Mesoscopic simulations of lipid membranes -...

Mesoscopicmembranes

Mingyang Humingyang@cmu.edu

IntroductionLipid Membrane

Why coarse-graining?

Solvent-free CG ModelSolvent-free

Code Implementation

PropertiesMembrane elasticity

Bending

Line tension

ApplicationsVesicles

Protein-induced budding

Lipid A-B–mixtures

stretching

Mesoscopic simulations of lipidmembranes

Mingyang Hu, Patrick Diggins, and Markus Deserno

Department of Physics, Carnegie Mellon University, Pittsburgh PA, USA

October 11, 2012

Mesoscopicmembranes

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Bending

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IntroductionLipid MembraneWhy coarse-graining?

Solvent-free CG ModelSolvent-freeModel

Code Implementation

PropertiesMembrane elasticityBendingLine tension

ApplicationsVesiclesProtein-induced buddingLipid A-B–mixturesstretching

Mesoscopicmembranes

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Bending

Line tension

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OutlineIntroduction

Lipid MembraneWhy coarse-graining?

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Mesoscopicmembranes

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What’s a lipid membrane?

I forms continuous barriers around cells, cell nucleiI made of two layers of lipid moleculesI controls the diffusion of molecules in and out of the cell

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Building blocks: Lipid molecules

I amphiphilic moleculeI hydrophilic head (polar: “attracted” by water)I hydrophobic tails (apolar: “repelled” by water)

I membranes form by spontaneous aggregation of lipidsI self-assembly process

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Building blocks: Lipid molecules

I amphiphilic moleculeI hydrophilic head (polar: “attracted” by water)I hydrophobic tails (apolar: “repelled” by water)

I membranes form by spontaneous aggregation of lipidsI self-assembly process

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Lindahl, E. & Edholm, O. Mesoscopic undulations and thickness fluctuations inlipid bilayers from molecular dynamics simulations. Biophys. J. 79, 426-433(2000)

All-atom lipid bilayer20 nm× 20 nm, 1024 lipidssimulation time: 10 ns

What if we want a boxlength of L = 200nm?

How does computing effort scale with L?

effort ∼ L2︸︷︷︸geometry

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× L4︸︷︷︸equilibration

∼ L6

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Efficiency

20 nm −→ 200 nmMillion times more

computationally expensive!

I amount of material ∼ membrane area A = L2.I domain decomposition scheme:

I increase # CPUs ∼ AI but simulation time ∼ A3

I uncompensated factor of A2

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Efficiency

20 nm −→ 200 nmMillion times more

computationally expensive!

I CG helps to understand the essence of the problemsI CG reduces the number of DOFI CG allows larger time stepsI CG smoothens the free energy surface⇒ Better sampling

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OutlineIntroduction

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The CG model1

Coarse-grain in order to probe themesoscopic regime of lipid bilayersI generic top-down bead-springI 3 beads: reasonable aspect ratioI only pair forcesI solvent free

I Note: several other similar models exist.I Here I’ll talk about the one used in our groupI It has been implemented in ESPResSo.

1I.R. Cooke, K. Kremer, M. Deserno, Phys. Rev. E 72, 011506 (2005)I.R. Cooke and M. Deserno, J. Chem. Phys. 123, 224710 (2005).

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The CG model1

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The CG model1

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Why is “solvent free” good?

membrane

solvent

surface

Examplea

I 16,000 DPD lipids, 4 beads per lipid.I 64,000 particles for lipids.I But in total 1,536,000 particles in box!

96% simulation time spent with solvent.

aM. Laradji & P.B. Sunil Kumar, Phys. Rev. Lett. 93, 198105 (2004).

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Why is “solvent free” good?

membrane

solvent

surface

Examplea

I 16,000 DPD lipids, 4 beads per lipid.I 64,000 particles for lipids.I But in total 1,536,000 particles in box!

They had a good reason for doing this:study of the dynamics of domain growth,where hydrodynamics is an important factor.

aM. Laradji & P.B. Sunil Kumar, Phys. Rev. Lett. 93, 198105 (2004).

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Difficulties

I Implicit solvent models are incredibly common and usefulin polymer physics.

I Why has it taken so long for them to appear in the field ofmembrane research?

I Polymers don’t first have to self assemble!I One needs additional cohesion to make the lipids come

together.I Fluidity has proven to be the major challenge.

Lennard-Jones interactions:I weak attraction→ gas phaseI no fluid phase in between?!I strong attraction→ solid (gel) phase

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I link three beads Vbond(r) = − 12 kbondr2

∞ ln[1− (r/r2

I make lipid stiff Vbend(r13) = 12 kbend(r13 − 4σ)2

I nonbonded Vrep(r) = 4ε[( rc

)12 −( rc

]Θ(rc−r)

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Overall phase behavior

I long-range attractions “save” the system some entropy!

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Self-assembly

movies/self_assembly_1.mpg

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OutlineIntroduction

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Implementation

I mbtools architecture:I engine (C)

I interactions (lj-cos2 interaction implemented as a FEATURE)I several analysis routines (e.g. modes, stress tensor)

I user-interface (Tcl)I system generation (e.g. initial particle positions)I parameters (configuration files)I register particles in ESPResSo (e.g. topology, interactions)I call to integrate commandI analysis/output

Program structure located in:

ESPRESSO/packages/mbtools

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Program structure

I mbtools is a Tcl package.I set of Tcl functionsI (crude) version control

I load package to get access to enclosed functionspackage require mbtools [1.0.0]

I ESPResSo loads mbtools during initialization (fileESPRESSO/scripts/init.tcl)lappend auto_path "[pwd]/packages/mbtools/"

I auto_path contains the list of (sub)directories checkedby the package loader

I use namespaces to avoid defining two functions with thesame name

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Program structure

# Espresso/packages/mbtools/mbtools.tcl#

package require ::mbtools::utilspackage require ::mbtools::system_generationpackage require ::mbtools::analysis

package provide mbtools 1.0.0

namespace eval mbtools {

# Espresso/packages/mbtools/utils/setup.tcl#

proc ::mbtools::utils::readcheckpoint { dir }

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Config file example (incomplete...)

# .../mbtools/examples/simplebilayer.tcl#

# define geometryset geometry { geometry "flat -fixz" }# time stepset main_time_step 0.01# analysislappend analysis_flags pressure

From command line:

Espresso scripts/main.tcl simplebilayer.tcl

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OutlineIntroduction

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Membrane elasticity

I Model membrane as a 2D elastic sheet (continuum theory)

∫dA{

12κK2 + σ

∫dxdy

{κ(∇2h)2 + σ(∇h)2}

I κ: bending modulusI K: total curvatureI σ: surface tensionI h(x, y): height function (Monge gauge)

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Bending modulus

I Model membrane as a 2D elastic sheet (continuum theory)

∫dA{

12κK2 + σ

∫dxdy

{κ(∇2h)2 + σ(∇h)2}

I κ: bending modulusI K: total curvatureI σ: surface tensionI h(x, y): height function (Monge gauge)

I Fourier expansion and equipartition theorem

〈|hq|2〉 =kBT

L2[κq4 + σq2︸︷︷︸set to 0

kBTL2κ

I determine bending modulus κ

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Fluctuation spectrum from continuum theory

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Fluctuation spectrum from continuum theory

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However...

I Equilibration time of Fourier modes scales like q−4

I Large bending modulus κ from small perturbation (kBT)→ small signal!

h(x) = hqeiqx → K = −h′′(x) = hqq2eiqx

〈K2〉 = 〈|h′′(x)|2〉 = q4〈|hq|2〉 =kBTL2κ

R̄ =1

〈K2〉1/2 =

kBTL ' 3 . . . 5L

I Result relevant for strong bending?

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κ from actively bent membranes

I first implementation:W.K. den Otter and W.J. Briels, J. Chem. Phys.118, 4712 (2003)

I Enforce large undulation mode,measure constraining force.

Simpler way: stretch a membrane tether!2

2V. A. Harmandaris and M. Deserno, J. Chem. Phys. 125, 204905 (2006)

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κ from actively bent membranes3

I Area:A = 2πRL

I Energy:

R2 × A

I Force:

(∂E∂L

= · · · = 2πκR

I Bending modulus:

κ =FR2π' F R

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κ from actively bent membranes4

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Simpler way of extracting the line tension

I simulate a periodicallyhalf-connected bilayer in abox

I stress tensor will beimbalanced precisely bytwice the line tension!

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Simpler way of extracting the line tension

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OutlineIntroduction

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Vesicles

I competition between bending rigidity and line tension 5

I sonicate vesicle solution: rip vesicles into bits and pieces!I these (flat) pieces will merge and grow biggerI at what point will they again close up and form vesicles?

5“The size of bilayer vesicles generated by sonication”, W. Helfrich, Physics LettersA, Vol 50, Issue 2, p. 115-116

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Vesicles

movies/c16sa.mpg

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Vesicles

Evesicle = 4πR2 ·

+ κ1R· 1

]= 4π(2κ+ κ)

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Vesicles

What values do we expect?

I κ = 20 kBT ' 80 pN · nmI κ ' −κ (very little is known about κ, come back tomorrow!)I γ ' 10 pN

I Rpancake =4(2κ+κ)

γ' 4κ

γ' 320 pNnm

10 pN = 32 nm

I This is then also the diameter of vesicles we expect to find!

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Intuitive, but no physical justification!

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Protein-induced budding6

I 36 curved caps, ∼50,000 lipidsI 160nm side-length, total time ∼1msI no lateral tensionI no explicit interaction between caps

6B.J. Reynwar et al., Nature 447, 461 (2007)

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movies/caps.avi

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wAB < wAA = wBB

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Lipid A-B–mixtures10

movies/budding.mpg

10B.J. Reynwar & M. Deserno, Biointerphases 3, FA118 (2009)

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Lipid A-B–mixtures11

movies/spinodal.avi

11from Sarah Veatch

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wAB < wAA = wBB

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stretching modulus12

12i.r. cooke and m. deserno, j. chem. phys. 123, 224710 (2005)

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stretching modulus

simple theory:farago, jcp, 2003; tolpekina/den otter/briels, jcp 2004; cooke/deserno, jcp 2005

membrane stretching plus line energy

rescaling of energy

equilibrium condition for pore radius

m(a− as − πr2)2

as+ 2πγr

λ3 =γas

πm, r̃ =

rλ, b =

a− as

r̃3 − br̃ + 1 = 0

Mesoscopic simulations of lipid membranes -...

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