A report on an experiment I did of doing electrophoresis with proteins

19
E l
∑i [u i]µ iu z iu
=
E u
∑
i [l i]µ il z il
Earlier it was argued that the velocities of the ions in the two gels have to be the same because of continuity and
hence here one uses that for the upper ion whose parameters are subscripted with u and for the lower ions whose
corresponding parameters are subscripted with l. Then one has from the velocity equality,
Substituting this back in the earlier equation we have,
E u µ u x u = E l µ l x l
µ u x u
∑i [u i]µ iu z iu
=
µ l x
∑ l
i [l i]µ il z il
The above is known as the Kohlrausch Regulating Function
One of ways to see what this function does is to consider a situation when µ l x l > µ u x u and in that case one can see
that if an upper ion finds itsel in the lower gel then it will move slower than the surroundings and will be eventually
be overtakn by the boundary and it will return to its original location. Similarly if a lower ion inds itself in the upper
gel it will move faster than the boundary and will soon overtake it and get back to its original point. Thus the system
maintains a clear separating layer between the two regions.
Let µ p and x p be the mobility and the dissociation fraction of the proteins and µ g and x g be for the glycine. From
the experimental set-up one knows that glycines will be in the upper layer and the proteins will be in the lower layer
and the chloride ions are common to both the layers but has different concentrations and hence Kohlrausch Regulating
function for this case says that,
µ g x g
[g]µ g z g + [cu]µ c z c
=
µ p x p
[p]µ p z p + [cl]µ c z c
(using c to denote the chloride ions)
By microscopic charge neutrality one has [g]z g = −[cu]z c and [p]z p = −[cl]z c .
Using that the crucial equation that emerges is that,
(
cg
x g
)
(
cp
x p
) =
(
µg
)
z g
( )
µp
z p
( )
µp − µ c
µ g − µ c
If the effects of the gel are not included then the above equation explains why a system of ions of different mobilities
should maintain distinct separating boundaries while flowing in the presence of a constant potential difference.
The analysis of the gel on this above effect is a further comlicated phenomenon which in someway can be thought of
as providing a viscous drag force which imepedes the motion. But the modelling of that effect runs into complications
ofthe fact that denatured protein molecules in the system tend to be cylindrical in shape and hence the direction of
their approach becomes a crucial factor after a certain molecular mass number. This is what is dubbed as “Non-
Newtonian” viscosity. Further also needs to model the effect that the static pH boundary in the gel does not disrupt
the moving pH boundary of the ion solution.
C. Some comment about the polymerization
It is a still harder problem to model how doe the pore size of the gel vary as a function of the concentration of
the polymer. As a minimal model consider that the gel consists of n × n × n cubical cells for evey unit cube in it
and network be formed of polymer chains of unit length. Then one can approximate the pore diameter by the length
of a side of the cell (say r) and then one can see that r = 1 n − ( 1 + 1 n)
d and the number of polymer chains in each
such cube is 3(n + 1) 2 . Then simple calculation gives that if the concentration of the polymer is increased by J then
number of squares per edge would becme (n + 1) √ J − 1 and hence from that one can calculate by how much the r
changes with J.