€

dxdykc

2∇ 2z+

( )2

+ kλ λ+2+ γ int ∇λ+

( )2

+ 2γ int∇z+ ⋅∇λ+ ⎧ ⎨ ⎩A

∫

+kA

2t02 z−

( )2

+ 2kcc0∇2z− + 2kc c0 − c0

' Σ0( )z−

t0

∇ 2z− +kc

2∇ 2z−

( )2

+kλ λ−2+ γ int ∇λ−

( )2

+ 2γ int∇z− ⋅∇λ−}

Undulations (bending & protrusion)

Peristaltic bending

Peristaltic protrusions (and coupling to bending)

€

kc

2: Monolayer bending modulus

kλ : Protrusion binding constant

γ int: Interfacial (water/hydrocarbon) tension

kA

2: Monolayer compressibility modulus

c0 : Monolayer spontaneous curvature

•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

A consistent elastic model

Marrink & Mark

Scott & coworkers

Lindahl &Edholm

€

hq

2=

kBT

kcq4

+kBT

2 kλ + γ intq2

( )

tq

2=

kBT

kcq4 − 4

kc

t0

c0 − c0' Σ0( )q

2 + kA / t02

+kBT

2 kλ + γ intq2

( )

Fits to fluctuation spectra

Positive spontaneous curvature & conservation of volume favors modulated thickness

Effect of spontaneous curvature

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:

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are needed to see this picture.

Lindahl, E. and O. Edholm. Biophysical Journal 79:426(2000)

gramicidin A channel lifetimes

€

kd (z0(1))

kd (z0(2))

≈e−β Felas (TS )−Felas (D )( )

1

e−β (Felas (TS )−Felas (D ))2

Rate vs. monolayer thickness

In progress…

Free energy calculations to establishpotential 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)

Lawrence LinGrace Brannigan

Adele TamboliPeter Phillips

Jay Groves, Larry Scott, Jay MashlSiewert-Jan Marrink

Olle Edholm

NSF, ACS-PRF, Sloan Foundation,UCSB

Acknowledgements