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
Biophysics
My
research
Computer
science
(HPC)
Emphasizing the interdisciplinary
nature of computational biophysics
Fact: Water Makes Life Possible
Fox
L. Freberg
Vander
Kass ‘05
A Crucial Consequence of Solvation
• Molecular binding involves sacrificing solute--solvent
interactions for solute--solute interactions:
d=0
d=1
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
Modeling ions in solution
is critical! But today’s
 Single water: sphere of radius 1.4 Angstrom
focus is on the simpler
 Highly polarizable:  = 80
math of “pure” water.
• In total: mixed-dielectric Poisson
Solving the PDE Directly is Possible, But…
The idea: Just throw down
a finite-difference grid or a
finite-element mesh and
go to town!
PDE Complications
1. Boundary conditions are at infinity
2. Point charges must be spread onto the grid
3. The dielectric interface is approximated
Green’s Representation Formula
• Well known: boundary values of a harmonic
function determine it uniquely
Ex:
D
Dirichlet: given
Neumann: given
(For exterior domain!)
• The challenge is determining
S
given BV.
 Separation of variables
 Numerical: finite elements, finite differences..
 In 3 dimensions, solve for 3-dimensional unknown
• Alternatively: if you knew BOTH conditions, (3 dimensions)
Potential
Surface integrals ONLY
anywhere in D Thus: finding the other
boundary
condition gives you the answer directly
Deriving a Boundary Integral Equation:
Exterior Neumann Problem
• Given
r is in the
domain D
, need to find
r’ is on the
boundary S
• Let r approach surface
Given data
D
S
Addressing the Singularities
• Single-layer potential
D
S
• Dipole-layer
Limit depends on WHICH SIDE of the
Also continuous
asis approaching!!
surface
your point
Continuous as
z
z
r
+
+
+ ++ + ++ +
- - - - -
x
a
+
y
r
a
+ ++
+
+++++ ++- -- -
x
y
Deriving a Boundary Integral Equation:
Exterior Neumann Problem
• Given
, need to find
D
S
Why Bother With Integral Equations?
Easy problem:
Medium problem:
Exterior problems?
To infinity
Problems with mostly
empty, uninteresting space
Practical Advantages of PDE Approaches
PDE
•Accessible (many codes)
•Reliable, durable
•Versatile, does OK job
1.
2.
3.
4.
BIE
•Less accessible (few codes)
•Hard to convince it to run
•Does one thing really well
Much more general (nonlinearity, etc)
Easier to parallelize (that’s different from “easy”)
Often easier to explore model space (see point 1)
PDE solvers give sparse systems; BIE, dense systems!
Similarity Between FEM and BEM
• Both weighted residual methods:
BEM
FEM
1 on panel i
0 elsewhere
Enforce
(Galerkin method)
Enforce
Galerkin:
Differences Between BEM and FEM
1. Extra freedom in choosing test functions
Collocation: test = delta functions
Centroids of
elements
2. Matrix elements are harder to compute
Galerkin FEM:
Smooth integrand:
Easily computed
with quadrature!
Galerkin BEM:
Double integral of a
singular function!!
Fast Solvers for Integral Equations
1. Solve Ax=b approximately using Krylov-subspace iterative methods such
as GMRES:
2) time and memory
Storing
matrix:
O(N
2. Compute dense matrix-vector product using O(N) method (fast multipole;
Each FFT;
multiply:
O(N2) time
tree code; precorrected
FFTSVD)
Storing compressed matrix: O(N) time and memory
3. Improve iterative convergence
withO(N)
preconditioning
Each multiply:
time
4. For many problems, use diagonal entries!
Iteration converges faster if matrix
eigenvalues are “well clustered”
P “looks like” A-1
A Boundary Integral Method For
Biomolecule Electrostatics
+
+ + + + +- - -
Conservation law
Constitutive relation
+
+
+
- -
-
-
1. Boundary conditions handled exactly
2. Point charges are treated exactly
3. Meshing emphasis can be placed directly on the interface
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:
Single +1 charge
+1, -1 charges 3 A apart
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
 Sphere: analytical
A hundred years of analysis!
• Eigenvalues are real
  in [-1/2,+1/2)
• -1/2 is always an EV
• Left, right eigenvectors of -1/2
are constants
-1/10
-1/6
-1/2
• BIBEE approximates E’s eigenvalues
 P uses 0 (limit for sphere, prolate spheroid)
 CFA uses -1/2 (known extremal)
i
BIBEE Clarifies an Empirical, Heuristic Model
+
+
BIBEE approx. charge includes
all contributions
R1
“Effective Born radius” - the
radius of a sphere with the same
solvation
R2energy
R3
Coulomb-field approximation:
corresponds exactly to ignoring
Stillthe
equation:
theoperator.
basis of totally
integral
nonphysical Generalized Born (GB) models
BIBEE/CFA is the extension of CFA to multiple charges!
No ad hoc parameters, no heuristic interpolation
Same approach taken by Borgis et al. in variational CFA
BIBEE/CFA Energy Is a Provable Upper Bound
Feig et al. test set, > 600 proteins
• 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)
BIBEE: Improve by Analyzing the Sphere

-1/10
i
-1/6
-1/2
• 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)
BIBEE: Accurate One-parameter Model

-1/10
-1/6
-1/2
i
Dominant energies come from
dominant modes: try to
capture dipole/quadrupole
modes approximately!
• This effective parameter is expected to be
rigorously determined by approximating protein
as ellipsoid (Onufriev+Sigalov, ‘06)
Bardhan+Knepley, J. Chem. Phys. (in press)
BIBEE: A New, Rigorous Model
Tripeptide
Protein-Drug
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)
SGB/CFA (heuristic) time
1.198
2.623
7.070
14.611
28.861
BIBEE time
2.974
6.540
18.125
39.066
77.205
# boundary elements
Total BEM time
Matrix compression time
• 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
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
-10
-20
-30
20
40
Eigenvalue Index
60
80
Cumulative Electrostatic Free Energy (kcal/mol)
Projection of charge distribution onto eigenvector
• Here, 25% of the total energy comes from modes with eigenvalues
smaller than 1% of the maximum eigenvalue
Eigenvalue Magnitude
104
102
100
10-2
20
40
60
Eigenvalue Index
80
BIBEE Is An Accurate, Parameter-Free Model
Snapshots from MD
• Peptide example
Met-enkephalin
BIBEE’s stronger “diagonal” appearance
indicates superior reproduction of the
All models
essentially the same here.
eigenvectors
of the look
operator.
SGB/CFA
GBMV
BIBEE/CFA
BIBEE: A New, Rigorous Model of
Continuum Electrostatics for Proteins
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
Applied
Math
Biophysics
BIBEE
Computer
science
(HPC)
Nonlocal Continuum Electrostatics:
Adding molecular realism “the right way”
KNOWN
weaknesses
of Poisson
model:
First
look for ways
to extend
1. Linearmodels--don’t
response assumption
existing
just give up
and reinvent everything!
Nonlinearity IS important for more highly charged species!
Test Caveat:
with allatom molecular
2. Violates continuum-length-scale assumption
dynamics
Oxygen
Relatively
small
deviation!
Lone pair electrons
Hydrogen bonds
y=x denotes exactly
Water molecules have finite size linear response
Hydrogens


Water molecules form semi-structured networks
Nina, Beglov, Roux ‘97
Nonlocal Continuum Electrostatics:
Demonstrating the Failure Mode
Run all-atom molecular dynamics: ion surrounded by water
Consequence: ion energies are wrong
Significant structuring
of charge density!
Data points: radii from molecular
simulation (Aqvist 1990) and
energies from experimental data
Ion radius in nanometers
Nonlocal Continuum Modeling:
A Classical Multiscale Theory
• Studied since the 1970s in numerous domains
Problems whose length
scales are NOT well
separated from those of the
constituent molecules!
de Abajo ‘08
Duan et al. ‘07
Schatz
et al. ‘01nonlocal
Expect
Park ‘06
theory to
play major rolesScott
in et al. ‘04
nanoscale science and
Gao et al.,
‘09
engineering
modeling…
Nonlocal Continuum Electrostatics:
Nonlocal Dielectric Response
• 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
Supported by
experiments
and
Local
response
detailed
simulations
Wave number
(inverse distance)
Smoothly interpolates
between known limits
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.
2.
3.
Define an auxiliary field: the displacement potential
Molecular surface
“Licorice”
“Cartoon”
Approximate the nonlocal boundary condition
Double reciprocity leads to a boundary-integral method
Nonlocal Continuum Electrostatics:
1. Introduce an Auxiliary Potential
• Use Helmholtz decomposition:
• Electrostatic potential now satisfies a Yukawa equation:
Yukawa/linearized PoissonBoltzmann 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
Nonlocal boundary condition: Choose to drop
• The actual PDEs complete the local formulation:
Nonlocal Continuum Electrostatics:
3. Green’s Theorem + Double Reciprocity
• Electric potential Green’s theorem gives a volume integral
• The displacement potential is harmonic:
0
• Defining single- and double-layer operators
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
Alloperators
of theseshare
are diagonal
For sphere, these
a
common eigenbasis: spherical harmonics
• 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
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
Local theory needs
unrealistically large
dielectric constants to
match experiment!
3
2
Error in pKa
value (RMSD)
1
Ion or charged chemical
group, buried in protein
0
Measured protein
dielectric constants
suggest  = 2-5
5
20
Demchuk+Wade, 1996
40
60
80
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)
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
# of boundary 1000 10,000 100,000 1,000,000
elements
Dense BEM
Fast BEM
Illustration of surface
Memory
7
700
70,000
representations
for 0.07
needed (GB)
memory-constrained
dense and fast BEM
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)
Required accuracy
Dense methods used
previously could not
achieve useful accuracy!
Bardhan and Hildebrandt, DAC ‘11
Nonlocal Continuum Electrostatics:
Fast BIE Solver Enables Tests on Proteins
Local 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”
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
Applied
Math
Biophysics
Nonlocal
model
Computer
science
(HPC)
Summary:
• Improve understanding
of existing models
• Develop new models on
strong foundations
Applied
Math
• Stringent tests of new
models
Biophysics
My
research
• Leverage HPC expertise by
re-using computational
primitives
• “Think computationally” to
gain new insights into model
development
Computer
science
(HPC)
• Identify critical model
weaknesses
• Explain previously
unresolved phenomena
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)