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 modelingC.M. Shephard, Biochem. J., 370, 233, 2003.

S.O. Nielsen e al., J. Phys.:Condens. Matter., 16, R481, 2004.

a

Validation of CG models

Multi-scale simulations

Coarse grain Atomistic

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

TM

T =M

M

M =

MLibrary structures from simulated annealing atomistic MD

0

( ) 0

0

z y

z x

y x

J

0( ) exp ( )R R J

At every point R0 on the manifold SO(3) we construct a continuous, differentiable

mapping between a neighborhood of R0 on the manifold and an open set in 3

3 , R

1s H g

where

The objective (energy) function can be expanded to quadratic order about R0

and the conjugate gradient incremental step is

HgRORO tt 0

12

ˆ(cos , sin ) ,q

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.

Computationally efficient algorithm because of the special relationship between SO(3) and the group of unit quaternions Sp(1)

Minimize an energy function

CHH

CCC

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 1 COM 2r

u v

Need to compute the gradient

11 0

1

ˆ( )x

R u R J x u

20122

121 )(),( duRvRrkRRO

O

x1

Bend

1 1 2 1

1 1 2 1

( ) ( )arccos

Ru R u r R v R u

R u R u r R v R u

COM 1 COM 2r

u vu’

202

121 )(),( kRRO

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)

CHH

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 atomsTaken 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:

Bonds T = 294 K

Bends T = 1125 K

Torsions T = 75 K

One-fours T = 97 K

100 consecutive CG frames with incremental updating

Final structure equipartition estimate:Bonds T = 316 KBends T = 1002 KTorsions T = 79 KOne-fours T = 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 126 9902 (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, 228301 (2005) and J. Chem. Phys. 123, 124907 (2005)

1) 2)

)()(21

21),( zUzU eezz P

z

zzUzU dzeezzzz2

0

1)2()(1

2111)2()2( P

The probability density and the potential are related by[normalization convention follows g(r)]

Ue P

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

z

I

z

IzzUzU

dzzzz

dzzzzeezzz 2

0 111

2

0 111)2()(

21

)2,(

)2,()2(

11

P

PP

),(),( 21)()(

2121 zzeezz IzUzU PP

doesn’t involve the surface. Can be obtained from liquid simulations.IP

Nanoscale organization: Experimental observation

Surfactant ethylene oxide units alkyl chain length StructureC10E3 3 10 monolayerC12E5 5 12 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.)