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

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?

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

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

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.

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…)

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.

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.

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

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).

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)

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

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.

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…)

1 Degree of polymerization 50 Motivation The 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 t ~ ns (NMR) * persistence length ~ 1.5 -3 nm (ho) (helical structures) DP4 DP1 DP38 water Dynamic and structural properties of Corn Syrup and solvated oligosaccharides

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

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…)

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

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…)

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.

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…)

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

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

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…)

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

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

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~5.5-7 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)

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)