Electrostatics Poisson-Boltzmann equation finite-difference see review by Sharp and Honig (1990) Delphi GRASP solvation energy –interactions –Generalized-Born

Poisson-Boltzmann equation Laplace equation: Poisson equation: –potentials must meet at dielectric boundary Poisson-Boltzmann equation –effect of ions in solvent on potential field –z i is charge of ion i, c i is concentration –salt/ionic effects: counter-ions move in solvent to adjust local concentration to local potential

for 1:1 salts, alternative form is

DELPHI (Honig) finite difference method: Jacobian relaxation Nicholls and Honig (1991, JCompChem) Honig and Nicholls (1995, Science)

How to use Delphi t2K/delphi/delphiTOC.html param files (copy to local directory): –parseres.siz, parseres.crg (Sitkoff, Sharp, Honig, 1994); polar H’s, vdw radii, and partial charges for aa’s and na’s) – note: HIS/HID/HIE/HIP –check hydrogen names script: –unix> delphi delphi.out output: –energies in log file –check net assigned charge –.phi (for GRASP or chimera) –potentials at specific coords dhfr.in gsize=65 scale=1.0 in(pdb,file="dhfr.pdb") in(siz,file="parseres.siz") in(crg,file="parseres.crg") indi=4.0 exdi=80.0 prbrad=1.4 salt=0.10 bndcon=2 maxc= !linit=800 nonit=800 energy(s,c,g) out(phi,file="dhfr-mesh.phi") in(frc,file="dhfr-mesh.pdb") out(frc,file="dhfr-mesh.pot") site(a,x,p,q) (1) total grid energy : kt (2) self-reaction field energy : kt (3) total s.charge,no epsin carrying : (4) corrected reaction field energy: kt (5) total reaction field energy : kt (6) coulombic energy : kt (7) All energy terms but grid and self_react.: kt 1 kT = kcal/mol for T = 298 K and k = kcal/molK 1 kT = kcal/mol for T = 298 K and k = kcal/molK

Uses of Delphi Calculation of pKa’s –place a test charge, evaluate potential, don’t forget to subtract solvation energy of test charge Calculation of binding energies (P-P complexes) –Do 3 runs: A (apo/solvated), B (apo/solv), A+B (complex) –reviews: Gilson and Honig (1988) Sheinerman, Norel, Honig (2000) –Sheinerman and Honig (2002, JMB) study of 4 complexes – barnase:barstar, human growth hormone: receptror, neuraminidase:antibody, Ras:kinase role of polar vs. non-polar interactions varies (show correlation plot of binding affinities with estimates via delphi)

examples of Delphi potentials mapped onto molecular surfaces (using GRASP) acetylcholine esterase DNA-binding proteins from DNA polymerase III subunit 

Solvation Energy important for interactions –free energy of binding involves desolvation of receptor and ligand (polar and non-polar contributions) total electrostatic energy of molecule includes –Coulombic interaction of charges (and dipoles), –plus energy due to solvent “reaction field” (charges attracted to surface) –“self energy” – int. charge with induced surface charges –cross terms– reduction in charge-charge interactions by attracted surface charges to other (“solvent screening”) –Gilson and Honig (1988)

reaction field energy in Delphi, total energy includes grid energy, must subtract out do calculations twice: –once for vacuum (e=1) and once for water (e=80) –take difference of potentials at each grid point alternatively: calculate charges at surface positions –mapping to fixed grid creates approximation error –can “scale” surface points to molecular surface to increase accuracy –these are the “corrected” reaction field energies in Delphi  i are surface charges q j are molecule charges

Non-polar term,  G solv,np cavity formation + VDW attraction –weak, typically proportional to surface area (SA) –Sitkoff Sharp Honig (1994) –fit for alkanes:  =5.0 ± 0.5 cal/mol Å 2 b=0.86 ± 0.1 kcal/mol –depends on curvature of cavity –Massova & Kollman (2000), Ferrari et al (2007)* use  =7.2 cal/mol Å 2 (b=0) or  =5.4 cal/mol Å 2 (b=0.92 kcal/mol) –  cav =-38,  vdw =+46 (Noskov; Friedman) Levy et al (JACS, 2003) – On the Non- polar Hydration Free Energy... * see footnote to Table 1

Interactions difference of energy of apo vs. complex in solvent vs. vacuum over half of complex have substantial changes between apo and complexed forms (Betts & Sternberg, 1999) energy related to induced fit (Noskov and Lim, 2001) Marilyn Gunner

Implicit Models of Solvation avoid solving PBE for potential – too slow for dynamics/docking model  G sol via scaling of charge-charge interactions according to depth of buriedness depends on solvent-accessible surface, shape of dielectric boundary

Generalized Born Approximation The goal of GB theory can be thought of as an effort to find a relatively simple analytical formula, resembling Equation 6, which for real molecular geometries will capture, as much as possible, the physics of the Poisson equation. Born approximation for ion (point charge in sphere of atomic radius) use effective Born radii Ri,Rj to scale charge-charge interactions (eqn. 6) radius a  G solv + a  =  wat =80  =  vac =1 q (1/f for Ri=Rj=1/2)

from Warshel, Russel, Churg (1984)

Effective Born radius calculation requires integration over volume of the molecule (shape) (show increase in effective Born radius with depth of burial...)

Methods to calculate Born radii replace volume integration (1/r 4 ) with atom-pairwise computation methods: –Still et al (1990) – numeric integration –Qui (1997) – add volumes of atoms –Ghosh Rapp Friesner (1998) – surface integral –Hawkins Cramer Truhlar (1996) analytic formula for 1/r 4 in sphere radii scaling params to account for overlaps –Liu Kuntz Zou (2004) – grid in DOCK –Dominy & Brooks (1999) – re-fit params for CHARMM bend: 1-3 connected atoms stretch: 1-2 connected atoms CCF: close-contact function

GB-solv can be added as term in AMBER FF: –calculation of solvation params (effective Born radii) changes with shape/conformation –see AMBER 10 manual also SASA term in CHARMM 19 (EFF1) Warshel, Russell, Churg (1984) – self-energy Onsager energy of buried dipole