Measuring the size and shape of macromolecules
Most of the commonly used techniques for measuring molecular weight of biomolecules are
based on observing the motion of the molecule in solution. To relate these observations to the
molecular properties, the theory of hydrodynamics is applied. Hydrodynamic theory, however, is
a macroscopic theory; it is based upon the motion of large particles (like marbles, or submarines)
in a continuum (i.e., the individual solvent molecules are ignored). It turns out that the same
relationships also apply fairly well to macromolecules and polymers, and also, to a large extent,
for small molecules whose size approximates the size of the solvent molecules. The analysis,
therefore, of the motion of macromolecules is based on models, e.g. a sphere, or an ellipsoid, or a
rod. One can not only obtain the molecular weight of a molecule, but can also tell whether the
molecule is asymmetric.
Hydrodynamics: study of the objects in water
How do they move?
Translation
Rotation
1) Movement with no external force
free diffusion
2) Movement under the influence of an external force
centrifuge
electrophoresis, etc.
Hydrodynamics techniques measure the frictional resistance
of the moving macromolecule in solution:
1. Intrinsic viscosity
the influence of the object on the solution properties
2. Free translational diffusion
3. Centrifugation/sedimentation
4. Electrophoresis
5. Gel filtration chromatography
6. Free rotational diffusion/fluorescence anisotropy
What do we learn here?
1. We will be examining the basis for several techniques that rely on the
movement of macromolecules in solution to reveal information about
their size and shape, or separate molecules based on these differences.
2. We will discuss the results of frictional energy dissipation due to the
solvent moving about a macromolecule (referred to as a “particle”),
which changes the solution viscosity, and the affect of this frictional drag
on the diffusion of macromolecules in solution.
3. We will consider methods that have become routine laboratory
techniques such as
SDS-polyacrylamide electrophoresis,
gel filtration chromatography, and
sedimentation.
Hydrodynamics
Measurements of the motion of macromolecules in solution form the basis of
most methods used to determine molecular
size , density , shape , molecular weight
Equations based on the behavior of macroscopic objects in water
can be used successfully to analyze molecular behavior in solution
sphere
ellipsoid
(prolate, oblate)
disk
rod
etc.
Techniques
Free Motion
Viscosity
translational diffusion
rotational diffusion
Mass Transport
sedimentation
electrophoresis
gel filtration
The hydrodynamic particle
hydrodynamically bound water
shear plane: frictional resistance to motion depends on
size
shape
A macromolecule in solution is hydrated and when it moves the shear plane that defines
the boundary of what is moving includes the associated water. For a globular protein the
water of hydration is approximately one layer of water. Recall that this is very rapidly
exchanging with the bulk solvent water, but nevertheless this associated water of
hydration defines the actual hydrodynamic particle.
Hydrodynamic Parameters
1.: Specific Volume:
(inverse of density)
volume
 Vi
gram
If there are g1 and g2 grams
of solvent and solute then
Volume = V1g1 + V2g2
total volume:
vol
gram
Consider a two
component system;
solvent (1) and
solute (2)
x grams
The specific volume is the inverse of the density of an anhydrous macromolecule. This
quantity which is more relevant from a thermodynamic viewpoint.
Proteins: V2 ~ 0.7 - 0.75 mL/g
Na+ DNA: V2 ~ 0.5 - 0.53 mL/g
Hydrodynamic Parameters
2. Partial Specific Volume:
change in volume per gram under specific conditions
V
Vi = (
)
gi T,P,gj
Volume = V1g1 + V2g2
Gibbs - Duham relationship
We will assume that specific volume is approximately equal
to the partial specific volume: Vi  Vi
Proteins: VP ~ 0.7 - 0.75 mL/g
Na+ DNA: VDNA ~ 0.5 - 0.53 mL/g
For proteins, the value of the specific volume can be accurately predicted by summing the
specific volumes of the component individual amino acids.
Note that V2 is the inverse of the density of the anhydrous macromolecule. This can be
measured directly by observing the volume change upon dissolving the macromolecule, or the
density change of the solution. The bar on top of the V denotes volume per gram.
Hydrodynamic Parameters
3. Effective mass
This is the molecular mass minus the
mass of the displaced solvent (the
buoyant factor).
Take into account buoyancy
by subtracting the mass of the displaced water
[mass - (mass of displaced water)]
[m - m(V2) = m (1 - V2)]
M (1 - V2)
N
mV2 = volume of particle
mV2 = mass of displaced
solvent
  solvent density
V2 = Vol of the molecule
g
m = (M/N) mass of particle
N = Avogadro number
M = molecular weight
Hydrodynamic Parameters
4. Hydration
1=
gH2O
gprotein
mh = m + m 1
mh = M ( 1 + 1)
N
Vh = M ( V2 + 1 V1)
N
Vh = (V2 + 1 V1)
Mass of hydrated protein
Volume of hydrated
protein
Volume of hydrated
protein per gram of
protein
Problems of hydrodynamic measurements in general:
the properties depend on
too many parameters!
1) size
2) shape
3) solvation
Some solutions to obtain useful information:
1 Use more than one property and eliminate one unknown.
Example: sedimentation (So) and diffusion (D) can eliminate shape factor
2 Work under denaturing conditions to eliminate empirically
the shape factor and solvation factors. These are constants for both standards
and the unknown sample - then find Molecular weight.
Examples:
1) gel filtration of proteins in GuHCl;
2) SDS gels.
3 If Molecular weight is known: compare results with predictions based on an
unhydrated sphere of mass M. Then use judgement to explain deviations
on the basis of shape or hydration.
Behavior of limiting forms – e.g; rods, spheres- are calculated for comparisons
All hydrodynamic measurements yield a
Stokes’ radius.
In general, the results of hydrodynamic measurements are interpreted in terms of the
Stokes’ radius, Rs, of the particle. The Stokes’ radius is the radius of a spherical
particle which would give the measured hydrodynamic property (intrinsic viscosity,
diffusion, sedimentation, electrophoretic mobility or gel filtration elution).
If the molecular weight (M) of the protein (or particle) is known along with its density,
then the radius can be readily calculated if it is assumed that the protein is spherical.
This is called the minimal radius, Rmin, which refers to the unhydrated protein.
One can then compare the Stokes radius (Rs) to the calculated Rmin.
If they are similar, then it is likely that the protein really is a sphere, since the calculated
and experimental quantities are in agreement.
If they are not similar, then one of the assumptions used to calculate the value of Rmin
must be incorrect. The assumptions are that the protein (or particle) is not hydrated
and that the particle is spherical. The inclusion of the water of hydration results in a
modest change in particle size for a compact particle such as a globular protein. On
the other hand, as we shall see, a highly asymmetric particle (e.g., very elongated)
can cause a major discrepancy. Essentially, a highly asymmetric molecule behaves
as a sphere with a size much larger than the value of Rmin. There is a large frictional
drag on the movement of a very asymmetric molecule in solution or through a gel
matrix.
The overall strategy is summarized in the next slide.
Next we consider:
1 What effect does the interaction of the hydrodynamic particle and
the solvent have on solution properties?
viscosity
2 What effect does the particle-solvent interaction have on the motion
of the particle?
translational motion
- free
- in a force field
rotational motion
Viscosity
Frictional interactions between “layers” of solution results in energy dissipation
u (velocity)
dy
u + du
A
Ff 
du • A
du
=•(
)•A
dy
dy
(shearing force)
  coefficient of viscosity
cgs units  Poise
 is related to the amount of energy dissipated per unit volume per unit time
Many ways to measure solution viscosity
capillary viscometer: measures the
rate of flow of solution through a
capillary with a pressure drop P
Effect of macromolecules on viscosity: only 2 parameters
plus solute
 =1+ •
o
solvent alone
fraction of volume
occupied by “particles”
“shape” factor
(also called the Simha factor)
20
Shape
factor

For a sphere:  = 2.5
Shape factor is very
large for highly
asymmetric
molecules
10
2.5
1
5
10
15
axial ratio = a/b
b
a
viscosity effects are very large for very elongated polymers (large ) (e.g. actin) or
molecules that “occupy” large volume (large ) (e.g. DNA)
Relative viscosity and Specific viscosity

o = r relative viscosity = 1 + 
 - o
= sp specific viscosity
o
sp = r - 1 = 
 = shape factor
 = fraction of solution
volume occupied by
solvent particles
c = g / mL solvent
But  = Vh • c
so
sp
=  • Vh
c
Vh = volume per gram of
protein occupied by solvated
particle
ideally, the increase in specific viscosity per gram of solute should be a constant
( • Vh ). Often it is not due to “non-ideal” behavior
Shape factor
vol/g of protein of hydrated particle
Non-ideal case (i.e.,reality):
sp
=  • Vh + (const.) • c + ....
c
sp
c
(ideal)
c
Extrapolate measurement to infinite dilution: Intrinsic viscosity
lim (sp /c) = []
limit at low concentration (intercept in graph above)
C0
Intrinsic viscosity: [], units = mL / g
[] =  • Vh
Intrinsic viscosity is dependent only on size
and properties of the isolated
macromolecule.
Intrinsic Viscosities of Proteins
in Aqueous Salt Solution
Molecular Dimensions of Proteins from Intrinsic Viscosities
Intrinsic viscosity
[] =  • Vh
monomer
 = 2.5
[] =  • Vh
 increases
polymer
if  = 25 then [] is 10x larger
[] =
 • Vh
if we assume a spherical shape,  = 2.5,
we will calculate an incorrect and
very large value for Vh
The Intrinsic Viscosity of Flexible Polymers
(DNA, Polysaccharides, etc.)
For flexible polymers, several theories have been offered to compute the
frictional interactions with the solvent. It is not clear how one could
decide what the value of ϕ should be for a large flexible polymer with
lots of solvent within the polymer matrix. It turns out that most of the
solvent within RG, the radius of gyration, from the center of mass must
be relatively immobile so that the flexible polymer behaves similar to
that expected for a compact sphere with R ≈ 0.8RG.
Hence, the flexible polymer occupies effectively a much larger fraction
of solution than with the same mass of a compact polymer.
For a flexible polymer like DNA
or denatured protein
Behaves like compact particles with R  0. 8 x radius of gyration
(radius of gyration measures mass distribution and can be obtained from light scattering)
example:
[]  4 mL / g
for a native, globular protein
But for a random coil, if there is ~ 100 g “bound” solvent per gram
solute
Vh  V2 + 100  100 mL/ g
[] = 2.5 • Vh
very large “coil”
Viscosity : examples
I. Hemocyanin (M = 106): native vs GuHCl denatured
sp
C
60
GuHCl denatured
40
[] =  • Vh = 2.5 Vh
coiled
20
0
cm3
g
Native
0
5
compact
10
15
C (mg / mL) of BSA
For a native, globular
protein, [η] ≈ 4 ml/g.
[η ] for same protein
denatured in Gu • HCl is
about 50 ml/g.
If the amount of water
within the flexible
polymer is about 100 g
per gram of solute
(δ1=100), then
Vh ≈ 100 ml/g.
II. T7 DNA (from phage T7, double strand): salt dependence
3000
[sp]
C
expanded coil
2000
1000
cm3
g
0
0
0.01
0.1
log [NaCl]
1.0
Note the huge values of
[η ] at low ionic
strengths. This results
from the expansion of
the DNA due to charge
repulsion of the
phosphates. A typical
value of [η] for double
strand DNA at 0.2 M
NaCl is about 100 ml/g.
Stokes Radius
Radius of the sphere which has the hydrodynamic properties consistent
with the hydrodynamic measurement
[] =  • Vh
1. measure
[] = 2.5 • Vh
2. assume
a sphere
3. calculate Stokes radius
4/3 Rs3
M/N
volume of the
molecule (spherical)
mass of the
molecule
Solve for the Stokes radius:
Rs = [
M 3
•
Vh]1/3
N 4
where Vh = []/2.5
MINIMUM RADIUS
The radius expected if the molecule is an anhydrous sphere
[4/3  Rmin3]
V2 =
(M/N)
Rmin = [
M 3
1/3
V
]
2
N 4
anhydrous sphere - point of reference
CALCULATED
Compare the Stokes radius (measured) to the “minimum radius”
expected assuming the molecule is an anhydrous sphere
Question: How close does the assumption of an anhydrous sphere come
to explaining the value of Rs?
If Rs/Rmin is not much larger than 1.0, then the assumption of the
molecule being spherical is likely reasonable.
Rs should be slightly larger than Rmin due to hydration
If Rs/Rmin is much larger than 1.0, then either the molecule is
not spherical ( >>2.5) or it is not compact (Vh>>V2)
For an anhydrous particle:
anhydrous vol/gram
[] =  • V2
for an anhydrous sphere:  = 2.5
[] = 2.5 • V2
For real macromolecules, the intrinsic viscosity will vary from the above due to
1. correction due to hydration
Vh (hydrated vol/g = V2 +H20)
 = 2.5
2. correction due to asymmetry
 > 2.5
Vh = V2 (anhydrous vol/g)
or both hydration and asymmetry
 > 2.5
and
Vh = V2 + H2O
Comparison of intrinsic viscosity values for two proteins:
1. Ribonuclease:
mol wt: 13, 683
V2 = 0.728 ml/g
Rmin = 17 Å
2. Collagen:
mol wt: 345,000
V2= 0.695 ml/g
Rmin = 59 Å
Interpreting Viscosity Data:
Does a reasonable amount of hydration explain the measured value of []?
protein
[] mL/g
Ribonuclease 3.3
Collagen
1150
maximum solvation maximum asymmetry
Rs (Å)
H2O (g/g)

(a/b)
19.3
0.59
4.5
3.9
400
460
1660
175
[] =  Vh =  [4/3  Rs3] N
M
Stokes Radius
Rs = (
Appropriate for collagen
Rs/Rmin = 6.8 (400/59)
3 [] M 1/3
)
4N
Vh = V2 (anhydrous value)
shape correction:
 > 2.5
hydration correction:
Vh = V2 + H2O
Appropriate for ribonuclease
Rs/Rmin = 1.14 (19.3/17)
2.5
 = 2.5