Measuring the size and shape of macromolecules

Hydrodynamics: study of the objects in waterHow do they move?

TranslationRotation

1) Movement with no external force free diffusion

2) Movement under the influence of an external forcecentrifugeelectrophoresis, etc.

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 techniques measure the frictional resistanceof the moving macromolecule in solution:

1. Intrinsic viscositythe 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.

HydrodynamicsMeasurements 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

The hydrodynamic particle

hydrodynamically bound water

shear plane: frictional resistance to motion depends onsize

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

: (inverse of density)

Volume = V1g1 + V2g2 vol x grams gram

iVgram

volume

Proteins: V2 ~ 0.7 - 0.75 mL/g Na+ DNA: V2 ~ 0.5 - 0.53 mL/g

1. Specific Volume:

Consider a two component system; solvent (1) and solute (2)

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.

Hydrodynamic Parameters

change in volume per gram under specific conditions

Vi = ( )

Volume = V1g1 + V2g2 Gibbs - Duham relationship

We will assume that specific volume is approximately equalto the partial specific volume: Vi Vi

Vgi T,P,gj

Proteins: VP ~ 0.7 - 0.75 mL/g Na+ DNA: VDNA ~ 0.5 - 0.53 mL/g

2. Partial Specific Volume:

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.

For proteins, the value of the specific volume can be accurately predicted by summing the specific volumes of the component individual amino acids.

Take into account buoyancyby subtracting the mass of the displaced water

[mass - (mass of displaced water)]

[m - m(V2) = m (1 - V2)]

M (1 - V2)

N

Volg

solvent density

of the moleculeV2 =

3. Effective mass

mV2 = volume of particle

mV2 = mass of displaced solvent

m = (M/N) mass of particle

Hydrodynamic ParametersThis is the molecular mass minus the mass of the displaced solvent (the

buoyant factor).

N = Avogadro number M = molecular weight

mh = m + m 1

M ( 1 + 1)N

M ( V2 + 1 V1)N

Vh = (V2 + 1 V1)

mh =

Vh =

Mass of hydrated protein

Volume of hydrated protein per gram of protein

Volume of hydratedprotein

gH2O

gprotein1=

4. Hydration

Hydrodynamic Parameters

Problems of hydrodynamic measurements in general:

the properties depend on 1) size too many parameters! 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

ViscosityFrictional interactions between “layers” of solution results in energy dissipation

A

u (velocity)

u + du 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

du • A du dy dy

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

viscosity effects are very large for very elongated polymers (large ) (e.g. actin) or molecules that “occupy” large volume (large ) (e.g. DNA)

b a

Shape factor is very large for highly

asymmetric molecules1 5 10 15

axial ratio = a/b

20

10

2.5

For a sphere: = 2.5

= 1 + • o

plus solute

solvent alone “shape” factor

fraction of volume occupied by “particles”

Shapefactor

(also called the Simha factor)

= 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

- o

o

sp

c

= shape factor

= fraction of solution volume occupied by solvent particles

c = g / mL solvent

Vh = volume per gram of protein occupied by solvated particle

Shape factor vol/g of protein of hydrated particle

= • Vh

Relative viscosity and Specific viscosity

Non-ideal case (i.e.,reality):

= • Vh + (const.) • c + ....sp

csp

c

c

(ideal)

Intrinsic viscosity: [], units = mL / g

[] = • Vh Intrinsic viscosity is dependent only on size and properties of the isolated macromolecule.

Extrapolate measurement to infinite dilution: Intrinsic viscositylim (sp /c) = [] limit at low concentration (intercept in graph above)

C0

Intrinsic Viscosities of Proteins in Aqueous Salt Solution

Molecular Dimensions of Proteins from Intrinsic Viscosities

[] = • Vh

[] = • Vh

= 2.5 if = 25 then [] is 10x larger

monomer polymer

[] = • Vh

if we assume a spherical shape, = 2.5,we will calculate an incorrect andvery large value for Vh

increases

Intrinsic viscosity

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 DNAor 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”

C (mg / mL) of BSA

Native

sp

C

cm3

g

GuHCl denatured

compact

coiled

I. Hemocyanin (M = 106): native vs GuHCl denatured

0 5 10 15

60

40

20

0

II. T7 DNA (from phage T7, double strand): salt dependence

[sp]

C

cm3

g

3000

2000

1000

0

0 0.01 0.1 1.0 log [NaCl]

expanded coil

[] = • Vh = 2.5 Vh

Viscosity : examples

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.

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.

Rs = [ • Vh]1/3M 3N 4

Solve for the Stokes radius:

where Vh = []/2.5

Stokes Radius Radius of the sphere which has the hydrodynamic properties consistent

with the hydrodynamic measurement

[] = 2.5 • Vh

volume of themolecule (spherical)

mass of themolecule

2. assume a sphere

4/3 Rs3

M/N

[] = • Vh

3. calculate Stokes radius1. measure

MINIMUM RADIUSThe radius expected if the molecule is an anhydrous sphere

V2 = anhydrous sphere - point of reference

Rmin = [ V2]1/3 CALCULATED

[4/3 Rmin3]

(M/N)

M 3 N 4

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 cometo explaining the value of Rs?

If Rs/Rmin is not much larger than 1.0, then the assumption of themolecule 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 isnot spherical ( >>2.5) or it is not compact (Vh>>V2)

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

= 2.5 Vh (hydrated vol/g = V2 H20

2. correction due to asymmetry

> 2.5 Vh = V2 (anhydrous vol/g) or both hydration and asymmetry

> 2.5 and Vh = V2 + H2O

anhydrous vol/gram

Comparison of intrinsic viscosity values for two proteins:

1. Ribonuclease:mol wt: 13, 683V2 = 0.728 ml/gRmin = 17 Å

2. Collagen:mol wt: 345,000V2= 0.695 ml/gRmin = 59 Å

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 3.3 19.3 0.59 4.5 3.9

Collagen 1150 400 460 1660 175

[] = Vh = [4/3 Rs3] N

M

Rs = ( )1/33 [] M4 N Stokes Radius

2.5

Vh = V2 (anhydrous value) shape correction: > 2.5

hydration correction: Vh = V2 + H2O = 2.5

Appropriate for collagenRs/Rmin = 6.8 (400/59)

Appropriate for ribonucleaseRs/Rmin = 1.14 (19.3/17)