Coarse Grained Methods for Simulation of
Percec and Frechet Dendrimers
Georgios Zamanakos, Nagarajan Vaidehi,
Dan Mainz, Guofeng Wang, Ryan Martin,
Tahir Cagin, and William Goddard III
Materials and Process Simulation Center
Beckman Institute (139-74)
California Institute of Technology
Pasadena, California 91125
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NEIMO dynamics
• Newton - Euler Inverse Mass Operator Method
A O(N) method for simulating constrained systems.
• Uses internal coordinates to directly incorporate the constraints
in the dynamical model.
• Instead of cartesian atoms, it uses rigid subunits composed of
atoms, clusters, whose relative motion is described by the
internal coordinates
When to use NEIMO?
•Large systems that need to be simulated for long time scales
•Eliminates fast degrees of freedom  Large time step
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NEIMO formalism
•Internal Variable Model
Within constrained MD, the dynamical equations of motion become:
M ( )  C ( ,)  T ( )
 is the vector of generalized coordinates
T is the vector of forces
M is the mass matrix
C denotes the Coriolis forces
The solution of these equations is a O(N 3) process
•Inverse Innovations Operator Factorization
By using spatial algebra formalism we can factorize the mass matrix.
The inverse mass matrix then can be expressed as a product of spatial operators:
M 1  [ I  HK ]t D[ I  HK ]
Without explicit inversion of the mass matrix we can solve the constrained equations of
motion (O(N) process).
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NEIMO algorithm
•Recursive Algorithm
1. Base to tip recursion to compute the orientation, location and velocities of each of
the clusters.
2. Tip to base recursion to compute the components of the mass matrix.
3. Final base to tip recursion to compute the accelerations of all the clusters.
NEIMO Chain
Tip
Base
cluster
Hinge
•Tree topology molecules
The current implementation of NEIMO allows only for tree topology molecular models.
Thus, the clusters cannot be allowed to form closed loops.
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Integration of NEIMO in MPSim
•MPSim: a molecular dynamics computer program efficient for high capacity
MD simulations, up to 10 million atoms.
Important features include:
• The Cell Multipole method (CMM) which dramatically reduces the cost of long range
Coulomb and Van der Waals interactions.
• Mixed mode dynamics, which include rigid body and cartesian dynamics.
• Generic forcefields.
• Constant temperature, constant pressure and microcanonical dynamics.
• Implicit solvents
• Portability in highly parallel shared memory systems.
•Goals
• Handle systems that require NEIMO dynamics due to their complexity in an environment
of other molecules which can be simulated using rigid body or flexible dynamics
• Some regions can be treated as rigid while others are treated with only the torsional
degrees of freedom (NEIMO).
• The solvent could be treated with cartesian dynamics, or even as an implicit solvent.
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Implicit Solvent Methods
•Continuum solvation calculations
 Methods that approximate the solvent molecules with a continuum description.
 They average the solvent shielding effect of forces between atom pairs.
 Based on Debye and Huckel’s theory of electrolyte solutions.
•Finite element method (PBF)
  (r ) (r )  4(r )
(Poisson-Boltzman equation)
 A high quality numerical mesh is set up in order to solve the above equation and a set of
linear equations are solved to get the potential .
 A effective charge density on the surface of the molecule is produced which corresponds
the effect of the solvent of the molecule (polarization term).
•Surface Generalized Born (SGB)
 Same continuum solvent picture as PBF but is based on simpler models.
 Much faster method which is easy to parallelize since atom calculations are distributed.
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Frechet Dendrimers
• Stimuli responsive macromolecules based on linear, star and dendritic blocks.
• They respond to a change in their environment through changes in shape, size or nature
of their exposed surface (e.g. hydrophilic or hydrophobic).
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• Using MPSim with NEIMO and implicit solvent dynamics, we can carry out long-time,
large scale MD simulations of this system in different solvents (THF and H COH), in
order to determine the equilibrium structure as a function of solvent and temperature.
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Initial
60ps
140ps
180ps
280ps
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Percec Dendrimers
• They form spherical aggregates that interact with each other
• The whole system forms a well-defined crystal with space group
Pm3 n
.
• Exhibits liquid crystal phase
Idea: Coarse grain description
• Mesoscopic dynamics: get an “average” interaction potential between the Percec “balls”
• Consider the energetics of a pair of spherical assemblies (Adiabatic motion)
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Percec Interaction Potential
•Two body potential curve: Flat bottom Morse potential
• A 216 pseudoatom system after being melted and quenched lead back to the
space group
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Summary
• Advanced, coarse-grained methods are necessary to model large systems such as
dendrimers.
• NEIMO molecular dynamics considers motions of clusters.
• Implicit solvent methods eliminate the need for a full-atom description of the solvent.
• An approximate two body potential can capture the qualitative behavior of the Percec
system.
Future Work
• Parallelization of SGB code.
• Three-body corrections to the two-body interactions
• Get more results!!!
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Acknowledgements
ARO-MURI (Doug Kiserow)
ARO-DURIP
References
• Rodriguez, G. IEEE J. Rob. Automat. 1987, 3, 624
• Jain, A.; Vaidehi, N.; Rodriguez, G. J. Comp. Phys. 1993, 106,
258
• Vaidehi, N.; Jain, A.; Goddard, W. III J. Phys. Chem. 1996,
100, 10509
• Tannor, D. et al J. Am. Chem. Soc., (1994), 116, 11875
• Frechet, J. M. et al Science, (1995), 269, 1080
• Balagurusamy, V.S. et al J. Am.Chem.Soc.,(1997),119,1539
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