From Atomistic to Coarse Grain Systems –Procedures & Methods

Frank Romer

Forschungszentrum Julich GmbHInstitute of Complex Systems & Institute for Advanced Simulation

Theoretical Soft Matter and Biophysics (ICS-2/IAS-2)

Mulliken Center Bonn27.11.2014

F. Romer Coarse Graining Recipes 1 / 44

Coarse Graining?

Granularity

(Redirected from Coarse grain)

“Granularity is the extent to which a system is broken down into smallparts, either the system itself or its description or observation. It is theextent to which a larger entity is subdivided. [..] Coarse-grained systemsconsist of fewer, larger components than fine-grained systems; acoarse-grained description of a system regards large subcomponents whilea fine-grained description regards smaller components of which the largerones are composed.”a

aWikipedia, Granularity, http://en.wikipedia.org/wiki/Granularity,(6.11.2014)

F. Romer Coarse Graining Recipes 2 / 44

Coarse Grainingreduce the degrees of freedom

first principles: e.g. CPAIMD-BLYP/DVR1

atomistic: e.g. 3-site model SPC/E2

coarse grain: e.g. 3TIP particle = 3 water molecules3

mesoscale: e.g. DPD particle = 107–109 water molecules4

fluid mechanics: continuum

1H.-S. Lee and M. E. Tuckerman, J. Chem. Phys. 125, 154507 (2006).2H. J. C. Berendsen et al., J. Phys. Chem. 91, 6269–6271 (1987).3J. Elezgaray and M. Laguerre, Comput. Phys. Commun. 175, 264 –268 (2006).4A. Kumar et al., Microfluidics and Nanofluidics 7, 467–477 (2009).

F. Romer Coarse Graining Recipes 3 / 44

Coarse Grainingtime & length scale

F. Romer Coarse Graining Recipes 4 / 44

1st Coarse Graining attempt

B. Smit et al., Nature 348, 624–625 (1990):“Computer simulations of a water/oil interface in the presence of micelles.”

a phenomenological model

water particle ,

oil particle , and

surfactant

Lennard-Jones potential5

o-o and w-w interactions are truncated at rc = 2.5σ

o-w interactions are truncated at rc = 21/6σ → completely repulsive6

5J. E. Lennard-Jones, Proc. Phys. Soc. London 43, 461 (1931).6J. D. Weeks et al., J. Chem. Phys. 54, 5237–5247 (1971).

F. Romer Coarse Graining Recipes 5 / 44

From Atomistic to CG Force Fields

F. Romer Coarse Graining Recipes 6 / 44

CG groups

dimyristoylphosphatidylcholine (DMPC) as coarse grained by J. Elezgaray and M. Laguerre7

define new objects → CG groups/particles:

mimic, at least partially, the behavior of a group of atoms

assignment have not to be mandatory bijective

bottom-up

Deconstruct the target moleculein groups of atoms, and thenfind a proper description foreach of this CG groups.

top-down

Defining several CG groupswith specific properties, andrebuild the target moleculeusing these CG groups.

7J. Elezgaray and M. Laguerre, Comput. Phys. Commun. 175, 264 –268 (2006).F. Romer Coarse Graining Recipes 7 / 44

Deriving Force Field

(a) class II force fields, e.g. CFF93 J. R. Maple et al., J. Comput. Chem. 15, 162–182 (1994)

(b) from thermodynamic data, e.g. OPLS W. J. Jorgensen and J. Tirado-Rives, J. Am.Chem. Soc. 110, 1657–1666 (1988)

(c) from thermodynamic data, e.g. MARTINI S. J. Marrink et al., J. Phys. Chem. B 111,7812–7824 (2007)

(d) the focus of this talk!

F. Romer Coarse Graining Recipes 8 / 44

atomistic to coarse-grain

atomistic reference system

MD or MC simulation of an all-atom representation:

atom coordinates/trajectories → distribution functions,structures

atomistic potentials → forces

Fit in the order of their relative contribution to the total forcefielda: Vstr → Vbend → Vnon-bonded → Vtors.Now we need a recipe!

aD. Reith et al., Macromolecules 34, 2335–2345 (2001).

coarse grain system

Vtot = (Vstr + Vbend + Vtors)︸ ︷︷ ︸Vbonded

+ (Vvdw + Ves)︸ ︷︷ ︸Vnon-bonded

F. Romer Coarse Graining Recipes 9 / 44

Bonded Forces

from all-atom simulation{−→f atom

}:

⇒ Force Matching

from all-atom simulation{−→r atom

}:

→ center of mass or geometrical center of the CG groups⇒ bond lengths

{rCG}

, angles{θCG}

and dihedral angels{ϕCG

}

F. Romer Coarse Graining Recipes 10 / 44

Bonded Forcesharmonic approximation

CG force field functions:

harmonic bond stretching

Vαβ(r) =kαβ

2

(r − r0

αβ

)2

and bending potential

Vαβγ(θ) =kαβγ

2

(θ − θ0

αβγ

)2

CG force field parameters:

equilibrium lengths/angels fromaveragesr0αβ = 〈rαβ〉, θ0

αβ = 〈θαβ〉force constants from standarddeviationkαβ = kBT/

⟨(r − rαβ)2

⟩,

kαβγ = kBT/⟨(θ − θαβγ)2

⟩

Non-harmonic potentials, conformational entropy?

F. Romer Coarse Graining Recipes 11 / 44

Bonded ForcesBoltzmann inversion method

Boltzmann inversion (BI) method from W. Tschop et al.8:

no restrictive functional form

conformational entropy included properly

A canonical ensemble with independent degrees of freedom q obey theBoltzmann distribution:

P(q) = Z−1 exp[−U(q)/kBT ].

If P(q) is known, one can invert and obtain:

U(q) = −kBT lnP(q).

8W. Tschop et al., Acta Polymerica 49, 61–74 (1998).F. Romer Coarse Graining Recipes 12 / 44

Bonded ForcesBoltzmann inversion method

Boltzmann inversion (BI) method procedure:

1 Gernerate data sets of CG group coordinates from all-atom NVTsimulations.

2 Build up histograms for bond lengths Hr (rαβ), bond angels Hθ(θαβγ)and torsion angles Hϕ(ϕαβγω).

3 Normalize distribution functions9:Pr (r) = Hr (r)

4πr2 , Pθ(θ) = Hθ(θ)sin θ , Pϕ(ϕ) = Hϕ(ϕ)

4 Assuming a canonical distribution and statistically independent DOFP(r , θ, ϕ) = exp[−U(r , θ, ϕ)/kBT ] = Pr (r) · Pθ(θ) · Pϕ(ϕ)interaction potentials for the CG model are given by:Uq(q) = −kBT lnPq(q) for q = r , θ, ϕ

9V. Ruhle et al., J. Chem. Theory Comput. 5, 3211–3223 (2009).F. Romer Coarse Graining Recipes 13 / 44

Non-Bonded Forces

Coarse graining methods using...Structural information:

Reverse Monte-Carlo (RMC) method

Iterative Boltzmann Inversion (IBI) method

Inverse Monte-Carlo (IMC) method

Forces:

Force Matching (FM) method

to gain non-bonded interaction potentials.

F. Romer Coarse Graining Recipes 14 / 44

Non-Bonded Forcesfrom structural information

derived from methods to determine atomistic potentials or structures.

structure factor S(q) ←Fourier→ pair distribution function g(r)10

10H. E. Fischer et al., Rep. Prog. Phys. 69, 233 (2006).F. Romer Coarse Graining Recipes 15 / 44

Radial distribution function (RDF)

radial/pair distribution/correlation function:

g(r) =1

4πr2

1

Nρ

N∑i=1

N∑j 6=i

〈δ (|rij | − r)〉

potential of mean force (PMF)11:

Uαβ(r) = −kT ln [gαβ(r)]

11J. Hansen and I. McDonald, Theory of simple liquids, 2nd ed. (Academic Press,London, 1986).

F. Romer Coarse Graining Recipes 16 / 44

Henderson Theorem

Is the pair potential derived from a RDF unique?R. L. Hendersona: “[...] The pair potential u(r) which gives rise to a givenradial distribution function g(r) is unique up to a constant.”

aR. Henderson, Physics Letters A 49, 197–198 (1974).

Gibbs-Bogoliubov inequation or Feynman-Kleinert variational principle12:F1 ≤ F2 + 〈H2 − H1〉1Consider two identical systems (g1 ≡ g2) except u1 6= u2.Assume u1 and u2 differs by more than a constant:f1 < f2 + 1

2n∫

d3r [u2(r)− u1(r)] g1(r) andf2 < f1 + 1

2n∫

d3r [u1(r)− u2(r)] g2(r).Combining these Eq. and with g1 ≡ g2 we get the contradiction 0 < 0!→ Assumption is wrong!

12R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080–5084 (1986).F. Romer Coarse Graining Recipes 17 / 44

Henderson Theorem

Is the pair potential derived from a RDF unique?R. L. Hendersona: Yes,“[...] the pair potential u(r) which gives rise to agiven radial distribution function g(r) is unique up to a constant.”

aR. Henderson, Physics Letters A 49, 197–198 (1974).

Okay, it’s unique, but does it exis?Chayes et al. have proven: Yes, if the given RDF is a two-particlereduction of any admissible N-particle probability distribution, there alwaysexists a pair potential that reproduces ita.

aJ. Chayes and L. Chayes, Journal of Statistical Physics 36, 471–488 (1984),J. Chayes et al., Communications in Mathematical Physics 93, 57–121 (1984).

F. Romer Coarse Graining Recipes 18 / 44

Reverse Monte-Carlo (RMC)method

F. Romer Coarse Graining Recipes 19 / 44

Reverse Monte-Carlo methodstructures in disordered materials

R. L. McGreevy and L. Pusztai utilized the Reverse Monte-Carlo methodto determine structures in disordered materials13:

matching RDF from experimental data gE (r) with MC data gS(r)

random initial MC configuration of N particles

MC step: random motion of one particle

acceptance criteria: comparing previous RDF gS(r) and new g ′S(r)with experimental gE (r):χ2 =

∑nri=1 (gE (ri )− gS(ri ))2 /σE

2(ri )

χ′2 =∑nr

i=1 (gE (ri )− g ′S(ri ))2 /σE2(ri )

P =

{1 if χ′2 < χ2

1√2πσ2

exp(−∆χ2

2σ2

)if χ′2 > χ2

13R. L. McGreevy and L. Pusztai, Mol. Simul. 1, 359–367 (1988).F. Romer Coarse Graining Recipes 20 / 44

Reverse Monte-Carlo methodstructures in disordered materials

Example: liquid Argon

N = 512

number of moves to converge (total/accepted) = 10697/2070

agreement of RDF χ2/nr = 0.075

Review: R. L. McGreevy, J. Phys.: Condens. Matter 13, R877 (2001)

Inherent shortcomings:

χ2 can not distinguish between one configuration with a largestatistical uncertainty but matches well the target RDF and aconfiguration with lower statistical uncertainty but misfits the peaks.

Because of constraints in the number of particles in MC ensemble andnumerical accuracy the relative uncertainty of gS(r) can become oneorder of magnitude larger than of diffraction data.

F. Romer Coarse Graining Recipes 21 / 44

Empirical Potential Monte-Carlo (EPMC) method

A. K Soper’s EPMC method14:

extentsion of the RMC (overcoming their shortcomings)

based on PMF: ψα,β(r) = −kT ln [gα,β(r)]

instead of comparing ∆χ2 a classical Markov-Chain-Monte-Carlo(MCMC) simulation is performed

EPMC is performed with potentials Uα,β(r)

Uα,β(r) can be later used in MD or MC simulation!

Input to the EPMC method:

set of target RDFs gDα,β(r)

reference pair potentials U refα,β(r)

hardcore limitationsconfigurational constraints

14A. K. Soper, Chem. Phys. 202, 295–306 (1996).F. Romer Coarse Graining Recipes 22 / 44

Empirical Potential Monte-Carlo method

The EPMC iteration procedure:

0 Set up system with correct T and ρ. Initial potentialsU0α,β(r) = U ref

α,β(r)

1 MCMC siumlation is performed → gα,β(r)

2 PMF is now used to generate a new potential energy functionUNα,β(r), as a perturbation of the initial/previous:

UNα,β(r) = U0

α,β(r) +[ψDα,β(r)− ψα,β(r)

]= U0

α,β(r) + kT ln[gα,β(r)/gD

α,β(r)]

3 update U0α,β(r)⇐ UN

α,β(r)

4 continue with step 1, until convergence:U0α,β(r) ≈ UN

α,β(r) = Uα,β(r)

F. Romer Coarse Graining Recipes 23 / 44

Empirical Potential Monte-Carlo method

Example: Water (experimental15, SPC/E16)

15A. K. Soper, J. Chem. Phys. 101, 6888–6901 (1994).16H. J. C. Berendsen et al., J. Phys. Chem. 91, 6269–6271 (1987).

F. Romer Coarse Graining Recipes 24 / 44

CG force field derived by the RMC/EPMC method

J. Elezgaray and M. Laguerre: dimyristoylphosphatidylcholine(DMPC)17:

four CG Groups (CHOL, PHOS, GLYC and CH23) plus water (3TIP)

charges: q = −1e on PHOS, q = +1e on CHOL

bonded interaction: harmonic approximation

potential update:

Un+1α,β (r) = Un

α,β(r) + ηkT ln[(

gnα,β(r) + δ

)/(g targetα,β (r) + δ

)]with η = 0.1 and δ = 10−3.

convergernce if εn < εmax with

εn = 1Npair

∑α,β,{r<rcut}

∣∣∣gnα,β(r)− g target

α,β (r)∣∣∣2

17J. Elezgaray and M. Laguerre, Comput. Phys. Commun. 175, 264 –268 (2006).F. Romer Coarse Graining Recipes 25 / 44

CG force field derived by the RMC/EPMC method

Initial potentials U refα,β(r):

all-atom NVT simulation for each CG group couple → g refαβ (r)

broken bonds were patched with hydrogen atoms

solute-solute: 10 of each CG group in water

solute-water: single CG group in water

if necessary with counter ions

all-atom water molecules were gathered in groups of three

⇒ U refα,β(r) = −kT ln

[g refαβ (r)

]

F. Romer Coarse Graining Recipes 26 / 44

CG force field derived by the RMC/EPMC method

DMPC molecule/bilayer:

Target RDFs were derived from an atomistic NPT simulation of2× 32 DMPC in a 40× 40× 70 A box filled with water.

The RMC reaches convergence (εmax = 10−2) after 20 iterations.

(a) CHOL-CHOL and (b) CHOL-3TIP. Continuous line:

data obtained with the optimized potentials. Dashed-line

data obtained from a coarse-grained version of the

reference (full-atom) simulation.

F. Romer Coarse Graining Recipes 27 / 44

Iterative Boltzmann Inversion (IBI)method

F. Romer Coarse Graining Recipes 28 / 44

Iterative Boltzmann Inversion (IBI) method

D. Reith et al. IBI method18:

natural extension of the Boltzmann inversion method19

Pq(q) = Hq(q)/4πr2 ≡ g(r)

potential update function:

Un+1 = Un + ∆Un

∆Un(r) = kBT ln

(gn(r)

gref(r)

)initial potential by PMF:

U(r) = −kBT ln (gref(r))

⇒ The IBI and the EPMC method are equivalent to each other!

18D. Reith et al., J. Comput. Chem. 24, 1624–1636 (2003).19W. Tschop et al., Acta Polymerica 49, 61–74 (1998).

F. Romer Coarse Graining Recipes 29 / 44

Inverse Monte-Carlo (IMC)method

F. Romer Coarse Graining Recipes 30 / 44

Inverse Monte-Carlo method

Lyubartsev and Laaksonen20 proposed a method to calculate effectiveinteraction potentials from the RDFs. They first called it “A reverseMonte-Carlo Approach”, but later they21 such as others (e.g.22) will referto it as inverse Monte-Carlo (IMC) method.

Inspired by the renormalization group Monte-Carlo method for phasetransition studies in the Ising model by R. H. Swendsen23, theyobserve the Hamiltonian of the system:H =

∑ij U(rij)

20A. P. Lyubartsev and A. Laaksonen, Phys. Rev. E 52, 3730–3737 (1995).21A. P. Lyubartsev et al., Soft Materials 1, 121–137 (2002).22V. Ruhle et al., J. Chem. Theory Comput. 5, 3211–3223 (2009), T. Murtola et al.,

Phys. Chem. Chem. Phys. 11, 1869–1892 (2009).23R. H. Swendsen, Phys. Rev. Lett. 42, 859–861 (1979).

F. Romer Coarse Graining Recipes 31 / 44

Inverse Monte-Carlo method

Hamiltonian of the system:

H =∑ij

U(rij) =∑α

UαSα

U(rij) = 0 if rij ≥ rcut

tabulated on a grid of M points:rα = α∆r , where α = [0, 1, ...,M], and ∆r = rcut/M

Sα is the number of all particle pairs at rij = rα:

〈Sα〉 = N(N−1)2

4πr2α∆rV g(rα)

F. Romer Coarse Graining Recipes 32 / 44

Inverse Monte-Carlo method

Number of all particle pairs at rij = rα:

〈Sα〉 =N(N − 1)

2

4πr2α∆r

Vg(rα)

Taylor→ ∆ 〈Sα〉 =∑γ

∂ 〈Sα〉∂Uγ

∆Uγ +O(∆U2)

where γ ≡ particle pair types. The derivatives can be obtained by usingthe chain rule:

A =∂ 〈Sα〉∂Uγ

=∂

∂Uγ

∫dqSα(q) exp

[−β∑

γ UγSγ(q)]

∫dq exp

[−β∑

γ UγSγ(q)]

= β (〈Sα〉 〈Sγ〉 − 〈SαSγ〉)

with β = 1/kBT and q number of degrees of freedom of the system.

F. Romer Coarse Graining Recipes 33 / 44

Inverse Monte-Carlo method

Correction term for the potentials Uγ

〈Sα〉 − S ref =∑γ

Aαγ∆Uγ

withAαγ = β (〈Sα〉 〈Sγ〉 − 〈SαSγ〉) ,

〈Sα〉 =N(N − 1)

2

4πr2α∆r

Vg(rα)

F. Romer Coarse Graining Recipes 34 / 44

Force Matching (FM) method

F. Romer Coarse Graining Recipes 35 / 44

Force Matching method

S. Izvekov’s and G. A. Voth’s FM method24:

based on F. Ercolessi and J. B. Adams FM method25:atomistic potentials ← ab initio

i = 1, ..,N atoms or CG sites

l = 1, ..., L configurations from atomistic or ab initio simulations

Frefil forces

objective function:

χ2 =1

3LN

L∑l=1

N∑i=1

∣∣∣Frefil − Fp

il(g1, ..., gM)∣∣∣2

24S. Izvekov and G. A. Voth, J. Chem. Phys. 123, 134105 (2005).25F. Ercolessi and J. B. Adams, Europhysics Letters 26, 583 (1994).

F. Romer Coarse Graining Recipes 36 / 44

Force Matching method

χ2 =1

3LN

L∑l=1

N∑i=1

∣∣∣Frefil − Fp

il(g1, ..., gM)∣∣∣2

Using cubic splines ensures a linear dependency of the force fields Fpil on its

parameters {gj} = (g1, ..., gM)26. Hence, minimization of χ2 can bewritten in a matrix notation:∥∥∥(Fp

il)′gj

∥∥∥T ∥∥∥(Fpil)′gj

∥∥∥ {gj} =∥∥∥(Fp

il)′gj

∥∥∥T Frefil

⇒ Fpil(g1, ..., gM) = Fref

il

i = [1,N], l = [1, L]

If M < N × L → overdetermined system of linear equations ⇒ solved inthe least-squares sense via QR or singular value decomposition method27.

26C. De Boor, A practical guide to splines, (Springer, New York, 1978).27C. L. Lawson and R. J. Hanson, Solving least squares problems, (Society for

Industrial and Applied Mathematics, 1995).F. Romer Coarse Graining Recipes 37 / 44

Force Matching methodImplementation

To fit pairwise central force field, the force fpi (rij) acting between particle i

and particle j is partitioned:

fpi (rij) = −

(f (rij) +

qiqjr2ij

)nij

The short ranged term f (r) is expressed by cubic splines:

f (r , {rk} , {fk} ,{f ′′k}

) =

A(r , {rk})fi + B(r , {rk})fi+1

+C (r , {rk})f ′′i + D(r , {rk})f ′′i+1

with r ∈ [ri , ri+1],

F. Romer Coarse Graining Recipes 38 / 44

Force Matching methodImplementation

Now we can express the known reference forces Frefαil for particles of species

α = [1,K ] and for a given configuration l = [1, L] in the following linearequations:

Frefαil = −

∑γ=nb,b

K∑β=1

Nβ∑j=1

(f +

qαβr2αil ,βjl

δγ,nb

)nαil ,βjl

with f = f(rαil ,βjl , {rαβ,γ,k} , {fαβ,γ,k} ,

{f ′′αβ,γ,k

})for each particle of species α : i = [1,Nα].

→ The parameters fαβ,γ,k , f′′αβ,γ,k and qαβ are subjected to the fit.

Charges qα are recovered by solving the system of nonlinear equations:qαqβ = 〈qαβ〉

F. Romer Coarse Graining Recipes 39 / 44

Force Matching methodCorrection

Why CG force fields often fail to maintain the proper internalpressure and as a result also predict wrong densities?

Pressure in MD simulations:

P =

(2

3

⟨E kin

⟩+ 〈W 〉

)/V

average kinetic energy:⟨E kin

⟩= NkBT/2

→ not conserved due to reduction of degrees of freedom N

system virial: 〈W 〉 =⟨

13

∑i<j fij · rij

⟩→ not conserved due to reduction/contraction of intramolecularcontributions.

F. Romer Coarse Graining Recipes 40 / 44

Force Matching methodCorrection

Pressure & density correction:Because

E kin ⊥⊥ fij

〈W 〉 ∼ fij

the FM force eld can be constrained by

3W atoml + 2∆E kin

l =∑

γ=nb,b

∑αβ

∑ij

(f · rαil ,βjl +

qαβrαil ,βjl

δγ,nb

)

to produce the correct pressure.

∆E kinl = E kin,atom

l − E kin,CGl ≈ E kin,atom

l

(1− NCG/Natom

)

F. Romer Coarse Graining Recipes 41 / 44

VOTCA

V. Ruhle et al., J. Chem. Theory Comput. 5, 3211–3223 (2009)http://www.votca.org

Supported methods:

BI for bonded potentials

Iterative Boltzmann Inversion

Inverse Monte Carlo

Force Matching

Supported file formats:

xtc, trr, tpr (all formats supported byGROMACS)

DLPOLY FIELD and HISTORY

LAMMPS dump files

pdb, xyz (to use with ESPResSo andESPResSo++)

F. Romer Coarse Graining Recipes 42 / 44

Conclusion

Basics on structure ⇔ pair potentials

Radial distribution function (RDF)Potential of mean force (PMF)Henderson theorem

Prominent coarse graining recipes:

Reverse Monte-Carlo (RMC) methodIterative Boltzmann Inversion (IBI) methodInverse Monte-Carlo (IMC) methodForce Matching (FM) method

I have skipped the MARTINI force field28. Why?

Because there is no straight forward recipe!

28S. J. Marrink et al., J. Phys. Chem. B 111, 7812–7824 (2007).F. Romer Coarse Graining Recipes 43 / 44

F. Romer Coarse Graining Recipes 44 / 44