1. Translational Diffusion: measuring the frictional force on the movement of a macromolecule in solution.
A particle under the influence of a constant applied force will accelerate. If it interacts with the medium (solvent) the frictional force (Ff) opposing that acceleration is proportional to the velocity (u). The proportionality coefficient which relates the frictional force (Ff) to the velocity (u) is called the frictional coefficient, f. This is a function of the molecular size and shape, and also contains the viscosity of the solution, η. Note that the η here is not a property of the molecule itself, as is intrinsic viscosity [η], but is just simply the viscosity of the solution, which is usually 0.01 Poise.
Hence, by measuring the speed with which a particle (protein) is moving through solution we are getting information about the limiting frictional resistance to that movement, which is contained within the frictional coefficient. It is useful here that molecules generally obey the same hydrodynamic rules that apply to macroscopic objects. In the case of macroscopic objects, such as a marble, the dependence of the frictional coefficient on molecular size and shape is known.

2. Translational diffusion: a sphere has the following frictional coefficient:
ftrans = 6πηRs.
The larger the radius of a sphere, the larger the frictional coefficient, and the slower will be the velocity under a given driving force.
Rotational diffusion: for a particle rotating in solution, there is also frictional resistance due to the shear force with the solvent. The hydrodynamics of this for a sphere is
frot = 8πηRs3.
Note that for rotational diffusion, the frictional coefficient increases in proportion to the volume of the sphere, whereas for translational diffusion, the dependence is proportional to the radius.

3. Diffusion Frictional Resistance to Macromolecule Motion
Frictional Force (Ff)
applied Force (Fap)
 velocity, u
Ff = f • u f  frictional coefficient
steady state - terminal velocity is reached u = (Fap / f)
For a sphere:
ftrans = 6RS translational motion
frot = 8RS3 rotational motion
Measure molecular motion  f  Rs  molecular size and shape information

4. Diffusion is a purely statistical (or entropic) concept
Diffusion is a purely statistical (or entropic) concept. It represents the net flow of matter from a region of high chemical potential (or concentration) to a region of low chemical potential. Diffusion can be considered as
1) the net displacement of a molecule after a large number of small random steps: random walk model.
Alternatively,
2) it can also be looked at in terms of the net flow of matter under the influence of a gradient in the chemical potential, which can be formally treated as a hypothetical "force" field.
Both approaches provide useful physical insight into how the diffusion of macromolecules can be utilized to yield molecular information. We will derive the concepts in terms of 1-dimension, and then generalize to 3-dimensions.

5. Phenomenological Equations
Translational diffusion
Phenomenological Equations
Measured classically by observing the rate of “spreading” of
the material: flux across a boundary
Two equations relate the rate of change of concentration
of particles as a function of position and time:
Fick’s first law
Fick’s second law

6. Flux of particles depends on the concentration gradient
Fick’s first law
A  area of reference plane
l l
n n2
(concentration)
Define the diffusion
coefficient: D (cm2/sec)
J = flux 
moles (net)
area • time
Fick’s 1st law:
J = -D dn dx

7. Change in concentration requires a difference in concentration gradient
Fick’s second law
area  A
Fick’s 2nd law:
= D
change in # particles
per unit time dn dt
d2n
dx2
dn (J1 - J2) A
dt dx A =
volume dx J1 J2
Constant gradient: same amount
leaves as enters the box
Gradient higher on left: more enters
the box than leaves

8. Translational Diffusion
Solve Fick’s Laws - differential equations ( relate concentration, position, time)
(define boundary and initial conditions)
diffusion in 1 - dimension
All (N) particles are at x = 0 at t = 0
Material spreads
c(x,t) = ( D t)1/2 • e  solution N 2
-x2 / 4 D t
(Gaussian)
Note: <x2> = average value of x2
<x2> = p(x) dx • (x2) where p(x) is the probability of the
molecules being at position x
<x2> = 2 D t mean square displacement + -

9. average displacement from the starting point = (6Dt)1/2
Mean square displacement - in 3-dimensions
isotropic diffusion Dx = Dy = Dz
<x2> = 2Dt
<y2> = 2Dt
<z2> = 2Dt
l2 = 6Dt
where l2 = <x2> + <y2> + <z2>
D = cm2/sec
NOTE: Mean displacement   time l2 6t
average displacement from the
starting point = (6Dt)1/2

10. Values of Diffusion Coefficients
1. Diffusion in a gas phase: D  1 cm2/sec (will depend on length of λ , the mean free path)
2. Diffusion within a solid matrix: D ≈ cm2/sec or much smaller
3. Diffusion of a small molecule in solution, such as sucrose in water at 25° C:
D ≈ 10-5 cm2/sec
(takes 3 days to go ~ 4 cm)
4. Diffusion of a macromolecule in solution, such as serum albumin (BSA, 70,000 mol weight) in water at 20° C:
D ≈ 6 · 10-7 cm2/sec
D  cm2/sec
(3 days  1 cm)
5. Very large molecules such as DNA diffuse so slowly that the measurements cannot even be made.
NOT a useful technique

11. Measuring the translational diffusion constant
The classical method of measuring the translational diffusion constant is to observe the broadening of a boundary that is initially prepared by layering the protein solution on a solution without protein. In practice, this is not done.
The preferred method for a purified protein is to use dynamic or quasi-elastic light scattering to get a value of the diffusion coefficient. The method measures the fluctuations in local concentrations of the protein within solution, and this depends on how fast the protein is randomly diffusing in the solution. There are commercial instruments to perform this measurement. Note that this does not yield a molecular weight but rather a Stokes radius. However, it is more common to use mass transport techniques to measure the Stokes radius of macromolecules.
As we saw with osmotic pressure and with intrinsic viscosity, it is often necessary to make a series of diffusion measurements at different concentrations of protein and then extrapolate to infinite dilution. When this is done a superscript “0” is added: Do. The next slide shows results obtained for serum albumin.

12. Translational Diffusion
Classical technique  boundary spread
(mg/mL)
start
end
x (distance)
BSA Do
3.22
3.28
3.24
Extrapolate to infinite dilution
(designated by Do)
107 x D
* values of D have NOT been
corrected for pure water
at 20oC
C (mg/mL)

13. Another way to measure the value of the
Diffusion Coefficient is by Fluorescence Correlation
Spectroscopy (FCS)
By measuring fluctuations in fluorescence, the residence time
of a fluorescent molecule within a very small measuring volume
(1 femtoliter, L) is determined.
This is related to the Diffusion Coefficient
excitation
emitted photons
molecules moving into
and out of the measuring
volume:
fast (left) vs slow (right)

14. Fluorescence Correlation Spectroscopy: FCS
the time-dependence of the fluorescence is expressed
as an autocorrelation function, G(),
the is the average value of the product of the
fluorescence intensity at time t versus the intensity at a
short time, , later. If the values fluctuate faster than time
 then the product will be zero.
deviation from
the average intensity

15. simulated autocorrelation functions of
An example of FCS:
simulated autocorrelation functions of
a free fluorescence ligand and the same
ligand bound to a protein
1:1 mix of
free/bound ligand
bound ligand on
slow moving protein
G() goes to zero
at long times
free ligand (small, fast diffusion)

16. Relating D to molecular properties
Rate of mass flux is inversely proportional to the frictional drag
on the diffusing particle kT f
D =
k = Boltzman constant
f = frictional coefficient
But f = 6RS
for a sphere or radius Rs
 = viscosity of solution
This expression appears to be valid for large objects such as marbles, and for macromolecules, and even for small molecules, at least to the extent that D·η is a constant for a molecule of fixed radius as the viscosity is changed.
Stokes-Einstein
Equation kT
6RS
D =

17. Stokes Radius obtained from Diffusion
kT Assumes a spherical shape
6RS Stokes Radius kT
6D
1 You need additional information to judge whether the particle is really spherical
-A highly asymmetric particle behaves like a larger sphere - higher frictional coefficient (f)
2 Deviations from the assumption of an anhydrous sphere (Rmin) are due to either
a) hydration
b) asymmetry
3 Stokes radius from different techniques need not be identical
D =
2. Calculate Rs
RS =
1. Measure D

18. Interpreting the meaning of the Stokes Radius
kT kT
f RS
f = 6Rs
Define: fmin = 6Rmin
so: Rs f
Rmin fmin
Hydrodynamic theory defines the shape dependence of f, frictional coefficient
D = =
1. Measure D and get Rs
2. Compare Rs to Rmin M N
vol = [V2 • ]
Rmin
Experimental: 4 3
= R3min
Theoretical: =
this allows one to estimate
effects due to molecular asymmetry
(a / b) b a
f / fmin

19. Shape factor for translational diffusion
for a prolate ellipsoid
frictional coefficient of ellipsoids
prolate
oblate
f/fmin
much larger effect
of shape on viscosity than
on diffusion
viscosity shape factor
frictional coefficient shape factor
(Cantor + Schimmel)

20. Protein M Do20,W x10-7 cm2/s RS(Å) (diffusion)
Interpreting Diffusion Experiments: does a reasonable amount of hydration explain the measured value of D?
Protein M Do20,W x10-7 cm2/s RS(Å) (diffusion)
RNAse , (Rmin=17Å)
Collagen , (Rmin=59Å)
protein Maximum solvation Maximum asymmetry
RNAse H2O = a/b = 3.4
Collagen  H2O = a/b = 300
RS (Vp + H2O) 1/3
Rmin Vp
solve for H2O
Volume per gram of anhydrous protein =

21. What is the Diffusion Coefficient of a Protein in the
bacterial cytoplasm?
Are proteins freely mobile?
Some proteins will be tethered
Some proteins will interact transiently with others
and appear to move slowly
Free diffusion will be slower due to “crowding” effect
excluded volume effect at high concentration of protein

22. Measuring the Diffusion of Proteins in the Cytoplasm of E. coli
Fluorescence Recovery After Photobleaching (FRAP)
Ready..
Aim...
Fire!
Diffusion of protein into the spot t1
E. coli cell t0 t2
1. Express a protein that is fluorescent: green fluorescent protein, GFP.
2. Use a laser to “photo-bleach” the fluorescent protein in part of a single
bacterial cell. This permanently destroys the fluorescence from
proteins in the target area.
3. Measure the intensity of fluorescence as the protein diffuses into
the region which was photo-bleached.

23. Diffusion of the Green Fluorescent Protein
inside E. coli
Single cell, expressing GFP
Bleach cell center with a laser, t0
t = 0.37 sec after flash
t = 1.8 sec after flash
one can observe the
molecules diffusing
back into the bleached
area
4 µm
J. Bacteriology (1999) 181,

24. Diffusion of the Green Fluorescent Protein
inside E. coli
Results: D = 7.7 µm2/sec (7.7 x 10-8 cm2/sec)
this is 11-fold less than the diffusion
coefficient in water = 87 µm2/sec
Slow translational diffusion is due to the crowding
resulting from the very high protein concentration in the
bacterial cytoplasm ( mg/ml)
J. Bacteriology (1999) 181,

25. Mass Transport Techniques
Measure the steady state velocity of hydrodynamic particles under the influence of an applied force
Fret
retardation force
Fap
applied force
1 Sedimentation velocity
2 Electrophoresis
3 Gel filtration chromatography

26. Sedimentation velocity
Fap = centrifugal force
Fret = frictional drag
measure: velocity = sedimentation
centrifugal acceleration coefficient S
Fret
retardation force
Fap
applied force

27. in steady state: Fap = Fret = f•(velocity)
Sedimentation Velocity .
 = circular velocity
radians / sec r
centrifuge
Fap = 2r (mh - h)
mass of particle corrected for buoyancy
mh = (1 + H2O) ; h = (V2 + VH2O H2O)
M M
N N
Fap = 2r M (1 - V2) N
Substitute:
Note: terms for bound water drop out of equation
in steady state: Fap = Fret = f•(velocity)
S = =
velocity M (1 - V2 )
2 r Nf
measure

28. Sedimentation Coefficient:
depends on three molecular variables: M, V2, and f
mol. wt inverse density of particle
shape dependence
S =
M (1 - V2) Nf
f = 6Rs
units : seconds
1 Svedberg = seconds

29. Sedimentation value depends on solution conditions:  and 
M (1 - V2) Nf
S =
f = 6Rs
infinite dilution
So20,w
20oC
in water (correct for viscosity and temperature from conditions of actual measurement)
S-values are
usually reported
for “standard conditions”

30. Types of Centrifuges used to measure the S-value
Analytical Ultracentrifuge (monitor the distribution of material by absorption or dispersion) as a function of time
Method of choice, but requires specialized equipment
Beckman “Optima” centrifuge
small sample, but must be pure - optical detection
used to determine sedimentation velocity  S
(frontal analysis (moving boundary method) - not zonal method)
Preparative Ultracentrifuge
common instrumentation
sedimentation coefficient obtained by a “zonal method”
 requires a density gradient to stabilize against turbulence / convection
 obtaining S usually requires comparison to a set of standards of known S value