Acurate determination of parameters for coarse grain model

Simulated systems:Simulated systems:

• Total 15 simulations using GROMACS of DPPC,

POPC, and DOPC bilayers with cholesterol at 0%,

10%, 20%, 25%, and 33% concentrations (total

200 lipids in each system).

• At 1 atm pressure (semi-isotropic). DPPC

mixtures at 323 K and DOPC, POPC mixtures at

303 K.

• PME with 1 nm real space cut off.

• VDW with 1.8 nm cut off.

• Force field parameters developed in our group.(Contact for parameters: [email protected],

[email protected])

• Simulated for 30 ns. Last 10 ns used in

averaging. One DPPC+cholesterol at 1:1 is simulated for 20 ns. Used last 5 ns for

volume computation.

mailto:[email protected]

Pure systems and comparison with experiments:Pure systems and comparison with experiments:

F(q)= ρe(z)−ρw*⎡⎣ ⎤⎦

−D/2

D/2

∫ cos(zq)dz

Experimental data from Prof. John Nagle.


• The system is divided into three regions. Water,

headgroup (HG) and hydrocarbon core.

• Component volumes (vi) are computed by the method

proposed by Feller et al.1, where the system is divided into

ns slabs and the function

is optimized.

• The hydrocarbon thickness (Dc) is calculated by Gibbs

surface method.

• The area per lipid (Al) is Vc/Dc.

G(vi )= 1− ni (zj )vii=1

3

∑⎛

⎝⎜

⎞

⎠⎟

2

zj

ns

∑

1Armen, R, Uitto, O, and Feller, S, Biophys. J., 75, 734 (1998)


DPPC POPC DOPC

MD Exp1 MD Exp2 MD Exp2

Temp. (K) 323 323 303 303 303 303

Vl (Å3) 1213 1228 1242 1256 1270 1303

Vc(Å3) 869 896 911 924 949 971

VHG (Å3) 344 333 330 331 320 331

2 Dc (Å) 27.0 27.9 27.4 27.1 27.6 26.8

DHH (Å) 36.7 37.8 36.6 37.0 37.2 36.7

Al (Å2) 64.4 64.2 66.5 68.3 68.8 72.5

Alg (Å2) 61.9 - 63.0 - 64.0 -

1 Kucerka, N. et al, Biophys. J., L83, (2006); 2 Kucerka, N. et al, J. Membrane Biol. 208, 193 (2005).


• The difference between Al and Alg is

approximately 4%, 5%, and 7.5% for DPPC, POPC

and DOPC bilayers respectively.

• We will consider Al as lipid area instead of Alg

because• It is experimentally determined

• The Alg depends on the parameters of

pressure coupling algorithm and other

simulation parameters.

Cholesterol mixtures and comparison with experiments:Cholesterol mixtures and comparison with experiments:

• Partial molecular volume

• Partial molecular area1

v(x)=Vbox(x)−nwVw

(Nl +Nc)=(1−x)Vl + xVc

a(x)= Abox(x)(Nl +Nc)

=(1−x)Al + xAc

a(x)(1−x)

=Al +x

(1−x)Ac

1Edholm, O and Nagle, J F, Biophys. J., 89, 1827 (2005)


Greenwood et. al,Chem. Phys. Lipids, 143, 1 (2006)


DPPC POPC DOPC

MD Expt1 MD Expt1 MD Expt1

Vl (Å3)

1214 1229

1239 1255 1269 1300(x < 0.25)

1197 1208

(x > 0.25)

Vchol(Å3)

594 574

608 623 607 633(x < 0.25)

645 637

(x > 0.25)

1Greenwood et. al,Chem. Phys. Lipids, 143, 1 (2006)

a(x)

(1−x)=Al +

x(1−x)

Ac

DPPC POPC DOPC

Almixture (Å2) 54.1 61.5 61.5

Acholmixture (Å2) 23.8 12.4 20.8

s

ide

sid

e

• The cholesterol packs preferentially with the -side toward Sn-1 chain of POPC and -side towards Sn-2 chain.

• Our force fields are accurately reproducing structural properties of the bilayers.

• Two sides of cholesterol play important role in structual organization in bilayers.

Coarse grain mapping

3D lipid bilayer 2D lattice field

lipid chain Lattice field si

Cholesterol Hard rod

Khelashvili, Pandit and Scott., J. Chem. Phys. (2005), 123, 034910.

si =−ntr

ns(ns −1)32cos2 m,i −

12

⎛⎝⎜

⎞⎠⎟m=1

ns

∑

Lipid chains as order parameter field

si is the order parameter of a chain.

H LIPID =− V0sls ′l<l ′l >∑

We consider average effect of the neighboring chains (Mean field approximation)

Zi = eΦisi

Allconformations

∑ Φi = V0 s jj =1

4

∑where


Lipid chains as order parameter field

We write a self-consistent equation as

si =

sceΦisc

Allconformations

∑

Zi

where the conformations of the chains are taken from a MD simulation of 1600 DPPC bilayer at 50 C (appropriately transformed).

Tune V0 to get correct phase transition temperature.


Binary mixture of DPPC and cholesterol

H =HLIPID − Vlc(rij ,θij )si − VCCr (rij )VCC

φ (φij )j=1

NChol

∑i=1

NChol

∑j=1

NChol

∑i=1

Nlip

∑

• Cholestrols are considered as rigid rods diffusing through the lattice field of the chain order parameters. • The coupling coefficients between DPPC and cholesterol are determined by fitting a straight line to cholesterol-DPPC chain interaction energy vs. si plot.

• The cholesterol positions and orientations are updated at the each step using Time dependent Ginzburg-Landau equation (TDGL) and then the lipid order parameters are updated by solving appropriate self-consistent equations.


Vlc (rij ,θij ) =Vlc 1−Δsin(θij )⎡⎣ ⎤⎦

Coarse grain dynamics


Initialize order parameter field, place cholesterol randomly, and

initialize MD chain libraries at given temperature

Solve self-consistent equation to equilibrateOrder parameter field around cholesterol positions.

Move cholesterol rods using TDGL equations


Order parameter fields after 20 s of simulation with 10000 chains and 7%, 11%, 18%, 25% and 33% cholesterols.



Pandit et al., Biophys. J. (2007), 123, 034910.

Modeling mixture of lipids (on going work)

Where C is the concentration of SM in the system.

H =− V0 (C)sisj<i, j>∑ − V1(C)siψi

i=1

Nlip

∑ − V2 (C)ψi<i, j>∑ ψ j

ψi = φi (DOPC) −φi (SM )

Composition field:

Where ’s are the volume fractions of the lipids.

V0 (C)=CV0SM +(1−C)V0

DOPC

V1(C)=C(1−C)V1 V2 (C)=C(1−C)V2

ψi ‘s are evolved using Cahn-Hilliard model

Heat capacity of ternary mixture system

Bond order interaction

1 van Duin et al , J. Phys. Chem. A, 105, 9396 (2001)

Bond order for C-C bond

BOij '(rij ) =exp a

rijr0

⎛

⎝⎜⎜

⎞

⎠⎟⎟

b⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

• Uncorrected bond order:

Where is for andbonds• The total uncorrected bond order is sum of three types of bonds• Bond order requires correction to account for the correct valency

• After correction, the bond order between a pair of atoms depends on the uncorrected bond orders of the neighbors of each atoms

• The uncorrected bond orders are stored in a tree structure for efficient access.

• The bond orders rapidly decay to zero as a function of distance so it is reasonable to construct a neighbor list for efficient computation of bond orders

Bond Order Interaction

Neighbor Lists for Bond Order

• Efficient implementation critical for performance

• Implementation based on an oct-tree decomposition of the domain

• For each particle, we traverse down to neighboring octs and collect neighboring atoms

• Has implications for parallelism (issues identical to parallelizing multipole methods)

Bond Order : Choline

Bond Order : Benzene

Other Local Energy Terms

• Other interaction terms common to classical simulations, e.g., bond energy, valence angle and torsion, are appropriately modified and contribute to non-zero bond order pairs of atoms

• These terms also become many body interactions as bond order itself depends on the neighbors and neighbor’s neighbors

• Due to variable bond structure there are other interaction terms, such as over/under coordination energy, lone pair interaction, 3 and 4 body conjugation, and three body penalty energy

Non Bonded van der Waals Interaction

• The van der Waals interactions are modeled using distance corrected Morse potential

Where R(rij) is the shielded distance given by

VvdW (rij )=Dij exp ij 1−R(rij )rvdW

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣⎢

⎤

⎦⎥−2Dij exp

12ij 1−

R(rij )rvdW

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣⎢

⎤

⎦⎥

R(rij )= rij +

1λij

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1

Electrostatics

• Shielded electrostatic interaction is used to account for orbital overlap of electrons at closer distances

• Long range electrostatics interactions are handled using the Fast Multipole Method (FMM).

VEle(rij )= fqiqj

rij3 +γij

−3⎡⎣ ⎤⎦13

Charge Equilibration (QEq) Method

• The fixed partial charge model used in classical simulations is inadequate for reacting systems.

• One must compute the partial charges on atoms at each time step using an ab-initio method.

• We compute the partial charges on atoms at each time step using a simplified approach call the Qeq method.

Charge Equilibration (QEq) Method

• Expand electrostatic energy as a Taylor series in charge around neutral charge.

• Identify the term linear in charge as electronegativity of the atom and the quadratic term as electrostatic potential and self energy.

• Using these, solve for self-term of partial derivative of electrostatic energy.

Qeq Method

We need to minimize:

subject to:

jii

ijii

iele qqHqXE ∑∑ +=21

0=∑i

iq

H ij =J iδij +1−δij

rij3 +γij

−3( )

1 3

where

Qeq Method

∑−=i

iieleiu quqEqE })({})({

0=−+−=∂∂

− ∑ jj

ijiui

qHuXEq

uXqH ijj

ij +−=∑

uXqH ~~~ +−=

Qeq Method

)1(1kk

k

iki uXHq +−=∑ −

∑∑∑∑∑ =+−= −−

i k

ik

ik

k

ik

ii uHXHq 011

From charge neutrality, we get:

∑∑∑∑

−

−

=

i kkik

ik

k

ik

H

XHu

11

1

Qeq Method

∑∑

=

ii

ii

t

su

Let

wherek

k

iki XHs ∑ −−= 1

kk

iki Ht 11∑ −−=

or ii

ikk sHX ∑=−

ii

iktH∑−=−1

Qeq Method

• Substituting back, we get:

i

ii

ii

iiii tt

ssutsq

∑∑

−=−=

We need to solve 2n equations with kernel H for si and ti.

Qeq Method

• Observations:– H is dense.

– The diagonal term is Ji

– The shielding term is short-range– Long range behavior of the kernel is 1/r

Hierarchical methods to the rescue! Multipole-accelerated matrix-vector products combined with GMRES and a preconditioner.

Hierarchical Methods

• Matrix-vector product with n x n matrix – O (n2)• Faster matrix-vector product

• Matrix-free approach• Appel’s algorithm, Barnes-Hut method

• Particle-cluster interactions – O (n lg n)• Fast Multipole method

• Cluster-cluster interactions – O (n)

• Hierarchical refinement of underlying domain• 2-D – quad-tree, 3-D – oct-tree

• Rely on decaying 1/r kernel functions• Compute approximate matrix-vector product at the

cost of accuracy

Hierarchical Methods …• Fast Multipole Method (FMM)

• Divides the domain recursively into 8 sub-domain• Up-traversal

• computes multipole coefficients to give the effects of all the points inside a node at a far-way point

• Down-traversal• computes local coefficients to get the effect of all far-away

points inside a node

• Direct interactions – for near by points• Computation complexity – O ((d+1)4n)

• d – multipole degree

Hierarchical Methods …

• Hierarchical Multipole Method (HMM)• Augmented Barnes-Hut method or variant of FMM• Up-traversal

• Same as FMM

• For each particle• Multipole-acceptance-criteria (MAC) - ratio of distance of the

particle from the center of the box to the dimension of the box

• use MAC to determine if multipole coefficients should be used to get the effect of all far-away points or not

• Direct interactions – for near by points• Computation complexity – O ((d+1)2n lg n)

Qeq: Parallel Implementation

• Key element is the parallel matrix-vector (multipole) operation

• Spatial decomposition using space-filling curves

• Domain is generally regular since domains are typically dense

• Data addressing handled by function shipping• Preconditioning via truncated kernel• GMRES never got to restart of 10!

Qeq Parallel Performance

Size Iterations P=1 P=4 P=32

43,051 7 104 32 (3.3) 4.95 (21.0)

108,092 9 316 87 (3.6) 13 (24.3)

255,391 9 718 187 (3.8) 27 (26.5)

Size corresponds to number of atoms; all times in seconds, speedups in parentheses. All runtimes on a cluster of Pentium Xeons connected over a Gb Ethernet.

Qeq Parallel Performance

Size P=1 P=4 P=16

43,051 73 21 (3.5) 3.2 (22.8)

108,092 228 61 (3.7) 9.2 (24.8)

255,391 508 132 (3.8) 19.1 (26.6)

Size corresponds to number of atoms; all times in seconds, speedups in parentheses. All runtimes on an IBM P590.

Parallel ReaX Performance

• ReaX potentials are near-field.• Primary parallel overhead is in multipole

operations.• Excellent performance obtained over

rest of the code.• Comprehensive integration and

resulting (integrated) speedups being evaluated.

Ongoing Work

• Comprehensive validation of parallel ReaX code

• System validation of code – from simple systems (small hydrocarbons) to complex molecules (larger proteins)

• Parametrization and tuning force fields.

Acurate determination of parameters for coarse grain model

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