Understanding Protein Electrostatics Using Boundary-

Integral Equations

Jaydeep P. Bardhan Dept. of Physiology and Molecular Biophysics

Rush University Medical Center, Chicago IL

Joint work with • M. Knepley (Computation Institute, U. Chicago)• P. Brune (Math and Computer Science Division, Argonne)• A. Hildebrandt (J. Gutenberg U., Mainz, Germany)

Outline:• Preliminaries:

Biomolecule electrostatics Continuum theory and

boundary-integral methods Numerical simulation

1.Fast Poisson approximation

2.Nonlocal continuum model

Applied Math

Computer science(HPC)

Biophysics

My research

Emphasizing the interdisciplinary nature of computational biophysics

Fact: Water Makes Life Possible

L. Freberg

Vander

Fox

Kass ‘05

A Crucial Consequence of Solvation

• Molecular binding involves sacrificing solute--solvent interactions for solute--solute interactions:

Basic Continuum Electrostatic Theory

• 100-1000 times faster than MD

• Protein model: Shape: “union of spheres” (atoms) Point charges at atom centers Not very polarizable: = 2-4

• Water model: no fixed charges Single water: sphere of radius 1.4

Angstrom Highly polarizable: = 80

• In total: mixed-dielectric Poisson

Modeling ions in solution is critical! But today’s focus is on the simpler math of “pure” water.

Solving the PDE Directly is Possible, But…

1. Boundary conditions are at infinity

2. Point charges must be spread onto the grid

3. The dielectric interface is approximated

PDE Complications

The idea: Just throw down a finite-difference grid or a finite-element mesh and go to town!

Green’s Representation Formula• Well known: boundary values of a

harmonic function determine it uniquely

• The challenge is determining given BV. Separation of variables Numerical: finite elements, finite differences.. In 3 dimensions, solve for 3-dimensional

unknown

• Alternatively: if you knew BOTH conditions,

D

Dirichlet: given

Thus: finding the other boundary condition gives you the answer

directly

Ex:S

Potential anywhere in

D

(3 dimensions)

Surface integrals ONLY

Neumann: given(For exterior domain!)

Deriving a Boundary Integral Equation: Exterior Neumann

Problem• Given , need to find

• Let r approach surface

r is in the domain D

Given datar ’ is on the boundary S

D

S

Addressing the Singularities• Single-layer potential

• Dipole-layer

ax

y

z

+ ++++ +

r

D

S

ax

y

zr

+ +++

++

Continuous as Also continuous as

-+

-+

-+

-+

-+

-+

-+ -

+-+ -

+

Limit depends on WHICH SIDE of the surface your point is approaching!!

Deriving a Boundary Integral Equation: Exterior Neumann

Problem• Given , need to find

D

S

Why Bother With Integral Equations?Easy problem:

Exterior problems?

To infinity

Problems with mostly empty, uninteresting

space

Medium problem:

Practical Advantages of PDE Approaches

1. Much more general (nonlinearity, etc)2. Easier to parallelize (that’s different from “easy”)3. Often easier to explore model space (see point 1)4. PDE solvers give sparse systems; BIE, dense systems!

•Less accessible (few codes)•Hard to convince it to run•Does one thing really well

•Accessible (many codes)•Reliable, durable•Versatile, does OK job

PDE BIE

Similarity Between FEM and BEM• Both weighted residual methods:

FEM BEM

1 on panel i0 elsewhere

Enforce Enforce

(Galerkin method) Galerkin:

Differences Between BEM and FEM

1. Extra freedom in choosing test functions

2. Matrix elements are harder to compute

Collocation: test = delta functions

Centroids of elements

Galerkin BEM:Galerkin FEM:

Smooth integrand: Easily computed with quadrature!

Double integral of a singular function!!

Fast Solvers for Integral Equations1. Solve Ax=b approximately using Krylov-subspace iterative

methods such as GMRES:

2. Compute dense matrix-vector product using O(N) method (fast

multipole; tree code; precorrected FFT; FFTSVD)

3. Improve iterative convergence with preconditioning

4. For many problems, use diagonal entries!

Storing matrix: O(N2) time and memoryEach multiply: O(N2) time

Storing compressed matrix: O(N) time and memoryEach multiply: O(N) time

P “looks like” A-1

Iteration converges faster if matrix eigenvalues are “well clustered”

A Boundary Integral Method For Biomolecule Electrostatics

+ -++

++ + + ++

---

-- -

1. Boundary conditions handled exactly

2. Point charges are treated exactly

3. Meshing emphasis can be placed directly on the interface

Conservation law

Constitutive relation

BIBEE: A New, Rigorous Model of Continuum Electrostatics for

Proteins“Boundary Integral Based Electrostatics Estimation”

• Idea: Use preconditioner to approximate inverse

No need to compute sparsified operator (saves time and memory)

No need for Krylov solve

• Test of elementary charges in a 20-Angstrom sphere:

+1, -1 charges 3 A apartSingle +1 charge

BIBEE: Introducing Different Variants• The preconditioning approximation takes into account the

singular character of the electric-field kernel:

• The Coulomb-field approximation ignores the operator entirely:

CFA seems better here… …and worse here.

BIBEE Approximates the Eigenvalues of the Boundary Integral Operator

• The integral operator has to be split into two terms

• BIBEE approximates E’s eigenvalues P uses 0 (limit for sphere, prolate

spheroid) CFA uses -1/2 (known extremal)

i

-1/2

-1/6

-1/10

• Eigenvalues are real in [-1/2,+1/2)• -1/2 is always an EV• Left, right eigenvectors of -1/2

are constants

A hundred years of analysis!

Sphere: analytical

BIBEE Clarifies an Empirical, Heuristic Model

R1 R2 R3

+ +BIBEE approx. charge

includes all contributions

Coulomb-field approximation: corresponds

exactly to ignoring the integral operator.

BIBEE/CFA is the extension of CFA to multiple charges!

No ad hoc parameters, no heuristic interpolation

Still equation: the basis of totally nonphysical Generalized Born (GB)

models

“Effective Born radius” - the radius of a sphere with the same solvation energy

Same approach taken by Borgis et al. in variational CFA

BIBEE/CFA Energy Is a Provable Upper Bound

• BIBEE/P is an effective lower bound, provable in some cases but not all• Another variant (BIBEE/LB) is a provable LB but too loose to be useful

Bardhan, Knepley, Anitescu (2009)

Feig et al. test set, > 600 proteins

BIBEE: Improve by Analyzing the Sphere

• Get first mode (monopole) analytically correct, other modes are bounded from below: tighter lower bound!

• Impact on sphere is better than impact on proteins (Feig et al. test set)

Bardhan+Knepley, J. Chem. Phys. (in press)

i

-1/2

-1/6

-1/10

BIBEE: Accurate One-parameter Model

• This effective parameter is expected to be rigorously determined by approximating protein as ellipsoid (Onufriev+Sigalov, ‘06)

Bardhan+Knepley, J. Chem. Phys. (in press)

i

-1/2

-1/6

-1/10

Dominant energies come from dominant modes: try

to capture dipole/quadrupole modes

approximately!

BIBEE: A New, Rigorous Model

• BIBEE is 3-5 times faster than full solve (including large setup time for both)

• Unoptimized implementation (will save big on setup time)

• Modern FMM implementation (Yokota, Knepley, Barba, et al.) gives 10-20X speedup

3968 7564 15,212 32,022 49,708

18.368 24.493 87.647 515.256 735.092

(3.271) (6.665) (18.274) (62.217) (109.040)

1.198 2.623 7.070 14.611 28.861

2.974 6.540 18.125 39.066 77.205

Tripeptide Protein-Drug

# boundary elements

Matrix compression time

Total BEM time

SGB/CFA (heuristic) time

BIBEE time

Reaction-Potential Operator Eigenvectors Have Physical Meaning

• Eigenvectors from distinct eigenvalues are orthogonal

• Thus: the eigenvectors correspond to charge distributions that do not interact via solvent polarization (weird, huh?)

• If an approximate method generates a solvation matrix , its eigenvectors should “line up” well with the actual eigenvectors, i.e.

i = j

“Getting the Modes Right” Is Important• Modes from small eigenvalues still contribute significantly

to the total energy

• Here, 25% of the total energy comes from modes with eigenvalues smaller than 1% of the maximum eigenvalue

Pro

jec

tio

n o

f ch

arg

e d

istr

ibu

tio

n o

nto

eig

enve

cto

r

Cu

mu

lati

ve E

lect

ros

tati

c F

ree

En

erg

y (k

cal/

mo

l)

102

100

10-2

104

Eigenvalue Magnitude

Eigenvalue Index

20 40 60 80

Eigenvalue Index

20 40 60 80-30

-20

-10

BIBEE Is An Accurate, Parameter-Free Model

• Peptide example

SGB/CFA GBMV BIBEE/CFA

Met-enkephalin

Snapshots from MD

All models look essentially the same here.

BIBEE’s stronger “diagonal” appearance indicates superior reproduction of the

eigenvectors of the operator.

BIBEE: A New, Rigorous Model of Continuum Electrostatics for

ProteinsApplied

Math

Computer science(HPC)

Biophysics

BIBEE

Design systematic approximation

Have proved that the model: • Gives upper and lower bounds• Preserves important physics

Relate empirical models to strong math

Leverages existing algorithms (e.g. fast multipole methods, parallel codes)

Next: Apply to other physics problems

Nonlocal Continuum Electrostatics: Adding molecular realism “the right

way”KNOWN weaknesses of Poisson model:

1. Linear response assumption

Caveat: Nonlinearity IS important for more highly charged species!

2. Violates continuum-length-scale assumption

Water molecules have finite size Water molecules form semi-structured networks

Oxygen

Hydrogens

Lone pair electrons

Hydrogen bonds

Nina, Beglov, Roux ‘97

Test with all-atom molecular dynamics

y=x denotes exactly linear response

Relatively small deviation!

First look for ways to extend existing models--don’t just give up

and reinvent everything!

Nonlocal Continuum Electrostatics: Demonstrating the Failure Mode

Run all-atom molecular dynamics: ion surrounded by water

Ion radius in nanometers

Data points: radii from molecular simulation (Aqvist 1990) and energies from experimental data

Consequence: ion energies are wrong

Significant structuring of

charge density!

Nonlocal Continuum Modeling:A Classical Multiscale Theory

• Studied since the 1970s in numerous domains

de Abajo ‘08Schatz et al. ‘01

Park ‘06

Scott et al. ‘04Duan et al. ‘07

Gao et al., ‘09

Problems whose length scales are NOT well

separated from those of the constituent

molecules!

Expect nonlocal theory to play major roles in

nanoscale science and engineering modeling…

Nonlocal Continuum Electrostatics:Nonlocal Dielectric Response

Smoothly interpolates between known limits

Supported by experiments and detailed simulations

Local response

Wave number (inverse distance)

• Polarization charge as a function of distance from the ion: not simple

Short-range: electronic response

Long-range: bulk behavior

• Local: bulk everywhere

• Nonlocal: simple function that captures asymptotes

Nonlocal Continuum Electrostatics:Lorentzian Model and Promising

Tests• Nonlocal response:

• Now

• Integrodifferential Poisson equation

Green’s function for

Single parameter fit for gives much better agreement with experiment!!

Nonlocal Continuum Electrostatics:Reformulation for Fast Simulations

• Integrodifferential equations in complex geometries?

• Result: No progress on nonlocal model for DECADES

Spherical ions, charges near planar half-spaces… nothing else.

• Breakthrough in 2004 (Hildebrandt et al.):1. Define an auxiliary field: the displacement potential2. Approximate the nonlocal boundary condition3. Double reciprocity leads to a boundary-integral

method

“Licorice” “Cartoon” Molecular surface

Nonlocal Continuum Electrostatics:1. Introduce an Auxiliary Potential

• Use Helmholtz decomposition:

• Electrostatic potential now satisfies a Yukawa equation:

Yukawa/linearized Poisson-Boltzmann

equation

Displacement potential acts as a volume source

Nonlocal Continuum Electrostatics:2. Approximate Nonlocal B.C.

• Original boundary conditions:

• Exact normal deriv. of solvent potentials satisfy

• The actual PDEs complete the local formulation:Nonlocal boundary condition: Choose to drop

Nonlocal Continuum Electrostatics:3. Green’s Theorem + Double

Reciprocity• Electric potential Green’s theorem gives a volume integral

• The displacement potential is harmonic:

• Defining single- and double-layer operators

0

Nonlocal Continuum Electrostatics: Purely BIE Formulation

• Three surface variables, two types of Green’s functions, and a mixed first-second kind problem

• Fasel et al. have recently derived a purely second-kind method

Hildebrandt et al. 2005, 2007

Nonlocal Continuum Electrostatics: Analytical Solution for Sphere

• Solve each mode independently and presto!

• Note: This is not about matching interior and exterior expansions--unlike the Kirkwood solution for local model

• This decomposition may provide further analytical insights (e.g., eigenvectors of reaction-potential operator)

Bardhan and Brune, to be submitted

For sphere, these operators share a common eigenbasis: spherical

harmonics

All of these are diagonal

Nonlocal Continuum Electrostatics: Charge Burial and the pKa Problem

• Understanding charge burial energetics is important! For protein folding, misfolding (Alzheimer’s), etc. For two molecules binding (drug-protein, protein-protein, etc.) For change in environment (pH, temperature, concentration,

etc.)

Ion or charged chemical group, alone in water

Ion or charged chemical group, buried in protein

Demchuk+Wade, 1996

Local theory needs unrealistically large dielectric constants to match experiment!

3

2

1

0

Error in pKa value

(RMSD)

20 40 60 805

Measured protein dielectric constants

suggest = 2-5

Nonlocal Continuum Electrostatics: Charge Burial and the pKa Problem

• Nonlocal theory with realistic dielectric constant predicts similar energies as (widely successful) local theories with unrealistic dielectric constants!

Bardhan, J. Chem. Phys. (in press)

Dense BEM Fast BEM

Nonlocal Continuum Electrostatics: Fast Solver is a Must for Accurate

Studies

• O(N2) memory limitation: big discretization errors• O(N) fast solver: only way to get accurate energies

Illustration of surface representations for memory-constrained dense and fast BEM

# of boundary elements

1000

10,000

100,000

1,000,000

Memory needed (GB)

0.07 7 700 70,000

Nonlocal Continuum Electrostatics: Fast BIE Solver Performance

• Time and memory scale linearly in the number of unknowns

• Unoptimized code still allows a laptop to solve 10X larger problems than is possible on a cluster

• Preconditioning is vital (use diagonal entries of blocks)

Dense methods used previously could not achieve useful accuracy!

Required accuracy

Bardhan and Hildebrandt, DAC ‘11

Nonlocal Continuum Electrostatics:Fast BIE Solver Enables Tests on

ProteinsLocal Model Nonlocal Model

• Observe reduced “electrostatic focusing” • Next step: compare to molecular dynamics

Bardhan and Hildebrandt, DAC ‘11

Nonlocal Continuum Electrostatics: Adding molecular realism “the right

way”Applied

Math

Computer science(HPC)

Biophysics

Nonlocalmodel

Extend the space of models that are supported by good theory

Derive fast analytical methods for testing the new theories

Test on important open questions

Build high-performance solvers for realistic, accurate simulations

Summary:

Applied Math

Computer science(HPC)

Biophysics

My research

• Improve understanding of existing models

• Develop new models on strong foundations

• Stringent tests of new models

• Identify critical model weaknesses

• Explain previously unresolved phenomena

• Leverage HPC expertise by re-using computational primitives

• “Think computationally” to gain new insights into model development

Acknowledgments• Support:

Wilkinson Fellowship at Argonne National Lab

Partial support from a Rush University New Investigator award

• Colleagues: Ridgway Scott (U. Chicago) Bob Eisenberg, Dirk Gillespie (Rush) Mala Radhakrishnan (Wellesley) Nathan Baker (Pacific Northwest Nat’l Lab)