Fast Methods for Simulation of Biomolecule Electrostatics

Shihhsien KuoMichael Altman

Jaydeep BardhanBruce TidorJacob White

November 12, 2002

Outline

Ø Problem statementØ Finite-difference approach and problemsØ Integral equation method and advantagesØ Fast solver implementationØ Computational resultsØ Conclusion and Future work

The Problem of Drug Design

Receptor Ligand

+

Complex

Designed drug molecule

Given protein molecule

• Electrostatics• Solvation• Van der Waals forces• Covalent bonding• Hydrophobic effect

Electrostatics

Salt water

Ligand(drug molecule)

++

-

--

-

+ +

+

Receptor (protein molecule)

+

+

+

+

+

+

- -

-

Electrostatics View

Minimize Electrostatic Binding Energy

+

+

+

+

+

+- -

-++-

---

+ +

++

+

+

+

+

+- -

-++-

---

+ +

+

Higher energy Lower energy

Determine the charge distribution in the ligandso that it is “Energetically Optimized” to bind

+

ligand receptorbinding desolvation desolvation interactionE E E E= + −

Ligand(drug molecule)

++

-

--

-

+ +

+

Receptor (protein molecule)

+

+

+

+

+

+

- -

-

An Electrostatic Analysis Problem

+_

-++

H2O

Na+

Cl_

_

_

_

_

_

_

+

+

+

+ ++

+

++

+

+ - ++

-++

- ++

-+

+

-++

-+

+

- ++

-++-+

+

-+

+

-++

-+

+

Salt water

Homogeneoussalt water

highε

Macromolecularsurface of ligand or ligand-receptor complex

lowε

A Simplified Physical Description

Point charges of ligand in homogeneous dielectric

( ) 2( ) ( ) ( ) ( ) ( ) ( ) 0r r r r r rε ϕ ε κ ϕ ρ∇ ∇ − + =r r r r r rilinearized Poisson-Boltzmann equation:

homogeneous dielectric

Molecular Surface Representation

macromolecular atoms

point charges at atom centers

probe solvent molecule

Simplified Mathematical Model: Inside Macromolecule

Macromolecularsurface

2

1

( ) ( )N

iin i

i low

qr r rϕ δ

ε=

∇ = − −∑r r r

Poisson equation

lowε

Simplified Mathematical Model: Salt Water Outside

Macromolecularsurface

2 2( ) ( )out outr rϕ κ ϕ∇ =r r

Linearized Poisson-Boltzmann equation

highε

Inverse Debye screening length to model ions

Interface Condition

Boundary Conditions:

( ) ( )in outo r or r

n nϕ ϕ

ε∂ ∂

=∂ ∂

r r( ) ( )in o out or rϕ ϕ=r r

highr

low

εε

ε=

lowε

highε

n

Why Use this Simplified Model?

n Atomistic Level Simulation is too expensiven Salt ions and water molecules treated

individually

n Continuum Model Matches Well with Experimental Data

Standard Finite-Difference Method

*

lowε lowε

lowεlowε

lowε

highε

highε

highε

highε

highε

lowε

lowε

Problem 3: Inexact Boundary Conditions

Problem 2: Poor Point Charge Approximation

Problem 1: Inaccurate Molecular Surface

set up boundary conditions andsolve for grid potentials

Integral equation: Interior Problem

MacromolecularSurface Ω

1

( ) ( ; ) ( ; ) ( ) ( ; )N

in in kin in in k

k low

G qr r r G r r r dr G r r

n nϕ

ϕε=Ω

∂ ∂ ′ ′ ′ ′ ′− = ∂ ∂ ∑∫

r r r r r r r r r

1( ; )

4inG r rr rπ

′ =′−

r r r r

Homogeneous low dielectric medium

lowε

Translation invariant Green’s function

( )inG r r′= −r r

Integral equation: Exterior Problem

( ) ( ; ) ( ; ) ( ) 0out outout out

Gr r r G r r r dr

n nϕ

ϕΩ

∂ ∂ ′ ′ ′ ′ ′− + = ∂ ∂ ∫r r r r r r r

; ( )( )4

r r

to t ou ue

G r rr

G r rr

κ

π

′− −

′ =′

′= −−

r r rr rr r r

NOTE: κ is real,electrostatic not fullwave problem

nMatch boundary conditions

( )in

rϕ ′r 1( )in

rrn

ϕε

∂′

∂r

Homogeneous high dielectric medium

highε

Advantages For Integral Equation Formulation

n Directly discretize surfaces

1( ) ( ; ) ( ; ) ( ) 0out in

in outr

Gr r r G r r r dr

n nϕ

ϕεΩ

∂ ∂′ ′ ′ ′ ′− + = ∂ ∂ ∫

r r r r r r r1

( ) ( ; ) ( ; ) ( ) ( ; )N

in in kin in in k

k low

G qr r r G r r r dr G r r

n nϕ

ϕε=Ω

∂ ∂ ′ ′ ′ ′ ′− = ∂ ∂ ∑∫

r r r r r r r r r

n Point charges treated exactly

n Handles infinite exterior

Standard piecewise constant collocation discretization method

( ) ( )in j jj

r a B rϕ ≈ ∑r r

( ) ( )inj j

j

r b B rn

ϕ∂≈

∂ ∑r r

r ∈Ωr

§ Piecewise constant basis functions

B10

B15

B2

§ Collocation points at panel centroids

Matrix Equation

1 4

0

Nk

k i k

in

o

in

tu out

j

j

aq

DD

S rb

rS

π=

=

−

∑ r r

14

j

inij

ipanel

S drr rπ

′= −′−∫

rr r

14

i

j

r routij

r ipanel

eS dr

r r

κ

ε π

′− −

′=′−∫

r r rr r

14

j

inij

ipanel

D drn r rπ

∂ ′= ′ ′∂ − ∫

rr r

4

i

j

r routij

ipanel

eD dr

n r r

κ

π

′− − ∂ ′= − ′ ′∂ − ∫

r r rr r

A sphere molecule: comparison with analytical result

100

10

1

ERROR

(%)

Iterative solver' ' '( ) ( ; ) ( )r K r r r drϕ σ≡ ∫

r r r r r

[ ],i j j iσ ϕ Κ =

Discretization

IterativeSolver

matrix-vector multiply black box

j guessσ ,i j j guess

σ Κ

improve guess solution

GaussianElimination

[ ]1

,j i j iσ ϕ−

= Κ

for i = 1:n-1 for j = i+1:n

Kj,i = Kj,i / Ki,ifor k = i+1:n

Kj,k = Kj,k - Kj,i / Ki,kend

endend

3( )NΟ 2( )NΟ per iteration

Use Fast Integral Equation Solver

n Multiple Green’s functions

n Translation Invariant kernel

( ; )4

r r

oute

G r rr r

κ

π

′− −

′ =′−

r rr r r r

1( ; )

4inG r rr rπ

′ =′−

r r r r

( log )N NΟ Matrix-vector multiply

Pre-corrected FFT algorithm

Picture courtesy of J. Phillips

matrix-vector multiply black box

j guessσ ,i j j guess

σ Κ

pre-corrected FFT

charge distribution

potential due to space-invariant kernel

4) direct interaction and correction among near neighbors

(4)(1)

1) projection of panel charges onto grid charges

(2)

2) grid potentials due to grid charges are computed by FFT

(3)3) potentials on panel centroids are

interpolated from grid potentials

[ ] [ ] [ ]11,i j ijP Pσ ϕ−− = Κ

[ ] ,i jP ≈ Κ Need to find a good preconditioner

And solve

hopefully better conditioned than [Ki,j]

Qsi molecule Ecm protein

Preconditioner on Two Examples

Preconditioner result: Qsi molecule

OO

O OO O

OO O

O OO

II

Preconditioner result: Ecm protein

O OO O

Accuracy comparison with DelPhi

18596

5842

9330

# of salt panels

82868

34114

17204

# of dielectric panels

ECM

TSA

Water

-653.88-646.42

-34.75-34.62

-3.17-3.14

DelPhipFFT

Esolvation (kcal/mol)

Convergence Results of Ecm Protein

pFFT DelPhi

Binding energy calculation of a protein-peptide complex

Energy calculated (kcal/mol)

14.51

14.52

Rdesolvation

131.03131.0324.47DelPhi

130.91130.8024.47pFFT

(L->R)interaction(R->L)interactionLdesolvation

Conclusions and Future work

ØWorking on coupling to charge optimization problem in drug design

ØExtending formulation to include more complicated geometry (inner cavities in macromolecule)

ØFine tuning existing pre-corrected FFT code

ØCarefully selected Integral Formulationresults in Fast Solver for BiomoleculeElectrostatics