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Inverse Monte Carlo Method for Determination of Effective Potentials for Coarse-Grained Models

Inverse Monte Carlo Method for Determination of Effective Potentials for Coarse-Grained Models. IPAM Workshop "Multiscale Modeling in Soft Matter and Biophysics September 26-30, 2005. Alexander Lyubartsev ( sasha@physc.su.se ) Division of Physical Chemistry

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Inverse Monte Carlo Method for Determination of Effective Potentials for Coarse-Grained Models

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  1. Inverse Monte Carlo Method for Determination of Effective Potentials for Coarse-Grained Models IPAM Workshop "Multiscale Modeling in Soft Matter and Biophysics September 26-30, 2005 Alexander Lyubartsev (sasha@physc.su.se) Division of Physical Chemistry Arrhenius Lab., Stockhom University

  2. Outline • Introduction • why do we need multiscale coarse-grained modeling • 2. Inverse Monte Carlo Method • how to build effective potentials for coarse-grained models • 3 Effective solvent-mediated potentials • ion-ion and ion-DNA • 4 Coarse-grained lipid model • large-scale simulations of lipid assemblies

  3. Why do we need coarse-grained modeling? a) polyelectrolyte problem: ions around DNA Atomistic MD: not really possible to sample distances 30-40 Å from DNA Primitive model (MC) - how good it is?

  4. b) Lipid bilayer in water Lipid (DMPC) All-atom MD: 1 lipid - more than 100 atoms (DMPC -118, DPPC - 130) ''minimal'' piece of bilayer: 6x6x2 = 72 lipids add at least 20 water molecules per lipid ⇒ about 13000 atoms (the picture above contains about 50000 atoms) A good object to waste CPU time....

  5. Multiscale approach All-atomic model Full information (but limited scale) MD simulation Coarse-graining – simplified model RDFs for selected degrees of freedom Effective potentials for selected sites Reconstruct potentials (inverse Monte Carlo) Increase scale Effective potentials Properties on a larger length/time scale Simulation of coarse grained model (MD,MC,BD,DPD...)

  6. Inverse Monte Carlo direct Properties Model inverse Interaction potential Radial distribution functions • Effective potentials for coarse-grained models from "lower level" simulations • (atomistic coarse grained; CPMD  atomistic) • Reconstruct interaction potential from experimental RDF • An interesting theoretical problem

  7. The method (A.Lyubartsev and A.Laaksonen, Phys.Rev.A.,52,3730 (1995)) Consider Hamiltonian with pair interaction: Va Make “grid approximation”: | | | | | | | Hamiltonian can be rewritten as: Rcut a=1,…,M Where Va=V(Rcuta/M) - potential within a-interval, Sa - number of particle’s pairs with distance between them within a-interval Note: Sais an estimator of RDF:

  8. Set of Va, a=1,…,M Space of Hamiltonians direct {Va}{<Sa>} inverse In the vicinity of an arbitrary point in the space of Hamiltonians one can write: where b = 1/kT

  9. Choose trial values Va(0) Algorithm: Direct MC Calculate <Sa>(n) and differences D<Sa>(n) = <Sa >(n) - Sa* Repeat until convergence Solve linear equations system Obtain DVa(n) New potential: Va(n+1) =Va(n) +DVa(n) An analogue- Newton method Initial approximation: mean force potential Va(0) =-kTln(g*(ra)) S S(V) S* V V* V1 V0

  10. Some comments • Solution of the inverse problem is unique for pair potentials (with exception of an additive constant) gik(r)  Vik(r)+const • There exist a simpler scheme to correct the potential: V(n+1)(r) = V(n)(r) + kT ln(g(n)(r)/gref(r)) (A.K.Soper, Chem.Phys.Lett, 202, 295 (1996)) Its convergence is however slower and may not work in multicomponent case • The precision of the inverse procedure can be defined by analysing eigen values and eigen vectors of the matrix

  11. Effective solvent-mediated potentials. Two levels of simulation of ionic, polymer or other solutions: 1) All-atom simulations (MD) with explicit water. 10000 atoms - box size ~ 40 Å 2) Continuum solvent, solutes - some effective potential, for example, ions - hard spheres interacting by Coulombic potential with suitable e. Ion radius - adjustable parameter (so called "primitive electrolyte model") The idea is to build effective solvent-mediated potential, which, maintaining simplicity of (2), takes into account molecular structure of the solvent

  12. A. Effective solvent-mediated potentials between Na+ and Cl- ions Reference MD simulations: H2Oflexible SPC model (K.Toukan, A.Rahman, Phys.Rev.B31, 2643 (1985) Na+s=2.35Å, e=0.544 kJ/M Cl-s=4.4Å, e=0.42 kJ/M (D.E.Smith, L.X.Dang, J.Chem.Phys., 100, 3757 (1994) Double time step algorithm, with short time step 0.2fs and long time step 2fs, was used NPT-ensemble, T=300K, P=1atm,

  13. Ion-ion effective potentials Ion-ion RDFs

  14. NaCl osmotic and activity coefficients Solvent-mediated effective potentials were applied to calculate osmotic and activity coefficients of Na+ and Cl- ions in the whole concentration range. MC simulations are carried out for 200 ion pairs using effective potentials Osmotic coefficient: Activity coefficient: Lines are calculated values and points are experimental data

  15. B. Ion-DNA effective solvent-mediated potentials Molecular dynamics: • One turn of DNA (dATGCAGTCAG): 635 atoms, • CHARMM force field • (A.D.MacKerell, J.Wiorkiewicz-Kuchera, M.Karplus, JACS, 117, 11946 (1995)) • flexible SPC water model + ions:

  16. All-atom model: Coarse-grained model Na+

  17. Ion - DNA effective potentials Ion - P Ion - C4 (base) Ion - C4’(sugar)

  18. MC simulation: a bigger DNA fragment (3 turns) in a box 100x100x102Å, ions interacting by effective solvent-mediated potentials; no explicit water. These are results for the density profile and integral charge

  19. Relative binding affinities of ions The order of relative binding affinities of alkali counterions to DNA, defined by MC simulation with effective potentials, is: Cs+ > Li+ > Na+ > K+ • The binding order was defined also in a number of experimental works: • P.D.Ross, R.L.Scruggs, Biopolymers, 2, 89 (1964) ; Electrophoresis: Li+>Na+>K+ • U.P.Strauss, C.Helfgott, H.Pink, J.Phys.Chem.,71,2550 (1967); Donnan equilibrium: Li+>Na+>K+ • S.Hallon et al, Biochemistry, 14, 1648 (1975); Circular dichroismCs+>Li+>K+>Na+ • P.Anderson, W.Bauer, Biochemistry, 17, 594 (1978), DNA supercoiling Cs+>Li+>K+>Na+ • M.L.Bleam, C.F.Anderson, M.T.Record, Proc. Natl.Acad.Sci USA,77,3085 (1980), NMR: Cs+>Li+>K+>Na+ • I.A.Kuznetsov et al, Reactive Polymers, 3, 37 (1984), Ion exchange Li+>K+Na+ Qualitative agreement with results of experiments of very different nature.

  20. Coarse-grained lipid model All-atom model 118 atoms Coarse-grained model 10 sites We need interaction potential for the coarse-grained model ! Use IMC and RDFs from atomistic MD.

  21. All-atomic molecular dynamics All-atomic MD simulation was carried out: • 16 lipid molecules (DMPC) dissolved in 1600 waters (6688 atoms) Box size: 40x40x40 Å • Initial state - randomly dissolved RDFs calculated during 12 ns after 2 ns equilibration • Force field: CHARMM 27, water - flexible SPC • T=313 K

  22. MD snapshot 16 DMPC lipids 1600 H2O

  23. R D F calculations N P 4 different groups -> 10 pairs 10 RDFs and eff. intermolecular potentials + 4 bond potentials CO C

  24. Inverse MC simulations: Purpose: find effective potentials which, for the coarse grained model, reproduce the same RDFs as the all-atomic model Intramolecular potentials: Bonded: from distance distribution between the atoms. Non-bonded - the same as intermolecular Total: 14 effective potentials Inverse MC - the box of the same size; the same number of lipids as in the corresponding MD; no solvent: charges +1 and -1 on "N" and "P" + dielectric constant e=70 (best fit to NN, NP and PP -potentials)

  25. Effective potentials: Bond potentials

  26. Coarse-grained simulations • Monte Carlo • Molecular Dynamics • Forces - from the potentials difference in the neigbouring grid points • Solvent is not present explicitly - MD may be considered only as another way to generate canonical ensemble • Time step 10-14 s + thermostat • Nose-Hoover • Local (Lowe-Andersen) • Langevine 3 cases : a periodic bilayer a finite piece of bilayer random initial state Equivalent all-atom simulations would correspond ~ 106-108 atoms

  27. Infinite bilayer Z Periodic box Density distribution Coarse-grained MC (392 lipids) All-atom MD (98 lipids)

  28. A sheet of bilayer The same initial state, but in a large simulation box: End of simulation: 109 - 1010 MC steps: View from the side View from the top (discoid shape)

  29. Vesicle formation Start from a square plain piece of membrane, 325x325 Å, 3592 lipids: cut plane

  30. Membrane self-assembly MD simulation of 392 CG lipids with Lowe-Andersen thermostat http://www.fos.su.se/physical/sasha/lipids

  31. Conclusions 1. The multiscale approach based on the inversion of radial distribution functions provides a straightforward way to build effective potentials for coarse-grained models 2. Examples of ionic solutions, ion-DNA interactions, lipid membranes show that effective potentials, derived exclusively from the atomistic model, provide realistic description for the coarse-grained model 3. Coarse-grained effective potentials may be plugged in into MC, MD, Brownian dynamics, DPD and used for simulation on larger length- and time scale Acknowledgements Aatto Laaksonen Martin Dahlberg Carl-Johan Högberg

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