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

2

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

3

1. Biomolecules: What and Why?

4

5

Biomolecules

6

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

7

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

€

ΔG = ?

2. Variational Implicit-Solvent Models

9

10

solvent

solute solvent

solute

Explicit vs. Implicit

Molecular dynamics (MD) simulations

Statistical mechanics

11

!   Solute-solvent interfacial property

€

γ 0

RSymbols: MD.

the Tolman length

€

γ 0 = 73mJ /m2

€

τ :

€

γ = γ 0 1− 2τH( )

Curvature effect

€

τ = 0.9 mean curvature

€

H :Huang et al., JCPB, 2001.

A

What to model with an implicit solvent?

!   Electrostatic interactions

€

∇ ⋅εε0∇ψ = −ρ

Poisson’s equation

12

solvent

solute

€

ε =1

€

ε = 80

Fermi repulsion vdW attraction

Solute

Water

!   Excluded volume and van der Waals dispersion

€

ULJ (r) = 4ε σr( )12 − σ

r( )6[ ]

σ rO

€

−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

13

14

Koishi et al., PRL, 2004. Liu et al., Nature, 2005. Sotomayor et al., Biophys. J. 2007

15

Dzubiella, Swanson, & McCammon: Phys. Rev. Lett. 96, 087802 (2006) J. Chem. Phys. 124, 084905 (2006)

Free-energy functional

€

r i

€

Ωm

Γ

€

Qi

€

Ωw

€

c j∞,

€

q j , wρ

€

G[Γ] = Pvol(Ωm ) + γ 0 (1− 2τH)dSΓ

∫

€

+ρw ULJ ,ii∑

Ωw

∫ (| r − r i |)dV + Gelec[Γ]

€

Gelec[Γ] : electrostatic free energy

!   The Poisson-Boltzmann (PB) theory

!   The Coulomb-field or Yukawa-field approximation

the Tolman length, a fitting parameter

€

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

16

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

€

V[ r 1,..., r N ] = Wbond

i, j∑ ( r i,

r j ) + Wbendi, j ,k∑ ( r i,

r j , r k )

€

+ WCoulombi, j∑ ( r i,Qi;

r j ,Qj )

€

H[Γ; r 1,..., r N ] = V[ r 1,...,

r N ]+ G[Γ; r 1,..., r N ]

€

minH[Γ; r 1,..., r N ] Equilibrium conformations

17

An effective total Hamiltonian

€

+ Wtorsion ( r i

i, j,k,l∑ , r j ,

r k, r l ) + WLJ

i, j∑ ( r i,

r j )

€

r i

€

Ωm

Γ

€

Qi

€

Ωw

Cheng, ..., Li, JCTC, 2009.

3. Dielectric Boundary Force

18

19

A shape derivative approach

Perturbation defined by

€

V :R3 → R3 :

€

˙ x = V (x)

€

x(0)= X{

€

x = x(X,t) = Tt (X)

€

Γt PBE:

€

ψt

€

Gelec[Γt ]

€

δΓGelec[Γ] =ddt$

% &

'

( ) t= 0

Gelec[Γt ]

Dielectric boundary force (DBF):

€

Fn = −δΓGelec[Γ]

€

r i

€

Ωm

Γ

€

Qi

€

Ωw

€

c j∞,

€

q j , wρ€

εm =1

€

εw = 80

Structure Theorem

Shape derivative

20

€

∇ ⋅εε0∇ψ − χwB'(ψ) = −ρ f

€

B(ψ) = β−1 c j∞ e−βq jψ −1( )j=1

M∑

€

r i

€

Ωm

Γ

€

Qi

€

Ωw

€

c j∞,

€

q j , wρ€

εm =1

€

εw = 80

Li,, SIMA, 2009 & 2011; Nonlinearity, 2009; Li, Cheng, & Zhang, SIAP, 2011.

€

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.

€

Gelec[Γ,•]

Proof. Direct methods in the calculus of variations.

§  Uniform bounds by comparison.

§  Regularity theory and routine calculations. Q.E.D. €

H1

€

L∞

21

€

∇ ⋅εε0∇ψ − χwB'(ψ) = −ρ f

Li, Cheng, & Zhang, SIAP, 2011. Luo, Private communications. Cai, Ye, & Luo, PCCP, 2012.

€

Gelec[Γ] = −εε02|∇ψ |2 +ρ fψ − χwB(ψ)

)

* + ,

- . ∫ dV

€

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

€

D 2 ≈

D 1

€

Gelec[Γ] =12∫ D 2 ⋅ E 2dV −

12∫ D 1 ⋅ E 1dV

Electric field: Electric displacement:

€

E = −∇ψ

€

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):

€

D 2 ≈

D 1

(κ = 0)(κ > 0)

26

The Yukawa-field approximation (YFA)

€

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

€

fi( r ,κ,Γ) =

1+κ | r − r i |1+κ | r i − Pi(

r ) |exp −κ( r − Pi(

r )( )

€

−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

27

€

Vn = Vn ( r ,t)

€

r ∈ Γ(t)

!   Level-set representation

€

Γ(t) = { r ∈ Ω :ϕ( r ,t) = 0}

!   The level-set equation

)(tΓ

n

€

r !   Interface motion

for

0|| =∇+ ϕϕ nt V )(tΓ

€

z =ϕ( r ,t)

0=z

28

The Level-Set Method

Topological changes

Application to variational solvation

€

δΓG[Γ]( r ) = P + 2γ 0[H(

r ) − τK( r )]− ρwU( r ) + δΓGelec[Γ]

Relaxation

€

Vn = −δΓH[Γ;, r 1,..., r N ] = −δΓG[Γ]

€

d r idt

= −∇ r iH[Γ; r 1,...,

r N ] = −∇ r iV[ r 1,...,

r N ]−∇ r iG[Γ]

29

0|| =∇+ ϕϕ nt V

JCP, 2007, 2009; JCTC, 2009, 2012; PRL, 2009; J. Comput. Phys., 2010.

30

31

32

33

34

35

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

39

40

BphC

41

The p53/MDM2 complex (PDB code: 1YCR)

42

Molecular surface (green) vs. VISM loose (red) and VISM tight initials (blue) at 12 A.

43

44

5. Move Forward: Solvent Fluctuations

45

!  Small inertia: !  Body force:

46

€

ρwDuDt

−µ∇2u +∇pw = f +η in

€

Ωw (t)

€

∇ ⋅ u = 0 in

€

Ωw (t)

€

pm (t)Ωm (t) = K(T)

€

(pm − pw )n + 2µD(u)n = (γ 0H − fele )n at

€

Γ(t)

General description

€

r i

€

Ωm

Γ

€

Qi

€

Ωw

€

c j∞,

€

q j , wρ€

εm =1

€

εw = 80

€

∇ ⋅εε0∇ψ − χwB'(ψ) = −ρ f

Assumptions

€

DuDt

≈ 0

€

f = −ρw∇U,

€

U( r ) = ULJ ,ii∑ (| r − r i |)

47

m

rO

Q R(t)

! !w

A charged sphere !  Linearized PBE !  Fluctuations with decay

A generalized Rayleigh-Plesset equation

€

dRdt

= F(R) +η0R4µα

e−αRWt

€

fele =Q2

32π 2ε0

1εw

−1εm

%

& '

(

) * 1R4

−κ 2

εw 1+κR( )2R2,

- . .

/

0 1 1

€

F(R) =R4µ

ULJ (R) +K(T)R3

−2γ 0R

− p∞ + fele%

& '

(

) *

The Euler-Maruyama method

€

Rn+1 = Rn + F(Rn )Δt +η0Rn

4µαe−αRnΔWn

€

ΔWn : iid Gaussians with mean 0 variance

€

Δt

48

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)

49

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

52

!  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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