Boundary-element methods in
biological science and engineering
Jaydeep P. Bardhan
Dept. of Electrical and Computer Engineering
Northeastern University, Boston MA
Four Points For Today
1. Cellular and molecular biomedical problems also
need efficient simulation methods.
2. Fast BEM solvers represent an appealing approach
even at the molecular scale!
1. Challenge: Persuading community to abandon
beloved ad hoc fast methods for systematic ones.
2. Strategy: Systematic methods are more flexible as
we add new physics and address inverse problems.
Point 1: Efficient solvers are needed not only
for macroscopic biomedical problems…
100 m
Human
body
10-1 m
10-2 m
Organs/
tissues
Electrical Impedance
Tomography (EIT)
Electrocardiography
(ECG)
10-5 m
10-9 m
Human
cells
Molecules
Tumor growth
Transport through
blood vessel walls
Human eye
for keratoplasty
Balsim et al. ‘10
Lowengrub et al. ‘09
Electroencephalography (EEG)
Brain-computer interfaces
Cochlea (ear)
Peratta et al. ‘08
Hip prostheses
... And MEG
Prof. Bin He, UMN
Briare et al. ‘00
Sfantos et al ‘07
… but for microscopic ones as well!
100 m
Human
body
10-1 m
10-2 m
Organs/
tissues
Blood flow
Quantum mechanics
10-5 m
10-9 m
Human
cells
Molecules
Biomolecule electrostatics and
hydrodynamics
Nanotechnology
(quantum dots)
Rahimian, Biros et al (2010)
• Drug binding
• Protein folding
• Cell physiology
• Molecular design
Cell locomotion using flagella (sperm, bacteria)
Gelbard ‘01
Cell “rolling” along tissue surface
Molecular flexibility
Ramia ‘91
Cell adhesion to surfaces under shear flow
Gaver and Kute ‘98
King and Hammer ‘01
M. Bathe ‘08
A central molecular-scale modeling
problem: water.
 Biology uses water to control molecular binding, protein
folding, etc. Binding example is simple:
Protein
d=0
d=1
Basic Continuum Electrostatic Theory
 100-1000 times faster than MD
 Protein model:
o Shape: “union of spheres” (atoms)
o Point charges at atom centers
o Not very polarizable:  = 2-4
 Water model: no fixed charges
Linearized PoissonBoltzmann equation
o Single water: sphere of radius 1.4 Angstrom
Modeling ions in solution is
o Highly polarizable:  = 80
 In total: mixed-dielectric Poisson
critical! But today’s focus
is on the simpler math of
“pure” water.
A Boundary Integral Method For the
Poisson Biomolecule Problem
+
+ + + + +- - -
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
Fast BEM Solvers are Essential
1. Solve Ax=b approximately using Krylov-subspace iterative methods such
as
GMRES:
Memory
growth is QUADRATIC
Time is CUBIC!!
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!
Iteration converges faster if matrix
eigenvalues are “well clustered”
PReplace
“looks like”
A-1
quadratic
memory and cubic time
requirements with LINEAR requirements!
Application-Specific Challenges
1. “Continuum-solvent dynamics”
o
o
2.
o
o
Replace water molecules with dielectric
Calculate forces at each time step and integrate
These lead to thousands, or even
millions, of electrostatic
Continuum post-processing
of molecular dynamics
simulations...
Sample structures from explicit-water MD
Compute average continuum energy from samples
Some with identical dielectric
3. Electrostatic
component
boundaries,
someanalysis
with changing
o Compute each atom’s interaction with every other
boundaries!
o Useful in drug design and protein engineering!
Community’s Solution:
Fast Ad Hoc Models
 Up to 100X faster than solving Poisson
 Define effective (nonphysical) parameters
A given charge q in
complex molecule gives
rise to an energy E
Find the radius R of a sphere
that would have the same
energy given a central
charge
 Plug in to ad hoc (nonphysical) formula
Distance between charges
Effective radii
Generalized Born theory
 Can give blatantly unphysical results …
 … exhibits incorrect dependence on
dielectric constants…
 … needs all manner of handwaving
justifications for improvements …
 … is VERY, VERY popular.
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: Natural, Rigorous Generalized Born
+
+
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
Complementary Regimes of Accuracy
•
Small molecule’s reaction potential
matrix and eigendecomposition (not
the integral operator)
•
Top right: the electric fields induced by
several eigenvectors of L, at the
dielectric boundary
•
Charge distributions that generate
uniform displacement fields are “like”
low-order multipoles: CFA does well
here and P does poorly
•
Small eigenvalues are associated with
charge distributions that generate
rapidly varying displacement fields;
these are approximated well by P, not
CFA
V1
V2
V20
Goal: Make Fast Models More Rigorous
Many advantages for chemists/biophysicists:
1. Enables systematic model improvement
2. Prove approximation properties
3. Leverage existing fast, scalable algorithms
4. Can add better physics as we learn them
5. Natural coupling to inverse problems
We really want to approximate the dominant
modes of the integral operator.
 The integral operator has to be split
into two terms

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
o P uses 0 (limit for sphere, prolate spheroid)
o CFA uses -1/2 (known extremal)
Sphere: analytical
i
Mathematical Rigor Enables
Systematic Improvements
Snapshots from MD
Many parameters and
ad hoc correction terms

-1/10
-1/6
-1/2
i
Dominant energies come from
dominant modes: try to capture
dipole/quadrupole modes
approximately!
BIBEE fluctuations track actual ones
very closely – possible applications in
uncertainty quantification
Mean absolute error: 4% !
• This effective parameter is expected to be rigorously
determined by approximating protein as ellipsoid
(Onufriev+Sigalov, ‘06)
Bardhan+Knepley, J. Chem. Phys. (in press)
Goal: Make Fast Models More Rigorous
Many advantages for chemists/biophysicists:
 Enables systematic model improvement
2. Prove approximation properties
3. Leverage existing fast, scalable algorithms
4. Can add better physics as we learn them
5. Natural coupling to inverse problems
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)
The Reaction-Potential Matrix
 A weighted combination of charge distributions in the
solute molecule produces a weighted combination of
the individual responses:
 The “canonical” basis is the natural, atom-based point
of view
 We can also use the eigenvector basis for analysis!
 In comparing models we don’t just have to use the total
electrostatic solvation free energy
Reaction-Potential Operator Eigenvectors
Have Physical Meaning
• Eigenvectors from distinct eigenvalues are orthogonal
• Eigenvectors correspond to charge distributions that do not
interact via solvent polarization (this confuses chemists)
• If an approximate method generates a solvation matrix
its eigenvectors should “line up” well with the actual
eigenvectors, i.e.
i=j
,
BIBEE in Separable Geometries
 For half-spaces, spheres, ellipsoids, BIBEE
exactly reproduces actual eigenvectors.
 Proof for spheres, ellipsoids: use appropriate
harmonics
 Question for future: What about near
separable geometries?
Bardhan and Knepley, 2011
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
Goal: Make Fast Models More Rigorous
Many advantages for chemists/biophysicists:
 Enables systematic model improvement
 Prove approximation properties
3. Leverage existing fast, scalable algorithms
4. Can add better physics as we learn them
5. Natural coupling to inverse problems
Pre-corrected FFT Algorithm
• Potential calculation is a convolution.
• Convolutions are “cheap” in frequency space
• Green’s function independent! (Laplace,
Helmholtz, Stokes, etc.)
1.
2.
3.
4.
Project charges to grid
Point-wise multiplication in frequency space
Interpolate grid potentials
“Pre-correct” so that local interactions are accurate
Proteins
Kuo, Altman,
Bardhan, Tidor,
White (2002)
Circuit Simulation
Aerodynamics
Bioelectromagnetics
Willis, Peraire, White
Cadence Design Systems
Phillips and White (1997)
Higher-order Protein BEM
A geometry representative of a protein:
The barnase-barstar protein complex:
27
Bardhan + Altman et al., 2007
Altman + Bardhan, White, Tidor 2009
Develop scalable protein simulations with
leaders in parallel computing + FMM
760-node GPU cluster Degima
Parallel GPU FMM code
Picture courtesy T. Hamada
Cost of cluster: ~ US $420,000
Sustained: 34.6 Tflops
Application to proteins with PetFMM code of
Yokota, Cruz, Barba, Knepley, Hamada
Performance/price: 80 Mflops/$
Scalable algorithms enable bigger science
Lysozyme: ~2K atom charges, ~15K surface charges
1000 copies
800 Å
1000 lysozyme
molecules: model of
a concentrated
protein solution
1 copy
100 copies
10 copies
 “How do proteins work in the crowded environment of
the cell?”
Yokota, Bardhan, et al. 2009
Goal: Make Fast Models More Rigorous
Many advantages for chemists/biophysicists:
 Enables systematic model improvement
 Prove approximation properties
 Leverage existing fast, scalable algorithms
4. Can add better physics as we learn them
5. Natural coupling to inverse problems
We are still adding physics to our models.
“Classical” modeling: one can
assume the model is right!
Circuit simulation: Maxwell equations
Solid mechanics: elasticity
Airplane simulation: Navier-Stokes
These are just the
models associated
with the molecular
scale!!
Speed
CAD tools
All simulate
same thing!
Accuracy
Bio-modeling: “All models
are wrong, some are
useful”*
Diverse set of flawed models.
To avoid flaws, use expert insight.
New models are always evolving!
We have to connect multiple
models (uncertainty
quantification).
--George Box
Adding physics to the continuum model
using nonlocal dielectric theory
KNOWN weaknesses of Poisson model:
1. Linear response assumption
Nonlinear dielectrics ARE important for some molecules!
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:
Lorentzian Model and Promising Tests
Nonlocal response:
Now
Integrodifferential
Poisson equation
A. Hildebrandt et al. 2004
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:
Purely BIE Formulation
 Three surface variables, two types of Green’s functions,
and a mixed first-second kind problem
 The derivation uses double reciprocity theory, which can
be applied to nonlinear problems as well!
Have derived exact solution for charges in a sphere
Hildebrandt et al. 2005, 2007
Just as fast, but now with better physics!
 Unoptimized code still allows a
laptop to solve 10X larger
problems than is possible on a
cluster with dense methods
 Current work: comparing to
molecular dynamics simulations
Required accuracy
Dense methods used
previously could not
achieve useful accuracy!
Local model
Nonlocal model
Bardhan and Hildebrandt, DAC ‘11
Nonlocal Continuum Electrostatics:
Charge Burial and the pKa Problem
Understanding charge burial energetics is important!
o For protein folding, misfolding (Alzheimer’s), etc.
o For two molecules binding (drug-protein, protein-protein, etc.)
o 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)
A Common Framework for Multiple Models
GB-like
fast nonlocal
BIBEE
provides a unifying,
approximateapproach
model
scalable
to testing and
Fast GB-like nonlinear
extending
new of
physics.
Analytical
solution
approximations
nonlocal model for sphere
Improved GB
models
Explain Coulombfield approx.
Linearized
PB models
Biomolecular
complexes
Full nonlinear PB via
boundary-integrals
Coupling to fast, scalable
algorithms
Advanced PB
models
(Bikerman, etc.)
Dynamics: hybrid
explicit/implicit, and fully
implicit
Popular quantum methods couple
to exactly our Poisson problem
(“polarizable continuum model”)
Protein 1
Protein 2
Goal: Make Fast Models More Rigorous
Many advantages for chemists/biophysicists:
 Enables systematic model improvement
 Prove approximation properties
 Leverage existing fast, scalable algorithms
 Can add better physics as we learn them
5. Natural coupling to inverse problems
The Value of Systematic Approximations in
Inverse Problems: Biomolecule Design
 The electrostatic contribution to binding is
 A total of three simulations is needed.
Electrostatic Optimization of Biomolecules:
Applications in Analysis and Design
 E. coli chorismate mutase
inhibitors:
o Analyzed by Kangas and Tidor
o Suggested substitution
experimentally verified: result is
the tightest-binding inhibitor yet
known
 Barnase/barstar protein
complex:
o Tight-binding complex
o Optimal charge distribution closely
matches “wild-type” charge
distribution
Mandal and
Hilvert, 2003
Lee and Tidor, 2001
Challenge: Optimization is SLOW.
10 min/simulation *
2000 simulations (protein) = 2 CPU
A Novel Method: The Reverse-Schur Approach
 For these PDE constraints, we really only need to
solve multiple systems simultaneously:
 The unconstrained problem is therefore
 Constraints can be handled using standard methods
(Lagrange multipliers, etc.)
New Approach is Dozens to Hundreds
of Times Faster, but Formally Exact
Formally exact calculation
Method scales comparably with
normal PDE-constrained approaches
10 min/simulation = 20 min/optimization
(no matter how many charges!)
Bardhan et al., 2004; Bardhan et al., 2005; Bardhan et
al., 2007; Bardhan et al., 2009
BIBEE as Inverse Problem Regularizer
 Regularization can be performed using
“approximate” penalty functions:
Approximated eigenvectors closely
match actual ones
 No linear solve: Accurate but 10-20X
faster than simulation!!
 BIBEE/P captures small eigenvalues
very accurately  identify number of
directions to penalize
Single +1 charge
+1, -1 charges 3 A apart
Application: Cyclin-Dependent Kinase 2 and Inhibitor
PDE-constrained optimization is almost 200 times faster for this small molecule
Red: Optimized charge values
Blue: “Wild-type” charges (from 6-31G*/RESP)
Anderson, et al. 2003 (not exactly the optimized ligand)
Bardhan et al., J. Chem.
Theory Comput. (2009)
Summary: Pushing On All Dimensions
1. Fast, Scalable
Numerical Methods
2. Add Realism
But Preserve Speed
3. Solve Inverse
Problems in Design
4. Unify Theories
For New Science
Four Points For Today
1. Cellular and molecular biomedical problems also
need efficient simulation methods.
2. Fast BEM solvers represent an appealing approach
even at the molecular scale!
1. Challenge: Persuading community to abandon
beloved ad hoc fast methods for systematic ones.
2. Strategy: Systematic methods are more flexible as
we add new physics and address inverse problems.
Collaborators and Acknowledgments
 Fast methods: Michael Altman (Merck), Matt Knepley (U. Chicago), Rio
Yokota (King Abdullah University of Science and Technology), Lorena
Barba (Boston U.), Tsuyoshi Hamada (Nagasaki U.)
 Nonlocal continuum theory: Andreas Hildebrandt (Johannes Gutenberg
U., Mainz), Peter Brune, David Green (SUNY Stony Brook)
 Fast optimization: Michael Altman, Bruce Tidor (MIT), Jacob White
(MIT), Jung Hoon Lee (Merck), Sven Leyffer (Argonne) , Steve Benson
(Argonne), David Green, Mala Radhakrishnan (Wellesley)
 Approximation method: Matt Knepley, Mihai Anitescu (Argonne), Mala
Radhakrishnan
Support from:
1.
2.
3.
4.
Department of Energy (DOE) Computational Science Graduate Fellowship
(CSGF)
Wilkinson Fellowship in Math and Computer Science Division of Argonne
National Lab
NIH Technology Development (EUREKA)
Rush New Investigator Award