Electrostatics Poisson-Boltzmann equation finite-difference see review by Sharp and Honig (1990)...
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![Page 1: Electrostatics Poisson-Boltzmann equation finite-difference see review by Sharp and Honig (1990) Delphi GRASP solvation energy –interactions –Generalized-Born.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649d6a5503460f94a4855b/html5/thumbnails/1.jpg)
Electrostatics• Poisson-Boltzmann equation• finite-difference• see review by Sharp and Honig (1990)• Delphi• GRASP• solvation energy
– interactions– Generalized-Born
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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
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for 1:1 salts, alternative form is
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DELPHI (Honig)• finite difference method:
Jacobian relaxation
Nicholls and Honig (1991, JCompChem)
Honig and Nicholls (1995, Science)
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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=65scale=1.0in(pdb,file="dhfr.pdb")in(siz,file="parseres.siz")in(crg,file="parseres.crg")indi=4.0exdi=80.0prbrad=1.4salt=0.10bndcon=2maxc=0.0001!linit=800nonit=800energy(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 : 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
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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 ofbinding affinities withestimates via delphi)
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• examples of Delphi potentials mapped onto molecular surfaces (using GRASP)
acetylcholine esterase DNA-binding proteinsfrom DNA polymerase IIIsubunit
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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)
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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 chargesqj are molecule charges
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Non-polar term, Gsolv,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 Å2
• b=0.86 ± 0.1 kcal/mol– depends on curvature of cavity– Massova & Kollman (2000), Ferrari et
al (2007)* use =7.2 cal/mol Å2 (b=0) or =5.4 cal/mol Å2 (b=0.92 kcal/mol)
– cav=-38, vdw=+46 (Noskov; Friedman)• Levy et al (JACS, 2003) – On the Non-
polar Hydration Free Energy...
*http://dx.doi.org/10.1016/j.bmc.2007.08.019see footnote to Table 1
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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
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Implicit Models of Solvation• avoid solving PBE for potential – too slow for
dynamics/docking
• model Gsol via scaling of charge-charge interactions according to depth of buriedness
• depends on solvent-accessible surface, shape of dielectric boundary
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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
Gsolv+a
=wat=80=vac=1q
- -
--
--
-
- (1/f for Ri=Rj=1/2)
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from Warshel, Russel, Churg (1984)
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Effective Born radius
calculation requires integration over volume of the molecule (shape)
(show increase in effectiveBorn radius with depth of burial...)
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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 atomsstretch: 1-2 connected atomsCCF: close-contact function
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• 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