Variational Implicit Solvation:
Empowering Mathematics and
Computation to Understand
Biological Building Blocks
Bo Li
Department of Mathematics and
NSF Center for Theoretical Biological Physics
UC San Diego
Funding: NIH, NSF, DOE, CTBP
UC Irvine, October 25, 2012
MBB (Math &
Biochem-Biophys)
group
Li-Tien Cheng (UCSD)
Zhongming Wang (Florida Intern’l Univ.)
Tony Kwan (UCSD)
Yanxiang Zhao (UCSD)
Shenggao Zhou (Zhejiang Univ. & UCSD)
Tim Banham (UCSD)
Maryann Hohn (UCSD)
Jiayi Wen (UCSD)
Michael White (UCSD)
Yang Xie (Georgia Tech)
Hsiao-Bing Cheng (UCLA)
Rishu Saxena (UCSD/MSU)
Collaborators
J. Andrew McCammon (UCSD)
Joachim Dzubiella (Humboldt Univ.)
Piotr Setny (Munich & Warsaw)
Jianwei Che (GNF)
Zuojun Guo (GNF)
Xiaoliang Cheng (Zhejiang Univ.)
Zhengfang Zhang (Zhejiang Univ.)
Zhenli Xu (Shanghai Jiaotong Univ.)
2
OUTLINE
1.  Biomolecules: What and Why?
2.  Variational Implicit-Solvent Models
3.  Dielectric Boundary Force
3.1 The Poisson-Boltzmann Theory
3.2 The Coulomb-Field and Yukawa-Field
Approximations
4.  Computation by the Level-Set Method
5.  Move Forward: Solvent Fluctuations
6.  Conclusions
3
1. Biomolecules: What and
Why?
4
Biomolecules
5
Protein structures
We have more than 100,000
proteins in our bodies.
Each protein is produced from a
set of only 20 building blocks.
Protein functions
Antibodies, enzymes, contractile,
structural, storage, transport, etc.
To function, proteins fold
into three-dimensional
compact structures.
Wiki
6
Protein Folding
Misfolding diseases
Alzheimer’s, Parkinson’s, etc.
Levinthal’s paradox
If a protein with 100 amino acids can try out 10^13
configurations per second, then it would take 10^27 years
to sample all the configurations. But proteins fold in seconds.
Free-energy landscape
Averagely, more than 10^100 local minima for a protein
with 100 amino acids each of which has 10 configurations.
7
water
Solvation
water
solvation
solute
ΔG = ?
conformational
change
solute
water
solute
binding
receptor ligand
€
protein folding
molecular recognition
8
2. Variational Implicit-Solvent
Models
9
Explicit vs. Implicit
solvent
solute
Molecular dynamics
(MD) simulations
solvent
solute
Statistical mechanics
10
What to model with an implicit solvent?
!   Solute-solvent interfacial property
γ0
Curvature effect
Symbols: MD.
R
γ = γ 0 (1− 2τH )
γ 0 = 73mJ /m 2

τ = 0.9 A
€
€
€
€€
the Tolman length
H : mean curvature
τ:
Huang et al., JCPB, 2001.
11
!   Excluded volume and van der Waals dispersion
The Lennard-Jones (LJ) potential
Solute
U LJ (r) = 4ε
[(
σ 12
r
)
−(
)]
σ 6
r
Fermi repulsion vdW attraction
Water
€
O
r
σ
−ε
solvent
!   Electrostatic interactions
Poisson’s equation
€
∇ ⋅ εε0∇ψ = −ρ
solute
ε =1
ε = 80
12
Commonly used implicit-solvent models
Surface energy PB/GB calcula1ons PB = Poisson-Boltzmann
GB = Generalized Born
solvent excluded surface (SES)
probing ball
vdW surface
Possible issues
!   Hydrophobic cavities
!   Curvature correction
solvent accessible surface (SAS)
!   Decoupling of polar and nonpolar contributions
13
Koishi et al., PRL, 2004.
Liu et al., Nature, 2005.
Sotomayor et al., Biophys. J. 2007
14
Variational Implicit-Solvent Model (VISM)
Dzubiella, Swanson, & McCammon: Phys. Rev. Lett. 96, 087802 (2006)
J. Chem. Phys. 124, 084905 (2006)
Free-energy functional
G[Γ] = Pvol(Ωm ) + γ 0 ∫ (1− 2τH)dS
Γ
 Qi
ri
Ωw
Ωm
Γ
 
€
+ ρ w ∫ ∑U LJ ,i (| r − ri |)dV + G€
elec [Γ]
τ:
Ωw
i
€
c ∞j , q j ,
ρw
the Tolman length, a fitting parameter
€
Gelec [Γ] : electrostatic free energy€
€
€
!   The Poisson-Boltzmann (PB) theory
!   The Coulomb-field or Yukawa-field approximation
€
15
Geometrical part:
Pvol(Ωm ) + γ 0 area(Γ) − 2γ 0τ ∫ H dS
Γ
(+c ∫
K
Γ
K dS
)
Hadwiger’s Theorem
Let C = 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 ,
!   conditionally continuous:
U j ,U ∈ C ,U j → U ⇒ F (U j ) → F (U ),
then
F (U ) = aVol (U ) + bArea(∂U ) + c ∫ HdS + d ∫ KdS ∀U ∈ M .
∂U
∂U
Application to nonpolar solvation
Roth, Harano, & Kinoshita, PRL, 2006.
Harano, Roth, & Kinoshita, Chem. Phys. Lett., 2006.
16
Coupling solute molecular mechanics
with implicit solvent
 
 
  
V[ r1,..., rN ] =∑W bond ( ri , rj ) + ∑W bend ( ri , rj , rk )
i, j
i, j,k
   
 
+ ∑W torsion ( ri , rj , rk , rl ) + ∑W LJ ( ri , rj )
i, j,k,l


+∑WCoulomb ( ri ,Qi ; rj ,Q j )
i, j
€ effective total Hamiltonian
An
i, j
Γ
 Qi
ri
Ωw
Ωm
€ €
 
 
 
H[Γ;
€ r1,..., rN ] = V[ r1,..., rN ] + G[Γ; r1,..., rN ]€
€
 
min H[Γ; r1,..., rN ]
Equilibrium conformations
Cheng, ..., Li, JCTC, 2009.
17
3. Dielectric Boundary Force
18
Dielectric boundary force (DBF): Fn = −δΓGelec [Γ]
εm = 1
A shape derivative approach
Perturbation defined by V : R 3 → R 3 :
{
€
€
€
x = x(X,t) =€T€t (X)€ €
x˙ = V (x)
x(0)= X
€
PBE: ψ t
Γt
Gelec [Γt ]
€
 Qi
ri
Ωw
Ωm
€
c ∞j , q j ,
ρw
€ δΓGelec [Γ] = $& d ') Gelec [Γt ]
% dt ( t= 0
€
€ Theorem
Structure
€
Γ
εw = 80
€
€
Shape derivative
19
4.1 The Poisson-Boltzmann Theory
εm = 1 εw = 80
∇ ⋅ εε0∇ψ − χ w B'(ψ ) = − ρ f
 Qi
) εε0
,
ri
2
Ωw
Gelec [Γ] = ∫ +−
| ∇ψ | + ρ f ψ − χ w B(ψ ).dV Γ
* 2
Ωm
M
− βq ψ
B(ψ ) = β −1 ∑ c ∞j e j −1
€
€
j=1
€ €
c ∞j , q j , ρ w
€
Theorem. Gelec [Γ,•] has a unique maximizer, uniformly
€
1
∞
bouded in H and L . It is the unique solution to the PBE.
€ €
Proof. Direct methods in the calculus of variations.
€
€§  Uniform
€ bounds by comparison.
§  Regularity theory and routine calculations. Q.E.D.
(
)
Li,, SIMA, 2009 & 2011; Nonlinearity, 2009; Li, Cheng, & Zhang, SIAP, 2011.
20
) εε0
,
2
Gelec [Γ] = ∫ +−
| ∇ψ | + ρ f ψ − χ w B(ψ ).dV
* 2
∇ ⋅ εε0∇ψ − χ w B'(ψ ) = − ρ f
Theorem. Let n point from Ωm to Ωw . Then
ε0 & 1 1 )
ε
2
δΓGelec [Γ] = ( − + | ε∂nψ |2 + 0 (εw − εm ) (I − n ⊗ n)∇ψ + B(ψ ).
2 ' εm εw *
2
Li, Cheng, & Zhang, SIAP, 2011. Luo, Private communications. Cai, Ye, & Luo,
PCCP, 2012.
Consequence: Since ε w > ε m , the force −δΓGelec [Γ] > 0.
Chu, Molecular Forces, based on Debye’s lectures, Wiley, 1967.
“Under the combined influence of electric field generated by
solute charges and their polarization in the surrounding medium
which is electrostatic neutral, an additional potential energy
emerges and drives the surrounding molecules to the solutes.”
21
22
23
24
4.2 The Coulomb-Field and Yukawa-Field
Approximations

E = −∇ψ


D = εε0 E
Electric field:
Electric displacement:
Electrostatic free energy:
Gelec [Γ] =
€
∫
1 
D2 ⋅ E 2 dV −
2
&"
&!
€
xi Q i
%
$G
€
∫
1 
D1 ⋅ E1dV
2
xi Q i
!
"m
#m
"m
#m
"w
#w
The Coulomb-field approximation (CFA):
The Yukawa-field approximation (CFA):


D2 ≈ D1


D2 ≈ D1
No need to solve partial differential
€ equations.
€
(κ = 0)
(κ > 0)
25
The Colulomb-field approximation (CFA)
2


1 #1 1&
Qi (r − ri )
Gelec [Γ] =
−
%
( ∫ Ω ∑   3 dV
2
32π ε 0 $ ε w ε m ' w i=1 r − ri
N
2



1 # 1 1 & Qi (r − ri )
−δΓGelec [Γ](r ) =
% − (∑  3
2
32π ε 0 $ ε w ε m ' i=1 r − ri
N
The Yukawa-field approximation (YFA)
€
€
€
(
  2
N
1
1

Qi ( r − ri )
1
*
Gelec [Γ] =
f
(
r
,
κ
,Γ)
−
∑ i
∫
 3
32π 2ε0 Ω w * εw i=1
εm
r − ri
)
 

1+ κ | r − ri |


f i ( r ,κ,Γ) =

 exp(−κ ( r − Pi ( r ))
1+ κ | ri − Pi ( r ) |
−δΓGelec [Γ] : Γ → R
Too complicated!
Cheng, Cheng, & Li, Nonlinearity, 2011.
x
  2+
Qi ( r − ri ) ∑ r − r 3 - dV
i=1
i
,
N
"
!m
pi (x)
xi
!w
26
4. Computation by the LevelSet Method
27
The Level-Set Method
n

r
!   Interface motion


Vn = Vn ( r ,t) for r ∈ Γ(t)
Γ(t )
!   Level-set representation
€
€


Γ(t) = {r ∈ Ω : ϕ ( r ,t) = 0}
€
€

z = ϕ ( r ,t)
!   The level-set equation
ϕt + Vn | ∇ϕ |= 0
z=0
Γ (t )
Topological changes €
28
Application to variational solvation
ϕt + Vn | ∇ϕ |= 0
Relaxation

dri
 
 
= −∇ ri H[Γ; r1,..., rN ] = −∇ ri V[ r1,..., rN ] − ∇ ri G[Γ]
dt


Vn = −δΓ H[Γ;, r1,..., rN ] = −δΓG[Γ]




δΓG[Γ]( r ) = P + 2γ 0 [H( r ) − τK( r )] − ρ wU( r ) + δΓGelec [Γ]
€
€
€
JCP, 2007, 2009; JCTC, 2009, 2012; PRL, 2009; J. Comput. Phys., 2010.
29
30
31
32
33
34
35
Two xenon atoms
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
PMF: Level-set (circles) vs. MD (solid line).
MD: Paschek, JCP, 2004.
12
36
Two paraffin plates
PMF: Level-set (circles) vs. MD (line).
MD: Koishi et al. PRL, 2004; JCP, 2005.
37
A hydrophobic receptor-ligand system
'()*+,23/240+5(627./010.(
"&$
0(010542.8219.2/:5442)(3)4.7)
0(010542.82.()245*+)2)(3)4.7)
"&!
"&"
'()*+,
PMF
"&#
"#%
"#$
"#!
!!
wall-particle distance
!"
#
"
!
-./010.(
$
%
&#
38
A benzene molecule
39
BphC
40
The p53/MDM2 complex (PDB code: 1YCR)
41
Molecular surface (green) vs. VISM loose (red) and VISM
tight initials (blue) at 12 A.
42
43
44
5. Move Forward: Solvent
Fluctuations
45
General description
Du
ρw
− µ∇ 2 u + ∇pw = f + η
Dt
∇ ⋅ u = 0 in Ωw (t)
pm (t) Ωm (t) = K(T)
∇ ⋅ εε0∇ψ − χ w B'(ψ ) = − ρ f
εm = 1
in Ωw (t)
€
at
€
€
Du
≈0
Dt
!   Small inertia:
€ 
!   Body force:
f = −ρ w∇U, U( r ) =
€
Ωw
Ωm
€
€
( pm − pw )n + 2µD(u)n = (γ 0 H − f ele €
)n
Assumptions

ri Qi
Γ
€ €€ €
εw = 80
c ∞j , q j ,
ρw
Γ(t)
 
∑U LJ ,i (| r − ri |)
i
46
A charged sphere
!   Linearized PBE
!   Fluctuations with decay
Q
O
!m
R(t)
r
!w
A generalized Rayleigh-Plesset equation
dR
η0 R −αR
= F(R) +
e Wt
dt
4 µα
(
R%
K(T) 2γ 0
F(R) =
− p∞ + f ele *
'U LJ (R) + 3 −
)
4µ &
R
R
€
€
f ele
€
€
/
Q2 ,% 1 1 ( 1
κ2
.' − * 4 −
1
=
2 2
2
32π ε0 .-& εw εm ) R εw (1+ κR) R 10
The Euler-Maruyama method
η0 Rn −αR n
Rn +1 = Rn + F(Rn )Δt +
e ΔW n
4 µα
ΔW n : iid Gaussians with mean 0 variance
Δt
47
0.4
0
0.35
−0.05
0.3
0.25
−0.1
0.2
0.15
−0.15
0.1
0.05
−0.2
0
−0.05
3
4
5
6
7
The force F(R) vs. R
8
−0.25
9
3
4
5
6
7
8
9
The potential U(R)
48
10
9
0.3
8
0.25
7
0.2
6
0.15
5
0.1
4
0.05
3
2
0
2000
4000
6000
R = R(t)
8000
10000
0
3
4
5
6
7
8
Probability density of R
49
9
6. Conclusions
50
Achivement
!   Level-set VISM with solute molecular mechanics; freeenergy functional; hydrophobic cavities, charge effects,
multiple states, etc.
!   Effective DBF: PB theory, CFA and YFA.
!   Initial work on the solvent dynamics with fluctuations.
Issues
!
!
!
!
  Efficiency: mimutes to hours.
  Parameters: similar to that for MD force fields.
  More details: charge asymmetry, hydration shells, etc.
  Coarse graining, coupling with other models.
51
Current and future work
!   Level-set VISM coupled with the full PBE.
!   Molecular recognition + drug design: host-guest
systems.
!   Solvent dynamics: hydrodynamics + fluctuation.
!   Brownian dynamics coupled with continuum diffusion.
!   Fast algorithms, GPU computing, software
development.
! Multiscale approach: solute MD + solvent fluid motion.
!   Mathematics and statistical mechanics of VISM.
52
Roles of mathematics and computation
!   Many mathematical concepts and methods are used:
differential geometry, PDE, stochastic processes,
numerical PDE, numerical optimization, etc.
!   More is needed: geometrical flows for protein folding;
stochastic methods for hydrodynamic interactions;
topological methods for DNA and RNA structures; etc.
!   Computation is essential: real biomolecular systems are
very complicated and the mathematical problems cannot
be solved analytically.
!   Collaboration between mathematics and biological
sciences is crucial.
53
Thank you!
54