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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
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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 • zi is charge of ion i, ci is concentration • salt/ionic effects: counter-ions move in solvent to adjust local concentration to local potential
DELPHI (Honig) Nicholls and Honig (1991, JCompChem) Honig and Nicholls (1995, Science) Jacobian relaxation • finite difference method:
How to use Delphi • https://www.scripps.edu/rc/softwaredocs/msi/insight2K/delphi/delphiTOC.html • http://bcr.musc.edu/manuals/delphi.htm • 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.in > delphi.out • output: • energies in log file • check net assigned charge • <potential_map>.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=0.0001 !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 kT = 0.592 kcal/mol for T = 298 K and k = 0.001986577 kcal/mol•K (1) total grid energy : 5168.769 kt (2) self-reaction field energy : -19088.44 kt (3) total s.charge,no epsin carrying : 1.4302 (4) corrected reaction field energy: -782.8139 kt (5) total reaction field energy : -19871.26 kt (6) coulombic energy : -8125.605 kt (7) All energy terms but grid and self_react.: -8908.419 kt 1 kT = 0.592 kcal/mol for T = 298 K and k = 0.001986577 kcal/mol•K
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) DNA-binding proteins from DNA polymerase III subunit b acetylcholine esterase
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 di are surface charges qj are molecule charges
Non-polar term, DGsolv,np • cavity formation + VDW attraction • weak, typically proportional to surface area (SA) • Sitkoff Sharp Honig (1994) • fit for alkanes: • g=5.0 ± 0.5 cal/mol Å2 • b=0.86 ± 0.1 kcal/mol • depends on curvature of cavity • Massova & Kollman (2000), Ferrari et al (2007)* use g=7.2 cal/mol Å2 (b=0) or g=5.4 cal/mol Å2 (b=0.92 kcal/mol) • gcav=-38, gvdw=+46 (Noskov; Friedman) • Levy et al (JACS, 2003) – On the Non-polar Hydration Free Energy... *http://dx.doi.org/10.1016/j.bmc.2007.08.019 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 DGsol via scaling of charge-charge interactions according to depth of buriedness • depends on solvent-accessible surface, shape of dielectric boundary
- - - + a - q DGsolv - e=evac=1 - e=ewat=80 - - radius a 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) (1/f for Ri=Rj=1/2)
Effective Born radius (show increase in effective Born radius with depth of burial...) calculation requires integration over volume of the molecule (shape)
Methods to calculate Born radii • replace volume integration (1/r4) 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/r4 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