coarse grained to atomistic mapping algorithm a tool for multiscale simulations
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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.)
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