Computational Bioelectrostatics

Transcription

1 Computational Bioelectrostatics Matthew Knepley Computational and Applied Mathematics Rice University Scientific Computing Seminar University of Houston Houston, TX April 20, 2017 M. Knepley (CAAM) MGK UH 1 / 44

2 Computational Science & Applied Mathematics Begins with the numerics of BIEs and PDEs, and mathematics of the computation, M. Knepley (CAAM) MGK UH 3 / 44

3 Computational Science & Applied Mathematics Begins with the numerics of BIEs and PDEs, and mathematics of the computation, is Distilled into high quality numerical libraries, and M. Knepley (CAAM) MGK UH 3 / 44

4 Computational Science & Applied Mathematics Begins with the numerics of BIEs and PDEs, and mathematics of the computation, is Distilled into high quality numerical libraries, and Culminates in scientific discovery. M. Knepley (CAAM) MGK UH 3 / 44

5 What is PETSc? PETSc is one of the most popular software libraries in scientific computing. As a principal architect since 2001, I developed unstructured meshes (model, algorithms, implementation), nonlinear preconditioning (model, algorithms), FEM discretizations (data structures, solvers optimization), optimizations for multicore and GPU architectures. M. Knepley (CAAM) MGK UH 4 / 44

6 What is PETSc? Knepley, Karpeev, Sci. Prog., Brune, Knepley, Scott, SISC, Vertices Depth 0 Edges Depth 1 Faces Depth 2 Cells 0 1 Depth 3 As a principal architect since 2001, I developed unstructured meshes (model, algorithms, implementation), nonlinear preconditioning (model, algorithms), FEM discretizations (data structures, solvers optimization), optimizations for multicore and GPU architectures. M. Knepley (CAAM) MGK UH 4 / 44

7 What is PETSc? Brune, Knepley, Smith, and Tu, SIAM Review, Type Sym Statement Abbreviation Additive + x + α(m(f, x, b) x) M + N + β(n (F, x, b) x) Multiplicative M(F, N (F, x, b), b) M N Left Prec. L M( x N (F, x, b), x, b) M L N Right Prec. R M(F(N (F, x, b)), x, b) M R N Inner Lin. Inv. \ y = J( x) 1 r( x) = K( J( x), y 0, b) N \K As a principal architect since 2001, I developed unstructured meshes (model, algorithms, implementation), nonlinear preconditioning (model, algorithms), FEM discretizations (data structures, solvers optimization), optimizations for multicore and GPU architectures. M. Knepley (CAAM) MGK UH 4 / 44

8 What is PETSc? Aagaard, Knepley, and Williams, J. of Geophysical Research, As a principal architect since 2001, I developed unstructured meshes (model, algorithms, implementation), nonlinear preconditioning (model, algorithms), FEM discretizations (data structures, solvers optimization), optimizations for multicore and GPU architectures. M. Knepley (CAAM) MGK UH 4 / 44

9 What is PETSc? Knepley and Terrel, Transactions on Mathematical Software, As a principal architect since 2001, I developed unstructured meshes (model, algorithms, implementation), nonlinear preconditioning (model, algorithms), FEM discretizations (data structures, solvers optimization), optimizations for multicore and GPU architectures. M. Knepley (CAAM) MGK UH 4 / 44

10 PETSc PETSc Citations, 4330 Total Manual Webpage M. Knepley (CAAM) MGK UH 5 / 44

11 Outline Bioelectrostatics 1 Bioelectrostatics 2 Approximate Operators 3 Approximate Boundary Conditions 4 Future Directions M. Knepley (CAAM) MGK UH 6 / 44

12 Bioelectrostatics The Natural World Bioelectrostatics Induced Surface Charge on Lysozyme M. Knepley (CAAM) MGK UH 7 / 44

13 Bioelectrostatics Physical Model Bioelectrostatics Electrostatic Potential φ Region I: protein Region II: solvent Surface M. Knepley (CAAM) MGK UH 8 / 44

14 Bioelectrostatics Mathematical Model Bioelectrostatics We can write a Boundary Integral Equation (BIE) for the induced surface charge σ, σ( r) + ˆɛ Γ σ( r )d 2 r Q n( r) 4π r r = ˆɛ q k n( r) 4π r r k (I + ˆɛD ) σ( r) = k=1 where we define ˆɛ = 2 ɛ I ɛ II ɛ I + ɛ II < 0 M. Knepley (CAAM) MGK UH 9 / 44

15 Problem Bioelectrostatics Boundary element discretizations of the solvation problem: can be expensive to solve are more accurate than required by intermediate design iterations M. Knepley (CAAM) MGK UH 10 / 44

16 Outline Approximate Operators 1 Bioelectrostatics 2 Approximate Operators 3 Approximate Boundary Conditions 4 Future Directions M. Knepley (CAAM) MGK UH 11 / 44

17 Approximate Operators Bioelectrostatics Mathematical Model The reaction potential is given by φ R σ( r )d 2 r ( r) = 4πɛ 1 r r = Cσ Γ which defines G es, the electrostatic part of the solvation free energy G es = 1 q, φ R 2 = 1 q, Lq 2 = 1 q, CA 1 Bq 2 where Bq = ˆɛ Ω n( r) Aσ = I + ˆɛD q( r )d 3 r 4π r r M. Knepley (CAAM) MGK UH 12 / 44

18 Approximate Operators BIBEE Approximate D by a diagonal operator Boundary Integral-Based Electrostatics Estimation Coulomb Field Approximation: uniform normal field ( 1 ˆɛ ) σ 2 CFA = Bq Eigenvectors: BEM e i e j BIBEE/P Lower Bound: no good physical motivation ( 1 + ˆɛ ) σ LB = Bq 2 M. Knepley (CAAM) MGK UH 13 / 44

19 Approximate Operators BIBEE Approximate D by a diagonal operator Boundary Integral-Based Electrostatics Estimation Coulomb Field Approximation: uniform normal field ( 1 ˆɛ ) σ 2 CFA = Bq Eigenvectors: BEM e i e j BIBEE/P Preconditioning: consider only local effects σ P = Bq M. Knepley (CAAM) MGK UH 13 / 44

20 Approximate Operators BIBEE Bounds on Solvation Energy Theorem: The electrostatic solvation energy G es has upper and lower bounds given by ( ˆɛ ) 1 q, CBq 1 q, CA 1 Bq 1 ( 1 ˆɛ 1 q, CBq, ) and for spheres and prolate spheroids, we have the improved lower bound, 1 2 q, CBq 1 q, CA 1 Bq, 2 and we note that ˆɛ < 1 2. M. Knepley (CAAM) MGK UH 14 / 44

21 Approximate Operators Energy Bounds: Proof: Bardhan, Knepley, Anitescu, JCP, 130(10), 2008 I will break the proof into three steps, Replace C with B Symmetrization Eigendecomposition shown in the following slides. We will need the single layer operator S for step 1, Sτ( r) = τ( r )d 2 r 4π r r M. Knepley (CAAM) MGK UH 15 / 44

22 Approximate Operators Energy Bounds: First Step Replace C with B The potential at the boundary Γ given by φ Coulomb ( r) = C T q can also be obtained by solving an exterior Neumann problem for τ, φ Coulomb ( r) = Sτ = S(I 2D ) 1 ( Bq) 2ˆɛ so that the solvation energy is given by = 2ˆɛ S(I 2D ) 1 Bq 1 q, CA 1 Bq = 1ˆɛ S(I 2D ) 1 Bq, (I + ˆɛD ) 1 Bq 2 M. Knepley (CAAM) MGK UH 16 / 44

23 Approximate Operators Energy Bounds: Second Step Quasi-Hermiticity Plemelj s symmetrization principle holds that and we have SD = DS S = S 1/2 S 1/2 which means that we can define a Hermitian operator H similar to D leading to an energy H = S 1/2 D S 1/2 1 q, CA 1 Bq = 1ˆɛ Bq, S 1/2 (I 2H) 1 (I + ˆɛH) 1 S 1/2 Bq 2 M. Knepley (CAAM) MGK UH 17 / 44

24 Approximate Operators Energy Bounds: Third Step Eigendecomposition The spectrum of D is in [ 1 2, 1 2 ), and the energy is 1 q, CA 1 Bq = 2 i 1 ˆɛ (1 2λ i) 1 (1 + ˆɛλ i ) 1 xi 2 where and H = V ΛV T x = V T S 1/2 Bq M. Knepley (CAAM) MGK UH 18 / 44

25 Approximate Operators Energy Bounds: Diagonal Approximations The BIBEE approximations yield the following bounds 1 q, CA 1 2 CFA Bq = i 1 2 where we note that q, CA 1 P Bq = i 1 q, CA 1 2 LB Bq = i ( 1 ˆɛ (1 2λ i) 1 1 ˆɛ ) 1 xi ˆɛ (1 2λ i) 1 xi 2 1 ˆɛ (1 2λ i) 1 ˆɛ < 1 2 ( 1 + ˆɛ ) 1 xi 2 2 M. Knepley (CAAM) MGK UH 19 / 44

26 Approximate Operators BIBEE Accuracy Electrostatic solvation free energies of met-enkephalin structures 20 Electrostatic Free Energy (kcal/mol) BEM SGB/CFA 140 GBMV BIBEE/CFA BIBEE/P Time (ps) Snapshots taken from a 500-ps MD simulation at 10-ps intervals. Bardhan, Knepley, Anitescu, JCP, ion free energies using met-enkephalin structures taken from a 500-ps MD simulation plotted as time rgies are inm. kcal/mol. Knepley (CAAM) a All estimates are plotted. b BIBEE/LB MGK has been omitted for clarity. UH 20 / 44 (b)

27 Approximate Operators Generalized Born Approximation The pairwise energy between charges is defined by the Still equation: Ges ij = 1 ( 1 1 ) N 8π ɛ II ɛ I i,j where the effective Born radius is R i = 1 ( 1 1 ) 1 8π ɛ II ɛ I E i q i q j r 2 ij + R i R j e r 2 ij /4R i R j where E i is the self-energy of the charge q i, the electrostatic energy when atom i has unit charge and all others are neutral. M. Knepley (CAAM) MGK UH 21 / 44

28 Approximate Operators Crowded Protein Solution Important for drug design of antibody therapies M. Knepley (CAAM) MGK UH 22 / 44

29 Approximate Operators BIBEE Scalability Yokota, Bardhan, Knepley, Barba, Hamada, CPC, M. Knepley (CAAM) MGK UH 23 / 44

30 Outline Approximate Boundary Conditions 1 Bioelectrostatics 2 Approximate Operators 3 Approximate Boundary Conditions 4 Future Directions M. Knepley (CAAM) MGK UH 24 / 44

31 Approximate Boundary Conditions Bioelectrostatics Physical Model Electrostatic Potential φ Region I: protein Region II: solvent Surface M. Knepley (CAAM) MGK UH 25 / 44

32 Approximate Boundary Conditions Kirkwood s Solution (1934) The potential inside Region I is given by Φ I = Q k=1 q k ɛ 1 r r k + ψ, and the potential in Region II is given by Φ II = n n=0 m= n C nm r n+1 Pm n (cos θ)e imφ. M. Knepley (CAAM) MGK UH 26 / 44

33 Approximate Boundary Conditions Kirkwood s Solution (1934) The reaction potential ψ is expanded in a series n ψ = B nm r n Pn m (cos θ)e imφ. n=0 m= n and the source distribution is also expanded Q k=1 q k ɛ 1 r r k = n n=0 m= n E nm ɛ 1 r n+1 Pm n (cos θ)e imφ. M. Knepley (CAAM) MGK UH 27 / 44

34 Approximate Boundary Conditions Kirkwood s Solution (1934) By applying the boundary conditions, letting the sphere have radius b, Φ I r=b = Φ II r=b ɛ I Φ I r r=b = ɛ II Φ II r r=b we can eliminate C nm, and determine the reaction potential coefficients in terms of the source distribution, B nm = 1 (ɛ I ɛ II )(n + 1) ɛ I b 2n+1 ɛ I n + ɛ II (n + 1) E nm. M. Knepley (CAAM) MGK UH 28 / 44

35 Approximate Boundary Conditions Approximate Boundary Conditions Theorem: The BIBEE boundary integral operator approximations ( A CFA = I 1 + ˆɛ ) 2 A P = I have an equivalent PDE formulation, Q ɛ I Φ CFA,P = q k δ( r r k ) ɛ II Φ CFA,P = 0 k=1 ɛ I Φ C I ɛ II r r=b = Φ II r or ψ CFA r=b r Φ I r=b = Φ II r=b 3ɛ I ɛ II ɛ I + ɛ II Φ C I r r=b = Φ II r ψ P r r=b, where Φ C 1 is the Coulomb field due to interior charges. M. Knepley (CAAM) MGK UH 29 / 44

36 Approximate Boundary Conditions Approximate Boundary Conditions Theorem: For spherical solute, the BIBEE boundary integral operator approximations have eigenspaces are identical to that of the original operator. BEM eigenvector e i e j BIBEE/P eigenvector M. Knepley (CAAM) MGK UH 29 / 44

37 Approximate Boundary Conditions Proof of PDE Equivalence Proof: Bardhan and Knepley, JCP, 135(12), In order to show that these PDEs are equivalent to the original BIEs, Start with the fundamental solution to Laplace s equation G(r, r ) Note that Γ G(r, r )σ(r )dγ satisfies the bulk equation and decay at infinity Insertion into the approximate BC gives the BIBEE boundary integral approximation M. Knepley (CAAM) MGK UH 30 / 44

38 Approximate Boundary Conditions Proof of PDE Equivalence Proof: Bardhan and Knepley, JCP, 135(12), In order to show that these PDEs are equivalent to the original BIEs, Start with the fundamental solution to Laplace s equation G(r, r ) Note that Γ G(r, r )σ(r )dγ satisfies the bulk equation and decay at infinity Insertion into the approximate BC gives the BIBEE boundary integral approximation M. Knepley (CAAM) MGK UH 30 / 44

39 Approximate Boundary Conditions Proof of PDE Equivalence Proof: Bardhan and Knepley, JCP, 135(12), In order to show that these PDEs are equivalent to the original BIEs, Start with the fundamental solution to Laplace s equation G(r, r ) Note that Γ G(r, r )σ(r )dγ satisfies the bulk equation and decay at infinity Insertion into the approximate BC gives the BIBEE boundary integral approximation M. Knepley (CAAM) MGK UH 30 / 44

40 Approximate Boundary Conditions Proof of PDE Equivalence Proof: Bardhan and Knepley, JCP, 135(12), In order to show that these PDEs are equivalent to the original BIEs, Start with the fundamental solution to Laplace s equation G(r, r ) Note that Γ G(r, r )σ(r )dγ satisfies the bulk equation and decay at infinity Insertion into the approximate BC gives the BIBEE boundary integral approximation M. Knepley (CAAM) MGK UH 30 / 44

41 Approximate Boundary Conditions Proof of Eigenspace Equivalence Proof: Bardhan and Knepley, JCP, 135(12), In order to show that these integral operators share a common eigenbasis, Note that, for a spherical boundary, D is compact and has a pure point spectrum Examine the effect of the operator on a unit spherical harmonic charge distribution Use completeness of the spherical harmonic basis M. Knepley (CAAM) MGK UH 31 / 44

42 Approximate Boundary Conditions Proof of Eigenspace Equivalence Proof: Bardhan and Knepley, JCP, 135(12), In order to show that these integral operators share a common eigenbasis, Note that, for a spherical boundary, D is compact and has a pure point spectrum Examine the effect of the operator on a unit spherical harmonic charge distribution Use completeness of the spherical harmonic basis M. Knepley (CAAM) MGK UH 31 / 44

43 Approximate Boundary Conditions Proof of Eigenspace Equivalence Proof: Bardhan and Knepley, JCP, 135(12), In order to show that these integral operators share a common eigenbasis, Note that, for a spherical boundary, D is compact and has a pure point spectrum Examine the effect of the operator on a unit spherical harmonic charge distribution Use completeness of the spherical harmonic basis M. Knepley (CAAM) MGK UH 31 / 44

44 Approximate Boundary Conditions Proof of Eigenspace Equivalence Proof: Bardhan and Knepley, JCP, 135(12), In order to show that these integral operators share a common eigenbasis, Note that, for a spherical boundary, D is compact and has a pure point spectrum Examine the effect of the operator on a unit spherical harmonic charge distribution Use completeness of the spherical harmonic basis M. Knepley (CAAM) MGK UH 31 / 44

45 Approximate Boundary Conditions Proof of Eigenspace Equivalence Proof: Bardhan and Knepley, JCP, 135(12), In order to show that these integral operators share a common eigenbasis, Note that, for a spherical boundary, D is compact and has a pure point spectrum Examine the effect of the operator on a unit spherical harmonic charge distribution Use completeness of the spherical harmonic basis The result does not hold for general boundaries. M. Knepley (CAAM) MGK UH 31 / 44

46 Approximate Boundary Conditions Series Solutions Note that the approximate solutions are separable: B nm = 1 ɛ 1 n + ɛ 2 (n + 1) γ nm B CFA nm = 1 ɛ 2 1 2n + 1 γ nm B P nm = 1 1 ɛ 1 + ɛ 2 n + 1 γ nm. 2 If ɛ I = ɛ II = ɛ, both approximations are exact: B nm = B CFA nm = B P nm = 1 ɛ(2n + 1) γ nm. M. Knepley (CAAM) MGK UH 32 / 44

47 Approximate Boundary Conditions Series Solutions Note that the approximate solutions are separable: B nm = 1 ɛ 1 n + ɛ 2 (n + 1) γ nm B CFA nm = 1 ɛ 2 1 2n + 1 γ nm B P nm = 1 1 ɛ 1 + ɛ 2 n + 1 γ nm. 2 If ɛ I = ɛ II = ɛ, both approximations are exact: B nm = B CFA nm = B P nm = 1 ɛ(2n + 1) γ nm. M. Knepley (CAAM) MGK UH 32 / 44

48 Approximate Boundary Conditions Asymptotics BIBEE/CFA is exact for the n = 0 mode, B 00 = B CFA 00 = γ 00 ɛ 2, whereas BIBEE/P approaches the exact response in the limit n : lim B nm = lim B P n n nm = 1 (ɛ 1 + ɛ 2 )n γ nm. M. Knepley (CAAM) MGK UH 33 / 44

49 Approximate Boundary Conditions Asymptotics BIBEE/CFA is exact for the n = 0 mode, B 00 = B CFA 00 = γ 00 ɛ 2, whereas BIBEE/P approaches the exact response in the limit n : lim B nm = lim B P n n nm = 1 (ɛ 1 + ɛ 2 )n γ nm. M. Knepley (CAAM) MGK UH 33 / 44

50 Approximate Boundary Conditions Asymptotics In the limit ɛ 1 /ɛ 2 0, lim B nm = ɛ 1 /ɛ 2 0 lim ɛ 1 /ɛ 2 0 BCFA nm = lim ɛ 1 /ɛ 2 0 BP nm = γ nm ɛ 2 (n + 1) γ nm ɛ 2 (2n + 1), γ nm ɛ 2 ( n so that the approximation ratios are given by ), B CFA nm B nm = n + 1 2n + 1, Bnm P = n + 1 B nm n M. Knepley (CAAM) MGK UH 34 / 44

51 Approximate Boundary Conditions Improved Accuracy BIBEE/I interpolates between BIBEE/CFA and BIBEE/P Bardhan, Knepley, JCP, M. Knepley (CAAM) MGK UH 35 / 44

52 Approximate Boundary Conditions Basis Augmentation We examined the more complex problem of protein-ligand binding using trypsin and bovine pancreatic trypsin inhibitor (BPTI), using electrostatic component analysis to identify residue contributions to binding and molecular recognition. M. Knepley (CAAM) MGK UH 36 / 44

53 Approximate Boundary Conditions Basis Augmentation Looking at an ensemble of synthetic proteins, we can see that BIBEE/CFA becomes more accurate as the monopole moment increases, and BIBEE/P more accurate as it decreases. BIBEE/I is accurate for spheres, but must be extended for ellipses. M. Knepley (CAAM) MGK UH 37 / 44

54 Approximate Boundary Conditions Basis Augmentation For ellipses, we add a few low order multipole moments, up to the octopole, to recover 5% accuracy for all synthetic proteins tested. M. Knepley (CAAM) MGK UH 38 / 44

55 Approximate Boundary Conditions Resolution Boundary element discretizations of the solvation problem: can be expensive to solve Bounding the electrostatic free energies associated with linear continuum models of molecular solvation, Bardhan, Knepley, Anitescu, JCP, 2009 are more accurate than required by intermediate design iterations Analysis of fast boundary-integral approximations for modeling electrostatic contributions of molecular binding, Kreienkamp, et al., Molecular-Based Mathematical Biology, 2013 M. Knepley (CAAM) MGK UH 39 / 44

56 Approximate Boundary Conditions Resolution Boundary element discretizations of the solvation problem: can be expensive to solve Bounding the electrostatic free energies associated with linear continuum models of molecular solvation, Bardhan, Knepley, Anitescu, JCP, 2009 are more accurate than required by intermediate design iterations Analysis of fast boundary-integral approximations for modeling electrostatic contributions of molecular binding, Kreienkamp, et al., Molecular-Based Mathematical Biology, 2013 M. Knepley (CAAM) MGK UH 39 / 44

57 Approximate Boundary Conditions Resolution Boundary element discretizations of the solvation problem: can be expensive to solve Bounding the electrostatic free energies associated with linear continuum models of molecular solvation, Bardhan, Knepley, Anitescu, JCP, 2009 are more accurate than required by intermediate design iterations Analysis of fast boundary-integral approximations for modeling electrostatic contributions of molecular binding, Kreienkamp, et al., Molecular-Based Mathematical Biology, 2013 M. Knepley (CAAM) MGK UH 39 / 44

58 Outline Future Directions 1 Bioelectrostatics 2 Approximate Operators 3 Approximate Boundary Conditions 4 Future Directions M. Knepley (CAAM) MGK UH 40 / 44

59 More Physics Future Directions New Phenomena: New Model: M. Knepley (CAAM) MGK UH 41 / 44

60 More Physics Future Directions New Phenomena: Charge Hydration Asymmetry New Model: M. Knepley (CAAM) MGK UH 41 / 44

61 More Physics Future Directions New Phenomena: Charge Hydration Asymmetry New Model: Nonlinear Boundary Condition M. Knepley (CAAM) MGK UH 41 / 44

62 More Physics Future Directions New Phenomena: Charge Hydration Asymmetry Solute Solvent Interface Potential New Model: Nonlinear Boundary Condition M. Knepley (CAAM) MGK UH 41 / 44

63 More Physics Future Directions New Phenomena: Charge Hydration Asymmetry Solute Solvent Interface Potential New Model: Nonlinear Boundary Condition Static Solvation Potential M. Knepley (CAAM) MGK UH 41 / 44

64 More Physics Future Directions New Phenomena: Charge Hydration Asymmetry Solute Solvent Interface Potential Solvent Thermodynamics New Model: Nonlinear Boundary Condition Static Solvation Potential M. Knepley (CAAM) MGK UH 41 / 44

65 More Physics Future Directions New Phenomena: Charge Hydration Asymmetry Solute Solvent Interface Potential Solvent Thermodynamics New Model: Nonlinear Boundary Condition Static Solvation Potential Solvation Layer Interface Condition (SLIC) M. Knepley (CAAM) MGK UH 41 / 44

66 More Physics Future Directions Predicting Solvation Free Energies and Thermodynamics in Polar Solvents and Mixtures using a Solvation-Layer Interface Condition, Tabrizi et.al., JCP 146(9), 2017 M. Knepley (CAAM) MGK UH 42 / 44

67 Future Directions Impact of Mathematics on Science Applied Mathematics Computational Leaders have always embraced the latest technology and been inspired by physical problems, M. Knepley (CAAM) MGK UH 43 / 44

68 Future Directions Impact of Mathematics on Science Applied Mathematics Computational Leaders have always embraced the latest technology and been inspired by physical problems, M. Knepley (CAAM) MGK UH 43 / 44

69 Future Directions Impact of Mathematics on Science Applied Mathematics Computational Leaders have always embraced the latest technology and been inspired by physical problems, M. Knepley (CAAM) MGK UH 43 / 44

70 Future Directions Impact of Mathematics on Science Applied Mathematics Computational Leaders have always embraced the latest technology and been inspired by physical problems, PETSc M. Knepley (CAAM) MGK UH 43 / 44

71 Future Directions Impact of Mathematics on Science Applied Mathematics Computational Leaders have always embraced the latest technology and been inspired by physical problems, Enabling Scientific Discovery M. Knepley (CAAM) MGK UH 43 / 44

72 Thank You!