1. Continuum Electrostatics
G Solvation
Continuum Electrostatics

2. G Solvation solG = solGVdW + solGcav + solGelecH
r = 1-5
r = 78.54
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

3. G Solvation
solGelec = difference in electrostatic work necessary to charge ion:
solGelec = NA wsoln – NA wideal
Work to transfer ion from vacuum to solution with the same electrostatic potential: Work = solGelec = 0Zie i dqi
i = electrostatic potential for ion i and its ionic atmosphere of neighbors j

4. Electrostatic Potential
i(r)
rij qi qj
uniform dielectric
r = relative dielectric constant
r = for water (attenuates interaction)

5. Electrostatic Potential
Difficult to solve in general.
Examples: two extreme cases
Spherical ions
Point charge in an electrolyte solution
Ionic atmosphere or ionic halo
Charged solute radius ri in very dilute solution or non-electrolyte solution

6. 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) r qi qj
uniform solvent dielectric + - rD

7. Boltzmann distribution
pj(r) r qi qj
uniform solvent dielectric + - rD
thermal jostling
collisions disrupt ionic halo
Noj = number of ions j k = R/NA

8. 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) r
i(r)
2i higher

9. Poisson Equation– Spherical Ion
the higher the charge density the faster the potential drops i i r j j i j j i j

10. Screened Coulomb Potential
Point charges, uniform solvent dielectric
(r) = ro
qj = zj e

11. Screened Coulomb Potential
Point charges, uniform solvent dielectric
pj(r) r qi qj
uniform solvent dielectric + - rD

12. Born Approximation Spherical solute with radius ri and charge zi
Very dilute solution or non-electrolyte
Electrostatic potential is caused by polarization of the solvent by the partial charges of the solute.
i(ri) =
r = 1 zi + ri
r = 78.54

13. Born Approximation – Self Potential
welec = = =
welec =

14. Born Approximation – Solvation Free Energy
elecG = NA welec(real) – NA welec(ideal)
elecG = NA wsolution – NA wvacuum
elecG =
= -

15. Poisson-Boltzmann Equation
Continuum Electrostatics with Background Electrolyte ) ( x u ε Ñ × -
sinh 2 κ π 4 i c δ z kT e å + =
*N. A. Baker

16. Poisson-Boltzmann Equation) ( π 4 2 i c x δ z kT e - å ) ( x u ε Ñ × - = + ) (
sinh 2 x u κ
*N. A. Baker

17. Poisson-Boltzmann Equation
Linearized ) ( x u ε Ñ × - 2 κ π 4 i c δ z kT e å + =

18. sinh

19. Electrostatic potential of the 30S ribosomal subunit
Top: face which contacts 50S subunit

20. Web links
Nathan A. Baker;
Jeffry D. Madura;

21. Boltzmann equation also useful:) ( x u ε Ñ × -
sinh 2 κ π 4 i c δ z kT e å + = W Î = x g u ) ( -
Linearized Poisson
Boltzmann equation also useful: å - = + Ñ × i c x δ z kT πe u κ ε ) ( 4 2 -
Free energies and forces obtained from integrals of u