5. Principles of Electrophoresis; Macromolecular Charges in Solution
Electrophoresis is the forced migration
of charged particles, usually macromolecules, in an electric field (Figure
16.18). If a macromolecule has a net
charge q and a constant, uniform external electric field is applied, there
will be a net force
on the molecule given
by
In general, the
macromolecule will accelerate rapidly and the electric force will be balanced
very rapidly by a growing frictional force,
due to collisions
with solvent molecules. After reaching
equilibrium, the molecule will migrate in the electric field with a constant
velocity, obtained from setting the net force
equal to zero and
solving for v,
(7)
The electrophoretic mobility U is defined as the
velocity normalized by the applied electric field, and using Equation (7) can
be written as
(8)
Mobility is an intrinsic property
of the macromolecule, depending only on its charge and frictional properties.
Figure
16.18. Electric and viscous drag forces
acting on a macromolecule.
For a real
macromolecule in solution, both the actual net charge q and the
frictional factor f will be difficult to ascertain. If electrophoretic mobility were to be
measured, one of these parameters would still need to be obtained independently
before the other could be found from the above equation. This fact, and difficulties in generating a
known uniform field, have made electrophoresis complex and little used as an
analytical tool to learn about the electrical properties of
macromolecules. However, there are a
number of electrophoresis methods in almost daily use in most biomolecular
laboratories. Before considering some
of these techniques in a bit of detail in the next Section, we need to gain a
basic understanding of the charge on a macromolecule in solution.
Figure
16.19 The three different ionic forms
of the amino acid alanine. Proteins,
made from hundreds of amino acids, will have large numbers of variable electric
charges, depending on the pH of the surroundings
Unlike
isolated ions, such as Na+ or Cl-, that have a definite
charge state, macromolecules have a variable net charge that depends on the pH
of their local environment.
Macromolecules such as proteins or nucleic acids, consist of many
subunits, each with multiple ionizable charged groups that may be neutral,
positive or negative, depending on the pH.
The term zwitterion or polyelectrolyte is used to describe
such macromolecules with numerous charged groups (Figure 16.19). By adjusting the pH, the net charge on a
macromolecule can thus be made positive, negative or neutral. That particular pH at which the
macromolecule is electrically neutral is called the isoelectric point. At pH
values below the isoelectric point the macromolecule has a net positive charge
while at higher pH’s its net charge is negative.
Figure 16.20 A region of a macromolecule with its
surrounding cloud of counterions.
Macromolecules
are rarely suspended in pure water.
Almost always they are found with salts, buffers, and often with many
other small and large molecules. While
the concentration of ions gives some measure of their effectiveness in
electrical shielding, a better measure is the ionic strength I, defined as
(9)
where the sum is over all ionic
species of concentration ci and valence zi.
It is important
to realize that although the Coulomb force is long-range, as we have discussed,
normally macromolecules in solution will be effectively electrically shielded
unless at very low ionic strengths (Figure 16.20). Because of the electrical
attraction of opposite charges, a charged macromolecule in solution will have
large numbers of small ions of opposite charge, called counterions, surrounding each of its charged groups. These counterions form a charge cloud that
tends to completely cancel the effects of the macromolecular charge beyond a
certain characteristic distance, known as the screening (or Debye) length. A calculation of the screening length LD finds
(10)
where k is a (dielectric) constant characteristic of the
electrical properties of the solvent (water has k = 80), kB
is Boltzmann’s constant, and T is the absolute temperature. Table 16.2 gives the screening lengths for
different concentrations of ions in water.
Effectively, at ion concentrations of above about 10 - 100 mM, the
macromolecular charges are fully screened and there are no electrical
interactions with other large molecules until they come within about 1 nm. At lower ion concentrations there may be
longer range electric interactions between macromolecules.
Table 16.2 Screening lengths at different ionic
strengths of solution.
Concentration (mM) |
Screening length (nm) for monovalent ions |
Screening length (nm) for divalent ions |
0.1 |
30.4 |
17.6 |
1.0 |
9.6 |
5.6 |
10 |
3.0 |
1.8 |
100 |
1.0 |
0.6 |
6. Modern Electrophoresis Methods
There is a
fundamental problem in using electrophoresis as described in the previous section. In order to maintain a buffer and solvent
system with a typical salt concentration of 0.1 M, even a very modest electric
field will create substantial heating of the solution, resulting in convection
currents that would completely distort the controlled migration of
macromolecules. We will study this
heating phenomenon when we study electric currents, but it is ultimately due to
the transformation of kinetic energy into internal energy through collisions and
it is a similar effect to that resulting in the heat generated by a toaster,
for example. Early in the history of
electrophoresis, the answer to the heating problem was to reduce the ionic
strength of the solution; but then, as we have seen, long-range interactions
are possible and in some cases the macromolecules may not be stable under those
conditions. Today all electrophoresis
is carried out not in solution, but in gels, to avoid overall convection
problems due to heating or vibrational disturbances.
One
of the most important electrophoresis techniques is SDS gel electrophoresis, used to measure molecular weights of
proteins. Since the conformations of
proteins are so diverse, substantial information about a protein would be
required to know how the friction factor in Equation (8) is related to
molecular weight. Instead, in this
technique the proteins are first denatured so that they lose all of their
secondary structure and become simply random coil backbone polymers. Then SDS (sodium dodecyl sulfate), a highly
charged reagent that binds to all proteins with a very similar mass of SDS per
unit length of protein backbone, and thus a very similar electric charge per
unit length of protein, is added to saturate the protein. These highly charged SDS molecules exert
strong internal repulsive forces that tend to stretch out the random coil
protein into a rod-like shape. In
essence, all proteins are made to look virtually the same, rods of the same
diameter but with lengths that are proportional to the molecular weight of the
protein.
The
technique involves placing a small amount of such denatured SDS-protein mixture
(with a colored dye or stain added so that one can see where the fastest
migrating protein is located) at the top of a slab or tube of a gel (typically
polyacrylamide), and turning on an electric field within the gel using
electrodes attached to a power supply (Figure 16.21). The proteins and dye migrate down the gel at a constant rate that
depends on the molecular weight of the protein with the smaller proteins
migrating faster. At a given
concentration of gel material and given electric field strength, standards of
known molecular weight are used to empirically construct a calibration curve of
molecular weight vs electrophoretic mobility (basically determined from the
distance traveled down the tube normalized between 0 and 1- see Figure
16.22). Molecular weights of unknown
samples can be determined from their mobilities and such a calibration
curve. Over a limited molecular weight
range, the electrophoretic mobility of proteins is found to be proportional to
the logarithm of their molecular weight, as shown in the figure. This technique, known as SDS-PAGE (PolyAcrylamide Gel Electrophoresis), can rapidly and
cheaply measure molecular weights with an accuracy of about 5% and can also
determine trace amounts of impurities in a sample. It is one of the most common tools in the study of proteins
today.
Figure 16.21 Gel electrophoresis being set up to
run. Plexiglass housing holds a slab gel
that is being loaded with a sample
Figure 16.22 Example of calibration plot for
SDS-polyacrylamide gel electrophoresis.
Precisely how macromolecules move through
the supporting gel material in gel electrophoresis is not well understood. Our description and the usual analysis of
electrophoretic mobility is totally empirical.
For very large macromolecules such as high molecular weight DNAs that
tend to get stuck in the pores of even very dilute gels, it has been experimentally
discovered that, by using a series of electric field pulses of short duration
and varying direction, DNA migration can be enhanced. These efforts have led to an increased understanding of the
migration of macromolecules in gels.
Such knowledge is also applicable to the motion of macromolecules
through networks of filamentous proteins within the cytoplasm of a cell.
Another
important gel electrophoresis method, using
the ideas developed above on the polyelectrolyte nature of macromolecules, is isoelectric focusing. Native proteins migrate in an electric field
through a gel in which a pH variation has been established. Proteins migrating in the gel will
constantly vary their electric charge as the local pH changes until they arrive
at the location corresponding to their isoelectric point (Figure 16.23). They remain there since, with their net
charge equal to zero, they experience no force. Since the isoelectric point of a protein is an intrinsic
property, a detailed map of proteins separated according to isoelectric points
can be obtained.
Figure
16.23 Schematic of isoelectric
focusing. Polyelectrolytes move until
they reach their isoelectric point and have zero net charge.
Often
isoelectric focusing is combined with SDS-PAGE in two-dimensional gel electrophoresis. In this case, the native proteins are first run in a pH gradient
gel slab along one direction. When
completed, the electric field is set at 90o to its initial
direction, a new gel slab saturated with SDS and denaturants is butted against
the original gel slab, and the proteins are made to migrate into the new
gel. There they denature, acquire an
SDS coat and migrate according to molecular weight along the new direction. When complete there is a two dimensional map
of proteins with isoelectric point along one direction and, by calibration,
molecular weight obtainable from the position along the other direction (Figure
16.24). Proteins with similar
isoelectric points or with similar molecular weights can be further separated
by this method so long as the other property is distinct. There are numerous other variations of these
techniques in one or two dimensions in current use with new methods being
developed.
Figure 15.24 Two-dimensional gel electrophoresis of the
proteins of the influenza virus. The
vertical scale is molecular weight while the horizontal scale is isoelectric
point.