G Solvation

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 wideal 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 ic

xxδzkTe

*N. A. Baker

)(π4 2

ii ic

xxδzkTe

Poisson-Boltzmann Equation

)()( xuxε )(sinh)(2 xuxκ

*N. A. Baker

Poisson-Boltzmann Equation Linearized

)()( xuxε )()(2 xuxκ )(π4 2

ii ic

xxδzkTe

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 2i

iic xxδz

kTe

Linearized Poisson

-

Boltzmann equation also useful:

iii

c xxδzkTπe

xuxκxuxε )(4

)()()()(2

2

-

xxxgxu )()(

Free energies and forces obtained from integrals of u