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

2. Outline 1.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. 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.... Lipid (DMPC) b) Lipid bilayer in water

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

6. Inverse Monte Carlo Model Properties direct 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 Consider Hamiltonian with pair interaction: Make “grid approximation”: Hamiltonian can be rewritten as: Where V  =V(R cut  /M) - potential within  -interval, S  - number of particle’s pairs with distance between them within  -interval  =1,…,M Note: S  is an estimator of RDF : ( A.Lyubartsev and A.Laaksonen, Phys.Rev.A.,52,3730 (1995)) VV | | | | | | | R cut

8. In the vicinity of an arbitrary point in the space of Hamiltonians one can write: where Set of V  ,  =1,…,M  Space of Hamiltonians {V  } {<S   } direct inverse  = 1/kT

9. Algorithm: Choose trial values V  (0) Direct MC Calculate (n) and differences  (n) = (n) - S  * Solve linear equations system Obtain  V  (n) New potential: V  (n+1) =V  (n) +  V  (n) Repeat until convergence An analogue- Newton method S V V * V 1 V 0 S*S* S(V) Initial approximation: mean force potential V  (0) =-kTln(g * (r  ))

10. Some comments Solution of the inverse problem is unique for pair potentials (with exception of an additive constant) g ik (r)  V ik (r)+const There exist a simpler scheme to correct the potential: V (n+1) (r) = V (n) (r) + kT ln(g (n) (r)/g ref (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 . 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: H 2 O flexible SPC model (K.Toukan, A.Rahman, Phys.Rev.B31, 2643 (1985) Na+  =2.35Å,  =0.544 kJ/M Cl-  =4.4Å,  =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 RDFs Ion-ion effective potentials

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 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: 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 H 2 O

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

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  =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 ~ 10 6 -10 8 atoms

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

28. A sheet of bilayer The same initial state, but in a large simulation box: End of simulation: 10 9 - 10 10 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