Variational Implicit Solvation: Empowering Mathematics and...
Transcript of Variational Implicit Solvation: Empowering Mathematics and...
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
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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) 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.)
Collaborators
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
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1. Biomolecules: What and Why?
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Biomolecules
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Antibodies, enzymes, contractile, structural, storage, transport, etc.
Protein functions
We have more than 100,000 proteins in our bodies.
Protein structures
Wiki
Each protein is produced from a set of only 20 building blocks.
To function, proteins fold into three-dimensional compact structures.
Protein Folding
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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.
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Solvation
protein folding molecular recognition
solvation
conformational change
water
water
solute solute
solute
water
receptor ligand
binding
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ΔG = ?
2. Variational Implicit-Solvent Models
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solvent
solute solvent
solute
Explicit vs. Implicit
Molecular dynamics (MD) simulations
Statistical mechanics
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! Solute-solvent interfacial property
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γ 0
RSymbols: MD.
the Tolman length
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γ 0 = 73mJ /m2
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τ :
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γ = γ 0 1− 2τH( )
Curvature effect
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τ = 0.9 mean curvature
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H :Huang et al., JCPB, 2001.
A
What to model with an implicit solvent?
! Electrostatic interactions
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∇ ⋅εε0∇ψ = −ρ
Poisson’s equation
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solvent
solute
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ε =1
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ε = 80
Fermi repulsion vdW attraction
Solute
Water
! Excluded volume and van der Waals dispersion
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ULJ (r) = 4ε σr( )12 − σ
r( )6[ ]
σ rO
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−ε
The Lennard-Jones (LJ) potential
Surface energy PB/GB calcula1ons
Commonly used implicit-solvent models
solvent accessible surface (SAS)
probing ball
vdW surface
solvent excluded surface (SES)
Possible issues ! Hydrophobic cavities ! Curvature correction ! Decoupling of polar and nonpolar contributions
PB = Poisson-Boltzmann GB = Generalized Born
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Koishi et al., PRL, 2004. Liu et al., Nature, 2005. Sotomayor et al., Biophys. J. 2007
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Dzubiella, Swanson, & McCammon: Phys. Rev. Lett. 96, 087802 (2006) J. Chem. Phys. 124, 084905 (2006)
Free-energy functional
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r i
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Ωm
Γ
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Qi
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Ωw
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c j∞,
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q j , wρ
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G[Γ] = Pvol(Ωm ) + γ 0 (1− 2τH)dSΓ
∫
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+ρw ULJ ,ii∑
Ωw
∫ (| r − r i |)dV + Gelec[Γ]
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Gelec[Γ] : electrostatic free energy
! The Poisson-Boltzmann (PB) theory
! The Coulomb-field or Yukawa-field approximation
the Tolman length, a fitting parameter
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τ :
Variational Implicit-Solvent Model (VISM)
Hadwiger’s Theorem
Pvol(Ωm )+γ0area(Γ)− 2γ0τ H dSΓ∫ +cK K
Γ∫ dS( )
Let C = the set of all convex bodies, M = the set of finite union of convex bodies. If is
! rotational and translational invariant, ! additive:
! conditionally continuous: ),()(,, UFUFUUCUU jjj →⇒→∈
RMF →:
,,)()()()( MVUVUFVFUFVUF ∈∀∩−+=∪
.)()()( MUKdSdHdScUbAreaUaVolUFUU
∈∀++∂+= ∫∫ ∂∂
then
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Application to nonpolar solvation Roth, Harano, & Kinoshita, PRL, 2006. Harano, Roth, & Kinoshita, Chem. Phys. Lett., 2006.
Geometrical part:
Coupling solute molecular mechanics with implicit solvent
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V[ r 1,..., r N ] = Wbond
i, j∑ ( r i,
r j ) + Wbendi, j ,k∑ ( r i,
r j , r k )
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+ WCoulombi, j∑ ( r i,Qi;
r j ,Qj )
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H[Γ; r 1,..., r N ] = V[ r 1,...,
r N ]+ G[Γ; r 1,..., r N ]
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minH[Γ; r 1,..., r N ] Equilibrium conformations
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An effective total Hamiltonian
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+ Wtorsion ( r i
i, j,k,l∑ , r j ,
r k, r l ) + WLJ
i, j∑ ( r i,
r j )
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r i
€
Ωm
Γ
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Qi
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Ωw
Cheng, ..., Li, JCTC, 2009.
3. Dielectric Boundary Force
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A shape derivative approach
Perturbation defined by
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V :R3 → R3 :
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˙ x = V (x)
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x(0)= X{
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x = x(X,t) = Tt (X)
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Γt PBE:
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ψt
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Gelec[Γt ]
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δΓGelec[Γ] =ddt$
% &
'
( ) t= 0
Gelec[Γt ]
Dielectric boundary force (DBF):
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Fn = −δΓGelec[Γ]
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r i
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Ωm
Γ
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Qi
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Ωw
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c j∞,
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q j , wρ€
εm =1
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εw = 80
Structure Theorem
Shape derivative
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∇ ⋅εε0∇ψ − χwB'(ψ) = −ρ f
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B(ψ) = β−1 c j∞ e−βq jψ −1( )j=1
M∑
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r i
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Ωm
Γ
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Qi
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Ωw
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c j∞,
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q j , wρ€
εm =1
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εw = 80
Li,, SIMA, 2009 & 2011; Nonlinearity, 2009; Li, Cheng, & Zhang, SIAP, 2011.
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Gelec[Γ] = −εε02|∇ψ |2 +ρ fψ − χwB(ψ)
)
* + ,
- . ∫ dV
4.1 The Poisson-Boltzmann Theory
Theorem. has a unique maximizer, uniformly bouded in and . It is the unique solution to the PBE.
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Gelec[Γ,•]
Proof. Direct methods in the calculus of variations.
§ Uniform bounds by comparison.
§ Regularity theory and routine calculations. Q.E.D. €
H1
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L∞
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∇ ⋅εε0∇ψ − χwB'(ψ) = −ρ f
Li, Cheng, & Zhang, SIAP, 2011. Luo, Private communications. Cai, Ye, & Luo, PCCP, 2012.
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Gelec[Γ] = −εε02|∇ψ |2 +ρ fψ − χwB(ψ)
)
* + ,
- . ∫ dV
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δΓGelec[Γ] =ε02
1εm
−1εw
&
' (
)
* + |ε∂nψ |
2 +ε02εw −εm( ) (I − n ⊗ n)∇ψ 2
+ B(ψ).
Theorem. Let point from to . Then n Ωm Ωw
Consequence: Since the force
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.”
εw > εm, −δΓGelec[Γ]> 0.
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4.2 The Coulomb-Field and Yukawa-Field Approximations
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D 2 ≈
D 1
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Gelec[Γ] =12∫ D 2 ⋅ E 2dV −
12∫ D 1 ⋅ E 1dV
Electric field: Electric displacement:
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E = −∇ψ
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D = εε0
E
No need to solve partial differential equations.
!xi
!"
"
#"#
iQi
#
"
Qx iG$
wwm
m
m
m
%
& &
Electrostatic free energy:
The Coulomb-field approximation (CFA):
The Yukawa-field approximation (CFA):
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D 2 ≈
D 1
(κ = 0)(κ > 0)
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The Yukawa-field approximation (YFA)
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Gelec[Γ] =1
32π 2ε0
1εw
f i( r ,κ,Γ)Qi(
r − r i) r − r i
3i=1
N
∑2
−1εm
Qi( r − r i) r − r i
3i=1
N
∑2(
)
* *
+
,
- - Ωw
∫ dV
(x)!w
m
"
! x i
xpi
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fi( r ,κ,Γ) =
1+κ | r − r i |1+κ | r i − Pi(
r ) |exp −κ( r − Pi(
r )( )
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−δΓGelec[Γ] :Γ→ R
Cheng, Cheng, & Li, Nonlinearity, 2011.
Too complicated!
The Colulomb-field approximation (CFA)
−δΓGelec[Γ](r ) = 1
32π 2ε0
1εw−1εm
#
$%
&
'(
Qi (r − ri )r − ri
3i=1
N
∑2
Gelec[Γ]=1
32π 2ε0
1εw−1εm
#
$%
&
'(
Qi (r − ri )r − ri
3i=1
N
∑Ωw∫
2
dV
4. Computation by the Level-Set Method
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€
Vn = Vn ( r ,t)
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r ∈ Γ(t)
! Level-set representation
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Γ(t) = { r ∈ Ω :ϕ( r ,t) = 0}
! The level-set equation
)(tΓ
n
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r ! Interface motion
for
0|| =∇+ ϕϕ nt V )(tΓ
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z =ϕ( r ,t)
0=z
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The Level-Set Method
Topological changes
Application to variational solvation
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δΓG[Γ]( r ) = P + 2γ 0[H(
r ) − τK( r )]− ρwU( r ) + δΓGelec[Γ]
Relaxation
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Vn = −δΓH[Γ;, r 1,..., r N ] = −δΓG[Γ]
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d r idt
= −∇ r iH[Γ; r 1,...,
r N ] = −∇ r iV[ r 1,...,
r N ]−∇ r iG[Γ]
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0|| =∇+ ϕϕ nt V
JCP, 2007, 2009; JCTC, 2009, 2012; PRL, 2009; J. Comput. Phys., 2010.
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PMF: Level-set (circles) vs. MD (solid line).
2 3 4 5 6 7 8 9 10 11 12d/
-2
-1
0
1
2
w(d
)/k
BT
3 4 5 6 7 8 9 10 11
-1
0
1
W(d
)/k
BT
Å
36 MD: Paschek, JCP, 2004.
Two xenon atoms
PMF: Level-set (circles) vs. MD (line).
37 MD: Koishi et al. PRL, 2004; JCP, 2005.
Two paraffin plates
38 !! !" # " ! $ % &#
"#!
"#$
"#%
"&#
"&"
"&!
"&$
'()*+,
-./010.(
'()*+,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
A hydrophobic receptor-ligand system
A benzene molecule
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BphC
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The p53/MDM2 complex (PDB code: 1YCR)
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Molecular surface (green) vs. VISM loose (red) and VISM tight initials (blue) at 12 A.
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5. Move Forward: Solvent Fluctuations
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! Small inertia: ! Body force:
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€
ρwDuDt
−µ∇2u +∇pw = f +η in
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Ωw (t)
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∇ ⋅ u = 0 in
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Ωw (t)
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pm (t)Ωm (t) = K(T)
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(pm − pw )n + 2µD(u)n = (γ 0H − fele )n at
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Γ(t)
General description
€
r i
€
Ωm
Γ
€
Qi
€
Ωw
€
c j∞,
€
q j , wρ€
εm =1
€
εw = 80
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∇ ⋅εε0∇ψ − χwB'(ψ) = −ρ f
Assumptions
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DuDt
≈ 0
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f = −ρw∇U,
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U( r ) = ULJ ,ii∑ (| r − r i |)
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m
rO
Q R(t)
! !w
A charged sphere ! Linearized PBE ! Fluctuations with decay
A generalized Rayleigh-Plesset equation
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dRdt
= F(R) +η0R4µα
e−αRWt
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fele =Q2
32π 2ε0
1εw
−1εm
%
& '
(
) * 1R4
−κ 2
εw 1+κR( )2R2,
- . .
/
0 1 1
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F(R) =R4µ
ULJ (R) +K(T)R3
−2γ 0R
− p∞ + fele%
& '
(
) *
The Euler-Maruyama method
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Rn+1 = Rn + F(Rn )Δt +η0Rn
4µαe−αRnΔWn
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ΔWn : iid Gaussians with mean 0 variance
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Δt
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The force F(R) vs. R
3 4 5 6 7 8 9!0.25
!0.2
!0.15
!0.1
!0.05
0
3 4 5 6 7 8 9!0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
The potential U(R)
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R = R(t)
0 2000 4000 6000 8000 100002
3
4
5
6
7
8
9
10
3 4 5 6 7 8 90
0.05
0.1
0.15
0.2
0.25
0.3
Probability density of R
6. Conclusions
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! Level-set VISM with solute molecular mechanics; free-
energy functional; hydrophobic cavities, charge effects, multiple states, etc.
! Effective DBF: PB theory, CFA and YFA. ! Initial work on the solvent dynamics with fluctuations.
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! 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.
Issues
Achivement
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! 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.
Current and future work
Roles of mathematics and computation
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! 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.
Thank you!
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