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

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• 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

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)

• examples of Delphi potentials mapped onto molecular surfaces (using GRASP)

acetylcholine esterase DNA-binding proteinsfrom DNA polymerase IIIsubunit

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 chargesqj are molecule charges

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

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 Gsol 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

Gsolv+a

=wat=80=vac=1q

- -

--

--

-

- (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 effectiveBorn radius with depth of burial...)

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

• 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