1. 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.

2. 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

3. 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.

4. 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 disk rod
(prolate, oblate)
etc.
Techniques
Free Motion Mass Transport
Viscosity translational diffusion sedimentation
rotational diffusion electrophoresis
gel filtration

5. 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.

6. Hydrodynamic Parameters
Consider a two component system; solvent (1) and solute (2)
1. Specific Volume:
: (inverse of density)
Volume = V1g1 + V2g vol x grams
gram
If there are g1 and g2 grams of solvent and solute then
total volume:
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 ~ mL/g Na+ DNA: V2 ~ mL/g

7. Hydrodynamic Parameters
2. Partial Specific Volume:
change in volume per gram under specific conditions
Vi = ( )
Volume = V1g1 + V2g Gibbs - Duham relationship
We will assume that specific volume is approximately equal
to the partial specific volume: Vi  Vi V
gi T,P,gj
Proteins: VP ~ mL/g Na+ DNA: VDNA ~ 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.

8. Hydrodynamic Parameters
This is the molecular mass minus the mass of the displaced solvent (the buoyant factor).
3. Effective mass
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
r = solvent density
Vol g
V2 =
of the molecule
m = (M/N) mass of particle
mV2 = volume of particle
mV2 = mass of displaced solvent
N = Avogadro number
M = molecular weight

9. Hydrodynamic Parameters
4. Hydration
gH2O
gprotein
1=
mh = m + m 1
M ( 1 + 1) N
M ( V2 + 1 V1)
Vh = (V2 + 1 V1)
Mass of hydrated protein
mh =
Volume of hydrated
protein
Vh =
Volume of hydrated protein per gram of protein

10. Some solutions to obtain useful information:
Problems of hydrodynamic measurements in general:
the properties depend on 1) size
too many parameters! ) 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

11. 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.

12. Stokes radius MEASURED
A general approach for interpreting hydrodynamic data is to interpret
the data in terms of what size of spherical particle would have the measured properties
Stokes radius MEASURED
Compare the measured hydrodynamic properties to that expected for an unhydrated sphere
Rmin (CALCULATED easily if you know M, V2)
Deviations due to
hydration non-spherical shape or both Rs
Rmin = Rs
Ratio measures the deviation from the expected behavior of the unhydrated sphere

13. viscosity Next we consider:
What effect does the interaction of the hydrodynamic particle and the solvent have on solution properties?
viscosity
What effect does the particle-solvent interaction have on the motion of the particle?
translational motion
- free
- in a force field
rotational motion

14. Viscosity
Frictional interactions between “layers” of solution results in energy dissipation
u (velocity) dy
u + du A
du • A du
dy dy
Ff  =  • ( ) • A
(shearing force)
  coefficient of viscosity
cgs units  Poise
 is related to the amount of energy dissipated per unit volume per unit time

15. Many ways to measure solution viscosity
capillary viscometer: measures the
rate of flow of solution through a
capillary with a pressure drop P

16. Effect of macromolecules on viscosity: only 2 parameters
plus solute  o
= 1 +  • 
fraction of volume occupied by “particles”
solvent alone
“shape” factor
(also called the Simha factor) 20 10
2.5
For a sphere:  = 2.5
Shape
factor 
Shape factor is very large for highly asymmetric molecules
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)

17. Relative viscosity and Specific viscosity
= r relative viscosity = 1 + 
= sp specific viscosity
sp = r - 1 = 
But  = Vh • c so
ideally, the increase in specific viscosity per gram of solute should be a constant ( • Vh ). Often it is not due to “non-ideal” behavior  o
 = shape factor
 = fraction of solution volume occupied by solvent particles
c = g / mL solvent
Vh = volume per gram of protein occupied by solvated particle
 - o o
sp c
=  • Vh
Shape factor vol/g of protein of hydrated particle

18. Non-ideal case (i.e.,reality):
=  • Vh + (const.) • c
sp 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.

19. Intrinsic Viscosities of Proteins in Aqueous Salt Solution

20. Molecular Dimensions of Proteins from Intrinsic Viscosities

21. Intrinsic viscosity [] =  • Vh  increases monomer polymer  = 2.5
if  = 25 then [] is 10x larger
[] =  • Vh
[] =  • Vh
if we assume a spherical shape,  = 2.5,
we will calculate an incorrect and
very large value for Vh

22. 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.

23. 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  V  mL/ g
[] = 2.5 • Vh
very large “coil”

24. Viscosity : examples
I. Hemocyanin (M = 106): native vs GuHCl denatured
For a native, globular protein, [η] ≈ 4 ml/g.
[η ] for same protein denatured in Gu • HCl is about 50 ml/g.
GuHCl denatured
coiled 60 40 20
sp C
cm3 g
[] =  • Vh = 2.5 Vh
If the amount of water within the flexible polymer is about 100 g per gram of solute (δ1=100), then
Vh ≈ 100 ml/g.
Native
compact
C (mg / mL) of BSA
II. T7 DNA (from phage T7, double strand): salt dependence
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.
3000
2000
1000
expanded coil
[sp] C
cm3 g
log [NaCl]

25. Stokes Radius 3. calculate Stokes radius 1. measure 2. assume a sphere
Radius of the sphere which has the hydrodynamic properties consistent
with the hydrodynamic measurement
[] =  • Vh
[] = 2.5 • Vh
3. calculate Stokes radius
1. measure
volume of the
molecule (spherical)
2. assume
a sphere
4/3 Rs3
M/N
mass of the
molecule
Solve for the Stokes radius:
M 3
N 4 
Rs = [ • Vh]1/3
where Vh = []/2.5

26. The radius expected if the molecule is an anhydrous sphere
V2 = anhydrous sphere - point of reference
Rmin = [ V2]1/ CALCULATED
[4/3  Rmin3]
(M/N)
M 3
N 4
MINIMUM RADIUS
The radius expected if the molecule is an anhydrous sphere

27. If Rs/Rmin is not much larger than 1.0, then the assumption of the
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)

28. For an anhydrous particle: [] =  • 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 +dH20)
2. correction due to asymmetry
 > Vh = V2 (anhydrous vol/g)
or both hydration and asymmetry
 > and Vh = V2 + H2O
anhydrous vol/gram

29. Comparison of intrinsic viscosity values for two proteins:
1. Ribonuclease:
mol wt: 13, 683
V2 = ml/g
Rmin = 17 Å
2. Collagen:
mol wt: 345,000
V2= ml/g
Rmin = 59 Å

30. Interpreting Viscosity Data:
Does a reasonable amount of hydration explain the measured value of []?
maximum solvation maximum asymmetry
protein [] mL/g Rs (Å) H2O (g/g)  (a/b)
Ribonuclease
Collagen
[] =  Vh =  [4/3  Rs3] N M
3 [] M
4  N 
Stokes Radius
Rs = ( )1/3
2.5
Appropriate for collagen
Rs/Rmin = 6.8 (400/59)
Vh = V2 (anhydrous value)
shape correction:  > 2.5
hydration correction: Vh = V2 + H2O
 = 2.5
Appropriate for ribonuclease
Rs/Rmin = (19.3/17)

31. Example: SDS-protein complexes
Can find M by comparing your unknown with a series of knowns - if for all cpds [] is a smooth function of M (ie both  and Vh)
Volume per mole
Vh, molar M
[] = ( ) = KMa You must use appropriate “standards”
1) compact sphere
[] = constant
a = 0
2) flexible polymer
[] = KM1/2
(since RGM) , a = 1/2
3) asymmetric molecules
[] = K Ma
a > 1
Example: SDS-protein
complexes

32. Example: Protein SDS complexes
BSA + saturating SDS (  23 ) 60 40 20
sp C
cm3 g
elongated complex
Native (  3)
compact
C (mg / mL) of BSA
For a series of proteins of varying Molecular Weight
2.0
log []
0.4
[] =  • Vh
[] = KMa = constant x na
so log [] = a log n + constant
slope  1.2  complex is highly asymmetric
log [# amino acids]  log (n)