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A consistent multiscale bridgeconnecting atomistic and

coarse-grained models

W. G. Noid

Department of ChemistryPenn State University

AcknowledgementsProf. Gregory A. VothDr.s Yanting Wang, Pu Liu, and Vinod Krishna

Prof.s Gary S. Ayton and Jhih-Wei Chu

Prof. H.C. Andersen

Funding: NIH Ruth L. Kirschstein NRSA postdoctoral fellowship

Collaborative Research in Chemistry grant (CHE-0628257)

Dr. Buddhadev Maiti

Daryl Mains, Wayne Mullinax

“Here we tackle the [protein folding] problem differently. First, we simplify the representation of a protein by averaging over fine details. This is done both to make the calculations much more efficient and also to avoid having to distinguish between many conformations that differ only in these finer details. Second, we simulate the folding of this simple structure …”

“Our method … is based on two assumptions: (1) that much of the protein’s fine structure can be eliminated by averaging, and (2) that the overall chain folding can be obtained by considering only the most effective variables (those that vary most slowly yet cause the greatest changes in conformation).”

Inspiration

Outline0. Motivation

1. The CG Mapping

2. Consistent CG Models

3. Multiscale Coarse-graining(MS-CG) Variational Principle

4. Many-body effects

5. Numerical Illustrations

6. Transferability

Noid Group

Voth Group

CG MappingAtomistic CG

MappingOperator

The mapping operator transforms an atomistic configuration onto a CG configuration by defining the coordinates of each site as a linear combinationof the coordinates defining each site.

MRN : rn a RN =M R

N(rn)

R I =M RI (rn)= cIiri

i∑

rn RN

R I

Consistent CG ModelsAtomistic Configuration Space CG Configuration Space

For a consistent CG model that reproduces the distribution of structures generated by the atomistic model, the appropriate CG potential is a many-body PMF.

Consistency

Noid, Chu, …, Voth, AndersenJ Chem Phys (2008)

pr (rn )∝ e−u(rn ) kBT PR (R

N )∝ e−U (RN ) kBT

PR (RN )=pR (R

N )

e−U (RN ) kBT ∝ drnpr(rn)d M R

N(rn)−RN( )∫

rn RN

MRN

Mean Force Field

In a consistent model, the CG force field is the conditioned expectation value of the atomistic force field (I.e., the mean force field).

RN

rN

r1n

r2n

r3n

R1N = MR

N (r1n) = MR

N (r2n) = L

fI (r1n)

fI (r2n)

fI (r3n)

FI (R1N )

R1N

FI (RN )=

−∂U(RN )∂R I

=fI (r

n )d M RN(rn)−RN( )

d M RN(rn)−RN( )

=E fI (rn) M R

N(rn)=RN⎡⎣ ⎤⎦

fI (rn )= fi(r

n)i∈I∑

Atomistic FF:

MS-CG Variational Principle for the PMF

c 2 ′F[ ] =13N

′FI MRN (rn )( ) − fI (r

n )2

I =1

N

∑= χ 2 F[ ] + ′F − F 2

c F[ ] = F − fχ ′F[ ] = ′F − f

The MS-CG variational principle determines the many-body PMF through a geometric optimization problem in the space of CG force fields.

Noid, Chu, …, Voth, AndersenJ Chem Phys (2008)

′F

F

f

Space of force fields of r n (atomistic)

Space of force fields of R N (CG)

c F[ ]c ′F[ ]

′F − F

Linear Least Squares ProblemFMS (RN )= FDG D (R

N )D∑

FMS (RN )

FDGD (R

N )

MS-CG FF

FF parameters

FF basis fcns

c 2 F( ) =13N

FDG I ,D (MRN (rn )) − fI (r

n )D∑

2

I =1

N

∑

′F

F

Space of CG FF

FMS

GD

G ′D

Space of CG FF spanned by GD{ }

The MS-CG variational principle determines by projecting the PMF onto the space of CG force fields spanned by the given basis.

FD

Simple Liquid AnalogFIJ =R̂ IJF(RIJ)

= FD R̂ IJd(RIJ −RD )D∑

FI = FDG I ,D (RN )

D∑

G I ,D = R̂ IJd(RIJ −RD )J∑

R [nm]

RD

FD

F(R)Pair force

RD

R ¢D

R̂IJ

GI , ¢D

GI ,D

R̂IK

I

J ¢J

K

¢K

For a simple liquid with central pair potentials, the basis function describes the distribution of sites at a given distance, , around a given site. RD

GI ,D

Role of Three-Body Correlations

GD ⋅f=G D ⋅F= G D ⋅G ′D( )F ′D′D

∑Projecting the atomistic FF onto the CG basis

− ′w (R) = F(R) + d ′R F( ′R )ρ [3]( ′R ;R)∫pair MF

CG pair force

direct indirect

conditioned 3-particle density

Projecting the atomistic force field onto a central pair potential results in the Yvon-Born-Green equation.

Yvon-Born-Green Equation

R ¢R

J K

I

Noid, Chu, Ayton, Voth. J Phys Chem B (2007)

Role of 3-Body Correlations II− ′w (R) = F(R) + d ′R F( ′R ) cosθ ( ′R ;R)∫

cosθ( ′R ;R)

F− ′w

R ¢R

J K

I

θ( ′R ;R)

The optimal approximation to the many-body PMF is obtained by treating the non-orthogonality (3-particle correlations) among the basis functions.

Noid, Chu, Ayton, Voth. J Phys Chem B (2007)

Results - MethanolCG pair potential CG pair structure

Calculations by Pu LiuNoid, Liu, Wang, …, Andersen, Voth. J Chem Phys (2008)

The approximate decomposition of the many-body PMF into central pair potentials quantitatively reproduces the pair distribution function.

Consistent Momenta Distribution

Restrictions upon the model:

1. CG Mapping: No atom can be involved in more than one site.

2. Site Masses:

In order to generate the correct momenta distribution, the mass of the CG site is equal to the net mass of the associated atoms only for the center-of-mass mapping.

Calculation by Pu Liu.Noid, Liu, Wang, …, Andersen, Voth. J Chem Phys (2008)

M I =cIi

2

m ii∑⎛⎝⎜

⎞⎠⎟

−1

EMIM+ cation

NO3- anion

Results - Ionic Liquids1-ethyl-3-methylimidazolium nitrate (EMIM+/NO3

-) ionic liquid (IL) pair

CG Potential

The many-body PMF is approximated with a CG potential including both bonded and pair-additive non-bonded terms.

Calculations by Yanting WangNoid, Liu, Wang, …, Andersen, Voth. J Chem Phys (2008)

U(RN )≈ U IJnb(RIJ)+

QIQJ

4pe0RIJ

⎡⎣⎢

⎤⎦⎥I >J

pairs

∑

+ U ib(di)+

i

bonds

∑ U iθ (θi)+

i

angles

∑ U iψ (ψ i)

i

dihedrals

∑

Intramolecular CG Interactions

EMIM+ cationBond-Stretch

Bond-Angle

A-C

A-C-E

B-A-C-E

Dihedral-Angle

Calculations by Yanting WangNoid, Liu, Wang, …, Andersen, Voth. J Chem Phys (2008)

Intramolecular CG bond-stretch and bond-angle interactions are remarkably well fit by harmonic forms.

Intermolecular CG InteractionsShort-ranged Nonbonded

A-A A-D

D-D E-E

EMIM+ cation

NO3- anion

Calculations by Yanting WangNoid, Liu, Wang, …, Andersen, Voth. J Chem Phys (2008)

CG pair potentials are primarily repulsive but not Lennard-Jones in form.

Bond-Stretch

Bond-Angle

Dihedral-Angle

A-C

A-C-E

B-A-C-E

EMIM+ cation

Intramolecular CG Distributions

CG

Atomistic

exp[−U iψ (ψ i) / kBT]

Calculations by Yanting WangNoid, Liu, Wang, …, Andersen, Voth. J Chem Phys (2008)

While the bond-stretch and bond-angle distributions are nearly quantitatively reproduced, the bond dihedral angle distribution is not well reproduced.

EMIM+ cation

NO3- anion

Intermolecular CG Distributions

CGAtomistic

Calculations by Yanting WangNoid, Liu, Wang, …, Andersen, Voth. J Chem Phys (2008)

A-A A-D

D-D E-E

Nonbonded pair distributions in the IL system are reproduced with reasonable accuracy.

Temperature Transferability

%cT2 F(RN ;T )⎡⎣ ⎤⎦=

13N

FI (RN ;T ) − λ (rn;T , ′T )fI (r

n )2

I =1

N

∑′T

Krishna, Noid, Voth (In preparation)

F(R)

R

dF(R)

Summary• CG models “liberate” us from the shackles of atomistic MD, but may be misleading

unless they are consistent with the underlying physics.

• The MS-CG variational principle, in principle, determines the many-body PMF, which

is the appropriate potential for a consistent CG model.

• When implemented with a finite basis set, the MS-CG variational principle

determines an optimal approximation to the many-body PMF.

• Many-body correlations must be considered when determining an optimal pair

decomposition of the many-body PMF.

• The MS-CG method systematically incorporates such correlations in determining a

molecular CG force field.

• A generalized MS-CG variational principle incorporates temperature transferrability.