Download Translational Diffusion

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.
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.
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
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.
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
Flux of particles depends on the concentration gradient
Fick’s first law
A  area of reference plane
l
l
n1
n2
(concentration)
Define the diffusion
coefficient: D (cm2/sec)
moles (net)
J = flux 
area • time
Fick’s 1st law:
dn
J = -D dx
Change in concentration requires a difference in concentration gradient
Fick’s second law
area  A
change in # particles
per unit time
Fick’s 2nd law:
dn
=D
dt
dn (J1 - J2) A
=
dt
dx A
d2n
dx2
volume
J1
dx
J2
Constant gradient: same amount
leaves as enters the box
Gradient higher on left: more enters
the box than leaves
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
-5
-4
-3
-2
-1
0
1
2/4Dt
N
-x
1/2
c(x,t) =
( D t) • e
2
2
3
4
5
 solution (Gaussian)
Note: <x2> = average value of x2
+
<x2> =
p(x) dx • (x2)
-
<x2> = 2 D t
where p(x) is the probability of the
molecules being at position x
mean square displacement
Mean square displacement - in 3-dimensions
isotropic diffusion
D x = Dy = Dz
<x2> = 2Dt
<y2> = 2Dt
<z2> = 2Dt
l2 = 6Dt
where l2 = <x2> + <y2> + <z2>
l2
D = 6t
cm2/sec
average displacement from the
starting point = (6Dt)1/2
NOTE: Mean displacement   time
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 ≈ 10-8 - 10-10 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  10-6 - 10-7 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
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.
Translational Diffusion
Classical technique  boundary spread
(mg/mL)
start
end
x (distance)
3.22
107 x D
BSA
Do
Extrapolate to infinite dilution
(designated by Do)
3.28
3.24
0
5
10
C (mg/mL)
15
* values of D have NOT been
corrected for pure water
at 20oC
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, 10-15 L) is determined.
This is related to the Diffusion Coefficient
excitation
emitted photons
http://www.probes.com/handbook/boxes/1571.html
molecules moving into
and out of the measuring
volume:
fast (left) vs slow (right)
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
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
G() goes to zero
at long times
free ligand (small, fast diffusion)
bound ligand on
slow moving
protein
Relating D to molecular properties
Rate of mass flux is inversely proportional to the frictional drag
on the diffusing particle
D=
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.
kT
f
But f = 6RS
k = Boltzman constant
f = frictional coefficient
for a sphere or radius Rs
 = viscosity of solution
Stokes-Einstein
Equation
D=
kT
6RS
Stokes Radius obtained from Diffusion
2. Calculate Rs
kT
D = 6R
S
kT
RS = 6D
Assumes a spherical shape
Stokes Radius
1. Measure 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
Interpreting the meaning of the Stokes Radius
D = kT = kT
f
6RS
1. Measure D and get Rs
2. Compare Rs to Rmin
f = 6Rs
Experimental:
Theoretical:
vol = [V2 •
Define: fmin = 6Rmin
so:
Rmin
=
M
]
N
4
R3min
3
Rs
= f
Rmin
fmin
Hydrodynamic theory defines the shape dependence of f, frictional coefficient
this allows one to estimate
effects due to molecular asymmetry
(a / b)
b
a
f / fmin
Shape factor for translational diffusion
for a prolate ellipsoid
frictional coefficient of ellipsoids
prolate
oblate
much larger effect
of shape on viscosity than
on diffusion
(Cantor + Schimmel)
viscosity shape
factor
frictional coefficient
shape factor
Interpreting Diffusion Experiments:
does a reasonable amount of hydration explain the measured value of D?
Protein
RNAse
Collagen
protein
RNAse
Collagen
Do20,W x10-7 cm2/s
M
13,683
345,000
RS(Å) (diffusion)
11.9
0.695
Maximum solvation
18 (Rmin=17Å)
310 (Rmin=59Å)
Maximum asymmetry
H2O = 0.35
 H2O = 218
a/b = 3.4
a/b = 300
Volume per gram of anhydrous protein
RS
Rmin
=
(Vp + H2O) 1/3
Vp
solve for H2O
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
Measuring the Diffusion of Proteins in the Cytoplasm of E. coli
Fluorescence Recovery After Photobleaching (FRAP)
Ready..
Aim...
Fire!
E. coli cell
t0
Diffusion of protein into the spot
t1
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.
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
4 µm
J. Bacteriology (1999) 181, 197-203
one can observe the
molecules diffusing
back into the bleached
area
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 (200 -300 mg/ml)
J. Bacteriology (1999) 181, 197-203