An efficient hybrid method for modelinglipid membranes with molecular resolution

Thanks to: C.D. Chau, A.V. Zvelindovsky, organizers!

Japan, 2010

G.J.A. (Agur) Sevink, M. Charlaganov & J.G.E.M Fraaije
Soft Matter Chemistry Group, Leiden University, The
Netherlands

Outline

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 Motivation
 Hybrid CG modeling (ongoing, conceptual) Enhanced sampling (quick) Conclusions/outlook

Motivation
Using/adapting dynamic mesoscopic methodology forblock copolymers to simulate life-mimicking (biomematic)
structures and structure formation
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• Veterinarian• Biologist• Physicist• Mathematician ?

Motivation

reduction

VW Project 2009-2012 ‘Multiscale hybrid modeling of(bio)membranes’ (Schmid, Zvelindovsky, Böker, AS)
Aim: Realistic computational modeling of liposome
formation, dynamics and (assisted) fusion

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“Cell membrane dynamics essentially lipidic” (100+ simulation papers)

Motivation: intriguing experiments in Leiden

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Vesicle fusion induced by coiled-coil motif (short peptide fragments)

Hana Robson Marsden et al, A reduced SNARE model for membrane fusion,Angew. Chem. 2330–2333, 2009.

General issues: length and time scales
Nm and mm: model for complete
vesicle and/or vesicle fusion requiresconsiderable coarse graining
Efficient, realistic, dynamic

fastclosure
fastcollapse

growth(coalescence)

fastslow
Ostwald ripening& vesicle fusionand fission
extremely slow

J. Leng, S. Egelhaaf, M. Cates(2002) Europhys. Lett.

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micellargrowthThe DNA of simulation
Methods
Based on SDSC Blue Horizon (SP3)512-1024 processors
1.728 Tflops peak performance
CPU time = 1 week / processor

100

Average, collective(µs) 10-6(ns) 10-9

Mesoscale methods
Thermodynamics

Semi-empiricalmethods

Monte Carlomolecular dynamics

Statistical physics(ps) 10-12

Ab initio
methods
Specific, detail(fs) 10-15

10-10Japan, 2010

Atomistic
Simulation
Methods(ms) 10-3

10-9

10-8

tight-binding
MNDO, INDO/S

10-7

10-6

10-5

10-4

Vesicle formation and fusion (2005)

METHOD: DDFT= mean-field
SCFT+diffusion

100 nmJapan, 2010

20% A2B2 in a selective bad solventMovie

200 nmAS., Zvelindovsky A.V. Macromolecules 38 7502-7513 (2005).

IssuesBeyond block copolymers: ‘Floppy’ Gaussian chains: onion vesicles
 Mean-field: concentrated systems

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 How to realistically represent lipids?
 Increasing complexity?

Hybrid particle-field model(ongoing work)

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Aim: flexibility, efficient andrealistic liposome simulation

DDFT: pattern formation dynamics in concentrated BCP
Enthalpic: mean-field interactions (FH)21
F[ ρ I ] = F ideal [ ρ I ,U I ] + F cohesive [ ρ I ] + κ H ∫ ∑ ρ I 2
V  I€dρ I (r)δF[ ρ,U]= M∇ ⋅ ρ I (r)∇(r) + ..........+ ηI (r)dtδρI
local kinetic model€

Pressure term, incompressible

noise

processing conditions

(quasi)equilibrium behavior, AB, ABC, branced
Phase transition under external fields (confinement, shear, E, etc)

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Entropic: Gaussian chains in selfconsistent field U

Synergetic validation: flat polymeric ‘membrane’
Structural transition due to thickness reduction : top view

ExperimentJapan, 2010

Δt sim ~ secCalculation€nucleation

annihilation

splittingHigh-speed SFM measurements of membrane dynamics: ~ sec ptf

Different representations of constituents

Variable compositionsolvent
chemical fragment

DDFT: Underlying harmonic spring, calculations and interactions
field-based
Particles (DPD): Harmonic spring, angle and torsion
potentials, soft core repulsive pair potentials

f repulsive
aij

aij = aij0 + Δaijliquid incompressibility€€distancerc€

Japan, 2010springHybrid modelF

DDFT
particles coupling[ ρ I , rk ] = F[ρI ] + U
[ rk ] + F[ ρ I , rk ] cK(r−r)ρ(r)drk∑ Ik ∫II ,kV

particles∂rk = Dk [ f kconserv − ∫ ∑ c IkK(r − r k )∇ρ I (r)dr]∂t + rkrandom (t)VI€dρ I ( r )= M∇ ⋅ ρ I ( r )∇[µ DDFT (r) + ∑ c IkK(r − r k )] + ηIdtk€€fields

Diffusion, timescales are more or less comparative (coupled update)Japan, 2010€hybridPhysical interpretationρΙ(r)Positive cr

Advantage is possibility to mix different representations on CG level forsame or different constituents: sparse (particles) + abundant (field)

Mapping: besides FH parameter (χ)/interaction strengths (a) we needcompressibility (κ ) and coupling ( Ik).

c€

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Coupling force: away from high density field values
Coupling chemical potential: field diffuses away from regions with many particles

Mapping particles and fields: binary system
Determine ‘free’ parameters by requiring thermodynamic consistency forsingle bead solvent in both representations.

: match either pressure orexcess chemical potential

use field partitioning toc Ik :determine
FH χ and Groot€€& Warren to convert tosoft-core potential strength→ c Ik = c Ik (a)€€Note: both particles and fields adapt dynamically€Japan, 2010c IkκκHybrid vs DPD lipid membrane simulation
Use these values and realistic DPD lipid parameters (163)solvent fieldsolvent particles

solvent field

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solvent particles

DPD, Shillcock and Lipowky 2002
(realistic)Hybrid calculation where the solvent is
replaced by a field, with the same S&L
parameters for the lipidHybrid membrane simulation

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Averaging over many initial condition and time frames

Additional benefits: implicit solvent

Preliminary: analytical equilibrium solution for solvent (field) can beconverted into an additional potential in particle description

 sol  lipid
 lipid lipid( rk , rk ) mapping →( ρ sol , rk ) analytic
→ rk€
CGMD, implicit solvent€
I.R. Cooke, K. Kremer, M. Deserno, Phys. Rev. E, 011506 (2005).

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V →A

Vesicle formation pathway following quench

solvent field
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300000200000323

Diffusion is patient (DPD – O(20000))
Experiments: slow process!

Solution? S-QN: accelerating collective modes

Accelerating collectivemodes in a CG particle
description

Japan, 2010Enhanced sampling:Stochastic Quasi-Newton method
Optimization in numerical mathemetics (objective function)Δxk = xk +1 − xk = −α k ∇Φ

Steepest descentDiffusion in statistical mechanics (potential function)€Δxk = −M ∇Φ(xk )Δt + 2MkBT Δt ΔWk
M(x)

√M(x)

Fluctuation-dissipation€

Curvature-dependent mobility

M(x) = (∇ 2Φ(x))−1+ spurious drift

Japan, 2010Δxk = xk +1 − xk = −α k H −1∇Φ Newton method−1B→HBkQuasi-Newton methodkStochastic Quasi-Newton method
Illustration: 1-D Harmonic oscillator
Φ(x) ~k 2x2€

k<1

for M = 1k T
dx = −xdt + 2 B dW (t)€kfor M = k −1 = (∇ 2Φ)−1drift term noise term ~ k-1€ slow modes

Sparse sampling€

fast modes

k>1
Dense sampling

Stability analysis: Δtmax

independent of kJapan, 2010€

dx = −kxdt + 2kB T dW (t)M(x) = M k (x k ) = M k (x k ,..., x 0 ) approximate of H (xk )−1

New factorized update method (equivalent to DFP) for Mk+1:€• Hereditary: minimal M k +1 − M k F• If M0 positive definite, Mk+1 positive definite (√M exists!)• Mk+1 is approximate of inverse Hessian (secant condition)• Efficiency: M k +1 = J k +1J kT+1update Jk+1€
Rouse chain

Additional costs per timestep but

Mk €→ H−Δt SQN >> Δt LD€

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Stochastic Quasi-Newton method

Stochastic Quasi-Newton method
Analysis for quadratic potential (Rouse chain): all modes evolve
equally fast (real-space Fourier acceleration)
Minimal model of a protein

11Φbond + Φbending + Φ dihedral + Φ LJ
22
Bead=amino acid (either neutral,hydrophobic or hydrophilic)

Conclusions (S-QN):
Enhanced sampling of energy landscape (many inherent states)
Hierarchical optimization (bond length, angles, torsions, non bonded)
Generic S-QN method: accelerated but no ’realistic’ dynamics

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Φ=Conclusions and outlook

To do:
Validate membrane material parameters in hybrid model
Concise derivation of implicit solvent
Implementation and parameterization of SNARE-like CG proteins
Outlook:Large scale simulations
Vesicle fusion
…

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Conclusions:New hybrid model for particle/field mixtures
Reuse DPD parameters for CG lipids
Possibility of implicit solvent (analytic)
Additional sparse constituents can be added as CG particle chains
New S-QN method to speed up formation kinetics

…

Questions?
(a.sevink@chem.leidenuniv.nl)

Japan, 2010

Thank you for yourattention

Japan, 2010

Minimal model of a protein (3D): sampling efficiency

T > T collapse

Standard LD (SLD)€One basin

Several basins

Our FSU method

Principles of SQN‘native state’29

Minimal model of a protein (3D): mode analysisLB8B(NL)2NBLB3LB

T << T fold

Native state: left, turn and right sub-domains€χχSLDnativeFSU€
Φ=11Φbond + Φbending + Φ dihedral + Φ LJ
2
2Equilibration order: bonds, angles, torsions, LJ (even for reduced spring constants)-> ‘soft’ RATTLE/SHAKE/LINCS
Principles of SQN30