From Atomistic to Coarse Grain Systems - Procedures & Methods

From Atomistic to Coarse Grain Systems –
Procedures & Methods
Frank Römer
Forschungszentrum Jülich GmbH
Institute of Complex Systems & Institute for Advanced Simulation
Theoretical Soft Matter and Biophysics (ICS-2/IAS-2)
Mulliken Center Bonn
27.11.2014
F. Römer Coarse Graining Recipes 1 / 44

Coarse Graining?
Granularity
(Redirected from Coarse grain)
“Granularity is the extent to which a system is broken down into small
parts, either the system itself or its description or observation. It is the
extent to which a larger entity is subdivided. [..] Coarse-grained systems
consist of fewer, larger components than ﬁne-grained systems; a
coarse-grained description of a system regards large subcomponents while
a ﬁne-grained description regards smaller components of which the larger
ones are composed.”a
a
Wikipedia, Granularity, http://en.wikipedia.org/wiki/Granularity,
(6.11.2014)

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

Coarse Graining
time & length scale

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

From Atomistic to CG Force Fields

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 molecule
in groups of atoms, and then
find a proper description for
each of this CG groups.
top-down
Defining several CG groups
with specific properties, and
rebuild the target molecule
using these CG groups.
7

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!

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 force
ﬁelda: Vstr → Vbend → Vnon-bonded → Vtors.
Now we need a recipe!
a
D. Reith et al., Macromolecules 34, 2335–2345 (2001).
coarse grain system
Vtot = (Vstr + Vbend + Vtors)
Vbonded
+ (Vvdw + Ves)
Vnon-bonded

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

Bonded Forces
harmonic 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 from
averages
r0
αβ = rαβ , θ0
αβ = θαβ
force constants from standard
deviation
kαβ = kBT/ (r − rαβ)2 ,
kαβγ = kBT/ (θ − θαβγ)2
Non-harmonic potentials, conformational entropy?

Bonded Forces
Boltzmann inversion method
Boltzmann inversion (BI) method from W. Tsch¨op et al.8:
no restrictive functional form
conformational entropy included properly
A canonical ensemble with independent degrees of freedom q obey the
Boltzmann distribution:
P(q) = Z−1 exp[−U(q)/kBT].
If P(q) is known, one can invert and obtain:
U(q) = −kBT ln P(q).
8
W. Tsch¨op et al., Acta Polymerica 49, 61–74 (1998).

Bonded Forces
Boltzmann inversion method
Boltzmann inversion (BI) method procedure:
1 Gernerate data sets of CG group coordinates from all-atom NVT
simulations.
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 DOF
P(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 ln Pq(q) for q = r, θ, ϕ
9
V. R¨uhle et al., J. Chem. Theory Comput. 5, 3211–3223 (2009).

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.

Non-Bonded Forces
from structural information
derived from methods to determine atomistic potentials or structures.
structure factor S(q) ←Fourier→ pair distribution function g(r)10
10
H. E. Fischer et al., Rep. Prog. Phys. 69, 233 (2006).

Radial distribution function (RDF)
radial/pair distribution/correlation function:
g(r) =
1
4πr2
1
Nρ
N
i=1
N
j=i
δ (|rij | − r)
potential of mean force (PMF)11:
Uαβ(r) = −kT ln [gαβ(r)]
11
J. Hansen and I. McDonald, Theory of simple liquids, 2nd ed. (Academic Press,
London, 1986).

Henderson Theorem
Is the pair potential derived from a RDF unique?
R. L. Hendersona: “[...] The pair potential u(r) which gives rise to a given
radial distribution function g(r) is unique up to a constant.”
a
R. Henderson, Physics Letters A 49, 197–198 (1974).
Gibbs-Bogoliubov inequation or Feynman-Kleinert variational principle12:
F1 ≤ F2 + H2 − H1 1
Consider two identical systems (g1 ≡ g2) except u1 = u2.
Assume u1 and u2 diﬀers by more than a constant:
f1 < f2 + 1
2n d3
r [u2(r) − u1(r)] g1(r) and
f2 < f1 + 1
2n d3
r [u1(r) − u2(r)] g2(r).
Combining these Eq. and with g1 ≡ g2 we get the contradiction 0 < 0!
→ Assumption is wrong!
12
R. P. Feynman and H. Kleinert, Phys. Rev. A 34, 5080–5084 (1986).

Henderson Theorem
Is the pair potential derived from a RDF unique?
R. L. Hendersona: Yes,“[...] the pair potential u(r) which gives rise to a
given radial distribution function g(r) is unique up to a constant.”
a
R. 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-particle
reduction of any admissible N-particle probability distribution, there always
exists a pair potential that reproduces ita.
a
J. Chayes and L. Chayes, Journal of Statistical Physics 36, 471–488 (1984),
J. Chayes et al., Communications in Mathematical Physics 93, 57–121 (1984).

Reverse Monte-Carlo (RMC)
method

Reverse Monte-Carlo method
structures in disordered materials
R. L. McGreevy and L. Pusztai utilized the Reverse Monte-Carlo method
to 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 gS (r)
with experimental gE (r):
χ2 = nr
i=1 (gE (ri ) − gS (ri ))2
/σE
2(ri )
χ 2 = nr
i=1 (gE (ri ) − gS (ri ))2
/σE
2(ri )
P =
1 if χ 2 < χ2
1√
2πσ2
exp −∆χ2
2σ2 if χ 2 > χ2
13
R. L. McGreevy and L. Pusztai, Mol. Simul. 1, 359–367 (1988).

Reverse Monte-Carlo method
structures 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 large
statistical uncertainty but matches well the target RDF and a
configuration with lower statistical uncertainty but misfits the peaks.
Because of constraints in the number of particles in MC ensemble and
numerical accuracy the relative uncertainty of gS (r) can become one
order of magnitude larger than of diffraction data.

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 gD
α,β(r)
reference pair potentials Uref
α,β(r)
hardcore limitations
conﬁgurational constraints
14
A. K. Soper, Chem. Phys. 202, 295–306 (1996).

Empirical Potential Monte-Carlo method
The EPMC iteration procedure:
0 Set up system with correct T and ρ. Initial potentials
U0
α,β(r) = Uref
α,β(r)
1 MCMC siumlation is performed → gα,β(r)
2 PMF is now used to generate a new potential energy function
UN
α,β(r), as a perturbation of the initial/previous:
UN
α,β(r) = U0
α,β(r) + ψD
α,β(r) − ψα,β(r)
= U0
α,β(r) + kT ln gα,β(r)/gD
α,β(r)
3 update U0
α,β(r) ⇐ UN
α,β(r)
4 continue with step 1, until convergence:
U0
α,β(r) ≈ UN
α,β(r) = Uα,β(r)

Empirical Potential Monte-Carlo method
Example: Water (experimental15, SPC/E16)
15
A. K. Soper, J. Chem. Phys. 101, 6888–6901 (1994).
16
H. J. C. Berendsen et al., J. Phys. Chem. 91, 6269–6271 (1987).

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+1
α,β (r) = Un
α,β(r) + ηkT ln gn
α,β(r) + δ / gtarget
α,β (r) + δ
with η = 0.1 and δ = 10−3.
convergernce if n < max with
n = 1
Npair α,β,{r<rcut} gn
α,β(r) − gtarget
α,β (r)
2
17

Initial potentials Uref
α,β(r):
all-atom NVT simulation for each CG group couple → gref
αβ (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
⇒ Uref
α,β(r) = −kT ln gref
αβ (r)

DMPC molecule/bilayer:
Target RDFs were derived from an atomistic NPT simulation of
2 × 32 DMPC in a 40 × 40 × 70 ˚A box ﬁlled with water.
The RMC reaches convergence ( max = 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.

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

Inverse Monte-Carlo (IMC)
method

Inverse Monte-Carlo method
Lyubartsev and Laaksonen20 proposed a method to calculate effective
interaction potentials from the RDFs. They first called it “A reverse
Monte-Carlo Approach”, but later they21 such as others (e.g.22) will refer
to it as inverse Monte-Carlo (IMC) method.
Inspired by the renormalization group Monte-Carlo method for phase
transition studies in the Ising model by R. H. Swendsen23, they
observe the Hamiltonian of the system:
H = ij U(rij )
20
A. P. Lyubartsev and A. Laaksonen, Phys. Rev. E 52, 3730–3737 (1995).
21
A. P. Lyubartsev et al., Soft Materials 1, 121–137 (2002).
22
V. Rühle et al., J. Chem. Theory Comput. 5, 3211–3223 (2009), T. Murtola et al.,
Phys. Chem. Chem. Phys. 11, 1869–1892 (2009).
23
R. H. Swendsen, Phys. Rev. Lett. 42, 859–861 (1979).

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
4πr2
α∆r
V g(rα)

Number of all particle pairs at rij = rα:
Sα =
N(N − 1)
2
4πr2
α∆r
V
g(rα)
Taylor
→ ∆ Sα =
γ
∂ Sα
∂Uγ
∆Uγ +O(∆U2
)
where γ ≡ particle pair types. The derivatives can be obtained by using
the 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.

Correction term for the potentials Uγ
Sα − Sref
=
γ
Aαγ∆Uγ
with
Aαγ = β ( Sα Sγ − SαSγ ) ,
Sα =
N(N − 1)
2
4πr2
α∆r
V
g(rα)

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
Fref
il forces
objective function:
χ2
=
1
3LN
L
l=1
N
i=1
Fref
il − Fp
il (g1, ..., gM)
2
24
S. Izvekov and G. A. Voth, J. Chem. Phys. 123, 134105 (2005).
25
F. Ercolessi and J. B. Adams, Europhysics Letters 26, 583 (1994).

χ2
=
1
3LN
L
l=1
N
i=1
Fref
il − Fp
il (g1, ..., gM)
2
Using cubic splines ensures a linear dependency of the force ﬁelds Fp
il on its
parameters {gj } = (g1, ..., gM)26. Hence, minimization of χ2 can be
written in a matrix notation:
(Fp
il )gj
T
(Fp
il )gj
{gj } = (Fp
il )gj
T
Fref
il
⇒ Fp
il (g1, ..., gM) = Fref
il
i = [1, N], l = [1, L]
If M < N × L → overdetermined system of linear equations ⇒ solved in
the least-squares sense via QR or singular value decomposition method27.
26
C. De Boor, A practical guide to splines, (Springer, New York, 1978).
27
C. L. Lawson and R. J. Hanson, Solving least squares problems, (Society for
Industrial and Applied Mathematics, 1995).

Implementation
To ﬁt pairwise central force ﬁeld, the force fp
i (rij ) acting between particle i
and particle j is partitioned:
fp
i (rij ) = − f (rij ) +
qi qj
r2
ij
nij
The short ranged term f (r) is expressed by cubic splines:
f (r, {rk} , {fk} , fk ) =
A(r, {rk})fi + B(r, {rk})fi+1
+C(r, {rk})fi + D(r, {rk})fi+1
with r ∈ [ri , ri+1],

Implementation
Now we can express the known reference forces Fref
αil for particles of species
α = [1, K] and for a given conﬁguration l = [1, L] in the following linear
equations:
Fref
αil = −
γ=nb,b
K
β=1
Nβ
j=1
f +
qαβ
r2
αil,βjl
δγ,nb nαil,βjl
with f = f rαil,βjl , {rαβ,γ,k} , {fαβ,γ,k} , fαβ,γ,k
for each particle of species α : i = [1, Nα].
→ The parameters fαβ,γ,k, fαβ,γ,k and qαβ are subjected to the ﬁt.
Charges qα are recovered by solving the system of nonlinear equations:
qαqβ = qαβ

Correction
Why CG force ﬁelds often fail to maintain the proper internal
pressure and as a result also predict wrong densities?
Pressure in MD simulations:
P =
2
3
Ekin
+ W /V
average kinetic energy: Ekin = NkBT/2
→ not conserved due to reduction of degrees of freedom N
system virial: W = 1
3 i<j fij · rij
→ not conserved due to reduction/contraction of intramolecular
contributions.

Correction
Pressure & density correction:
Because
Ekin ⊥⊥ fij
W ∼ fij
the FM force eld can be constrained by
3W atom
l + 2∆Ekin
l =
γ=nb,b αβ ij
f · rαil,βjl +
qαβ
rαil,βjl
δγ,nb
to produce the correct pressure.
∆Ekin
l = Ekin,atom
l − Ekin,CG
l ≈ Ekin,atom
l 1 − NCG
/Natom

VOTCA
V. Rühle 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 by
GROMACS)
DLPOLY FIELD and HISTORY
LAMMPS dump files
pdb, xyz (to use with ESPResSo and
ESPResSo++)

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) method
Inverse Monte-Carlo (IMC) method
I have skipped the MARTINI force ﬁeld28. Why?
Because there is no straight forward recipe!
28
S. J. Marrink et al., J. Phys. Chem. B 111, 7812–7824 (2007).

From Atomistic to Coarse Grain Systems - Procedures & Methods

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