Variational theory of solvation

1. Variational theory of solvation Joachim Dzubiella Physics Department@Technical University Munich CTBP summer school 2008 „Coarse-Grained Physical Modeling of Biological Systems“

2. Outline • Brief intro on implicit solvation. • A few (explicit) systems, for which established implicit solvent models fail. • Basics of a ‚variational implicit solvent model‘ (VISM).

3. I. Brief intro on implicit solvation.

4. Motivation: biomolecular (equilibrium) processes are described by changes in the solvation free energy G: vapor or „vacuum“ Important in the modeling/description of protein assemblies, protein-ligand docking, rational drug design, etc. Quick and accurate estimates desirable.

5. Explicit vs. implicit solvation some protein (solute) continuum approach: (solute-solvent interface, macroscopic constants) Explicit water, e.g., molecular dynamics (MD) simulation Explicit solvent Implicit solvent

6. Implicit solvation: What are the dominant contributions? • ‚Hydrophobicity‘: Simple picture: water does not like to go to a (planar, nonpolar) interface; it‘s missing some of its attractive interactions; it has to rearrange. Macroscopically this gives rise to a (vapor-liquid) surface tension lv. Surface tension units: energy / surface area (A) energy penalty ~ Hydrophobic effects/interactions. A hydrophobic particle is not well solvated.

7. Curvature effects play an important role on small (~ 0.1-2 nm) scales: example:solvation of a hard sphere in water R Empirical expression for weak curvatures (expansion in curvature): liquid-vapor surface tension (planar interface) Symbols: MD results from Huang etal. JCP B 105, 6704 (2001) (SPC/E water, P=1bar, T=300K) Correction coefficient (historically: Tolman length)

8. More dominant contributions: Solute Water • Excluded volume, dispersion (van der Waals) In classical calculations: Lennard-Jones interaction • Electrostatic interactions: With a given solute charge distribution rsolve Poisson‘s equation: The energy of a charge q is Fermi repulsion van der Waals (vdW) attraction (With counterions and salt: Poisson-Boltzmann) local dielectric screening by water

9. Macroscopic interface thermodynamics: Free energy (grand potential) of an interfacial system: water oil R for instance an oil droplet: Note: in macroscopic considerations vdW and electrostatic attractions are all absorbed in the surface tension .

10. Macroscopic interface thermodynamics: Minimization of  w.r.t. R yields the Young- Laplace equation: force-balance equation In general: : local mean curvature Note: example: minimal surface around two hard spheres (3-dimensional) Minimal surface equation H = 0

11. Established implicit solvent models Interface is defined beforehand Free energy is split into nonpolar (np) and polar (p) parts SAS MS which are evaluated separately • In general, no surface scaling on microscopic scales. • Solute-solvent interface is a prescribed input and ill-defined. • Nonpolar solvation is decoupled from electrostatics. (Many) fit-parameters depend on surface definition and solute conformations. with atomic surface area Poisson-Boltzmann/Generalized Born models

12. Established implicit solvent models Interface is defined beforehand Free energy is split into nonpolar (np) and polar parts SAS MS which are evaluated separately • In general, no surface scaling on microscopic scales. • Solute-solvent interface is a prescribed input and ill-defined. • Nonpolar solvation is decoupled from electrostatics. Fit-parameters depend on surface definition and solute conformations. with atomic surface area Poisson-Boltzmann/Generalized Born models Improvement by explicit consideration of vdW and volume terms Gallichio & Levy (2004), Wagoner & Baker (2006)

13. II. Systems, for which established implicit models conceptually fail.(selected examples)

14. Concave nonpolar pocket: Model for protein-ligand binding: concave pocket and one methane atom. MD simulation reveal a weakly solvated pocket and a strong hydrophobic attraction. P. Setny, JCP (2006).

15. Comparison to solvent accessible surface area (SASA) and molecular surface area (MSA) models: Onset of attraction is wrong by 2-4 Angstroms!

16. Capillary evaporation in hydrophobic confinement surface tension  L~nm D T. Koishi et al., Phys. Rev. Lett. 93, 185791 (2004)

17. But: drastic changes to interface location and ‚nanobubble‘ stability due to strong and local water-solute attraction: • - dispersion (van der Waals) or • hydrophilic (polar) patches!

18. Evaporation between hydrophobic nanospheres:influence of charging. MD simulation of two hard spheres (R=1nm) in explicit water R Mean force vs distance: Catenoidal shape Surface area: JD and J.-P. Hansen, J. Chem. Phys. 121, 5514 (2004) Surface-to-surface distance

19. Evaporation in hydrophobic channels pore length~1nm, radius R~ 0.5nm Explicit water MD shows that a narrow hydrophobic pore can be empty of water: Nanobubble blocks ion permeation. But: pore fills when a critical electric field (E>Ec) is applied across the pore! Ions can permeate! E >Ec JD, Rosalind Allen, and J.-P. Hansen, J. Chem. Phys. 120, 5001 (2004)

20. Voltage-gating in biological ion channels Consistent with recent MD simulations of the MscS-channel: Narrow hydrophobic region induces evaporation. Channel ist not permeable to ions! Pore fills with application of electrostatic potential. Channel conducts! M Sotomayor et al. Ion conduction through MscS as determined by electrophysiology and simulation Biophys. J. 92, 886-902 (2007)

21. Evaporation in other proteins? Explicit water MD simulations of the bee venom melittin: Water in hydrophobic core. Stable nanobubble P. Liu et al., Nature 437, 159 (2005)

22. J. Phys. Chem. B (2007)

23. Established implicit solvent models Energy is split into nonpolar (np) and polar parts Interface is defined beforehand which are evaluated separately with atom surface area Poisson-Boltzmann/Generalized Born models Obviously, established models fail as interface location is predefined ! Geometric, dispersion, and electrostatic contributions have to be coupled/balanced somehow…

24. III. Variational implicit solvent model

25. Variational implicit solvent model (VISM) excluded volume Define volume exclusion function: Volume Interface area Sharp kink density: Liquid water bulk density JD, J. Swanson, J. A. McCammon: J. Chem. Phys. 124, 084905 (2006); Phys. Rev. Lett. 96, 087802 (2006).

26. VISM free energy functional Basic idea: write the free energy G as a functional of and obtain the solute/water interface by minimization: Possible form:

27. Free energy functional: pressure term Energy penalty for expanding a cavity of volume V against the liquid bulk pressure P.

28. Free energy functional: interfacial term Separate cavity formation effects from dispersion and electrostatics Scaled particle theory or Tolman correction R liquid-vapor surface tension(planar interface) Tolman length Approximation: Symbols: MD results from Huang etal. JCP B 105, 6704 (2001) (SPC/E water, P=1bar, T=300K) with mean curvature H

29. Free energy functional: nonelectrostatic solute-solvent contribution accounts for excluded volumes and dispersion (van der Waals) interactions is typically modeled by the Lennard-Jones (LJ) interaction

30. Free energy functional: electrostatics with Variation of with respect to the electrostatic potential  yields the Poisson-Boltzmann (PB) equation: solute charge density Sharp kink approximation for the position-dependent dielectric constant: dielectric constant in no-water region dielectric constant in bulk liquid

31. Minimization yields Gaussian curvature K=1/(R1R2) • second order partial differential equation (PDE) in terms of pressure, curvatures, dispersion, electrostatics (PB) • Generalized Laplace-Young eq. • force-per-surface-area balance equation • no analytical solution possible level-set method • ! solute is explicitly/atomistically considered in U(r) - term. • Note: electrostatics is still on a PB-level.

32. Example: solvation of one LJ sphere The functional reduces to a function of the radius R of the sphere: Inset: charge contribution Example: Lennard-Jones parameter  = 0.2 kJ/mol and  = 2.85 A.

33. Solvation of one uncharged LJ sphere Tolman length  = fit parameter estimated from MD For SPC water:

34. Effective interaction between two LJ-spheres (modeling two Xe-atoms, radius~0.3nm) Water mediated interaction Best fit Tolman length: Symbols: Implicit model Red lines: Explicit water MD from D. Paschek, JCP 120, 6674 (2004) full interaction (including Xe-Xe LJ part)

35. Hydrophobic nanospheres (R~1nm) + charging Implicit model interface q the only fit parameter is the Tolman length  Symbols: explict MD simulation Lines: theory using

36. Solvent evaporation between nanoplates Interfaces from Implicit model (no electrostatics here, just 2x6x6 nonpolar LJ spheres) Calculated using Effective interaction: Symbols: Implicit model Red line: Explicit water MD MD results from : T. Koishi et al., PRL 93, 185791 (2004)

37. Solvation of a C60 fullerene (nonpolar) Mean curvature Solvation free energy from MD: Best fit Tolman length Side note: enthalpy-entropie compensation in solvation: Solvation free energy is a difference of big numbers: Solvation entropy: Solvation enthalpy: Big problem for solvation free energy calculations!

38. Helical (toy) alkanes ( ~30 nonpolar LJ spheres) Mean curvature Solvation free energy from MD: Best fit Tolman length

39. Summary, questions, work in progress • Solute-solvent interface location at extended nonpolar surfaces sensitively depends on local geometry, dispersion, electrostatics: established implicit models fail, VISM captures the physics. but many things to do: • Numerical solution with the level-set method, marriage to PB solvers, and more benchmarking to MD simulations. • Tolman/curvature correction: testing of higher order expansion in curvature; concave/convex geometry. • Refined dielectric boundary definition to capture asymmetric solvation of cationic/anionic groups. • Interface dynamics, interface fluctuations, and coupling to molecular motions of the solute atoms. What is the interface mobility?