1. How to make coarse grain force fields from atomistic simulations.
Vale Molinero
Materials and Process Simulation Center,
Caltech

2. outline What is a coarse grain model Developing a model from atomistic
What to be reproduced by the cg
Scales in cg / time in cg
Parameterization vs numerical functions
Possible targets to reproduce
Optimization
Transferability of cg parameters
Can we use the same parameters always?

3. What’s a coarse grain model?
Polymers and many materials show a hierarchy of length scales and associated time scales.
Coarsening is limiting the number of degrees of freedom and the frequency of their motion.
What are the links between scales?
No clear definition of CG, but is a scale in which particles represent atoms in the order of a monomer of polymer chain (5-50 atoms, approx).
Preserve connectivity.
Difference between generic (toy) models and coarse grain is that cg are derived to represent a specific material.
Terminology:
Coarse grain simulations 
Multiscale simulations

4. Comparing QMAtomistic & Atomistic  CG
the degrees of freedom in the atomistic model not always are well separated.
How much is retained is a measure of the coarseness of the model.
averages over degrees of freedom (electronic) that are usually well separated from the one retained (nuclear)
For n = 1800 cm-1, t = 55 fs
To integrate MD equations of motions, the time step shouldn’t be longer than ~ t /10 = 5.5 fs

5. Comparing QMAtomistic & Atomistic  CG
the atomistic interaction sites are usually located on the nuclei of the QM atoms.
Symmetry of the molecule is preserved while averaging electronic degrees of freedom.
the particles should be positioned to describe the lowest frequency modes of the molecule & to represent the excluded volume interaction (shape).
Low frequency modes of a molecule are usually not localized… so, trimming the number of particles usually change significantly the shape of the power spectrum.
the CG model and the atomistic one do not have the same symmetry.

6. Atomistic  CG The steps of the coarse graining machinery
0) Define your goals
1) Degree of coarsening
2) Mapping atomistic into coarse grain
3) Interaction between the coarse grain particles
4) Atomistic target functions to be reproduced by the CG model
5) Parameter/function optimization
6) Enjoy! (but check first…)

7. Atomistic  CG Decisions to make:
1) Degree of coarsening (how many particles per monomer/molecule): this is application driven.  What are the minimal features of the atomistic model that should be retained to reproduce the desired properties?
Examples of features that may be sought to be preserved:
interaction energies,
shape of the molecule or total volume (density),
flexibility
connectivity in a polymer chain
ability to form a given phase (crystalline or amorphous)
handedness or asymmetry in the chains
This defines the number of beads (superatoms/coarse-grain particles)
Examples:
PEO polymer modeling: -(CH2-CH2-O)n we wanted to represent the helicity of the overall chain, the flexibility, and the excluded volume. We choose one coarse grain particle per monomer.
Glucose monomer and oligomer: we wanted to represent the helicity of the chain, its segmental motion, shape, and to retain the exceptional glass forming abilities of glucose: we choose 3 particles per monomer.

8. Atomistic  CG 2) Mapping of the atomistic into the coarse grain.
Where are the beads?
This is crucial in defining the shape of the molecule.
In a polymer chain is relevant for the connectivity and branching.
This defines the position of beads (superatoms/coarse-grain particles) and affects the parameterization of the cg force field.

9. Graphical Examples 1 bead per monomer 2 beads per monomer
Bisphenol-A polycarbonate
a-glucose molecule
1 bead per monomer
2 beads per monomer
Kremer et al.
1 bead per monomer
polymer
1 bead per monomer in different positions
R1=C1
E6=C6C6
R4=C4 C1
3 beads per monomer

10. Atomistic  CG 3) How the coarse grain particles interact?
two posibilities: 1) analytical functions (like LJ, harmonic potentials, etc)
2) numerical functions of the bead coordinates.
analytical f. are easier to handle in standard molecular simulation software
are less versatile
have analytical derivatives!
need to be parameterized
numerical f. can represent “whatever” but sometimes at the cost of introducing back high frequencies. The derivatives should be obtained and listed numerically (interpolation). (I think MC is better suited for numerical than MD).

11. Atomistic  CG
4) Obtain the atomistic target function to be reproduced with the CG model.
Two general possibilities:
1) use minimized structures, T=0 properties.
2) use thermalized systems at the T that the CG is going to be used.
Is easier because does not require running MD for the atomistic target, nor for the CG to check the data.
Is more correct, cause the coarse grain parameterizations are state dependent
Targets: radial distribution function (T>0)
potential of mean force (T>0)
density, cell parameters, RMS displacements (T=0)
cohesive energies, compressibility (T>0 or T=0)
dynamics, power spectrum (T>0)

12. All the important decisions are made here (and once!)
The only iterative part is here… in the optimization of the CG parameters or functions that reproduce the atomistic
Transferability
of parameters should be checked

13. Illustration of CG development from atomistic simulations
M3B: a coarse grain model for the simulation of malto-oligosaccharides and their water mixtures. 
V. Molinero; W.A. Goddard III,
J. Phys. Chem. B 2004, 108,

14. Atomistic  CG The steps of the coarse graining machinery
0) Define your goals
1) Degree of coarsening
2) Mapping atomistic into coarse grain
3) Interaction between the coarse grain particles
4) Atomistic target functions to be reproduced by the CG model
5) Parameter/function optimization
6) Enjoy! (but check first…)

15. 1 Degree of polymerization 50
Motivation
The goal: to model polydisperse mixtures of oligosaccharides, and glucose glasses
1 Degree of polymerization %
Molecule size: 3 to 1000 atoms. Minimum formulation atoms.
Broad distribution of length and timescale: Dilute solutions of amylose, experimental: * segmental dynamics t ~ ns (NMR) * persistence length ~ nm (ho) (helical structures)
DP4
DP1
DP38
water
Dynamic and structural properties of Corn Syrup and solvated oligosaccharides

16. Unsolved questions that we could not answer through atomistic simulations and we aimed to study with the CG model.
Motivation
Structure -Conformation of chains in the mixture: Coil? Helix? Hybrid? (persistence length) Water distribution in the structure: Pockets? Channels? Scattered? Dynamics -Diffusion in supercooled mixtures (hopping?) -How water diffuses in glasses of carbohydrates
Dynamic and structural properties of Corn Syrup and solvated oligosaccharides

17. Atomistic  CG The steps of the coarse graining machinery
0) Define your goals
1) Degree of coarsening
2) Mapping atomistic into coarse grain
3) Interaction between the coarse grain particles
4) Atomistic target functions to be reproduced by the CG model
5) Parameter/function optimization
6) Enjoy! (but check first…)

18. M3B model: Reconstruction: M3B  atomistic
Water molecule  1 bead
monomer  3 beads
M3B model:
a-glucose residue:
monomer unit
B1-B4’
glycosidic bond
B1=C1
B6=C6
B4=C4
Molecular shape is well captured
24 atoms  3 beads
Can represent the chain conformation around glycosidic bonds
Reconstruction: M3B  atomistic
the 3 beads completely define the orientation of the glucose monomer.
Bonus!
DP11 RMS=0.34Å

19. Atomistic  CG The steps of the coarse graining machinery
0) Define your goals
1) Degree of coarsening
2) Mapping atomistic into coarse grain
3) Interaction between the coarse grain particles
4) Atomistic target functions to be reproduced by the CG model
5) Parameter/function optimization
6) Enjoy! (but check first…)

20. Parameterization scheme
M3B energy expression
Harmonic bonds
Harmonic angles
Shift dihedral torsions
Morse nonbond
(all pairs, except 1,2 & 1,3 bonded)
NVT
641’6’
641’ 64 +
E =
Parameterization scheme
Step 3 - VALENCE POTENTIAL
THAT MATCH GAS PHASE DISTRIBUTIONS
Step 1- INITIAL GUESS FOR NONBONDING POTENTIAL
Step 2-NONBONDING POTENTIAL REFINEMENT WITH MCSA FOR FIXED GEOMETRY
Step 4 - VALENCE & NONBOND JOINT OPTIMIZATION
FOR A WIDE RANGE OF STRESSES AND ALLOWING RELAXATION OF THE M3B STRUCTURES.
Morse parameters of water chosen to reproduce E, r and D of liquid water at 300 K.

21. Atomistic  CG The steps of the coarse graining machinery
0) Define your goals
1) Degree of coarsening
2) Mapping atomistic into coarse grain
3) Interaction between the coarse grain particles
4) Atomistic target functions to be reproduced by the CG model
5) Parameter/function optimization
6) Enjoy! (but check first…)

22. Fidelity of the parameterization
Density Cohesive Energy
Equation of state (0 K)
Bond distances
Angles
Structural
Final bond constants are ~2 parameterized by gas phase simulations.
Results for glucose, minimization results.
Different colors correspond to different reference samples.
Glucose shape is very well represented

23. Chain conformation
141’4’ L R
Handedness
141’4’

24. Atomistic  CG The steps of the coarse graining machinery
0) Define your goals
1) Degree of coarsening
2) Mapping atomistic into coarse grain
3) Interaction between the coarse grain particles
4) Atomistic target functions to be reproduced by the CG model
5) Parameter/function optimization
6) Enjoy! (but check first…)

25. Comparison of CPU time cerius2 in 1 processor sgi origin RS10000
Atomistic model
1 fs time step
21-24 particles per monomer
ewald for the nonbond
Coarse Grain Model
10 fs time step
3 particles per monomer
spline for the nonbond
Timing of MD for the same DP4 bulk system shows that the bead model is ~7000 times faster

26. Water distribution in sugar mixtures
Water structure is heterogenous in a length-scale of a few water molecular diameters
Water structure percolates between 17-20%w/w for all the atomistic & coarse grain models studied.
(clustering distance= 4 A)
M3B gives the same water distribution (percolation, water-water coordination distribution) than the atomistic model.
Water content increases
16.5%
20%w

27. 40 ns simulation 300 K, starting from helix
Helical structures
Left-hand single helices
L- Double helices
Parallel & antiparallel have comparable energy
Vh-amylose structure
M3B cell parameters between 3-6% of Xray data Density within 1% of experimental value.
n~ h~7-8.3 Å
40 ns simulation 300 K, starting from helix
Anti Parallel
M3B can form a variety of helical structures without having directional interaction (hydrogen bonds)

28. successes of M3B
It reproduced the atomistic structure of water in sugar mixtures without using any HB or directional interaction. (packing/shape and right E)
Was able to form all the helical structures of polysaccharides: left and right hand single helices, parallel and anti-parallel double helices. Predicted relative stabilities and structures in agreement with atomistic simulations and experimental observations. (segmental modes kept/ well parameterized torsions).
Predicted glass transition temperatures in excellent agreement with the experiment. (surprise! Energetics/shape).
Was used to unravel the mechanism of water and glucose diffusion in supercooled mixtures, all the predictions in quantitative agreement with experiments. (shape/energetics)
Was used to explain how water diffusion continues below the glass transition temperature in carbohydrate mixtures. (shape/energetics)