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

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

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

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

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

At every point R 0 on the manifold SO(3) we construct a continuous, differentiable mapping between a neighborhood of R 0 on the manifold and an open set in R 3 where The objective (energy) function can be expanded to quadratic order about R 0 and the conjugate gradient incremental step is

Updated rotation is obtained by quaternion multiplication q 0 q s. The other source of efficiency comes from working at the coarser level: there are only three variables (one rotation matrix) per coarse grained site. Computationally efficient algorithm because of the special relationship between SO(3) and the group of unit quaternions Sp(1)

Minimize an energy function C H H C C C H H H H H H 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)

Bond COM 1COM 2 r uv Need to compute the gradient

Bend COM 1COM 2 r u v u’u’ 

Coarse grain to atomistic mapping Minimize over SO(3) with fixed center of mass 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.

liquid 20 dodecane molecules shown in a box of 1050 molecules (bulk density = 0.74 g/mL) C H H C C H H H H 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

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: BondsT = 294 K BendsT = 1125 K TorsionsT = 75 K One-foursT = 97 K

100 consecutive CG frames with incremental updating Final structure equipartition estimate: BondsT = 316 K BendsT = 1002 K TorsionsT = 79 K One-foursT = 247 K Very fine convergence tolerance

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.

M. F. Islam et. al., Nano Lett. 3, 269 (2003) SDS Solubilization of Single-Wall Carbon Nanotubes in Water JACS (2004) Islam -- Would explain difference between SDS and NaDDBS Smalley – Science 297, 593 (2002) JACS 126, 9902 (2004): SANS data C. Mioskowski, Science 300, 775 (2003)

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

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

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. Two interacting particles doesn’t involve the surface. Can be obtained from liquid simulations.

Nanoscale organization: Experimental observation Surfactantethylene oxide unitsalkyl chain length Structure C10E monolayer C12E5 512 hemi-spheres L. M. Grant et. al. J. Phys. Chem. B 102, 4288 (1998) C12E5 on graphite C10E3 on graphite AFM images Schematic illustration

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

Extension to curved surfaces Triton X-100 adsorbing on carbon nanotube 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

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