An efficient hybrid method for modeling lipid membranes with molecular resolution

Transcription

1 An efficient hybrid method for modeling lipid membranes with molecular resolution G.J.A. (Agur) Sevink, M. Charlaganov & J.G.E.M Fraaije Soft Matter Chemistry Group, Leiden University, The Netherlands Thanks to: C.D. Chau, A.V. Zvelindovsky, organizers!

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

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

4 Motivation reduction Cell membrane dynamics essentially lipidic (100+ simulation papers) VW Project Multiscale hybrid modeling of (bio)membranes (Schmid, Zvelindovsky, Böker, AS) Aim: Realistic computational modeling of liposome formation, dynamics and (assisted) fusion

5 Motivation: intriguing experiments in Leiden Vesicle fusion induced by coiled-coil motif (short peptide fragments) Hana Robson Marsden et al, A reduced SNARE model for membrane fusion, Angew. Chem , 2009.

6 General issues: length and time scales Nm and mm: model for complete vesicle and/or vesicle fusion requires considerable coarse graining Efficient, realistic, dynamic micellar growth fast collapse fast growth (coalescence) slow closure fast Ostwald ripening & vesicle fusion and fission extremely slow J. Leng, S. Egelhaaf, M. Cates (2002) Europhys. Lett.

7 10 0 The DNA of simulation Based on SDSC Blue Horizon (SP3) processors Tflops peak performance CPU time = 1 week / processor Methods (ms)10-3 (µs)10-6 (ns)10-9 (ps)10-12 (fs)10-15 Atomistic Simulation Methods Semi-empirical methods Statistical physics Ab initio methods Specific, detail Mesoscale methods Thermodynamics Average, collective tight-binding MNDO, INDO/S Monte Carlo molecular dynamics

8 Vesicle formation and fusion (2005) METHOD: DDFT= mean-field SCFT+diffusion 100 nm 20% A 2 B 2 in a selective bad solvent Movie 200 nm AS., Zvelindovsky A.V. Macromolecules (2005).

9 Issues Beyond block copolymers: How to realistically represent lipids? Increasing complexity? Floppy Gaussian chains: onion vesicles Mean-field: concentrated systems

10 Hybrid particle-field model Aim: flexibility, efficient and realistic liposome simulation (ongoing work)

11 DDFT: pattern formation dynamics in concentrated BCP Enthalpic: mean-field interactions (FH) F[ρ I ] = F ideal [ρ I,U I ] + F cohesive [ρ I ] κ H V I ρ I 2 dρ I (r) dt Entropic: Gaussian chains in selfconsistent field U = M ρ I (r) δf[ρ,u] δρ I Pressure term, incompressible (r) η I (r) noise local kinetic model processing conditions (quasi)equilibrium behavior, AB, ABC, branced Phase transition under external fields (confinement, shear, E, etc)

12 Synergetic validation: flat polymeric membrane Structural transition due to thickness reduction : top view Experiment Δt sim ~ sec Calculation nucleation annihilation splitting High-speed SFM measurements of membrane dynamics: ~ sec ptf

13 Different representations of constituents solvent Variable composition chemical fragment spring DDFT: Underlying harmonic spring, calculations and interactions field-based Particles (DPD): Harmonic spring, angle and torsion potentials, soft core repulsive pair potentials f repulsive a ij a ij = a ij 0 + Δa ij liquid incompressibility distance r c

14 Hybrid model F hybrid [ρ I, r k ] = F DDFT [ρ I ] + U particles [ r k ] + F coupling [ρ I, r k ] particles r k = D k [ f k conserv dρ I ( r ) dt V I = M ρ I ( r ) [µ DDFT (r) + I,k c Ik V K( r rk)ρ I ( r ) dr c Ik K(r rk) ρ I (r)dr] t + r random k (t) k c Ik K(r rk) ] + η I fields Diffusion, timescales are more or less comparative (coupled update)

15 Physical interpretation ρ Ι (r) Positive c r Coupling force: away from high density field values Coupling chemical potential: field diffuses away from regions with many particles Advantage is possibility to mix different representations on CG level for same or different constituents: sparse (particles) + abundant (field) Mapping: besides FH parameter (χ)/interaction strengths (a) we need compressibility ( κ ) and coupling ( c Ik ).

16 Mapping particles and fields: binary system Determine free parameters by requiring thermodynamic consistency for single bead solvent in both representations. κ c Ik κ : match either pressure or excess chemical potential c Ik : use field partitioning to determine FH χ and Groot & Warren to convert to soft-core potential strength c Ik = c Ik (a) Note: both particles and fields adapt dynamically

17 Hybrid vs DPD lipid membrane simulation Use these values and realistic DPD lipid parameters (163) solvent field solvent particles solvent field 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

18 Hybrid membrane simulation Averaging over many initial condition and time frames

19 Additional benefits: implicit solvent Preliminary: analytical equilibrium solution for solvent (field) can be converted into an additional potential in particle description ( r k sol, r k lipid ) (ρ sol, r lipid k ) mapping analytic V A r k lipid CGMD, implicit solvent I.R. Cooke, K. Kremer, M. Deserno, Phys. Rev. E, (2005).

20 Vesicle formation pathway following quench solvent field Diffusion is patient (DPD O(20000)) Experiments: slow process! Solution? S-QN: accelerating collective modes

21 Enhanced sampling: Accelerating collective modes in a CG particle description

22 Stochastic Quasi-Newton method Optimization in numerical mathemetics (objective function) Δx k = x k +1 x k = α k Φ Δx k = x k +1 x k = α k H 1 Φ Quasi-Newton method B k Steepest descent Newton method B k H 1 Diffusion in statistical mechanics (potential function) Δx k = M Φ(x k )Δt + 2Mk B T ΔtΔW k M(x) Fluctuation-dissipation M(x) Curvature-dependent mobility M(x) = ( 2 Φ(x)) 1 + spurious drift

23 Stochastic Quasi-Newton method Illustration: 1-D Harmonic oscillator Φ(x) ~ k 2 x 2 dx = kxdt + 2k B TdW (t) for M =1 k<1 dx = xdt + 2 k BT dw (t) for M = k 1 = ( 2 Φ) 1 k drift term noise term ~ k -1 slow modes fast modes k>1 Sparse sampling Dense sampling Stability analysis: Δt max independent of k

24 Stochastic Quasi-Newton method M(x) = M k (x k ) = M k (x k,..., x 0 ) approximate of H (x k ) 1 New factorized update method (equivalent to DFP) for M k+1 : M k +1 M k Hereditary: minimal F If M 0 positive definite, M k+1 positive definite ( M exists!) M k+1 is approximate of inverse Hessian (secant condition) T Efficiency: update J k+1 M k +1 = J k +1 J k +1 Rouse chain M k H Additional costs per timestep but Δt SQN >> Δt LD

25 Stochastic Quasi-Newton method Analysis for quadratic potential (Rouse chain): all modes evolve equally fast (real-space Fourier acceleration) Minimal model of a protein Φ = 1 2 Φ bond Φ bending + Φ dihedral + Φ LJ 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

26 Conclusions and outlook 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 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

27 Thank you for your attention Questions?

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29 Minimal model of a protein (3D): sampling efficiency Standard LD (SLD) T > T collapse One basin Several basins Our FSU method Principles of SQN native state 29

30 Minimal model of a protein (3D): mode analysis T << T fold LB 8 B(NL) 2 NBLB 3 LB Native state: left, turn and right sub-domains χ χ SLD native FSU Φ = 1 2 Φ bond Φ bending + Φ dihedral + Φ LJ Equilibration order: bonds, angles, torsions, LJ (even for reduced spring constants) -> soft RATTLE/SHAKE/LINCS Principles of SQN 30