boundary-element methods in biological science and engineering jaydeep p. bardhan dept. of...

Boundary-element methods in biological science and

engineering

Jaydeep P. Bardhan Dept. of Electrical and Computer Engineering

Northeastern University, Boston MA

Four Points For Today1. 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 10-1 m 10-5 m 10-9 m10-2 m

Humanbody

Organs/tissues

Humancells

Molecules

Electrical ImpedanceTomography (EIT)

... And MEG

Prof. Bin He, UMN

Electroencephalography (EEG)Brain-computer interfaces

Electrocardiography (ECG)

Human eye for keratoplasty

Peratta et al. ‘08

Transport through blood vessel walls

Balsim et al. ‘10

Tumor growth

Lowengrub et al. ‘09

Cochlea (ear)

Briare et al. ‘00

Sfantos et al ‘07

Hip prostheses

… but for microscopic ones as well!

100 m 10-1 m 10-5 m 10-9 m10-2 m

Humanbody

Organs/tissues

Humancells

Molecules

Rahimian, Biros et al (2010)

Blood flow

Cell locomotion using flagella (sperm, bacteria)

Ramia ‘91

Gaver and Kute ‘98

Cell adhesion to surfaces under shear flow

Cell “rolling” along tissue surface

King and Hammer ‘01 M. Bathe ‘08

Molecular flexibility

Quantum mechanics

Nanotechnology(quantum dots)

Gelbard ‘01

Biomolecule electrostatics and hydrodynamics

• Drug binding• Protein folding• Cell physiology• Molecular design

Biology uses water to control molecular binding, protein folding, etc. Binding example is simple:

Protein

Protein

A central molecular-scale modeling problem: water.

Basic Continuum Electrostatic Theory

100-1000 times faster than MD

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

Water model: no fixed chargeso Single water: sphere of radius 1.4

Angstromo 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.

Linearized Poisson-Boltzmann equation

A Boundary Integral Method For the Poisson Biomolecule Problem

+ -++

++ + + ++

---

-- -

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

Fast BEM Solvers are Essential1. 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!

P “looks like” A-1

Iteration converges faster if matrix eigenvalues are “well clustered”

Memory growth is QUADRATIC Time is CUBIC!!

Replace quadratic memory and cubic time requirements with LINEAR

requirements!

Application-Specific Challenges

1. “Continuum-solvent dynamics”o Replace water molecules with dielectrico Calculate forces at each time step and integrate

2. Continuum post-processing of molecular dynamics

o Sample structures from explicit-water MDo Compute average continuum energy from samples

3. Electrostatic component analysiso Compute each atom’s interaction with every othero Useful in drug design and protein engineering!

These lead to thousands, or even millions, of

electrostatic simulations...

Some with identical dielectric boundaries, some with changing boundaries!

Community’s Solution: Fast Ad Hoc Models

Up to 100X faster than solving Poisson

Define effective (nonphysical) parameters

Plug in to ad hoc (nonphysical) formula

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

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:+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: Natural, Rigorous Generalized Born

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

Complementary Regimes of Accuracy

V1 V2V20

• 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

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

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

spheroid)o 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

Mathematical Rigor Enables Systematic Improvements

• 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

-1/10Dominant energies come from

dominant modes: try to capture dipole/quadrupole modes

approximately!

Mean absolute error: 4% !

BIBEE fluctuations track actual ones very closely – possible applications in uncertainty quantification

Many parameters and ad hoc correction terms

Snapshots from MD



Enables systematic model improvement

2.Prove approximation properties




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

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

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.




Prove approximation properties




Pre-corrected FFT Algorithm

1. Project charges to grid2. Point-wise multiplication in frequency space3. Interpolate grid potentials4. “Pre-correct” so that local interactions are accurate

Phillips and White (1997)

• Potential calculation is a convolution. • Convolutions are “cheap” in frequency

space• Green’s function independent! (Laplace,

Helmholtz, Stokes, etc.)

Circuit Simulation

Cadence Design Systems

Proteins

Kuo, Altman, Bardhan, Tidor, White (2002)

Willis, Peraire, White

Aerodynamics Bioelectromagnetics

27

A geometry representative of a protein:

The barnase-barstar protein complex:

Bardhan + Altman et al., 2007Altman + Bardhan, White, Tidor 2009

Higher-order Protein BEM

Develop scalable protein simulations with leaders in parallel computing +

FMM760-node GPU cluster Degima

Cost of cluster: ~ US $420,000

Sustained: 34.6 Tflops

Performance/price: 80 Mflops/$Application to proteins with PetFMM code ofYokota, Cruz, Barba, Knepley, Hamada

Picture courtesy T. Hamada

Parallel GPU FMM code

800 Å

Scalable algorithms enable bigger science

“How do proteins work in the crowded environment of the cell?”

Lysozyme: ~2K atom charges, ~15K surface charges

1000 lysozyme molecules: model of a concentrated protein solution

10 copies

1 copy 100 copies

1000 copies

Yokota, Bardhan, et al. 2009





Leverage existing fast, scalable algorithms



We are still adding physics to our models.

Circuit simulation: Maxwell equationsSolid mechanics: elasticityAirplane simulation: Navier-Stokes

“Classical” modeling: one can assume the model is right!

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).

All simulate same thing!

Accuracy

Speed

CAD tools

--George Box

These are just the models

associated with the molecular

scale!!

Adding physics to the continuum model using nonlocal dielectric

theoryKNOWN weaknesses of Poisson model:

1. Linear response assumption

Caveat: Nonlinear dielectrics ARE important for some molecules!

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!

Nonlocal Continuum Electrostatics:Lorentzian Model and Promising

TestsNonlocal response:

Now

Integrodifferential Poisson equation

Green’s function for

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

A. Hildebrandt et al. 2004

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: 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!

Dense methods used previously could not achieve useful accuracy!

Required accuracy

Bardhan and Hildebrandt, DAC ‘11Local model Nonlocal model

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

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

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)

A Common Framework for Multiple Models

Biomolecular complexes

Biomolecular complexes

Linearized PB modelsLinearized PB models

Protein 1

Protein 1

Protein 2

Protein 2

Explain Coulomb-field

approx.

Explain Coulomb-field

approx.

Analytical solution of nonlocal model for

sphere

Analytical solution of nonlocal model for

sphere

GB-like fast nonlocal approximate modelGB-like fast nonlocal approximate model

Full nonlinear PB via boundary-integrals

Advanced PB models (Bikerman, etc.)

Fast GB-like nonlinear approximations

Dynamics: hybrid explicit/implicit, and fully

implicit

Dynamics: hybrid explicit/implicit, and fully

implicitPopular quantum methods

couple to exactly our Poisson problem (“polarizable continuum model”)

Popular quantum methods couple to exactly our Poisson

problem (“polarizable continuum model”)

Improved GB models

Improved GB models

Coupling to fast, scalable algorithms

BIBEE provides a unifying, scalable approach to

testing and extending new physics.





Leverage existing fast, scalable algorithms

Can add better physics as we learn them


The electrostatic contribution to binding is

A total of three simulations is needed.

The Value of Systematic Approximations in Inverse Problems:

Biomolecule Design

Electrostatic Optimization of Biomolecules:

Applications in Analysis and Design

Mandal and Hilvert, 2003 E. coli chorismate mutase

inhibitors:o Analyzed by Kangas and Tidoro Suggested substitution

experimentally verified: result is the tightest-binding inhibitor yet known

Barnase/barstar protein complex:o Tight-binding complexo Optimal charge distribution

closely matches “wild-type” charge distribution

Lee and Tidor, 2001

Challenge: Optimization is SLOW.

10 min/simulation * 2000 simulations (protein) = 2 CPU

weeks!!

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 ExactFormally exact calculation

Bardhan et al., 2004; Bardhan et al., 2005; Bardhan et al., 2007; Bardhan et al., 2009

10 min/simulation = 20 min/optimization (no matter how many charges!)

Method scales comparably with normal PDE-constrained approaches

BIBEE as Inverse Problem Regularizer Approximated eigenvectors closely

match actual ones Regularization can be performed

using “approximate” penalty functions:

No linear solve: Accurate but 10-20X faster than simulation!!

BIBEE/P captures small eigenvalues very accurately identify number of directions to penalize

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

Application: Cyclin-Dependent Kinase 2 and Inhibitor

Anderson, et al. 2003 (not exactly the optimized ligand)

Red: Optimized charge valuesBlue: “Wild-type” charges (from 6-31G*/RESP)

PDE-constrained optimization is almost 200 times faster for this small molecule

Bardhan et al., J. Chem. Theory Comput. (2009)

Summary: Pushing On All Dimensions

2. Add RealismBut Preserve Speed

3. Solve InverseProblems in Design

4. Unify TheoriesFor New Science

1. Fast, ScalableNumerical Methods

Four Points For Today1. 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. Department of Energy (DOE) Computational Science Graduate

Fellowship (CSGF)2. Wilkinson Fellowship in Math and Computer Science Division of

Argonne National Lab3. NIH Technology Development (EUREKA) 4. Rush New Investigator Award

boundary-element methods in biological science and engineering jaydeep p. bardhan dept. of...

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fast methods

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