Electrostatics

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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 11 salts, alternative form is
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DELPHI (Honig)
Nicholls and Honig (1991, JCompChem)
Honig and Nicholls (1995, Science)
Jacobian relaxation

• finite difference method

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How to use Delphi

• https//www.scripps.edu/rc/softwaredocs/msi/insigh
  t2K/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 Hs, vdw radii, and partial
  charges for aas and nas) note
  HIS/HID/HIE/HIP
• check hydrogen names
• script
• unixgt delphi lt delphi.in gt delphi.out
• output
• energies in log file
• check net assigned charge
• ltpotential_mapgt.phi (for GRASP or chimera)
• potentials at specific coords

dhfr.in ------- gsize65 scale1.0 in(pdb,file"dh
fr.pdb") in(siz,file"parseres.siz") in(crg,file"
parseres.crg") indi4.0 exdi80.0 prbrad1.4 salt
0.10 bndcon2 maxc0.0001 !linit800 nonit800 ene
rgy(s,c,g) out(phi,file"dhfr-mesh.phi") in(frc,fi
le"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/molK
(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/molK
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Uses of Delphi

• Calculation of pKas
• place a test charge, evaluate potential, dont
  forget to subtract solvation energy of test
  charge
• Calculation of binding energies (P-P complexes)
• Do 3 runs A (apo/solvated), B (apo/solv), AB
  (complex)
• reviews
• Gilson and Honig (1988)
• Sheinerman, Norel, Honig (2000)
• Sheinerman and Honig (2002, JMB)
• study of 4 complexes barnasebarstar, human
  growth hormone receptror, neuraminidaseantibody,
  Raskinase
• role of polar vs. non-polar interactions varies

(show correlation plot of binding affinities
with estimates via delphi)
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• examples of Delphi potentials mapped onto
  molecular surfaces (using GRASP)

acetylcholine esterase
DNA-binding proteins from DNA polymerase
III subunit b
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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 (e1) and once for water (e80)
• 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
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Non-polar term, DGsolv,np

• cavity formation VDW attraction
• weak, typically proportional to surface area (SA)
• Sitkoff Sharp Honig (1994)
• fit for alkanes
• g5.0 0.5 cal/mol Å2
• b0.86 0.1 kcal/mol
• depends on curvature of cavity
• Massova Kollman (2000), Ferrari et al (2007)
  use g7.2 cal/mol Å2 (b0) or g5.4 cal/mol Å2
  (b0.92 kcal/mol)
• gcav-38, gvdw46 (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
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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 DGsol 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)
(1/f for RiRj1/2)
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from Warshel, Russel, Churg (1984)
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Effective Born radius
(show increase in effective Born radius with
depth of burial...)
calculation requires integration over volume of
the molecule (shape)
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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 atoms stretch 1-2 connected
atoms CCF 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