Lecture : Inhomogeneous Solvation Theory with Application to WaterMap

1. Statistical Thermodynamics Lecture : Inhomogeneous Solvation Theory with Application to WaterMap Di Cui tug55642@temple.edu

2. Overview of the Lecture Part I: Basic theory of Inhomogeneous Solvation Theory (IST) Solvation Energy One Body Solvation Entropy Two Body Solvation Entropy Part II: Application of IST – WaterMap Factor Xa – ligand binding Part III: Application of IST – GIST Solvation Free Energy of Simple Solutes

3. Solvation Free Energy ΔG pure solvent solution Solvation free energy is the amount of free energy associated with dissolving a solute molecule into solvent. It corresponds to the free energy change for turning on the solute-solvent interaction potential. It can be obtained from free energy perturbation (FEP), a number of intermediate states needs to be introduced to connect the end-points. It can also be obtained using solution theory with simulation only at end-points.

4. Framework for Solvation Free Energy Calculation Molecular dynamics simulation using potential functions Distribution functions of solutions Solution theory to calculate solvation free energy

5. Inhomogeneous Solvation Theory (IST) Viewing the solution as inhomogeneous with solute considered “fixed” at the origin, generating an external field that creates solvent density fluctuations around. A formalism that isolates the effect of the solute on the structure of the the solvent next to it without having to deal with the large amounts of unperturbed solvent. ΔGsolv is solvation free energy, ΔEsolv is solvation energy, ΔSsolv is solvation entropy. The next step is to write the solvation energy and entropy in terms of correlation functions of end-points. Lazaridis T (1998) J Phys Chem B 102:3531-3541

6. Solvation Energy Esw is the total solute-solvent interaction in solution, Eww is the total solvent-solvent interaction in solution, Epure is the total interaction in pure water. Usw and Uww are the solute-solvent and solvent-solvent interaction potentials, x refers to the position and orientation of a solvent molecule relative to the solute, ρ(x) is density distribution from solute-solvent, ρ(2)(x,x1) is density distribution from solute-solvent-solvent.

7. Solvation Entropy Ssw accounts for the solute-water correlations, Sww for water-water correlations in solution and Spure for water-water correlations in pure water. Entropy density at location x, can be expressed based on Green-Wallace expansion: Wallace D (1987) J Chem Phys 87:2282-2284

8. Entropy Terms Continue Gilson M (2012) J Chem Phys 137, 044101

9. Application of IST - WaterMap WaterMap: mapping thermodynamic properties of water molecules that solvate protein binding sites and using this data to understand binding affinities. This tool was developed by Friesner’s group in Columbia University, who is founder of a software company called Schrödinger. Water displacement is a key step that affects protein-ligand binding. Usually, there is a gain in entropy as releasing binding site water molecules into bulk. Ligand + Protein Friesner A (2007) PNAS 104:808-813 Friesner A (2008) JACS 130:2817-2831

10. Displacement of Unstable Water Enantiomers bind to protein Bcl-xL, R-enantiomer binds much more strongly R S For R, Kd = 0.0008(μM), displacement of 3 unstable water makes the binding affinity larger For S, Kd = 0.252 (μM), without displacement of unstable water, binding affinity is smaller

11. Mapping Thermodynamic Properties of Water Binding Cavity: The starting points for each simulation was protein-ligand complex. The ligand was then removed and the vacated volumes that water can fill is binding cavity. Density Profile: Throughout the MD simulation, monitoring the water entered binding cavity and calculated the water density profile. Clustering Algorithm: Identifying subvolumes of the binding cavities with high densities, defined as hydration sites. IST: Excess energy and excess entropy for each hydration sites can be calculated based on IST.

12. Example: Solvent Density Distribution and Clustering Binding cavity of streptavidin and typical solvating water configuration Solvent density shown in green and the clustering of the density in wireframe

13. More on Five-membered Water Ring The formation of five-membered water ring is energetically favorable but entropically unfavorable. Only fleetingly observed in bulk water, but always observed in the streptavidin binding cavity due to the topographical characteristics. The ring is enclosed above and below by hydrophobic groups, the only orientations for water can maintain maximal number of hydrogen bonds are those consistent with ring formation. Due to the high order of the water in the ring, displacement of the water may contribute as large as -7 kcal/mol to the free energy.

14. WaterMap Example: Factor Xa Ligand Binding Factor Xa: an important drug target in the thrombosis pathway, several inhibitors are currently on clinical trials. Factor Xa inhibitors generally bind in an L-shaped conformation Using IST to calculate binding free energy differences ΔΔG between pairs of Factor Xa ligands, not to compute the absolute binding free energy ΔG of a given ligand and receptor. Displacement of water molecules is not the only factor determining absolute binding free energy. (loss of entropy of binding the ligand; interaction between ligand and protein) For congeneric ligands that differ by only small chemical modifications, these additional contributions are small, properties of the excluded solvent are dominant. Friesner A (2008) JACS 130:2817-2831

15. Factor Xa Hydration Site contribute both energetically and entropically contribute energetically contribute entropically Friesner A (2008) JACS 130:2817-2831

16. Scoring Function If a heavy atom of a ligand overlapped with a hydration site, it displayed the water from that site. The less energetically or entropically favorable the expelled water, the more favorable its contributions to the binding free energy. A hydration site would contribute to the binding free energy if Shs or Ehs were beyond the fitted entropy and energy cutoff parameters, SCO and ECO. A flat reward value was given for such hydration site as Srwd and Erwd. A fit cutoff distance (Rco) was used to determine whether ligand atom displaced water from a hydration site. Friesner A (2008) JACS 130:2817-2831

17. Congeneric Ligand Pairs XLC XLD -CH3 group here displaced water in hydration 13 ΔΔGscore = -2.85 kcal/mol ΔΔGexp = -2.94 kcal/mol

18. Comparison with Experiment R2 = 0.81 Computed relative free energy difference using the five-parameter form of scoring function vs experimental results of the 31 congeneric inhibitor pairs. Friesner A (2008) JACS 130:2817-2831

19. Choice of Subvolumes in IST An important decision in IST is the choice of subvolumes over which to perform the calculations. In protein binding sites, water molecules commonly cluster in distinct locations and the concept of hydration site is useful. For small molecule solvation, two approaches have been used to account for the volume around the solute: the region surrounding the solute was split into subshells at different distances or split into cubic voxels on a Cartesian grid (GIST).

20. Application of IST – GIST Grid Inhomogeneous Solvation Theory (GIST): discretization of inhomogeneous solvation theory on a 3D grid, the spatial integrals in the IST expressions are replaced by discrete sums over the voxels. Specifically, a spatial region R is discretized into voxels indexed by k, centered at locations rk and having volumes Vk, each voxel has the same volume and the density is treated as uniform over each voxel k. Gilson M (2012) J Chem Phys 137, 044101

21. GIST Example: CB7 Contour of ΔEww cyan: more favorable, orange: less favorable Contour of ΔEsw orange: more favorable, blue: less favorable Cucurbit[7]uril Contour of -TΔSswtrans tan: more favorable, red: less favorable Contour of -TΔSsworient yellow: more favorable, violet: less favorable Gilson M (2012) J Chem Phys 137, 044101

22. GIST Example : Results

23. GIST Example: Comparison with FEP Comparison of the solvation free energy for several small molecule solutes calculated from GIST and FEP. While a direct comparison with experiment is interesting, the results rely on the force field and particularly the water model that is used. A more useful comparison is with an equivalent computational technique, decoupling test of the method from test of the parameters. Huggins D (2013) JPCB 117, 8232-8244

24. Results of GIST Calculations on Simple Solutes Huggins D (2013) JPCB 117, 8232-8244

25. Solvation Free Energy Comparison Huggins D (2013) JPCB 117, 8232-8244

26. Solvation Free Energy Correlations Huggins D (2013) JPCB 117, 8232-8244

27. Summary Inhomogeneous Solvation Theory (IST) provide a framework for relating distribution functions of solutions to their thermodynamic properties. IST can be applied to calculate the solvation free energy of solute without introduction of intermediate states. In IST, systems are spatially decomposed to consider the contribution of specific regions to the total solvation free energy. Water molecules at a protein receptor site that are most advantageous to replace by a ligand can be identified to aid in the process of structure based drug design.