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, 2003.
Validation of
CG models
a
S.O. Nielsen e al., J. Phys.:Condens. Matter., 16,
R481, 2004.
Multi-scale simulations
Coarse grain
Atomistic
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.
Idea: rotate frozen library structures
T
M
T
M
Library structures from
simulated annealing
atomistic MD
T
M
T=
M
M=
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 R3
R( )  R0 exp J ( )
where
 0

J ( )    z
  y

The objective (energy)
function can be expanded to
quadratic order about R0
z
0
x
 R ,   
3
y 

 x 
0 
OR   OR0   g t   t H
and the conjugate gradient incremental
step is
s   H 1 g
Computationally efficient algorithm because of the special
relationship between SO(3) and the group of unit quaternions
Sp(1)
q  (cos , ˆ sin  ) ,  12 
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.
Minimize an energy function
C
H
H
C
H
H
H
C
H
C
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
O( R1 , R2 )  k ( r  R 2 v  R1u  d0 )
1
2
u
COM 1
2
v
r
Need to compute the gradient
( r  R2v  R1u  d 0 )


Os k
(r  R2v  R1u )
R1u
O
r  R2v  R1u
x1
x1x1

x1
R1u  R01 J ( xˆ ) u
COM 2
Bend
O( R1 , R2 )  k (   0 )
1
2
2
( R1u  R1u )  (r  R2v  R1u )
  arccos
R1u  R1u r  R2v  R1u

u’
COM 1
v
u
r
COM 2
Coarse grain to atomistic mapping
Optimized library
structure from a
simulated
annealing
atomistic MD run
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.
One molecule of dodecane
Minimize over SO(3) with
fixed center of mass
liquid
C
H
H
C
H
20 dodecane molecules shown in
a box of 1050 molecules (bulk
density = 0.74 g/mL)
H
H
C
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:
Bonds
T = 294 K
Bends
T = 1125 K
Torsions
T = 75 K
One-fours
T = 97 K
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
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.
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 – Science 297, 593 (2002)
Islam -- Would explain difference
between SDS and NaDDBS
JACS 126, 9902 (2004): SANS data
JACS 126 9902 (2004)
Strategy
1) Derive an effective interaction between a
liquid particle and the entire solid object
2) Coarse grain the liquid particles
1)
2)
1)
2)
1)
Is an old idea from colloid science : Hammaker
summation
My contribution : Phys. Rev. Lett. 94, 228301 (2005)
and J. Chem. Phys. 123, 124907 (2005)
2)
Fundamental idea:
two non-interacting particles
P ( z1 , z2 )  e
P ( z1  z2  2 z )  (2 z )
1
2z
e
 U ( z1 )  U ( z2 )
e
 U ( z1 )  U ( 2 z  z1 )
e
0
The probability density and the potential are related by
[normalization convention follows g(r)]
P  e  U
dz1
Two interacting particles
P ( z1 , z2 )  e
 U ( z1 )  U ( z2 )
e
PI ( z1 , z2 )
PI
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:
P ( z1  z2

 2 z) 
2z
0
e
 U ( z1 )  U ( 2 z  z1 )
e

2z
0
PI ( z1 ,2 z  z1 ) dz1
PI ( z1 ,2 z  z1 ) dz1
where the normalization constant is the numerator with U = 0,
namely with no surface.
Nanoscale organization: Experimental observation
Surfactant
C10E3
C12E5
ethylene oxide units
3
5
C12E5 on
graphite
AFM images
Schematic
illustration
L. M. Grant et. al. J. Phys. Chem. B 102, 4288 (1998)
alkyl chain length
10
12
Structure
monolayer
hemi-spheres
C10E3 on
graphite
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
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
Acknowledgements
• Bernd Ensing (ETH Zurich)
• Preston B. Moore (USP, Philadelphia)
• Michael L. Klein (U. Penn.)
Funding
National Institutes of Health