Transfer of charged molecules

1
Transfer of charged molecules
Electric field does work on ion as it passes
between solutions. DµA µA(solution 2) -
µA(solution 1) ZFV V (f2 - f1) potential
difference in volts between the two solutions. F
Faraday 96,485 (eV-1) charge of mole of
electrons. Z charge of the ion (e.g.
1) Note this is true for any f(x).
Convenient to write µA,tot µA ZFf
Na(2) Cl-(2) f2
Na(1) Cl-(1) f1
f1
Electrical Potential (f)
f2
Position (x)
2
Electrostatics in Water


Counter charges (in solution)




- - - - - -
y f(s)
Charged surface s charge density
y
Potential
Chemical potentials of molecules on the surface
are influenced by the surface potential, y.
3
Gouy-Chapman Theory


LD




- - - - - -
y0
Potential
4
Surface Potential


LD




- - - - - -
s
y0
Potential
Gouy Equation
5
Linearized Poisson-Boltzmann Equation


LD


Good for y0 25 mV


- - - - - -
s
y0
Potential
6
Mobility and Chemical Potential

• Molecular motion and transport disucssed in TSWP
  Ch. 6. We can address this using chemical
  potentials.
• Consider electrophoresis
• We apply an electric field to a charged molecule
  in water.
• The molecule experiences a force (F qE)
• It moves with a constant velocity (drift velocity
  u)
• It obtains a speed such that the drag exactly
  opposes the electrophoretic force. No
  acceleration steady motion.
• u qE1/f where f is a frictional coefficient
  with units of kg s-1.
• qE is a force with units of kg m s-2 giving u
  with the expected ms-1 velocity units.
• Think of (1/f) as a mobility coefficient,
  sometimes written as µ.
• u mobility force
• We can determine the mobility by applying known
  force (qE) and measuring the drift velocity, u.

7
Mobility and Chemical Potential

• Gradient of the chemical potential is a force.
• Think about gradient of electrical potential
  energy
• Extending this to the total chemical potential
• where f is a frictional coefficient
• (1/f) as a mobility coefficient

8
Mobility and Chemical Potential Example
-

Write down chemical potential as a function of
position in this electrophoresis If
concentration (c) is constant througout And
the drift velocity is
Potential (f)
Position (x)
What if c is not constant? Can the entropy term
give rise to an effective force that drives
motion? This is diffusion, and we can derive
Ficks Law (TSWP p. 269) from chemical potentials
in this way.
9
Equilibrium Dialysis Example
At equilibrium O2(out) O2(in,
aq) MbO2/MbO2(aq) Keq If we are able to
asses the total ligand concentration in the
dialysis bag O2(aq) MbO2 O2 (in,
total) Then MbO2 O2 (in, total) -
O2(aq) (these are measurable) Can compute
Keq. If we have direct a probe for MbO2, then
we dont need the dialysis, can read of
concentrations and compute Keq. Dialysis can
also be used to exchange solution (eg. change
salt)
H2O (l) O2(aq), N2(aq) etc.
H2O (l) O2(aq), N2(aq) Mb(aq), MbO2(aq)
Semipermeable membrane (cellulose) allows water
and dissolved small solutes to pass, blocks
passage of large proteins such as myoglobin (Mb)
10
Scatchard Equation
General version M A MA Keq
MA/(MA) Simplify by introducing n, the
average number of ligand molecules (A) bound to
the macromolecule (M) at equilibrium
Scatchard plot
NKeq
Slope -Keq
n/A
Scatchard equation N independent binding sites
per macromolecule.
For one ligand binding site per macromolecule
n
N
11
Cooperative Binding
For a macromolecule with multiple binding sites,
binding to one site can influence binding
properties of other sites. Failure of data
plotted in a Scatchard plot to give a straight
line indicates cooperative or anticooperative
binding among binding sites. Cooperative
binding of second ligand is made
easier Anticooperative binding of second ligand
is made more difficult Hemoglobin is a favorite
example of a protein with cooperative binding
behavior.

• Binds up to 4 O2
• Cooperative most O2 released in tissue while
  binding O2 maximally in the lungs
• Binding curve shows characteristic sigmoidal shape

1
Myoglobin
Hemoglobin
f
p50 1.5 Torr
p50 16.6 Torr
f fraction of sites bound
0
PO2(Torr)
0
40
12
Hill Plot
Scatchard equation (non-cooperative binding) For
cooperative binding. n Hill coefficient K a
constant, not the Keq for a single ligand Slope
of each line give the Hill cooperativity
coefficient. Slope 1 no cooperativity Slope
N maximum (all-or-nothing) cooperativity See
Example 5.4 (TSWP p. 204 - 207) for a detailed
study of Hemoglobin
1.5
Myoglobin n 1.0
logf /(1-f)
Hemoglobin n 2.8
-1.5
-0.5
2
logP02 (Torr)