A Level-Set Variational Implicit Solvent
Approach for Nonpolar Molecules
Li-Tien Cheng and Bo Li
Mathematics, UCSD
Joachim Dzubiella
Physics, Technical University Munich, Germany
John Che
Genomics Institute of Novartis Research Foundation
J. Andrew McCammon
Chemistry and Biochemistry, Pharmacology, UCSD
Support from NSF, DFG, Sloan, NIH, HHMI, CTBP, etc.
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OUTLINE
1.
2.
3.
4.
5.
Introduction
Variational Implicit Solvent Models
The Level-Set Method
Results of Level-Set Calculations
Conclusions and Outlook
2
1. Introduction
3
Molecules
Bond, angle, torsion
Polarity, interaction
Complex structure
Bohr’s model
Ball and stick model
(http://ghs.gresham.k12.or.us/)
Like-charge attraction of (-) actin protein rods
mediated by (+) barium ions. Wong et al., UIUC.
4
Biomolecules (proteins, nucleic acids, lipids, ...)
U. Conn
UCSB
Kyushu Inst. of
Tech., Japan
UCSB
5
solvent
Solvation
solute
Attraction and association of molecules of a solvent
(e.g., water) with molecules or ions of a solute
Molecular surface: geometry
Solvent and solute: polar and nonpolar determined
by dielectrics (water 80, ethanol 24, hexane 2)
Stable solvent-solute interaction: small and large
scales, hydrophobicity, etc.
Energy landscape – structure, function, dynamics, etc.
6
This work:
understand solvent-solute interaction
Solvation free energy of biomolecules
Implicit solvent models
Numerical methods for equilibriums
Efficient calculations of electrostatics
D. Cooper, U. Virginia
Water inside a protein. Imai
et al., J. Am. Chem. Soc.
7
2. Variational Implicit Solvent Models
8
Implicit vs. explicit
Explicit: treat each atom individually
First principle
Accurate
Small systems
solvent
solute
Implicit: coarse-grained
Approximation
Efficient
solvent
solute
9
Solvent accessible surface area (SASA) approach
surface area + Poisson-Boltzmann Æ free energy
Richards-Lee surface = level-set of distance function
10
A variational implicit solvent model coupling
polar and nonpolar solvation free energies
(Dzubiella, Swanson, & McCammon, 2006)
Γ
x i qi
solvent
Ω
solute
solute region
Ω
Γ = ∂ Ω solute-solvent dielectric boundary
xi
qi
ρj
qj
ρw
center of the i-th solute atom
fixed point charge at the i-th solute particle
bulk density of the j-th ion species
bulk charge of the j-th ion species
solvent density
11
Free energy of solvation
G[Γ] = Pvol(Ω) + ∫ γdS + ρw ∑ ∫ U LJ (| x − xi |)dV
Γ
Ω
c
i
[
]
+ 12 ∑ qi (ψ −ψ ref )(xi ) +kBT ∑ ρ j ∫ c exp(−q jψ / kBT ) −1 dx
i
j
Ω
Pvol (Ω) : Creation of a cavity in the solvent
P = pressure difference
∫ γdS :
Γ
Molecule rearrangement near the interface
γ = surface energy density
γ = γ 0 (1− 2δH ) (Tolman 1949)
δ = Tolman length
H = mean curvature
12
Geometrical part of the free energy
(
Pvol (Ω) + γ 0 area(Γ) − 2δ ∫ HdS + cK ∫ K dS
Γ
Γ
)
Hadwiger’s Theorem
Let K = the set of all convex bodies,
M = the set of finite union of convex bodies.
If F : M → R is
rotational and translational invariant,
additive:
F (U ∪ V ) = F (U ) + F (V ) − F (U ∩ V ) ∀U ,V ∈ M ,
and conditionally continuous:
U j ,U ∈ K ,U j → U ⇒ F (U j ) → F (U ),
then
F(U) = aVol(U) + bArea(∂U) + c∫ HdS+ d ∫ KdS ∀U ∈ M.
∂U
∂U
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ρw ∑ ∫ U LJ (| x − xi |)dV : Non-electrostatic, van der Waals
i
Ωc
ρw =
type, solute-solvent interaction
solvent density
U LJ ( r ) = 4ε 0
[ ( r ) −( r ) ]
σ 12
σ 6
Lennard-Jones potential
U LJ
O
σ
r
− ε0
14
1
2
∑ q (ψ −ψ
i
i
ref
[
]
)( xi ) +k BT ∑ ρ j ∫ c exp(−q jψ / k BT ) − 1 dx :
j
Ω
Electrostatic energy
The Poisson-Boltzmann equation
∇ ⋅ (ε∇ψ ) + 4πχΩ ∑ ρ j q j exp(−q jψ / k BT ) = −4π ∑ qiδ ( x − xi )
c
ψ = electrostatic potential
ψ ref = reference electrostatic potential
∑i qiδ ( x − xi ) = point charge distribution of solute
j
∑
j
i
ρ j q j exp(−q jψ / k BT ) = ionic charge distribution
χΩ =
c
characteristic function of Ω c
∇ ⋅ (ε solute∇ψ ref ) = −4π ∑i qiδ ( x − xi )
ε = dielectrics =
{
ε solute
ε solvent
in solute region Ω
c
in solvent region Ω
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Remarks
min{Pvol (Ω) + γ 0 area(Γ)}
The Laplace-Young relation: P (= Pl − Ps ) = 2γ 0 H
min
Γ
∫
Ωc
U LJ (| x − x1 |) dV
Ω = B ( x1 , σ )
1
4π
ULJ
σ
r
O
−ε0
Minimize
⎞
⎛ε
2
∫ ⎜⎜⎝ 2 | ∇ψ | + χΩc kBT ∑j ρ j exp(−q jψ / kBT ) ⎟⎟⎠dx − ∑i qiψ ( xi )
The Poisson-Boltzmann equation
∇ ⋅ (ε∇ψ ) + 4πχΩ ∑ ρ j q j exp(−q jψ / k BT ) = −4π ∑ qiδ ( x − xi )
c
j
i
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3. The Level-Set Method
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n
The level-set method
x
Interface motion
Vn = Vn (x, t)
for
x ∈ Γ(t )
Level-set representation
Γ(t ) = {x ∈ Ω : ϕ ( x, t ) = 0}
The level-set equation
ϕt + Vn | ∇ϕ |= 0
[
Γ(t )
z = ϕ ( x, t )
z=0
Γ (t )
]
ϕt + ∇ϕ ⋅ x& = 0
ϕ ( x(t ), t ) = 0
∇ϕ
∇ϕ ⋅ x& = (|∇ϕ| ⋅ x& ) | ∇ϕ |= (n ⋅ x&) | ∇ϕ |= Vn | ∇ϕ |
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Examples of normal velocity
Geometrically based motion
Motion by mean curvature
Vn = H
Motion by the surface Laplacian of mean curvature
Vn = − ∆ s H
External field
ut −∆u = 0
⎧u=H
⎨ ∂∂un = 0
⎩V =[ ]
n
∂u
∂n
in Ω− ∪Ω+
on Γ
on ∂Ω
on Γ
n
Ω−
Γ
Ω+
n
∂Ω
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Level-set formulas of geometrical quantities
Normal
n=
∇ϕ
|∇ϕ |
Mean curvature
1
H = ∇⋅n
2
Gaussian curvature
K = n ⋅ adj ( He (ϕ )) n
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Topological changes
Merging
Break-up
Disappearing
Nucleation?
Accuracy issues
Interface approximation
Conservation of mass
Rigorous analysis
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Application to solvation of nonpolar molecules
(Cheng, Dzubiella, McCammon, & Li, J. Chem. Phys., 2007,
sbumitted)
Free-energy
G[Γ] = Pvol(Ω) + ∫ γdS + ρw ∑ ∫ c U LJ (| x − xi |)dV
Γ
i
Ω
First variation
δG [ Γ ] = P + 2γ 0 [ H ( x ) − δK ( x )] − ρ w ∑ U LJ (| x − xi |)
δ ∫ dV = 1
i
Ω
δ ∫ dS = −2 H
Γ
δ ∫ HdS = − K
Γ
Normal velocity
V n = − δ G [Γ ]
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Algorithm
Step 1. Input parameters and initialize level-set function
Step 2. Calculate the normal and curvatures
Step 3. Calculate and extend the normal velocity
Step 4. Solve the level-set equation
Step 5. Reinitialize the level-set function
Step 6. Set
t := t + ∆t
and go to Step 2
New level-set techniques
Regularization of degenerated geometrically based
motion of interface.
A two grid method for calculating the interaction
energy with a Lennard-Jones potential.
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4. Results of Level-Set Calculations
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Numerical calculations
Units:
o
Energy = k BT , Length = A
Example 1. Two xenon atoms
P = 0, γ 0 = 0.174, δ = 1
ρ w = 0.033, σ = 3 .57 ,
ε 0= 0.431
Example 2. Two paraffin plates
P = 0, γ 0 = 0.174, δ = 0.9
ρ w = 0.033, σ = 3 .538 , ε 0= 0.2654
Example 3. Two alkane helical chains
P = 0, γ 0 = 0.176, δ = 1.3
ρ w = 0.033, σ = 3 .538 , ε 0= 0.2654
Example 4. A C60 molecule – buckyball
Example 5. A large molecule of 800 atoms.
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Example 1. Two xenon atoms
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2
0
-1
W(d)/kBT
w(d)/kBT
1
-2
1
0
-1
3 4 5 6 7 8 9 10 11
2
3
4
5
6
7
8
d/Å
9
10
11
12
Comparison of PMF by the level-set (circles) and by
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molecular dynamics (solid line) calculations.
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Example 2. Two paraffin plates
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50
effective interaction energy
0
−50
−100
−150
−200
MD simulation
level−set calculation
−250
2
4
6
8
10
12
14
16
18
distance between the plates
Comparison of the level-set and molecular dynamics
calculations (Koishi et al., 2004 and 2005) for the two
paraffin plates.
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Example 3. Two helical alkane chains
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A Richards-Lee type surface
Level-set calculation
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0.3
0.3
0.2
0.2
0.1
0.1
0
0
−0.1
−0.1
−0.2
−0.2
Mean curvature field
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Example 4. A C60 molecule – the buckyball
C60.mpg
34
0.2
0.1
0
Mean curvature field
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Example 5. A large biomolecule of 800 atoms
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5. Conclusions and Outlook
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Accomplishments
Examined and improved a class of variational
implicit solvent models.
Developed a level-set method for solvation of
nonpolar molecules.
Level-set calculations captured dewetting, local
minima, etc., and agree well with MD simulations.
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Current and future work
Revisit the Tolman length and surface tension
Coupling the Poisson-Boltzmann calculations with
the level-set relaxation
energy variations
numerical methods for elliptic interface
adaptive finite-element level-set method
Improvement of generalized Born models
analysis of geometrical effect
fast multipole summation
39
Diffuse-interface modeling
curvature effect
sharp-interface limit
Dynamics
efficient force calculations
stochastic effect
multiscale modeling
40