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],
• 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.
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.
Pure systems and comparison with experiments:Pure systems and comparison with experiments:
• 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)
Pure systems and comparison with experiments:Pure systems and comparison with experiments:
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).
Pure systems and comparison with experiments:Pure systems and comparison with experiments:
• 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)
Cholesterol mixtures and comparison with experiments:Cholesterol mixtures and comparison with experiments:
Greenwood et. al,Chem. Phys. Lipids, 143, 1 (2006)
Cholesterol mixtures and comparison with experiments:Cholesterol mixtures and comparison with experiments:
Cholesterol mixtures and comparison with experiments:Cholesterol mixtures and comparison with experiments:
Greenwood et. al,Chem. Phys. Lipids, 143, 1 (2006)
Cholesterol mixtures and comparison with experiments:Cholesterol mixtures and comparison with experiments:
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
Khelashvili, Pandit and Scott., J. Chem. Phys. (2005), 123, 034910.
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.
Khelashvili, Pandit and Scott., J. Chem. Phys. (2005), 123, 034910.
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.
Khelashvili, Pandit and Scott., J. Chem. Phys. (2005), 123, 034910.
Vlc (rij ,θij ) =Vlc 1−Δsin(θij )⎡⎣ ⎤⎦
Coarse grain dynamics
Khelashvili, Pandit and Scott., J. Chem. Phys. (2005), 123, 034910.
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
Binary mixture of DPPC and cholesterol
Order parameter fields after 20 s of simulation with 10000 chains and 7%, 11%, 18%, 25% and 33% cholesterols.
Khelashvili, Pandit and Scott., J. Chem. Phys. (2005), 123, 034910.
Binary mixture of DPPC and cholesterol
Pandit et al., Biophys. J. (2007), 123, 034910.
Binary mixture of DPPC and cholesterol
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.