Systematic Coarse-Graining of Molecular Models by the Inverse Monte Carlo and Newton Inversion

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Systematic Coarse-Graining of Molecular Models by the Inverse Monte Carlo and Newton Inversion Alexander Lyubartsev (sasha@physc.su.

1 Systematic Coarse-Graining of Molecular Models by the Inverse Monte Carlo and Newton Inversion Alexander Lyubartsev Division of Physical Chemistry Department of Material and Environmental Chemistry Stockholm University NSASM10 Dreseden September 2010

Computer modeling from 1st principles to mesoscale Three levels of molecular modeling: First-principles (Quantum Mechanics) Classical Molecular Mechanics / Dynamics nuclei electrons nano scale tape

2 Computer modeling from 1st principles to mesoscale Three levels of molecular modeling: First-principles (Quantum Mechanics) Classical Molecular Mechanics / Dynamics nuclei electrons nano scale tape measure 1 nm = 10-9 m atoms coarse-grained molecules atoms molecules atoms molecules 0.1 nm Meso-scale soft matter 1.0 nm small molecules 10 nm soft matter 100 nm large biomolecules nm organelles, bacteria The problem: Larger scale more approximations

Coares-graining: reduction degrees of freedom All-atom model 118 atoms Coarse-grained model 10 sites Large-scale simulations We need to: 1) Keep only important degrees of freedom and remove

3 Coares-graining: reduction degrees of freedom All-atom model 118 atoms Coarse-grained model 10 sites Large-scale simulations We need to: 1) Keep only important degrees of freedom and remove unimportant (solvent; local details of large molecules) 2) For important degrees of freedom we need interaction potential Question: what is the interaction potential for the coarse-grained model?

4 Formal solution: N-body mean force potential Hamiltonian (potential energy) of the original (FG) system: H(R1,R2,...RN, r1,r2,...,rn) important unimportant Ri can be combinations of original degrees of freedom (e.g. centres of mass) Partition function: Z = dr1... dr N dr 1... dr n exp H R1,..., R N, r 1,... r n = dr1... drn exp βh CG R1,..., R N 1 kt 1 CG H R 1,..., R N = ln dr 1... dr n exp βh R 1,..., R N,r 1,..., r n where = is the effective N-body coarse-grained potential = potential of mean force = free energy of the non-important degrees of freedom

5 HCG(R1,...RN) : provides the same structural properties for the important (CG) degrees of freedom R1,...RN as the FG system. Even thermodynamics can be restored (but: HCG(R1,...RN) is temperature and density dependent ). Problem with HCG(R1,...RN) : simulation with N-body potential is unrealistic. We need something usable e.g. pair potentials! H CG R 1, R2,..., R N V ij R ij i j Criteria for approximation: microscopical : - best fit for the energy - best fit for the force (force matching) ensemble averages (observables) - RDF (can be fit exactly) - other ensemble properties Rij = R i R j

6 Newton Inversion (Inverse Monte Carlo) Model Interaction potential direct inverse Properties radial distribution functions, E, P,... Effective potentials for coarse-grained models from "lower level" simulations (ab-initio atomistic ; atomistic coarse grained;) Reconstruct interaction potential from experimental RDF or experimental thermodynamic properties Use both data of fine grained simulations and experiment

7 The method (A.P.Lyubartsev, A.Mirzoev, L.J. Chen, A.Laaksonen, Faraday Discussions, 144, 2010) Assume: H(λ1,λ2,...λM): Hamiltonian (potential energy) depending on a set of parameters λ1,λ2,...λm. We wish to reproduce M canonical averages <S1>,<S2>,...,<SM> Set of λα, α=1,,m Space of Hamiltonians direct {<Sα>} {λα} inverse

8 In the vicinity of an arbitrary point in the space of Hamiltonians one can write: Δ S α = γ S α λγ Δλ γ O Δλ 2 where S S = λγ H H S γ S γ λα λα β = 1/kT This allows us to solve the inverse problem iteratively

9 Choose trial values λα(0) Direct MC or MD Calculate <Sα> (n) and differences <Sα>(n) = <Sα >(n) - Sα* Solve linear equations system Obtain λα(n) Repeat until convergence Algorithm: New Hamiltonian: λα(n1) =λα(n) λα(n) Newton method of solving non-linear equation: S S(λ) S λ S* λ* λ 1 λ0 λ if no convergence, a regularization procedure can be applied: <Sα>(n) = a(<sα >(n) - Sα*) with a<1

10 Reconstruction of interaction parameters from thermodynamic data LJ potential σ, ε direct inverse Pressure P, evaporation heat Ev Derivatives in the matrix: formulas like : 2 2 P 3 6 PV NkT 8 E V P P = σ σ V kt V PE P E E 3 = NkT PV σ σ kt

11 Toy example: Single site LJ water model Can we fit water molecule by a particle with LJ potential? Experimentally: at T=298K, ρ = 1g/cm3 P=1 bar E = kj/mol Start NI from LJ parameters of oxygen in SPC water... ε (kj/mol) 0.65 P(bar) E(kJ/mol) 0.2 σ (Å) Iteration Reg.param 1 But no use of this, with σ=3.121 Å and ε=5.192 kj/mol this is a solid...

12 Inverse Monte Carlo: Reconstruction of pair potentials from RDFs A.Lyubartsev, A.Laaksonen,Phys,Rev.E, 52, 1995) Consider Hamiltonian with pair interaction: H= V r ij Vα i, j Make grid approximation : H= V α S α α α=1,,m Here: Vα=V(Rcutα/M) - potential within α-interval = λ- parameters Sα - number of particle s pairs with distance between them within α-interval = 1 V g r = Sα estimator of RDF: α 2 2 4πr α Δ r N /2 Inversion matrix became: S = S S S S V So: we can reconstruct pair potential from RDF Rcut

13 Some comments Solution of the inverse problem is unique for pair potentials reconstructed from the corresponding RDFs gik(r) Vik(r)const A simpler scheme to correct the potential: V(n1)(r) = V(n)(r) kt ln(g(n)(r)/gref(r)) (A.K.Soper, Chem.Phys.Lett, 202, 295 (1996)) - also known as Iterative Boltzmann inversion May not converge in multicomponent case; May be reasonable to use in the beginning of the iteration process Intramolecular potentials (bonds, angles, torsions) can be easily included Origin: Renormalization Group method

Single site water model: reconstructing from atomistic RDF -0.82 0.

14 Single site water model: reconstructing from atomistic RDF O-O RDF Potential Problems: P 9000 bar ; not a liquid state,...

Let us generate g(r) at lower density, e.g. 0.

15 Let us generate g(r) at lower density, e.g g/cm3 OO RDF at different densities; for 500 water molecules

16 Effective potentials generated from different densities Potential obtained in inhomogeneous system: keep correct RDF and provides realistic pressure RDFs at density ρ=1 g/cm3

17 Computer modeling from 1st principles to mesoscale Three levels of molecular modeling: First-principles (Quantum Mechanics) Classical Molecular Mechanics / Dynamics nuclei electrons nano scale tape measure 1 nm = 10-9 m atoms coarse-grained molecules atoms molecules atoms molecules 0.1 nm Meso-scale soft matter 1.0 nm small molecules 10 nm soft matter 100 nm large biomolecules nm organelles, bacteria

18 From ab-initio to atomistic: ab-initio derived 3-site water model Reference simulation: ab-initio (CPMD) simulation of 32 water 3.0 Box sixe 9.86 Å, BLYP functional Vanderbilt pseudopotentials 25 Ha cutoff timestep 0.15 fs 25 ps simulation at 300 C (Chem.Phys.Lett, 325, 15 (2000)) O-O CPMD Exp SPC 2.5 RDF O-H H-H O-O H-H O-H r (Å) Ab-initio eff. pot SPC model 200 Eff. potential / kt Some details: r (Å) 6

19 Some insight Average energy is -25 kj/mol (flexible model ) Experimental: -41 kj/mol. E(32 H2O) 32 E(H2O,opt) = -29 kj/mol CPMD energy Comparison with CPMD: Difference about 4 kj/mol due to many body effects (?) effective potential energy

20 Li - water ab-initio potentials Input RDF-s obtained from 520 ps Car-Parrinello MD for 1 Li ion in 32 water molecules (A.P.Lyubartsev, K.Laasonen and A.Laaksonen, J.Chem.Phys., 114,3120,2001) IMC procedure: SPC model for water; only LiO potential varied to match ab-initio LiO RDF 60 Inversion RDF Fit Dang, J.Chem.Phys.96,6970(1992) Periole et al, J.Phys.Chem.B,102,8579(1998) Heinzinger, Physica BC, 131, 196 (1985) Eff.pot (kj/m) 40 Non-electrostatic part of the LiO potential may be well fitted by: ULiO = A exp(-br) where A = kj/m b = 3.63 Å-1 r(å) Atomistic simulations: Egorov et al, J.Phys.Chem. B, 107, 3234(2003)

21 From atomistic to coarse-grained: solvent-mediated potentials. Atomistic: All-atom simulations (MD) with explicit solvent (water). State of art: < atoms - box size ~ Å Time scale problem CG: Coarse-grain solutes; use continuum solvent (McMillan-Mayer level), From 60-ties: primitive electrolyte model: ions - hard spheres interacting by Coulombic potential with suitable ε, ion radius - adjustable parameter We can now build effective solvent-mediated potential, which takes into account molecular structure of the solvent. RDF from atomistic MD Implicit solvent effective potentials

22 NaCl ion solution Ion-ion RDFs (from atomistic MD) Ion-ion effective potentials 4,0 6 3,0 RDF 2,5 2,0 1,5 1,0 0,5 0,0 NaCl, L=39Å NaCl, L=24Å NaNa, L=39Å NaNa, L=24Å ClCl, L=39Å ClCl, L=24Å Prim. model 4 Eff. Potential / kt NaCl, L=39Å NaCl, L=24Å NaNa, L=39Å NaNa, L=24Å ClCl, L=39Å ClCl, L=24Å 3, r (Å) 10 5 r (Å) 10 15

23 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: osm. coef activity coef P osm V 1 F / c = T = kt c NkT Activity coefficient: μ ex λ=exp kt E Concentration (M) Lines are calculated values and points are experimental data Important to have water model with correct permittivity 1

24 Temperature dependence increase T U short r =U r qi q j 4 0 r ij

25 From tail asymptotic extract dielectric permitttivity

26 Short range effective potential Temperature dependence can be effectively included in ε

27 Ion-DNA solvent-mediated potentials All-atom model: Coarse-grained model Na

28 Atomistic MD RDFs for coarse-grained sites Ion - DNA effective potentials Ion - P Ion - C4 (base) 6 Li 6 Na Na * K 2 Cs Li Na Eff. potential / kt Eff. potential / kt 4 Li 4 Na * K* Cs 2 0 Na Eff. potential / kt 6 Ion - C4 (sugar) 4 Na * K* Cs r (Å) r (Å) r (Å)

29 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 1.0 Na NA * Integral charge Density profile (M/l) Li K* Cs PB Li Na r (Å) Na * 0.4 K* Cs PB r (Å) 40 50

30 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 dichroism Cs>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: I.A.Kuznetsov et al, Reactive Polymers, 3, 37 (1984), Ion exchange Cs>Li>K>Na Li>K Na Qualitative agreement with results of experiments of very different nature.

31 Proline in DMSO (A.P.Lyubartsev, A.Mirzoev, L.J. Chen, A.Laaksonen, Faraday Discussions, 144, 2010) D proline L-proline DMSO solvent Atomistic CG implicit

32 Experimentally: prolines in DMSO self-assemble into different structures, depending on ratio R- and D- forms. This can be also observed in atomistic simulations: but these are tedious slow Still, RDF can be computed and converted to effective potentials for CG model

33 Spatial distribution functions: From atomistic MD From CG simulations

34 Coarse-grained lipid model Lipid bilayer simulations: atoms Possible, but... Coarse-graining:

35 Atomistic MD: 16 DMPC lipids in 1600 H2O CHARMM 27 ff 25 ns simulations

36 R D F calculations N P 4 different groups -> 10 pairs 10 RDFs and eff. intermolecular potentials 5 bond potentials (N-P, P-CO, CO CO, CO-C, C-C) CO C 7 NP 6 6 solid - 16 lipids dashed - 64 lipids PP 3 RDF RDF 5 NN r(a) CC 2 PC NC 5 10 r(a) 15 20

37 Effective potentials: N - CO N - CH P - CO P - CH NN 10 Veff (kj/m) Veff (kj/m) 15 PP r(a) r (A) 30 CO - CO CO - CH CH - CH 10 Veff (kj/m) Veff (kj/m) Bond potentials NP N-P P - CO CO - CH CH - CH r(a) r(a) 6 7 8

38 Bilayer structure: Z Periodic box Density distribution Coarse-grained MC All-atom MD

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

40 Membrane self-assembly MD simulation of 392 CG lipids with Lowe-Andersen thermostat

41 Larger systems (1000 or more lipids) can form vesicles: spherical vesicle (cut in the middle) formed by self-assembly of 1000 DMPC lipids Sometimes: double bicell d-spacing 60-65Å

42 Formation of multilammelar vesicles Start from random configuration, 5000 lipids in 300Å After After 200 ns After After ns ns Phase separation on multilammelar phase with d-spacing ~ 60Å (20 Å water layer between lipids = 30 H2O per lipid) and bulk phase (here nothing) 60-62Å

43 Conclusions The multiscale approach based on the inversion of radial distribution functions provides a straightforward way to build effective potentials starting from ab-inition level to atomistic models and further to coarsegrained models, Effective potentials as good as underlying detailed model (or worse) Effective potentials derived from inhomogeneous setup seems to provide better transferability Use of Newton invertion combining fitting to RDF from detailed simulations and to experimentally known thermodynamical propeties may be an option (to do).

44 Acknowledgements Aatto Laaksonen Alexander Mirzoev Andrei Egorov LiJun Chen $$: Vetenskapsrådet CPU: SNIC resources at PDC, NSC, HPC2N

Multiscale Simulations of DNA and Nucleosome Core Particles

Multiscale Simulations of DNA and Nucleosome Core Particles Alexander Lyubartsev Department of Materials and Environmental Chemistry, Stockholm University Molecular Mobility and Order in Polymer Systems