Improved Solvation Models using Boundary Integral Equations

Transcription

1 Improved Solvation Models using Boundary Integral Equations Matthew Knepley and Jaydeep Bardhan Computational and Applied Mathematics Rice University SIAM Conference on the Life Sciences Minneapolis, MN August 9, 2018 M. Knepley (Rice) Solvation LS18 1 / 52

2 Collaborators Amir Molvai Tabrizi (postdoc, NE) Tom Klotz (grad student, Rice) Spencer Goossens (grad student, NE) Ali Rahimi (grad student, NE) M. Knepley (Rice) Solvation LS18 3 / 52

3 Main Point Solvation computation can benefit from non-poisson models. M. Knepley (Rice) Solvation LS18 4 / 52

4 Bioelectrostatics The Natural World Induced Surface Charge on Lysozyme M. Knepley (Rice) Solvation LS18 5 / 52

5 Bioelectrostatics Physical Model Electrostatic Potential φ Region I: protein Region II: solvent Surface M. Knepley (Rice) Solvation LS18 6 / 52

6 Bioelectrostatics Mathematical Model 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 (Rice) Solvation LS18 7 / 52

7 Bioelectrostatics Mathematical Model This is equivalent to a PDE model for the potentials Φ I,II in the two regions, and boundary conditions at the solute surface: ɛ I Φ I = ɛ II Φ II = 0 Q q k δ( r r k ) k=1 Φ I r=b = Φ II r=b ɛ I Φ I r r=b = ɛ II Φ II r r=b M. Knepley (Rice) Solvation LS18 8 / 52

8 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 (Rice) Solvation LS18 9 / 52

9 Outline Some History 1 Some History 2 Improving the Poisson Operator M. Knepley (Rice) Solvation LS18 10 / 52

10 Some History 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 (Rice) Solvation LS18 11 / 52

11 GB Problems Some History No global potential solution, only energy No analysis of the error For example, Salsbury 2006 consists of parameter tuning No path for systematic improvement For example, Sigalov 2006 changes the model The same atoms have different radii in different molecules, solvents temperatures LOTS of parameters Nina, Beglov, Roux 1997 M. Knepley (Rice) Solvation LS18 12 / 52

12 GB Problems Some History No global potential solution, only energy No analysis of the error For example, Salsbury 2006 consists of parameter tuning No path for systematic improvement For example, Sigalov 2006 changes the model The same atoms have different radii in different molecules, solvents temperatures LOTS of parameters Nina, Beglov, Roux 1997 M. Knepley (Rice) Solvation LS18 12 / 52

13 Some History Implicit Solvent Models State-of-the-art solvation models still use the same variation in radii Biomolecular electrostatics I want your solvation (model), J. Bardhan, Comp. Sci. & Disc., 5(1), (2012) M. Knepley (Rice) Solvation LS18 13 / 52

14 Outline Improving the Poisson Operator 1 Some History 2 Improving the Poisson Operator M. Knepley (Rice) Solvation LS18 14 / 52

15 Improving the Poisson Operator Origins of Electrostatic Asymmetry Explicit solvent molecular dynamics FEP Standard Maxwell boundary condition Proposed nonlinear boundary condition 1.4 / σ E n Coul E n Coul M. Knepley (Rice) Solvation LS18 15 / 52

16 Improving the Poisson Operator Origins of Electrostatic Asymmetry M. Knepley (Rice) Solvation LS18 16 / 52

17 Improving the Poisson Operator Origins of Electrostatic Asymmetry 10 Deeply buried charge The nonzero slope at q=0 is the static potential 0 G charging (kcal/mol) Solvent-exposed charge Symbols: explicit-solvent freeenergy perturbation calculations Lines: piecewise-linear model plus static potential q Asymmetry for a deeply buried charge is exclusively due to the static potential Asymmetry for a solvent-exposed charge results from both the static potential and the hydrogen-oxygen size difference M. Knepley (Rice) Solvation LS18 17 / 52

18 Main Idea Improving the Poisson Operator Maxwell Boundary Condition Assume the model and energy, and derive the radii, Φ I ɛ I n = ɛ Φ II II n M. Knepley (Rice) Solvation LS18 18 / 52

19 Main Idea Improving the Poisson Operator Solvation-Layer Interface Condition (SLIC) Assume the energy and radii, and derive the model. (ɛ I ɛh(e n )) Φ I n = (ɛ II ɛh(e n )) Φ II n M. Knepley (Rice) Solvation LS18 18 / 52

20 Main Idea Improving the Poisson Operator Using our correspondence with the BIE form, ( I + h(e n ) + ˆɛ ( 12 )) I + D σ = ˆɛ Q k=1 G n where h is a diagonal nonlinear integral operator. M. Knepley (Rice) Solvation LS18 18 / 52

21 SLIC Boundary Perturbation Improving the Poisson Operator h(e n ) = α tanh (βe n γ) + µ where α Size of the asymmetry β Width of the transition region γ The transition field strength µ Assures h(0) = 0, so µ = α tanh( γ) M. Knepley (Rice) Solvation LS18 19 / 52

22 Improving the Poisson Operator Accuracy of SLIC Residues 0 20 Charging free energy (kcal/mol) MD FEP Poisson, Roux radii NLBC, extrapolated NLBC, 4 vertices/a 2 NLBC, 2 vertices/a 2 ARG ASP CYS GLU HIS LYS TYR M. Knepley (Rice) Solvation LS18 20 / 52

23 Improving the Poisson Operator Accuracy of SLIC Protonation Symmetric LPB NLBC LPB G prot,salt (kcal/mol) ARG ASP CYS GLU HIS LYS TYR M. Knepley (Rice) Solvation LS18 21 / 52

24 Improving the Poisson Operator Accuracy of SLIC Synthetic Molecules 20 Hydration Free Energy (kcal/mol) Negative (MD, Mobley et al) Positive (MD, Mobley et al) Negative (NLBC, this work) Positive (NLBC, this work) Problem Index M. Knepley (Rice) Solvation LS18 22 / 52

25 Improving the Poisson Operator Accuracy of SLIC Synthetic Molecules Asymmetry in Electrostatic Free Energy (kcal/mol) MD, Mobley et al NLBC, this work Number of Atoms in Ring M. Knepley (Rice) Solvation LS18 23 / 52

26 Improving the Poisson Operator Accuracy of SLIC Synthetic Molecules 0 Electrostatic Solvation Free Energy (kcal/mol) Negative (MD, Mobley et al) 60 Positive (MD, Mobley et al) Negative (NLBC, this work) Positive (NLBC, this work) Number of Atoms in Ring M. Knepley (Rice) Solvation LS18 24 / 52

27 Improving the Poisson Operator Thermodynamics The parameters show linear temperature dependence α (Angstroms) W 0.6 MeOH F 15 AN DMF M. Knepley (Rice) Solvation LS18 25 / AN F

28 Improving the Poisson Operator Model Validation Courtesy A. Molvai Tabrizi Model validation and verification using experiment Water H 2 O Formamide CH 3 NO Dimethyl sulfoxide C 2 H 6 OS Methanol CH 3 OH Acetonitrile C 2 H 3 N Nitromethane CH 3 NO 2 Ethanol C 2 H 5 OH Dimethyl formamide C 3 H 7 NO Propylene carbonate CH 3 C 2 H 3 O 2 CO Northeastern University Mechanical and Industrial Engineering Bardhan Lab M. Knepley (Rice) Solvation LS18 26 / 52

29 Improving the Poisson Operator Model Validation Courtesy A. Molvai Tabrizi M. Knepley (Rice) Solvation LS18 27 / 52

30 Improving the Poisson Operator Model Validation Courtesy A. Molvai Tabrizi M. Knepley (Rice) Solvation LS18 28 / 52

31 Improving the Poisson Operator Model Validation Courtesy A. Molvai Tabrizi Model validation and verification using experiment Dimethyl 25 o C C 3 H 7 NO Computed Experimental Northeastern University Mechanical and Industrial Engineering Bardhan Lab M. Knepley (Rice) Solvation LS18 29 / 52

32 Improving the Poisson Operator Model Validation Courtesy A. Molvai Tabrizi A. Molavi Tabrizi, M.G. Knepley, and J.P. Bardhan, Generalising the mean spherical approximation as a multiscale, nonlinear boundary condition at the solute-solvent interface, Molecular Physics (2016). M. Knepley (Rice) Solvation LS18 30 / 52

33 Improving the Poisson Operator Thermodynamic Predictions Courtesy A. Molvai Tabrizi Experimental Data in Parentheses M. Knepley (Rice) Solvation LS18 31 / 52

34 Improving the Poisson Operator Thermodynamic Predictions Courtesy A. Molvai Tabrizi A. Molavi Tabrizi, S. Goossens, A. Rahimi, M.G. Knepley, and J.P. Bardhan, Predicting solvation free energies and thermodynamics in polar solvents and mixtures using a solvation-layer interface condition. Journal of Chemical Physical (2017). M. Knepley (Rice) Solvation LS18 32 / 52

35 Improving the Poisson Operator Main Successes of SLIC Accurate charging free energy using crystal radii (no fitting/temp dep) for (de-)protonation for individual atoms for mixtures M. Knepley (Rice) Solvation LS18 33 / 52

36 Improving the Poisson Operator Main Successes of SLIC Accurate charging free energy using crystal radii (no fitting/temp dep) for (de-)protonation for individual atoms for mixtures M. Knepley (Rice) Solvation LS18 33 / 52

37 Improving the Poisson Operator Main Successes of SLIC Accurate charging free energy using crystal radii (no fitting/temp dep) for (de-)protonation for individual atoms for mixtures M. Knepley (Rice) Solvation LS18 33 / 52

38 Improving the Poisson Operator Main Successes of SLIC Accurate charging free energy using crystal radii (no fitting/temp dep) for (de-)protonation for individual atoms for mixtures M. Knepley (Rice) Solvation LS18 33 / 52

39 Improving the Poisson Operator Main Successes of SLIC Accurate charging free energy using crystal radii (no fitting/temp dep) for (de-)protonation for individual atoms for mixtures M. Knepley (Rice) Solvation LS18 33 / 52

40 Improving the Poisson Operator Main Successes of SLIC Accurate transfer free energy for water-octanol system on par with explicit-solvent MD Reinterpretation of Mean Spherical Approximation Explains temperature dependence of model coefficients M. Knepley (Rice) Solvation LS18 34 / 52

41 Improving the Poisson Operator Main Successes of SLIC Accurate transfer free energy for water-octanol system on par with explicit-solvent MD Reinterpretation of Mean Spherical Approximation Explains temperature dependence of model coefficients M. Knepley (Rice) Solvation LS18 34 / 52

42 Improving the Poisson Operator What is missing from SLIC? Large packing fraction No charge oscillation or overcharging Classical DFT? (Gillespie, Microfluidics and Nanofluidics, 2015) No dielectric saturation Possible with different condition No long range correlations Use nonlocal dielectric M. Knepley (Rice) Solvation LS18 35 / 52

43 Future Work Future Work More complex solutes Better nonlinear solvers Mixtures Preliminary work is accurate Integration into community code Psi4, QChem, APBS M. Knepley (Rice) Solvation LS18 36 / 52

44 Thank You!

45 Outline Future Work Approximate Boundary Conditions Approximate Boundary Conditions M. Knepley (Rice) Solvation LS18 38 / 52

46 Bioelectrostatics Physical Model Future Work Approximate Boundary Conditions Electrostatic Potential φ Region I: protein Region II: solvent Surface M. Knepley (Rice) Solvation LS18 39 / 52

47 Future Work Kirkwood s Solution (1934) Approximate Boundary Conditions 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 (Rice) Solvation LS18 40 / 52

48 Future Work Kirkwood s Solution (1934) Approximate Boundary Conditions 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 (Rice) Solvation LS18 41 / 52

49 Future Work Kirkwood s Solution (1934) Approximate Boundary Conditions 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 (Rice) Solvation LS18 42 / 52

50 Future Work 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 (Rice) Solvation LS18 43 / 52

51 Future Work 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 (Rice) Solvation LS18 43 / 52

52 Future Work Proof of PDE Equivalence Approximate Boundary Conditions 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 (Rice) Solvation LS18 44 / 52

53 Future Work Proof of PDE Equivalence Approximate Boundary Conditions 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 (Rice) Solvation LS18 44 / 52

54 Future Work Proof of PDE Equivalence Approximate Boundary Conditions 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 (Rice) Solvation LS18 44 / 52

55 Future Work Proof of PDE Equivalence Approximate Boundary Conditions 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 (Rice) Solvation LS18 44 / 52

56 Future Work Proof of Eigenspace Equivalence Approximate Boundary Conditions 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 (Rice) Solvation LS18 45 / 52

57 Future Work Proof of Eigenspace Equivalence Approximate Boundary Conditions 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 (Rice) Solvation LS18 45 / 52

58 Future Work Proof of Eigenspace Equivalence Approximate Boundary Conditions 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 (Rice) Solvation LS18 45 / 52

59 Future Work Proof of Eigenspace Equivalence Approximate Boundary Conditions 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 (Rice) Solvation LS18 45 / 52

60 Future Work Proof of Eigenspace Equivalence Approximate Boundary Conditions 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 (Rice) Solvation LS18 45 / 52

61 Series Solutions Future Work Approximate Boundary Conditions 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 (Rice) Solvation LS18 46 / 52

62 Series Solutions Future Work Approximate Boundary Conditions 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 (Rice) Solvation LS18 46 / 52

63 Asymptotics Future Work Approximate Boundary Conditions 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 (Rice) Solvation LS18 47 / 52

64 Asymptotics Future Work Approximate Boundary Conditions 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 (Rice) Solvation LS18 47 / 52

65 Asymptotics Future Work Approximate Boundary Conditions 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 (Rice) Solvation LS18 48 / 52

66 Improved Accuracy Future Work Approximate Boundary Conditions BIBEE/I interpolates between BIBEE/CFA and BIBEE/P Bardhan, Knepley, JCP, M. Knepley (Rice) Solvation LS18 49 / 52

67 Basis Augmentation Future Work Approximate Boundary Conditions 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 (Rice) Solvation LS18 50 / 52

68 Basis Augmentation Future Work Approximate Boundary Conditions 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 (Rice) Solvation LS18 51 / 52

69 Basis Augmentation Future Work Approximate Boundary Conditions 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 (Rice) Solvation LS18 52 / 52