Continuum Electrostatics G Solvation Continuum Electrostatics

G Solvation solG = solGVdW + solGcav + solGelec H 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

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

Electrostatic Potential i(r) rij qi qj uniform dielectric r = relative dielectric constant r = 78.54 for water (attenuates interaction)

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

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

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

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

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

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

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

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

Born Approximation – Self Potential welec = = = welec =

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

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

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

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

sinh

Electrostatic potential of the 30S ribosomal subunit Top: face which contacts 50S subunit http://agave.wustl.edu/apbs/images/images/30S-canonical.html

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

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