Fast Methods for Simulation of Biomolecule Electrostatics · 2019-11-20 · Biomolecule...

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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

+ - ++

Salt water

Homogeneoussalt water

highε

Macromolecularsurface of ligand or ligand-receptor complex

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

( ) ( )N

qr r rϕ δ

∇ = − −∑r r r

Poisson equation

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

highε

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

highε

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

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

in in kin in in k

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

ϕε=Ω

∂ ∂ ′ ′ ′ ′ ′− = ∂ ∂ ∑∫

r r r r r r r r r

1( ; )

4inG r rr rπ

′ =′−

r r r r

Homogeneous low dielectric medium

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

∂ ∂ ′ ′ ′ ′ ′− + = ∂ ∂ ∫r r r r r r r

; ( )( )4

to t ou ue

G r rr

′− −

′ =′

′= −−

r r rr rr r r

NOTE: κ is real,electrostatic not fullwave problem

nMatch boundary conditions

rϕ ′r 1( )in

∂′

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

ϕεΩ

∂ ∂′ ′ ′ ′ ′− + = ∂ ∂ ∫

r r r r r r r1

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

in in kin in in k

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

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

r b B rn

ϕ∂≈

∂ ∑r r

r ∈Ωr

§ Piecewise constant basis functions

§ Collocation points at panel centroids

Matrix Equation

tu out

∑ r r

ipanel

S drr rπ

′= −′−∫

r routij

r ipanel

′− −

′=′−∫

r r rr r

ipanel

D drn r rπ

∂ ′= ′ ′∂ − ∫

r routij

ipanel

′− − ∂ ′= − ′ ′∂ − ∫

r r rr r

A sphere molecule: comparison with analytical result

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

improve guess solution

GaussianElimination

,j i j iσ ϕ−

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

G r rr r

′− −

′ =′−

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

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) 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

O OO O

Preconditioner result: Ecm protein

O OO O

Accuracy comparison with DelPhi

# of salt panels

# of dielectric panels

-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)

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

Fast Methods for Simulation of Biomolecule Electrostatics · 2019-11-20 · Biomolecule...

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