Boundary-element methods in biological science and engineering Jaydeep P. Bardhan Dept. of...
Transcript of 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
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
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.
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
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
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.
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.
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 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