Implicit solvent simulations

1. Implicit solvent simulations Nathan Baker (baker@biochem.wustl.edu) BME 540

2. Introduction to biomolecular electrostatics • Highly relevant to biological function • Important tools in interpretation of structure and function • Electrostatics pose one of the most challenging aspects of biomolecular simulation • Long range • Divergent • Existing methods limit size of systems to be studied Acetylcholinesterase Fasciculin-2

3. Implicit solvent simulations: background • Solute typically only accounts for 5-10% of atoms in explicit solvent simulation • Implicit methods: • Solvent treated as continuum of infinitesimal dipoles • Ions treated as continuum of charge • Some deficiencies: • Polarization response is linear and local • Mean field ion distribution ignores fluctuations and correlations • Apolar effects treated by various, heuristic methods

4. Modeling biomolecule-solvent interactions • Solvent models • Explicit • Molecular dynamics • Monte Carlo • Integral equation • RISM • 3D methods • DFT • Primitive • Poisson equation • Phenomenological • Generalized Born • Modified Coulomb’s law • Ion models • Explicit • Molecular dynamics • Monte Carlo • Integral equation • RISM • 3D methods • DFT • Field theoretic • Poisson-Boltzmann • Extended PB, etc. • Phenomenological • Generalized Born • Debye-Hückel Computational cost Level of detail

5. Explicit solvent simulations • Sample the configuration space of the system: ions, atomically-detailed water, solute • Sampling performed with respect to an ensemble: NpT, NVT, etc. • Algorithms: molecular dynamics and Monte Carlo • Advantages: • High levels of detail • Easy inclusion of additional degrees of freedom • All interactions considered explicitly • Disadvantages: • Slow (and uncertain) convergence • Time-consuming • Boundary effects • Poor scaling to larger systems • Some effects still not considered in many force fields…

6. Implicit solvent simulations • Free energy evaluations: • Usually based on static solute structures or small number of conformational “snapshots” • Solvent effects included in: • Implicit solvent electrostatics • Surface area-dependent apolar terms • Useful for: • Solvation energies • Binding energies • Mutagenesis studies • pKa calculations

7. Implicit solvent simulations • Stochastic dynamics • Usually based on Langevin or Brownian equations of motion • Solvent effects included in: • Implicit solvent electrostatics forces • Hydrodynamics • Random solvent forces • Useful for: • Bimolecular rate constants • Conformational sampling • Dynamical properties Animation courtesy of Dave Sept

8. Analytical models • Include: • Coulomb • Debye-Hückel • Generalized Born • Other • Simple and fast • Do not accurately capture solvation behavior • Require parameterization…

9. Coulomb law • Simplest implicit solvent model • Assumptions: • Solvent = homogeneous dielectric • Point charges • No mobile ions • Infinite domain (no boundaries) Charge magnitudes Solvent dielectric Charge locations

10. Coulomb law • Simplest implicit solvent model • Assumptions: • Solvent = homogeneous dielectric • Point charges • No mobile ions • Infinite domain (no boundaries) • Solution to Poisson equation Point charge distribution Boundary condition

11. Coulomb law • Simplest implicit solvent model • Assumptions: • Solvent = homogeneous dielectric • Point charges • No mobile ions • Infinite domain (no boundaries) • Solution to Poisson equation • Very simple energy evaluation

12. Debye-Hückel law • Similar to Coulomb’s law • Assumptions: • Solvent = homogeneous dielectric • Point charges • Non-interacting mobile ions with linear response • Infinite domain (no boundaries) Inverse screening length Mobile ion bulk density

13. Debye-Hückel law

14. Debye-Hückel law • Similar to Coulomb’s law • Assumptions: • Solvent = homogeneous dielectric • Point charges • Non-interacting mobile ions with linear response • Infinite domain (no boundaries) • Solution to Helmholtz equation

15. Debye-Hückel law • Similar to Coulomb’s law • Assumptions: • Solvent = homogeneous dielectric • Point charges • Non-interacting mobile ions with linear response • Infinite domain (no boundaries) • Solution to Helmholtz equation • Simple energy evaluation

16. Generalized Born • Used to calculate solvation energies (forces) • Modification of Born ion solvation energy: • Adjust effective radii of atoms based on environment • Differences between buried and exposed atoms • Fast to evaluate • Lots of variations • Hard to parameterize

17. Non-analytical continuum models • Include: • Poisson • Poisson-Boltzmann • More realistic description of biomolecules: • Allow for variable dielectrics: • Interior (2-20) • Solvent (80) • Define regions of inaccessibility for ions • Complicated geometries require numerical solution • More computationally demanding

18. Poisson equation • Describes electrostatic potential due to: • Inhomogeneous dielectric • Charge distribution • Assumes: • Linear and local solvent response • No mobile ions Dielectric function

19. Poisson equation: energies • Total energies obtained from • Integral of polarization energy

20. Poisson equation: energies • Total energies obtained from • Integral of polarization energy • Sum of charge-potential interactions

21. Poisson equation: energies • Total energies obtained from • Integral of polarization energy • Sum of charge-potential interactions • Energies contain self-interaction terms: • Infinite (for analytic solution) • Very unstable (for numerical solution) • Self-interactions must be removed

22. The reaction field • The potential due to inhomogeneous polarization of the solvent • The difference of potentials with: • Inhomogeneous dielectric • Homogeneous dielectric • Implicitly removes terms due to self-interactions: • Non-singular • Numerically-stable • Not available via simpler models… Reaction field

23. Reaction field example • Potentials near low dielectric bodies do not superimpose • Contain: • Coulombic term • Reaction field term Total electrostatic potential Reaction field

24. Solvation energy • Solvation energies obtained directly from reaction field • Difference of • Homogeneous • Inhomogeneous dielectric calculations • Self-energies removed in this process: • Numerical stability • Non-infinite results

25. A continuum descriptionof ion desolvation • Two Born ions at varying separations • Solve Poisson equation at each separation • Increase in energy as “water” is squeezed out of interface • Desolvation effect • Less volume of polarized water • Important points • Non-superposition of Born ion potentials • Reaction field causes repulsion at short distances • Dielectric medium “focuses” field

26. Low dielectric High dielectric Point charge A continuum descriptionof ion solvation • Born ion model • Non-polarizable ion • Point charge • Higher polarizability medium • “Reaction field” effects • Non-Coulombic potential inside ion due to polarization of solvent • Solvation energy • Simple model with analytical solutions

27. A continuum descriptionof ion solvation

28. A continuum descriptionof ion desolvation

29. Poisson-Boltzmann equation • Abbreviation = PBE • Describes electrostatic potential due to: • Inhomogeneous dielectric • Mobile counterions • “Fixed” (biomolecular) charge distribution • Assumes: • Linear and local solvent response • No explicit interaction between mobile ions

30. Poisson-Boltzmann derivation: step 1 • Start with Poisson equation to describe solvation • Supplement biomolecular charge distribution with mobile ion term Dielectric function Biomolecular charge distribution Mobile charge distribution

31. Poisson-Boltzmann derivation: step 2 • Choose mobile ion charge distribution form: • Boltzmann distribution  no explicit ion-ion interaction • No detailed structure for atom (de)solvation Ion charges Ion bulk densities Ion-protein steric interactions

32. Poisson-Boltzmann derivation: step 3 • Substitute mobile charge distribution back into Poisson equation • Result: Nonlinear partial differential equation

33. Equation coefficients: charge distribution • Charges are delta functions: hard to model • Often discretized as splines to “smooth” the problem • What about higher-order charge distributions?

34. Equation coefficients: mobile ion distribution • Provides: • Bulk ionic strength • Ion accessibility • Usually constructed based on “inflated van der Waals radii”

35. Equation coefficients: dielectric function • Describes change in dielectric response: • Low dielectric interior (2-20) • High dielectric solvent (80) • Many definitions: • Molecular (solid line) • Solvent-accessible (dotted line) • van der Waals (gray circles) • Inflated van der Waals (previous slide) • Smoothed definitions (spline-based and Gaussian) • Results can be very sensitive to the choice of surface!!!

36. Poisson-Boltzmann special cases • 1:1 electrolyte (NaCl) • Assume similar steric interactions for each species with protein • Simplify two-term series to hyperbolic sine Modified screening coefficient: zero inside biomolecule 1:1 electrolyte charge distribution

37. Poisson-Boltzmann special cases • 1:1 electrolyte (NaCl) • Assume similar steric interactions for each species with protein • Simplify two-term series to hyperbolic sine • Small charge-potential interaction • Linearized Poisson-Boltzmann

38. Non-specific salt effects: screening • Lots of types of non-specific ion screening: • Variable solvation effects (Hofmeister) • Ion “clouds” damping electrostatc potential • Changes in co-ion and ligand activity coefficients • Condensation • Not all ion effects are non-specific! • Generally reduces effective range of electrostatic potential • Shown here for acetylcholinesterase • Illustrated by potential isocontours • Observed experimentally in reduced binding rate constants

39. mAChE at 0 mM NaCl mAChE at 150 mM NaCl Non-specific salt effects: screening

40. Poisson-Boltzmann energies • Similar to Poisson equation • Functional = integral over solution domain • Solution extremizes free energy Fixed charge- potential interactions Dielectric polarization Mobile charge energy

41. PBE: removing “self energies” and calculating interesting stuff • Energy calculations must be performed with respect to reference system with same discretization: • Same differential operator: • Same charge representation • Reference systems implicit in • Solvation energies • Binding energies

42. Electrostatic influenceson ligand binding • Examine inhibitor binding to protein kinase A: • Part of drug design project by McCammon and co-workers • Illustrates how electrostatics governs specificity and affinity • Look at complementarity between ligand and protein electrostatics • Verify with experimental data (relative binding affinities) • Use to guide design of improved inhibitors

43. Balanol Protein Kinase A Electrostatic influenceson ligand binding

44. Electrostatic influenceson ligand binding

45. Poisson-Boltzmann equation:force evaluation • Integral of electrostatic potential over solution domain • Assume solution fixed over atomic displacements • Differentiate with respect to atomic positions • Contains contributions from Osmotic pressure Dielectric boundary Reaction field

46. PBE: considerations with force evaluation • Remove self-energies: two PB calculations to give “reaction field forces” • Inhomogeneous dielectric: non-zero fixed charge, dielectric boundary, and osmotic pressure forces • Homogeneous dielectric: only non-zero fixed charge forces • Coulombic interactions added in analytically • Uses: • Minimization • Single-point force evaluation • Dynamics • Need fast setup and calculation • Currently ~8 sec/calc for Ala2  1 day/ns with 10 fs steps

47. Solving the PE or PBE • Determine the coefficients based on the biomolecular structure • Discretize the problem • Solve the resulting linear or nonlinear algebraic equations

48. Discretization • Choose your problem domain: finite or infinite? • Usually finite domain • Requires relatively large domain • Uses asymptotically-correct boundary condition (e.g., Debye-Hückel, Coulomb, etc.) • Infinite domain requires appropriate basis functions • Choose your basis functions: global or local? • Usually local: map problem onto some sort of grid or mesh • Global basis functions (e.g. spherical harmonics) can cause numerical difficulties

49. Discretization: local methods • Polynomial basis functions (defined on interval) • “Locally supported” on a few grid points • Only overlap with nearest-neighbors  sparse matrices Boundary element (Surface discretization) Finite element (Volume discretization) Finite difference (Volume discretization)

50. Discretization: pros & cons • Finite difference: • Sparse numerical systems and efficient solvers • Handles linear and nonlinear PBE • Easy to setup and analyze • Non-adaptive representation of problem • Finite element: • Sparse numerical systems • Handles linear and nonlinear PBE • Adaptive representation of problem • Not easy to setup and analyze • Less efficient solvers • Boundary element: • Very adaptive representation of problem • Surface discretization instead of volume • Not easy to setup and analyze • Less efficient solvers • Dense numerical system • Only handles linear PBE