Variational Implicit Solvation of Biomolecules: From Theory to

Numerical Computations

Bo Li Department of Mathematics and

Center for Theoretical Biological Physics UC San Diego

CECAM Workshop: New Perspectives in Liquid State Theories for Complex Molecular Systems

Institut Henri Poincare, Paris, June 20-22, 2013

OUTLINE

1.  Introduction 2.  Variational Implicit-Solvent Models 3.  The Level-Set Method 4.  Test and Applications 5.  Conclusions

2

Solvation

protein folding

molecular recognition

solvation

conformational change water

water

solute solute

solute

water

receptor ligand binding

€

ΔG = ?

1. Introduction Biomolecular interactions §  Fundamental in biological structures and functions §  Molecular mechanical, charge-charge, and van der Waals interactions §  Water is essential §  Fluctuations, multiple scales, complex energy landscapes, and

explosive data §  Applications: protein folding, molecular recognition, etc.

3

solvent

solute solvent

solute

Biomolecular Modeling

explicit solvent vs. implicit solvent §  Solvent molecules are treated

explicitly as in molecular dynamics (MD) simulations.

§  First principle and atomic resolution §  Sampled statistical information §  Relatively small systems and long

computation: less efficient

§  Solvent molecules are treated implicitly.

§  Mean-field descriptions: efficient §  Large systems §  Direct thermodynamic data §  Coupling with fluid motion,

fluctuations, etc.

Statistical mechanics of implicit-solvent modeling X =Y =

solute deg. of freedom solvent deg. of freedom

total interaction potential U(X,Y ) =Um (X)+Uw (Y )+Umw (X,Y )

Probability P(X,Y ) = P0e−βU (X,Y )

P(X) = P(X,Y )dY =∫ P0e−βW (X )

potential of mean force W (X) :4

§  Solute-solvent interfacial energy

€

γ = γ 0 1− 2τH( )

Curvature effect (

What to model with an implicit solvent?

§  Electrostatic interactions

€

∇ ⋅εε0∇ψ = −ρPoisson’s equation

solvent

solute

€

ε =1

€

ε = 80

Fermi repulsion vdW attraction

Solute

Water

§  Excluded volume and solute-solvent van der Waals (vdW) interactions

€

ULJ (r) = 4ε σr( )12 − σ

r( )6[ ] The Lennard-Jones (LJ) potential

is mean curvature) H

symbol: MD solid line:

€

γ 0

dielectric coefficient 5

Surface  energy  

PB/GB  calcula1ons  

Commonly used surface-type implicit-solvent models

solvent accessible surface (SAS)

probing ball

vdW surface

solvent excluded surface (SES)

Possible issues §  No curvature correction §  Unable to describe hydrophobic cavities §  Decoupling of polar and nonpolar

contributions

PB = Poisson-Boltzmann GB = Generalized Born

This work §  Develop a robust implicit-solvent model for biomolecules interactions. §  Design and implement highly efficient and accurate computational

methods. §  Apply to molecular recognition, protein-protein interaction, protein

folding, and many other processes. 6

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

7

§  : pressure difference between inside/outside solutes §  : surface tension for flat solute-solvent interface §  : coefficient of curvature expansion §  : bulk solvent density

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]

ρw

2. Variational Implicit-Solvent Model (VISM)

dielectric boundary volumetric and surface energies

solute-solvent vdW interaction electrostatic free energy

P

τγ0

Dzubiella, Swanson, & McCammon (PRL 2006; JCP 2006) §  Free-energy minimization determines stable equilibrium conformations. §  Different interactions are coupled in the free-energy functional.

8

Electrostatic free energy and dielectric boundary force

The Poisson-Boltzmann (PB) theory

∇⋅εε0∇ψ − χwB '(ψ) = − Qiδrii∑

€

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

€

Gelec[Γ] = −εε02|∇ψ |2 +ρ fψ − χwB(ψ)

)

* + ,

- . ∫ dV

dielectric coefficient

The (normal component of ) dielectric boundary force Fn = −δΓGelec[Γ]Theorem. If is the unit normal at the dielectric boundary pointing from the solute region to solvent region then

δΓGelec[Γ]=ε02

1εm

−1εw

#

$%

&

'( |ε∂nψ |

2 +ε02εw −εm( ) (I − n⊗ n)∇ψ 2

+B(ψ).

nΩm Ωw

Corollary. Since the force in the direction . εw > εm,

Γ

The electrostatic force always points from the solvent to the solute region.

−δΓGelec[Γ]< 0 n

9 Che, Dzubiella, Li, & McCammon, JPCB, 2008. Li,Cheng, & Zhang, SIAM J. Applied Math, 2011.

The Coulomb-Field Approximations (CFA)

€

D 2 ≈

D 1

€

Gelec[Γ] =12∫ D 2 ⋅ E 2dV −

12∫ D 1 ⋅ E 1dV

Electric field and electric displacement : E

D = εε0

E

No need to solve partial differential equations.

!xi

!"

"

#"#

iQi

#

"

Qx iG$

wwm

m

m

m

%

& &

Electrostatic free energy (Born cycle):

−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

D1 =

Qi (r − ri )

4π r − ri3

i=1

N

∑The Coulomb field

D

State 1: before immersion State 2: after immersion, creating a dielectric boundary. The CFA:

Electrostatic energy

Dielectric boundary force

10 Cheng, Cheng, & Li, Nonlinearity, 2011. Wang, Che, Cheng, Dzubiella, Li, & McCammon, JCTC, 2012.

Coupling solute molecular mechanics with VISM

€

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

An effective total Hamiltonian

+ WLJi, j∑ (ri,

rj )

Force field of solute mechanical interactions

+ Wtorsion (ri

i, j,k,l∑ , rj,

rk,rl )

11

Cheng, Xie, Dzubiella, McCammon, Che, & Li, JCTC, 2009.

€

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 by the normal velocity

for

0|| =∇+ ϕϕ nt V)(tΓ

€

z =ϕ( r ,t)

0=z

3. The Level-Set Method

§  Easy handle of topological changes

§  Level-set formulas of geometrical quantities Unit normal Mean curvature Gaussian curvature Surface integral Volume integral

H =∇⋅n / 2

n =∇ϕ / |∇ϕ |

€

K = n ⋅ adj(He(ϕ)) n

€

f ( r )dS =Γ∫ f ( r )δ(ϕ)dV

R 3∫

€

f ( r )dV =Ω∫ f ( r )[1−H(ϕ)]dV

R 3∫ 12

Application to variational solvation

δΓG[Γ](r ) = P + 2γ0[H (

r )−τK(r )]− ρw ULJ ,ii∑ (| r − ri |)+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]

0|| =∇+ ϕϕ nt V

Initial surfaces: tight wraps, loose wraps, or their combinations

13

Discretization of the level-set equation

0|| =∇+ ϕϕ nt V

€

Vn = −P − 2γ 0[H( r ) − τK( r )]+ ρwU( r )

€

ϕ k+1(x) −ϕ k (x) = −ΔtVnk (x) |∇ϕ k (x) |Forward Euler

Decomposition

€

ϕ t = A + B

€

B = [P − ρwU( r )] |∇ϕ |

€

A = 2γ 0[H( r ) − τK( r )] |∇ϕ |Central differencing

Upwinding

€

τ = 0

€

τ > 0

Central differencing + FFT or Cholesky decomposition

Semi-implicit

§  Special case:

§  General case: €

ϕ t = 2γ 0Δϕ + N(∇ϕ,∇2ϕ)

14 Cheng, Li, & Wang, J. Comput. Phys., 2010.

Potential of mean forces (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

two xenon atoms two paraffin plates

4. Test and Applications

15 Cheng, Dzubiella, McCammon, & Li, JCP, 2007.

Helical alkanes and C60 fullerene

16

17

Two charged paraffin plates

Plate-plate separation d = 10 A. Left: no charges. Middle: charges (0.2 e, 0.2 e). Right: charges (0.2 e, -0.2 e). Color code represents mean curvature.

Effect of charges to hydration

VISM surfaces vs. other fixed surfaces

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

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

18

Wang, Che, Cheng, Dzubiella, Li, & McCammon, JCTC, 2012.

Guo, Li, Dzubiella, Cheng, McCammon, Che, JCTC, 2013.

BphC: a two-domain protein

Upper row: uncharged. Lower row: charged. The domain separations are: 8 (left), 14 (middle), and 16 (right) A.

19 Wang, Che, Cheng, Dzubiella, Li, & McCammon, JCTC, 2012.

20

!! !" # " ! $ % &#"#!

"#$

"#%

"&#

"&"

"&!

"&$

'()*+,

-./010.(

'()*+,23/240+5(627./010.(

0(010542.8219.2/:5442)(3)4.7)0(010542.82.()245*+)2)(3)4.7)

PMF

particle-wall distance

A receptor-ligand system p53/MDM2

uncharged charged

A host-guest system CB[7]-B2

Cheng, Wang, Setny,Dzubiella, Li, & McCammon, JCP, 2009. Setny, Wang, Cheng, Li, McCammon, & Dzubiella, PRL, 2009. Zhou, Rogers, de Oliveira, Baron, Cheng, Dzubiella, Li, & McCammon, 2013.

21

22

"  Coupling solute molecular mechanics "  Effective dielectric boundary force "  The Coulomb-field approximation of electrostatic energy

"  Estimates of solvation free energy "  Hydrophobic cavities and multiple states of hydration "  Charge effect

Summary

€

G[Γ] = Pvol(Ωm ) + γ 0 (1− 2τH)dSΓ

∫

€

+ρw ULJ ,ii∑

Ωw

∫ (| r − r i |)dV + Gelec[Γ]

VISM free-energy functional

The level-set method: algorithm and coding

Numerical computations

5. Conclusions

23

"  Parameters: MD force fields; fit-parameters "  Efficiency: minutes to hours "  Molecular details: charge asymmetry, hydration shells, etc. "  Entropy calculations

Issues

SAS/SES vs. VISM vs. MD simulations

Application of VISM "  Estimate solvation free energy "  Describe equilibrium conformations

24

"  Level-set VISM coupled with the full PBE "  Application: host-guest systems, protein-protein

interactions "  Fast algorithms, GPU computing, software

development "  Phase-field VISM implementation "  Solvent dynamics: hydrodynamics + fluctuation "  Multiscale approach: solute MD + solvent fluid motion "  Mathematics and statistical mechanics of VISM

Current and future work

25

Funding: NSF, DOE, NIH, CTBP

Joachim Dzubiella (Humboldt Univ.) J. Andrew McCammon (UCSD) Li-Tien Cheng (UCSD) Jianwei Che (GNF) Zhongming Wang (Florida Intern’l Univ.) Piotr Setny (Munich & Warsaw) Zuojun Guo (GNF)

Main Collaborators

Acknowledgment

26

Thank you!

27

References

[1] Dzubiella, Swanson, McCammon, PRL, 96, 087802, 2006. [2] Dzubiella, Swanson, McCammon, JCP, 124, 084905, 2006. [3] Cheng, Dzubiella, McCammon, & Li, JCP, 127, 084503, 2007. [4] Che, Dzubiella, Li, and McCammon, JPC-B, 112, 2008. [5] Cheng, Xie, Dzubiella, McCammon, Che, & Li, JCTC, 5, 257, 2009. [6] Cheng, Wang, Setny, Dzubiella, Li, & McCammon, JPC, 113, 144102, 2009. [7] Setny, Wang, Cheng, Li, McCammon, & Dzubliella, PRL, 103, 187801, 2009. [8] Cheng, Li, & Wang, J. Comput. Phys., 229, 8497, 2010. [9] Cheng, Cheng, & Li, Nonlinearity, 24, 3215, 2011. [10] Li, Cheng, & Zhang, SIAM J. Applied Math, 71, 2093, 2011. [11] Wang, Che, Cheng, Dzubiella, Li, & McCammon, JCTC, 8, 386, 2012. [12] Cheng, Li, White, & Zhou, SIAM J. Applied Math, 73, 594, 2013. [13] Guo, Li, Dzubiella, Cheng, McCammon, & Che, JCTC, 9, 1778, 2013. [14] Zhao, Kwan, Che, Li, & McCammon, JCP, 2013 (accepted). [15] Zhou, Rogers, Oliveira, Baron, Cheng, Dzubiella, Li, & McCammon, JCTC,

2013 (submitted)

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