Lecture 31
Diffusion and transport
Reading for today: Chapter 17.21 to the end
Reading for Wednesday: Reading: Chapter 3, section B to 3.19
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Today’s Goals
Define the description of the diffusion coefficient for
molecules diffusing in solution
When diffusion is not enough: example of transport
mechanisms in cells
Kinesin moving on microtubules
Techniques that make use of diffusion to experimentally
determine molecular mass and shape using centrifugation
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Some numbers exemplify diffusion in solution
sucrose
Small molecule in water D ~ 10-5 cm2 s-1
rr .m.s.d .  6 Dt ~ 70 μm in 1 sec
myoglobin
Myoglobin (~17 kD) D ~ 1.1x10-8 cm2 s-1
rr .m.s.d .  6 Dt ~ 16 μm in 1 sec
~ 1 cm in 100 hours
virus
Bacteria are ~1 micron; eukaryotic cells are ~10s of micron
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Friction force resists diffusion driving force
Big molecules move slower than small ones
Due to friction, a force dependent on size, shape and velocity
Friction force:
Ffriction   fv
f is the friction factor
v is the velocity
Resists motion: (-) sign indicates direction opposite velocity
Molecules
 have a terminal velocity, which depends on the friction
and the driving force F
  fv  F
friction
term
drive
The flux of diffusion depends on the local concentration and the
velocity:

And:
Fdrive
J ( x) 
 c( x)
f
J ( x)  v  c( x)
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Driving force for diffusion comes from the
chemical potential
Ffriction   fv  Fdrive
Forces are derivatives of potentials
Remember that the chemical potential describes the
concentration
difference

c
    k BT ln  
1

The force from a concentration gradient is the derivative of
the chemical potential:
Fdrive
d
d
1 dc(x)

 k B T ln(c(x))  kB T
dx
dx
c(x) dx
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Defining D – Einstein’s PhD thesis
Fdrive
1 dc(x)
 k B T
c(x) dx
Fdrive
J ( x) 
 c( x)
f
Combining the two equations above gives:
Fdrive
k BT dc( x)
J ( x) 
c( x)  
f
f
dx
dc(x)
Comparing to Fick’s 1st law: J(x)  D
dx
Leads to:
k BT
D
f

f is called the friction factor – encoding size, shape and viscosity
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Viscosity (h)
Viscosity can be thought of as the fluid’s
flow rate in response to a force
Viscosity also describes resistance of a
fluid to particles moving through it
It is a proportionality constant between
momentum flux (Jpx) and the velocity
gradient :
J px
J px
dvx
 h
dz
Figure from The Molecules of Life (© Garland Science 2008)
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Viscosity
Viscosity, h (eta) is expressed in Poise units:
1 P = 1 g cm-1 s-1
1 cP = 1 g m-1 s-1
Water has a viscosity of ~1 cP at room temperature (from 1.79 cP
at 0 °C to 0.28 cP at 100 °C)
For comparison:
Glycerol ~1000 cP
Blood ~3-4 cP
Olive oil ~81 cP
Maple syrup ~3200 cP
Acetone ~0.31 cP
Cytosol ~5-20 cP
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f encodes particle size: Friction factor for a sphere
The friction factor for arbitrary shapes cannot be described
with an equation, but for simple spheres:
k BT
f sphere  6hr
Dsphere 
6hr
Makes sense: larger molecules or molecules in a higher
viscosity solution will diffuse slower
This equation for the diffusion coefficient is the Stokes-Einstein
equation
For proteins, average density r ~ 1.35 g/cm3
3MW
r3
4 r
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Diffusion is also dependent on shape
Sphere – same from all directions
Prolate – i.e. “football” shapes
Oblate – i.e. “disk” shapes
Both prolates and oblates of the same volume as a sphere will
encounter more drag in some directions, and less in others
Overall, they will diffuse slower than spheres
This is reflected in their friction factor, f
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Friction factor for other shapes
The friction factor is smallest for spheres
Prolate (football)
Oblate (disk)
Axial ratios of 2 lead to ~4% increase in f and corresponding decrease
in D
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Figure from The Molecules of Life (© Garland Science 2008)
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Explaining why the oblates have higher f than
prolates
view in x
view in y
view in z
Prolate:
Oblate:
Both have two oval “profiles” and one round. These are
the profiles that matter in terms of movement/friction in
the respective directions. The round profile for the oblate
is big, but it is small for the prolate
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Some concepts to remember
The motion of molecules follows a random walk
From Fick’s laws of diffusion, the particles spread in solution in 3D
according to:
rr .m.s.d .  6 Dt
Where
k BT
D
f
and for a sphere:
Dsphere
k BT

6hr
Diffusion coefficient is affected by solution viscosity and molecule
size and shape
Important result: the net distance traveled is in a random direction
and increases with t e.g.:
If rr.m.s.d. = 10 m in 1 sec
Then rr.m.s.d. = 77 m in 1 min (60 sec)
And rr.m.s.d. = 600 m in 1 hr (3600 sec)
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Diffusion in 1D, 2D and 3D
Diffusion in 3D: rr .m.s .d .  6 Dt
General case, applicable to cytosol, extracellular space, etc
Diffusion in 2D: rr .m.s .d .  4 Dt
Biological membranes form planar structures with 2D diffusion in
the bilayer plane
Diffusion in 1D: rr .m.s .d .  2 Dt
DNA-binding proteins simplify their search for target sequences
through electrostatics - ~1D search
Figures from The Molecules of Life (© Garland Science 2008)
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DNA-binding proteins “slide” along DNA
Rate constant for Lac repressor binding to DNA is ~1010 M-1 s-1
Faster than 3D diffusion-limited rate constants (~109 M-1 s-1 or less)
Use electrostatic non-specific interactions to reduce search space
sliding
oligomer = multiple binding sites
hopping
Figure from The Molecules of Life (© Garland Science 2008)
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Some numbers exemplify diffusion in solution
sucrose
Small molecule in water D ~ 10-5 cm2 s-1
rr .m.s.d .  6 Dt ~ 70 μm in 1 sec
myoglobin
Myoglobin (~17 kD) D ~ 11x10-7 cm2 s-1
rr .m.s.d .  6 Dt ~ 16 μm in 1 sec
~ 1 cm in 100 hours
virus
Bacteria are ~1 micron; eukaryotic cells are ~10s of micron
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Diffusion vs. transport in cells
Diffusion can be fast enough for some cellular processes (e.g.
maintaining concentrations of small molecules like sugars and ATP
within the cytosol) but:
Cells have viscosity on the order of ~20 cP, or 20x that of water
Cells have diameters in 10s of m, and neurons have much longer
axons and dendrites, so some molecules have to travel long
distances
Some of the particles that need to migrate within cells are large
vesicles – some with radii ~100 nm
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Diffusion of a vesicle
Assuming a radius of 100 nm:
Dsphere
k BT
(1.38 10-23 J K -1 )(300 K)


6hr 6(3.14)(0.2 g cm -1 s -1 )(10-5 cm)
 1.110-16 J g -1 s
 1.110-16 (103 g m 2 s -2 ) g -1 s
Dsphere  1.110-13 m 2 s -1
rr .m.s.d .  6 Dt  6(1.110 13 m 2 s -1 )( 1 s)
 0.8 μm in 1 second
That might not seem so bad, but…
Increases with t1/2 (only 6.3 m in 1 min; 49 m in 1 hr)
In random direction
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Actin, tubulin and motor proteins
Myosin moves along actin
filaments
Transport cargo
Powering muscle contraction
Kinesin and dynein move along
microtubules
Transport of cargo, like vesicles
http://www.bscb.org/softcell/images/mp_tripple.gif
Red – actin
Green – tubulin
Blue - DNA
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Kinesin molecular mechanism
ATP binding to the leading subunit changes the
conformation to dock its linker
Causes release of the lagging ADP-bound subunit
The lagging subunit steps over, becomes the
leading subunit
Linker conformational change favors forward movement
ATP is hydrolyzed in the lagging subunit, and ADP
is released from the leading subunit
ATP binding to the leading subunit starts the
next cycle
Figure from The Molecules of Life (© Garland Science 2008)
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Summary: Active transport
Vesicles often need to accumulate
at specific locations, e.g.:
Secretory vesicles
Synaptic vesicles
Active transport:
Can go faster than diffusion
Basic Neurochemistry (1999) Lippincott Raven Eds
Can go in defined direction
Can go against a gradient
Needs a source of energy (e.g. ATP)
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Lab techniques that make use of diffusion
The diffusion coefficient (D) and friction factor (f) can provide
information about the three-dimensional structure of
macromolecules
Shape
Apparent molecular weight
i.e. oligomerization, quaternary structure
One way to determine D, f is to set up a “competition” between
the diffusive force (Fdiff) and another force
Electrophoresis – electrical force
Centrifugation – centrifugal force
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Centrifugation
From the side:
From the top:
Swinging bucket
Fixed angle
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Centrifugal force
w
Fcent  mw r
2
m is the mass of the particle subjected to the force
w is the angular velocity
rate of rotation in radians per second
r is the distance from the axis of rotation
r
Fcent
Centrifugation protocols typically use rpm, or “revolutions per
minute” to state the velocity of the rotor. For the units to work out,
i.e. to obtain a proper centrifugal force value, you need to convert to
radians per second:
1 rpm = 2 radians / 60 seconds
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Centrifugal force
w
Fcent  mw r
2
m is the mass of the particle subjected to the force
w is the angular velocity
rate of rotation in radians per second
r is the distance from the axis of rotation
r
Fcent
This force is opposed by viscous drag, and when the two are equal,
the molecules reach a constant (“terminal” velocity)
We need to take into account that solvent will also feel the
centrifugal force, leading to buoyancy. Therefore the difference in
mass between the protein and solvent leads to the net force:
F  meff w r
2
where
meff
 r

solvent
 m
1

 r


protein 
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Velocity centrifugation
meff
 r

solvent
 m
1

 r


protein 
rsolvent ~ 1.0 g/cm3 for water
rprotein ~ 1.35 g/cm3
rnucleic acid ~ 1.7 g/cm3
We had defined the friction force as fv, so the terminal velocity is:
meff w r  fvterm
2
meff w r

f
2
vterm
These experimental results are expressed as sedimentation
coefficients (S), normalizing for the centrifugal force applied:
vterm meff
S 2 
wr
f
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Sedimentation velocity centrifugation in practice
Time
Air
Spinning
Force
Solution
r
r
r
r
vterm dr / dt
S 2  2
wr
wr
dr
Sw 2dt 
 Sw 2t  ln r
r
ln r
Concentration
Concentration
c0
Concentration
c0
c0
r
r
Slope = Sw2
Time
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Sedimentation velocity in practice
Sample data:
Pellet
Meniscus
Time
vterm meff
S 2 
wr
f
Velocity in cm/s
Acceleration cm/s2
S in seconds
Svedberg units 1 S = 10-13 s
If a 100 kD protein (measured at 10S)
interacts with a 200 kD protein
(measured at 16S) to form a dimer of
35S, what can you infer about the
quaternary structure?
Biochem. Soc. Trans. (2008) 36, 766-770
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Some examples of sedimentation coefficients
Non-additive S values – highlight non-spherical shapes
Ribosome:
vterm meff
S 2 
w x
f
Proteasome:
20S
26S
Figure from The Molecules of Life (© Garland Science 2008)
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Equilibrium centrifugation
The premise of equilibrium centrifugation experiments:
When the concentration gradient no longer changes, it is in
thermodynamic equilibrium
area A of plane
at position r
rotor center
r
At equilibrium:
Jcentrifugation = Jdiffusion
Jcentrifugation
Jdiffusion
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Equilibrium centrifugation
The premise of equilibrium centrifugation experiments:
When the concentration gradient no longer changes, it is in
thermodynamic equilibrium
Flux from centrifugation = flux from diffusion
dc( r )
c( r ) Sw rAdt  D
Adt
dr
2
A is the area of the plane of the sample
Substituting S and D from previous equations:
meff 2
k BT dc( r )
c( r )
w r
f
f
dr
Important: equilibrium centrifugation does NOT depend on shape
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Equilibrium sedimentation
Switching effective mass to molar units and kB to R

 2
r
dc( r )
solvent


c( r ) M 1 
w
r  RT


r
dr
protein


Isolating M:
Equilibrium
sedimentation
experiments give the
true molecular mass
of the native form of
the protein
RT
dc( r ) 1
M 

 c( r ) rdr
r
2
solvent

w 1 

r
protein


d ln c( r )
d ln c( r )
M 
2
d
(
r
)


r
2
solvent

w 1 

r
protein


2
d (r 2 )
2 RT
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Actual example from the first published experiment
The very first equilibrium sedimentation experiment was already
mentioned earlier in the semester:
Mass of hemoglobin in the native state = 68 kD
Hemoglobin is a tetramer of 17 kD subunits!
Concentration
Each curve
is displaced
to the left
Note how it takes
~36 hrs to reach
equilibrium
Diffusion is slow!
r
Force
Spinning
Svedberg and Fahraeus JACS 1926
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Equilibrium centrifugation in practice
2 RT
d ln c( r )
M 
2
d
(
r
)


r
2
solvent

w 1 

r
protein




r
solvent

w 1 

r
protein

 d (r 2 )
d ln c( r )  M
2 RT
2
Plot ln(c) vs r2 should give a straight line if there is a single species of
uniform molecular mass in the sample
From these experiments,
Molecular mass can be
obtained from the slope
Figure from The Molecules of Life (© Garland Science 2008)
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Measuring a KD
If a protein forms an oligomer with a KD in
the range of the protein concentration
used
4A ↔ A4
Both the monomeric and oligomeric
species will be populated significantly
The KD can be determined by fitting the
shape of the concentration vs. position
curve
The concentration of oligomer will vary
throughout the cell
4A ← A4
Figure from The Molecules of Life (© Garland Science 2008)
4A → A4
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Some concepts to remember
The spread of molecules through diffusion is proportional to t1/2
Diffusion is a slow process
Diffusion of molecules (characterized by the diffusion coefficient, D)
is affected by solution viscosity and molecule size and shape
Cells use active transport mechanisms to rapidly transport cargo
motors like kinesin moving along microtubule
Centrifugation methods :
Velocity sedimentation provides
information about shape
Equilibrium sedimentation provides
the molecular mass of native oligomers
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