Coarse grained to atomistic mapping algorithm A tool for multiscale simulations

1. Coarse grained to atomistic mapping algorithmA tool for multiscale simulations Steven O. Nielsen Department of Chemistry University of Texas at Dallas

2. Outline • Role of inverse mapping in • Multiscale simulations • Validation of coarse grained (CG) models • CG force field development • Schematic picture • Some mathematical details • Application to molecular systems • Illustrative example : bulk dodecane • Conclusions Coarse grained strategies for aqueous surfactant adsorption onto hydrophobic solids

3. Spatial / Temporal scales in computational modeling C.M. Shephard, Biochem. J., 370, 233, 2003. Validation of CG models S.O. Nielsen e al., J. Phys.:Condens. Matter., 16, R481, 2004.

4. Coarse grain Atomistic Multi-scale simulations Wholesale mapping Mixed CG/AA representation On-the-fly mapping Automated CG force field construction Can switch back and forth repeatedly and refine the coarse grain potentials by force matching or other algorithms.

5. Idea: rotate frozen library structures M T T M Library structures from simulated annealing atomistic MD T = M M T M =

6. At every point R0 on the manifold SO(3) we construct a continuous, differentiable mapping between a neighborhood of R0on the manifold and an open set in R3 where The objective (energy) function can be expanded to quadratic order about R0 and the conjugate gradient incremental step is

7. Computationally efficient algorithm because of the special relationship between SO(3) and the group of unit quaternions Sp(1) Updated rotation is obtained by quaternion multiplication q0qs. The other source of efficiency comes from working at the coarser level: there are only three variables (one rotation matrix) per coarse grained site.

8. C H H H H H C C H C H H Minimize an energy function • interactions are only between atoms belonging to different coarse grained units • Bonds • Bends • Torsions, 1-4 • Non-bonded (intermolecular and within the same long-chain molecule)

9. Bond COM 1 u v COM 2 r Need to compute the gradient

10. Bend COM 1 COM 2 q u’ v u r

11. Coarse grain to atomistic mapping Optimized library structure from a simulated annealing atomistic MD run One molecule of dodecane Anticipate performing the inverse mapping at each coarse grain time step. The SO(3) conjugate gradient method should be efficient this way because each subsequent time step is close to optimized. Minimize over SO(3) with fixed center of mass

12. C H H H H C C H H liquid Energy function consists of: • 1 bond, 4 bends, 4 torsions, and 4 one-fours per “join” between intramolecular CG sites • All L-J repulsions between H atoms Taken directly from the CHARMM force field 20 dodecane molecules shown in a box of 1050 molecules (bulk density = 0.74 g/mL)

13. Single snapshot – fully converged Calculate the fully atomistic CHARMM energy on the SO(3) converged structure From the equipartition theorem, expect to have ½ kT energy per degree of freedom: Bonds T = 294 K Bends T = 1125 K Torsions T = 75 K One-fours T = 97 K

14. 100 consecutive CG frames with incremental updating Very fine convergence tolerance Final structure equipartition estimate: Bonds T = 316 K Bends T = 1002 K Torsions T = 79 K One-fours T = 247 K

15. Conclusions • The coarse grained to atomistic mapping algorithm presented here uses SO(3) optimization to align optimized molecular fragments corresponding to coarse grained sites • The algorithm’s efficiency comes from using quaternion arithmetic and from optimizing at the coarse grained level • The mapping algorithm will play an important role in multiscale simulations and in the development and validation of coarse grained force fields.

16. SDS Solubilization of Single-Wall Carbon Nanotubes in Water C. Mioskowski, Science 300, 775 (2003) M. F. Islam et. al., Nano Lett. 3, 269 (2003) Smalley – Science297, 593 (2002) Islam -- Would explain difference between SDS and NaDDBS JACS126, 9902 (2004): SANS data JACS 126 9902 (2004)

17. 1) 2) Strategy • Derive an effective interaction between a liquid particle and the entire solid object • Coarse grain the liquid particles

18. 1) 2) • Is an old idea from colloid science : Hammaker summation • My contribution : Phys. Rev. Lett. 94, 228301 (2005) and J. Chem. Phys. 123, 124907 (2005) Fundamental idea: two non-interacting particles The probability density and the potential are related by [normalization convention follows g(r)]

19. Two interacting particles doesn’t involve the surface. Can be obtained from liquid simulations. The probability of the center of mass being at height z is given by: where the normalization constant is the numerator with U = 0, namely with no surface.

20. Nanoscale organization: Experimental observation Surfactant ethylene oxide units alkyl chain length Structure C10E3 3 10 monolayer C12E5 5 12 hemi-spheres C10E3 on graphite C12E5 on graphite AFM images Schematic illustration L. M. Grant et. al. J. Phys. Chem. B 102, 4288 (1998)

21. Snapshots of C12E5 Self-Assembly on Graphite Surface t=0ns t=0.64ns t=3.3ns d=5.0 nm t=3.75ns t=4.3ns t=6.0ns

22. Extension to curved surfaces Theory for cylinders and spheres is done. Applications are being carried out for the solubilization of carbon nanotubes and for the (colloidal) solubilization of quantum dots Triton X-100 adsorbing on carbon nanotube

23. Acknowledgements • Bernd Ensing (ETH Zurich) • Preston B. Moore (USP, Philadelphia) • Michael L. Klein (U. Penn.) Funding National Institutes of Health