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How to make coarse grain force fields from atomistic simulations.

Vale Molinero

Materials and Process Simulation Center,

Caltech

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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?

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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

QMAtomistic & Atomistic CG

averages over degrees of freedom (electronic) that are usually well separated from the one retained (nuclear)

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.

c /1

For = 1800 cm-1, = 55 fs

To integrate MD equations of motions, the

time step shouldn’t be longer than ~ /10 =

5.5 fs

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QMAtomistic & 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.

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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…)

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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.

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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.

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Graphical Examples

a-glucose molecule

1 bead per monomer

polymer

Bisphenol-A polycarbonate

Kremer et al.

R1=C1

E6=C6C6R4=C4 C1

1 bead per monomer

2 beads per monomer

3 beads per

monomer

1 bead per monomer in different positions

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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).

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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.

(1) Is easier because does not require running MD for the atomistic target, nor for the CG to check the data.

(2) 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)

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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

All the important decisions are made here

(and once!)

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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, 1414-1427.

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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 MotivationThe goal: to model polydisperse

mixtures of oligosaccharides, and

glucose glasses

1 Degree of polymerization 50

%

Molecule size: 3 to 1000 atoms.

Minimum formulation 104-105 atoms.

Broad distribution of length and timescale:

Dilute solutions of amylose, experimental:

* segmental dynamics ~ ns (NMR)

* persistence length ~ 1.5 -3 nm ()

(helical structures)DP4 DP1

DP38

water

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Motivation

Unsolved questions that we could not answer through atomistic simulations and we aimed to study with the CG

model.

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

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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…)

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Reconstruction: M3B atomistic

the 3 beads completely define the orientation of the glucose

monomer.

B1=C1

B6=C6B4=C4

-glucose residue: monomer unit

B1-B4’glycosidic bond

Molecular shape is well captured

24 atoms 3 beads

M3B model:

Can represent the chain conformation around glycosidic

bonds

DP11 RMS=0.34Å

Water molecule 1 bead

monomer 3 beads

Bonus!

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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 M3B energy

expression

Parameterization

scheme Step 3 - VALENCE

POTENTIALTHAT 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 OPTIMIZATIONFOR A WIDE RANGE OF STRESSES AND ALLOWING

RELAXATION OF THE M3B STRUCTURES.

Harmonic bonds

20 )(

2

1)( ddkdE b

)}(2){()( )1/(5.02)1/(5.0 oijoij RRRRoij eeDRV

20 )(

2

1)( kE )cos(1(

2

1)( 0

kkkk

nbE

Harmonic angles Shift dihedral torsions Morse nonbond

(all pairs, except 1,2 & 1,3 bonded)NV

T

641’6’

NVT

641’64

+ ++E =

Morse parameters of water chosen to reproduce E, and D

of liquid water at 300 K.

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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

Glucose shape is very well represented

Results for glucose, minimization results.

Different colors correspond to different reference samples.

Bond distances

Angles

Structural

Final bond constants are ~2 parameterized by gas

phase simulations.

Density Cohesive Energy

Equation of state (0 K)

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141’4’

Chain conformation

141’4’

L R

Handedness

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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…)

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Comparison of CPU timecerius2 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.

Wate

r conte

nt in

crease

s

8%w

16.5%

20%w

27Helical 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.

M3B can form a variety of helical structures without having directional interaction (hydrogen bonds)

n~5.5-7 h~7-8.3 Å

40 ns simulation 300 K, starting from helix

Anti Parallel

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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)