Coarse grained to atomistic mapping algorithm A tool for multiscale simulations

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Coarse grained to atomistic mapping algorithmA
tool for multiscale simulations

• Steven O. Nielsen
• Department of Chemistry
• University of Texas at Dallas

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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
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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.
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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.
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Idea rotate frozen library structures
M
T
T
M
Library structures from simulated annealing
atomistic MD
T
M
M
T
M
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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
where
The objective (energy) function can be expanded
to quadratic order about R0
and the conjugate gradient incremental step is
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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.
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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)

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Bond
Need to compute the gradient
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Bend
q
u
v
u
r
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Coarse grain to atomistic mapping
Optimized library structure from a simulated
annealing atomistic MD run
One molecule of dodecane
Minimize over SO(3) with fixed center of mass
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.
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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)
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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
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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
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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 algorithms 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.

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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)
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Strategy

• Derive an effective interaction between a liquid
  particle and the entire solid object
• Coarse grain the liquid particles

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• 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)
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Two interacting particles
doesnt 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.
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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)
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Snapshots of C12E5 Self-Assembly on Graphite
Surface
t0ns
t3.3ns
t0.64ns
d5.0 nm
t6.0ns
t4.3ns
t3.75ns
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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
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Acknowledgements

• Bernd Ensing (ETH Zurich)
• Preston B. Moore (USP, Philadelphia)
• Michael L. Klein (U. Penn.)

Funding
National Institutes of Health