Inverse Monte Carlo Method for Determination of Effective Potentials

for Coarse-Grained Models

Alexander Lyubartsev (sasha@physc.su.se)

Division of Physical ChemistryArrhenius Lab., Stockhom University

IPAM Workshop "Multiscale Modeling in Soft Matter and BiophysicsSeptember 26-30, 2005

Outline

1. Introduction why do we need multiscale coarse-grained modeling

2. Inverse Monte Carlo Method how to build effective potentials for coarse-grained models

3 Effective solvent-mediated potentialsion-ion and ion-DNA

4 Coarse-grained lipid modellarge-scale simulations of lipid assemblies

Why do we need coarse-grained modeling?a) polyelectrolyte problem: ions around DNA

10 20 30 40

0.1

1

(+2) MC (+2) PB (+1) MC (+1) PB (-1) MC (-1) PB

Ion

dens

ity p

rofil

e (

M/l

)

r ( Å )

Atomistic MD: not really possible to sample distances 30-40 Å from DNAPrimitive model (MC) - how good it is?

All-atom MD:

1 lipid - more than 100 atoms (DMPC -118, DPPC - 130)''minimal'' piece of bilayer: 6x6x2 = 72 lipidsadd at least 20 water molecules per lipid ⇒ about 13000 atoms(the picture above contains about 50000 atoms)

A good object to waste CPU time....

Lipid (DMPC)

b) Lipid bilayer in water

Multiscale approach

All-atomic model Full information(but limited scale)

RDFs for selected degrees of freedom

Effective potentialsfor selected sites

Effectivepotentials

Properties on a larger length/time scale

MD simulationCoarse-graining – simplified model

Reconstruct potentials(inverse Monte Carlo)

Increasescale

Simulation of coarse grained model(MD,MC,BD,DPD...)

Inverse Monte Carlo

Model Propertiesdirect

inverseInteractionpotential

Radial distributionfunctions

•Effective potentials for coarse-grained models from "lower level" simulations (atomistic coarse grained; CPMD atomistic)

•Reconstruct interaction potential from experimental RDF

•An interesting theoretical problem

The method

Consider Hamiltonian with pair interaction:

)(,

jiijrVH

Make “grid approximation”:

Hamiltonian can be rewritten as:

SVH

Where V=V(Rcut/M) - potential within -interval, S - number of particle’s pairs with distance between them within -interval

=1,…,M

Note: S is an estimator of RDF:

S

NV

rrrg

2/41

)( 22

(A.Lyubartsev and A.Laaksonen, Phys.Rev.A.,52,3730 (1995))

V

| | | | | | |Rcut

In the vicinity of an arbitrary point in the space of Hamiltonians one can write:

)( 2VOVV

SS

SSSS

qSVdq

qSVqdqS

VV

S

)(exp

)(exp)(

where

Set of V, =1,…,M Space of Hamiltonians

{V} {<S}direct

inverse

= 1/kT Nrrq

,...,1

Algorithm:Choose trial values V

(0)

Direct MC

Calculate <S> (n) and differences

<S>(n) = <S>(n) - S*

Solve linear equations system

Obtain V(n)

New potential: V(n+1) =V

(n) +V(n)

Rep

eat u

ntil

conv

erge

nce

An analogue-Newton method

S

VV* V1 V0

S*

S(V)V

S

Initial approximation:

mean force potentialV

(0) =-kTln(g*(r))

Some comments• Solution of the inverse problem is unique for pair potentials

(with exception of an additive constant)

gik(r) Vik(r)+const

• There exist a simpler scheme to correct the potential: V(n+1)(r) = V(n)(r) + kT ln(g(n)(r)/gref(r)) (A.K.Soper, Chem.Phys.Lett, 202, 295 (1996))

Its convergence is however slower and may not work in multicomponent case

• The precision of the inverse procedure can be defined by analysing eigen values and eigen vectors of the matrix

V

S

Effective solvent-mediated potentials.

Two levels of simulation of ionic, polymer or other solutions:

1) All-atom simulations (MD) with explicit water. 10000 atoms - box size ~ 40 Å

2) Continuum solvent, solutes - some effective potential, for example, ions - hard spheres interacting by Coulombic potential with suitable . Ion radius - adjustable parameter (so called "primitive electrolyte model")

The idea is to build effective solvent-mediated potential, which, maintaining simplicity of (2), takes into account molecular structure of the solvent

A. Effective solvent-mediated potentials between Na+ and Cl- ions

Reference MD simulations:

H2O flexible SPC model

(K.Toukan, A.Rahman, Phys.Rev.B31, 2643 (1985)

Na+ =2.35Å, =0.544 kJ/M

Cl- =4.4Å, =0.42 kJ/M(D.E.Smith, L.X.Dang, J.Chem.Phys., 100, 3757 (1994)

Double time step algorithm, with short time step 0.2fs and long time step 2fs, was used NPT-ensemble, T=300K, P=1atm,

Ion-ion RDFs Ion-ion effective potentials

5 100,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

NaCl, L=39Å NaCl, L=24Å NaNa, L=39Å NaNa, L=24Å ClCl, L=39Å ClCl, L=24Å

R D

F

r (Å)5 10 15

-2

0

2

4

6

NaCl, L=39Å NaCl, L=24Å NaNa, L=39Å NaNa, L=24Å ClCl, L=39Å ClCl, L=24Å Prim. model

Eff

. Pot

entia

l / k

T

r (Å)

NaCl osmotic and activity coefficientsSolvent-mediated effective potentials were applied to calculate osmotic andactivity coefficients of Na+ and Cl- ions in the whole concentration range. MC simulations are carried out for 200 ion pairs using effective potentials

Osmotic coefficient:

NkT

VP

c

cF

kTosm

T

/1

Activity coefficient:

kTex exp

Lines are calculated values and points are experimental data

1E-3 0.01 0.1 10.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3 osm. coef activity coef

Concentration (M)

B. Ion-DNA effective solvent-mediated potentials

Molecular dynamics:

• One turn of DNA (dATGCAGTCAG): 635 atoms,CHARMM force field (A.D.MacKerell, J.Wiorkiewicz-Kuchera, M.Karplus, JACS, 117, 11946 (1995))• flexible SPC water model + ions: Run 1 2 3 4 5

No. of H2O 500 500 1050 1050 500

Counterions 20 Li+ 20 Na+ 30 Na+ 30 K+ 20 Cs+

Coions - - 10 Cl- 10 Cl- -

Simulation time(ns)

2.5 2 2.5 2.5 1.5

All-atom model: Coarse-grained model

Na+

Ion - DNA effective potentials

2 4 6 8 10 12 14 16

-6

-4

-2

0

2

4

6 Li+

Na+

Na+*

K+

Cs+

Eff.

po

ten

tial /

kT

r (Å)

2 4 6 8 10 12 14 16

0

2

4

6 Li

+

Na+

Na+*

K+*

Cs+

Eff.

pot

entia

l / k

T

r (Å)2 4 6 8 10 12 14 16

-2

0

2

4

6 Li

+

Na+

Na+*

K+*

Cs+

Eff.

pot

entia

l / k

T

r (Å)

Ion - P Ion - C4 (base) Ion - C4’(sugar)

MC simulation: a bigger DNA fragment (3 turns) in a box 100x100x102Å, ions interacting by effective solvent-mediated potentials; no explicit water.These are results for the density profile and integral charge

0 10 20 30 40 500.0

0.2

0.4

0.6

0.8

1.0

Li+

Na+

Na+*

K+*

Cs+

PBInte

gra

l ch

arg

e

r (Å)

0 10 20 30 40 50

0.01

0.1

1

Li+

Na+

NA+*

K+*

Cs+

PB

De

nsi

ty p

rofil

e (

M/l)

r (Å)

Relative binding affinities of ions

The order of relative binding affinities of alkali counterions to DNA, defined by MC simulation with effective potentials, is:

Cs+ > Li+ > Na+ > K+

The binding order was defined also in a number of experimental works:

•P.D.Ross, R.L.Scruggs, Biopolymers, 2, 89 (1964) ; Electrophoresis: Li+>Na+>K+

•U.P.Strauss, C.Helfgott, H.Pink, J.Phys.Chem.,71,2550 (1967); Donnan equilibrium: Li+>Na+>K+

•S.Hallon et al, Biochemistry, 14, 1648 (1975); Circular dichroism Cs+>Li+>K+>Na+

•P.Anderson, W.Bauer, Biochemistry, 17, 594 (1978), DNA supercoiling Cs+>Li+>K+>Na+

•M.L.Bleam, C.F.Anderson, M.T.Record, Proc. Natl.Acad.Sci USA,77,3085 (1980), NMR:Cs+>Li+>K+>Na+

•I.A.Kuznetsov et al, Reactive Polymers, 3, 37 (1984), Ion exchange Li+>K+Na+

Qualitative agreement with results of experiments of very different nature.

Coarse-grained lipid model

All-atom model118 atoms

Coarse-grained model10 sites

We need interaction potential for the coarse-grained model !

Use IMC and RDFs from atomistic MD.

All-atomic molecular dynamics

All-atomic MD simulation was carried out:

● 16 lipid molecules (DMPC) dissolved in 1600 waters (6688 atoms)Box size: 40x40x40 Å

● Initial state - randomly dissolvedRDFs calculated during 12 ns after 2 ns equilibration

● Force field: CHARMM 27, water - flexible SPC

● T=313 K

MD snapshot

16 DMPC lipids1600 H2O

5 10 15 200

1

2

3

4

5

6

7solid - 16 lipidsdashed - 64 lipids

NN

PP

NP

R D

F

r ( A )5 10 15 20

0

1

2

3

4

5

6

N C

P CC C R

D F

r ( A )

R D F calculations

P

N

CO

C

4 different groups -> 10 pairs10 RDFs and eff. intermolecular potentials+ 4 bond potentials

Inverse MC simulations:

Purpose: find effective potentials which, for the coarse grained model, reproduce the same RDFs as the all-atomic model

Intramolecular potentials: Bonded: from distance distribution between the atoms.Non-bonded - the same as intermolecular

Total: 14 effective potentials

Inverse MC - the box of the same size; the same number of lipids as inthe corresponding MD; no solvent: charges +1 and -1 on "N" and "P" +dielectric constant =70 (best fit to NN, NP and PP -potentials)

Effective potentials:

Bond potentials

0 5 10 15 20-10

0

10

20

P P

N P

N N

Ve

ff (k

J/M

)

r ( A )5 10 15 20

-5

0

5

10

15

20

N - CO N - CH P - CO P - CH

Veff (k

J/M

)

r (A)

5 10 15 20-10

0

10

20

CO - CO CO - CH CH - CH

Veff (

kJ/M

)

r ( A )3 4 5 6 7 8

0

10

20

30 N - P P - CO CO - CH CH - CH

Veff (

kJ/M

)

r ( A )

Coarse-grained simulations● Monte Carlo

● Molecular Dynamics Forces - from the potentials difference in the neigbouring grid pointsSolvent is not present explicitly - MD may be considered only as another

way to generate canonical ensembleTime step 10-14 s + thermostat● Nose-Hoover ● Local (Lowe-Andersen)● Langevine

3 cases : a periodic bilayera finite piece of bilayerrandom initial state

Equivalent all-atom simulations would correspond ~ 106-108 atoms

Infinite bilayer

Coarse-grained MC (392 lipids) All-atom MD (98 lipids)

Z

Periodic box

Density distribution

A sheet of bilayer

The same initial state, but in a large simulation box:

End of simulation: 109 - 1010 MC steps:

View from the side View from the top

(discoid shape)

Vesicle formationStart from a square plain piece of membrane, 325x325 Å, 3592 lipids:

cut plane

Membrane self-assemblyMD simulation of 392 CG lipids with Lowe-Andersen thermostat

http://www.fos.su.se/physical/sasha/lipids

Conclusions1. The multiscale approach based on the inversion of radial distribution functions provides a straightforward way to build effective potentialsfor coarse-grained models

2. Examples of ionic solutions, ion-DNA interactions, lipid membranes show that effective potentials, derived exclusively from the atomistic model, provide realistic description for the coarse-grained model

3. Coarse-grained effective potentials may be plugged in into MC, MD, Brownian dynamics, DPD and used for simulation on larger length- and time scale

Acknowledgements

Aatto Laaksonen Martin Dahlberg Carl-Johan Högberg