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G Solvation
Continuum Electrostatics
G Solvation
solG = solGVdW + solGcav + solGelec
solGVdW = solute-solvent Van der Waals
solGcav = work to create cavity in solvent
= surface tension x surface area Entropy penalty for rearrangement of
water molecules Evaluate from a series of alkanes
NH
H H
r = 1-5
r = 78.54
G Solvation solGelec = difference in electrostatic
work necessary to charge ion: solGelec = NA wsoln – NA wgas
Work to transfer ion from vacuum to solution with the same electrostatic potential Work = solGelec = 0
Zie i dqi
i = electrostatic potential for ion i and its ionic atmosphere of neighbors j
Electrostatic Potential
r = relative dielectric constant
r = 78.54 for water (attenuates interaction)
V(r) = i qi
V(r) = qj qi
4ro rij
i = qj
4ro rij
i(r)
rij
qiqj
uniform dielectric
0
Screening caused by ionic atmosphere
pj(r) dr = probability of finding an ion j at r to r+dr
rmp = rD = Debye length thickness of ionic atmosphere
pj(r)
rqi qj
uniform solvent dielectric
+
-
-
-
-
-
-
-
-
- -
+
+
+
+
rD
rD = 305 pm(m/m )½ =
1
Boltzmann distribution
thermal jostling collisions disrupt ionic halo
Nj = No e-i(r)qj
kT
pj(r) dr = 4r2Noj e-i(r)qj
kT dr
Noj = number of ions j in volume V k = R/NA
pj(r)
rqi qj
uniform solvent dielectric
+
-
-
-
-
-
-
-
-
- -
+
+
+
+
rD
Poisson Equation
Non-electrolyte Solutions or Dilute Solution Limit for Electrolyte Solutions
i(r) = qi pi(r) = charge density i(r) = charge per unit volume (r) (r) =o r(r)
2 i(r) = – i(r)(r)
2 = 2
x2 + 2
y2 + 2
z2
r
i(r)
2i higher
Poisson Equation– Spherical Ion
the higher the charge density the faster the potential drops
1r 2(r i(r))
r2 = – i(r)(r)
i
ri
i
ij
j
j
j
j
Screened Coulomb Potential
Point charges, uniform solvent dielectric (r) = ro
qj = zj e
i(r) = +(r) + -(r) = q+N+
V e–i(r)q+
kT + q-N-
V e–i(r)q-
kT
e-i(r)qj
kT 1 – i(r)qj
kT 2 =
j=1
s
q2j
ro kT
Nj
V
Screened Coulomb Potential
Point charges, uniform solvent dielectric
2(r i(r))r2 = 2 (r i(r))
i(r) = Ar e
-r =
qj
4ro r e
-r
rD = 1
pj(r)
rqi qj
uniform solvent dielectric
+
-
-
-
-
-
-
-
-
- -
+
+
+
+
rD
Poisson-Boltzmann Equation Continuum Electrostatics with
Background Electrolyte
)()( xuxε )(sinh)(2 xuxκ )(π4 2
ii i
cxxδz
kT
e
*N. A. Baker
)(π4 2
ii i
cxxδz
kT
e
Poisson-Boltzmann Equation
)()( xuxε )(sinh)(2 xuxκ
*N. A. Baker
Poisson-Boltzmann Equation Linearized
)()( xuxε )()(2 xuxκ )(π4 2
ii i
cxxδz
kT
e
sinh
Electrostatic potential of the 30S ribosomal subunit
http://agave.wustl.edu/apbs/images/images/30S-canonical.html
Top: face which contacts 50S subunit
Web links http://ashtoret.tau.ac.il/Homepage/courses/Poisson-Boltzmann.pdf http://www.biophysics.org/btol/img/Gilson.M.pdf Nathan A. Baker;
http://www.npaci.edu/ahm2002/ahm_ppt/Parallel_methods_cellular.ppt
Jeffry D. Madura; http://www.ccbb.pitt.edu/BBSI/6-11_class_jm.pdf
)()( xuxε )(sinh)(2 xuxκ )(π4 2
iii
c xxδzkTe
Linearized Poisson
-
Boltzmann equation also useful:
iii
c xxδzkT
πexuxκxuxε )(
4)()()()(
22
-
xxxgxu )()(
Free energies and forces obtained from integrals of u
