I. Transport Processes
Study the movement of molecules in
response to an applied
Force ==> F = m a (m = mass and a = acceleration)
A. Goals:
 Gain information on size, shape and
conformation of a macromolecule
 Separate mixtures of macromolecules
B. Types of Forces
 Concentration Gradient >
Diffusion
 Gravitational >
Centrifugation
 Electric Field >
Electrophoresis
 Mobile vs. Stationary Phases >
Chromatography
First let's look at general properties of
macromolecules
II. Volume and Hydration
A. Macromolecules (e.g. proteins, nucleic acids, etc.) interact
with H_{2}O
Proteins, nucleic acids, polysaccharides all have polar
functional groups capable of forming Hydrogen Bonds with H2O
 can consider them to have some amount of bound water
==> H_{2}O molecules whose motion is restricted
 most H_{2}O's aren't "stuck" to the macromolecule for
a long time; they can trade places with bulk H_{2}O
solvent
 How does bound H_{2}O affect molecules during
transport?
B. How much H_{2}O? Roughly, a monomolecular layer around
a protein is strongly affected. H_{2}O's in the next layer
out are affected somewhat.
 0.3 gm H_{2}O/gm protein = d
 0.6 gm H_{2}O/gm DNA
> Nucleic acids have many more polar groups
than proteins and a larger surface area ==> more bound
water
C. Volume of a Hydrated Molecule  must understand this to
explain Transport Properties
 here mass of solvent (m_{1}), temperature (T), and
pressure (P) are held constant
 solute is component 2 of a 2 component system;
component 1 is solvent
 units of V_{2} are cm3/gm  this looks like the
inverse of the units of density, but V_{2} is not the
inverse of density (even though this approximation works for
some molecules) MgSO4 has a negative V_{2} ==>
solution volume decreases when MgSO_{4} is dissolved in
H2O
 Partial Molar Volume  a similar property with
units of cm^{3}/mole
 V_{2} depends upon: pH, concentrations of other
solutes (salts etc.); won't worry about these effects in this
course.
 What is the volume of a hydrated molecule,
V_{h}?
V_{h} = here M = molecular weight, No =
Avogadro's number, d_{1} is the hydration (grams of H_{2}O bound per gram of solute),
and V_{1} is the partial specific
volume of H_{2}O (approx. 1)
 Measuring V_{2}'s
 measure densitites of solutions with different
solute concentrations ==> volume of solution with known gms
of H2O and gms solute
 Pyncnometer  vessel with very
accurately known volume
 Density Gradient  drop of solution will float
at a point of equal density
 Microbalance  weigh small volumes very
accurately
 sum V's for each amino acid residue (or nucleotide in
nucleic acids) ==> must know the amino acid composition.
Assumes V's of amino acids don't change for different proteins.
Actually works pretty well
 Sedimentation Equilibrium  talk aobut this later
 Typical Values:
Proteins: 0.69 
0.75 cm^{3}/gm
Nucleic Acids: 0.54
cm^{3}/gm with
Na^{+}
counterion
0.44 cm^{3}/gm with Cs^{+} counterion (for CsCl density
gradients)
III. Frictional Properties
A. What happens when a force, F, is applied to solute molecules?
 F causes an
acceleration, a: F
= m a = m
(dv/dt)
(letters in bold are vectors)
 Macromolecules moving through solution
experience Frictional Drag > a force proportional to
v but in the
opposite direction.
Frictional Drag = f v (f is a constant of proportionality)
 v increases
rapidly until the two forces are equal in magnitude but opposite
in direction
==> F  f
v = 0 ==>
v =
F / f. This
is like a skydivers "terminal velocity"
 f is called the Frictional Coefficient and is
determined by the size and shape of the macromolecule; these are
things we might wish to know.
 We will use this relationship, F = f v, repeatedly in studying
different transport processes.
B. Relationship Between Friction and Size ==> use Dimensional
Analysis to learn something about this relationship through common
sense.
 Friction must be propotional to:
 viscosity, h
 size, r = radius of hydrated sphere
 therefore  f is propotional to
h^{x} r^{y} (the question is what are x
and y?)
 f must have units of gm/s from F (gmcm/s^{2}) = f (gm/s) v
(cm/s)
 h has units, gm/cms
 Therefore  f is proportional to h^{1} r^{1} (gm/cms) cm =
gm/s
 For strong interactions between molecules and
solvent:
f_{sphere} =
6 p h
r ==> Stokes Law and r = Stokes
Radius
 Often see Stokes
Radius reported for molecular sizes
measured by Gel Filtration
Chromatography
 In terms of volume:
where V_{h}
is the volume of the hydrated molecule
==> we need to know d_{1} to calculate M
this shows the relationship between f and M
C. Relationship Between Friction and Shape  macromolecules
aren't perfect spheres
 Ellipsoids of Revolution  solids produced by rotating ellipses around one of
their two axes
 Oblate Ellipsoid  rotation around semiminor
axis, b ==> discus shaped. Vol = (4/3)pa^{2}b
 Prolate Ellipsoid  rotation around semimajor axis,
a ==> cigarshaped. Vol = (4/3)pab^{2}
Many macromolecules can often be approximated by
an Ellipsoid of Revolution, either Prolate or Oblate.
 For either, their surface area is greater than
for a sphere of the same volume
==> f_{ellipsoid} > f_{sphere}
 Perrin Shape Factor: F = f /
f_{sphere}
 can calculate F for either oblate or prolate ellipsoids of
known axial ration, a/b.
 Stokes Law
becomes: f = 6 p h r_{sphere} F
Prolate ellipsoids always have larger F's than
oblate ellipsoids of the same volume because they have larger surface
areas
IV. Macromolecular Diffusion
A. Mass Transort as a Flux of Particles (2 components)
 General Equation
 will be used in discussion of other Transport Processes:
J = L
F
[J = flux or
no. particles passing through area A per second; L = Conductivity;
F = force
causing the flow of molecules]
This equation applies to all sorts of
movements; for example
Ohms's Law: i
= V / R Here
i = current =
flux of electrons;
R = resistance ==> 1/R = conductivity; V = voltage, the applied
force
 What types of forces cause Mass
Transport?
 A system which is not at equilibrium will move
toward equilibrium.
 A system which is not at equilibrium has a gradient of
potential energy: dU/dr
 Net movement of particles is down the potential energy
gradient (like a ball rolls down a hill) to lower U
Thus: F =  (dU / dr)
and J = L
(dU /
dr)
B. What is L? Does it tell us something about the macromolecule?
 During a length of time, dt, how many
particles will pass through the area, A?
 Let's say that during dt, particles travel on
average, a distance dr.
 Then, the number of particles which pass through area, A, will
be the number in the volume V = A dr
 Let's work with this equation
 V = A dr = A (dr/dt) dt = A v
dt = (1 cm^{2}) v dt ==> cm^{3}
 If N_{i} = number of particles/cm^{3} and
if dt = 1 s; then J = N_{i} v = L
F
Here we have a general expression relating Flux to particle
density (concentration)
 Recall that: F = f v ==> v = F
/ f then substituting into 2.b gives:
J = (N_{i} / f)F = L F and ==>
L = N_{i} / f
 Let's change units to moles of particles. Divide twice by
Avogadro's number; first time to change concentration to moles
and second time to convert frictional coefficient, f, to mole
units:
(This eq.can be used for different transport
processes arising from different F's)
C. What is F for diffusion?
 The driving force in diffusion is a
concentration gradient, (dc/dr) which is described
thermodynamically as a Gradient of
Chemical Potential, µ.
 The chemical potential of a solute is defined
as the change in free energy, G, of the system caused by a
small change in the amount of that solute: µ_{2} = (dG / dn_{2}) with everything else
held constant
 µ_{2} is, therefore, a function
of concentration
µ =
µo + R T
ln (g
c): c is the concentration of solute; g is the activity coefficient which
corrects for the nonideal behavior of solutes in solution, the
fact that molecules of solute interact with one another
Recall: DG = D G_{o} + R T ln [products] /
[reactants] This is just like the equation above for chemical
potential summed for all species in a chemical reaction.
 We want to know: F =  (dU / dr) =  (dU / dc) (dc /
dr)
< applying the chain rule
or, when U is
chemical potential: (dµ / dr) =  (dµ / dc) (dc /
dr)

 D_{o} = (R T) /
(N_{o}f)
= (k T) / f; Boltzman's Constant, k = R/No ==> R for a
single molecule
 D_{o} is the diffusion coefficient when the
concentration is very small because
g > 1
and (d ln g / dc)>0; D_{o} is what we want to know
==> measure D and different concentrations and extrapolate to c
= 0
 from Do we can calculate f = and f gives us
information about size and shape
==> f_{sphere} = 6 p h r_{sphere}
 What does D_{o} tell us? ==> how fast
molecules move.
D_{o} has units cm^{2}/s ==> during time t s, a
molecule will move (diffuse) on average cm
Typical Values:
Glycine

75 gm/mole

D_{20,w} = 93.3 x 107
cm^{2}/s

Hemoglobin

68,000 gm/mole

D_{20,w} = 6.9 x 107
cm^{2}/s

Tomato Bushy Stunt Virus

10,700,000 gm/mole

D_{20,w} = 1.15 x 107
cm^{2}/s

D. How does one actually measure D?
 most obvious way is to layer solvent on top of
a macromolecular solution and watch diffusion occur
How do we interpret this? Fick's Second Law of Diffusion:
(dc / dt)_{r} = D (d^{2}c / dr^{2})
a differential equation which can be solved to
give: c(r,t) = c_{o} (4 p D t)^{1/2} exp (r^{2}/4Dt)
c_{o} = initial concentration. Thus, one can measure
dc/dr for a
particular time, t, and determine D from this equation. The
experiment is more difficult than it looks because:
 It takes days for appreciable diffusion to occur. and
 The system is very sensitive to changes in temperature and to
vibration which causes mixing
 QuasiElastic LightScattering or Dynamical
Light Scattering: Measure wavelengths
of scattered laser light
 The incident light is very monochromatic (a
characteristic of laser light)
 The scattered light has a broader distribution of wavelengths
(is less monochromatic) owing to the Doppler Effect. Light
scattered from moving particles has its wavelength shifted with
respect to a stationary detector; photons scattered from particles
moving away from the detector have a lower frequency (longer
wavelength) than the incident photon and photons scattered from
particles moving toward the detector have a higher frequency
(shorter wavelength).