all of the numerical values that we have calculated have been deduced analytically by
applying some equation: there is a direct method to get from the data to a precise answer.
However, it is commonplace to come across problems where the values we are interested
in are not directly accessible. In reality such problems may range from those for which it
is genuinely difficult to imagine any formulaic method (hard problems: see
Chapter 25
) to
those where a direct formulation is merely inconvenient or slow.
The topic of this example is the estimation of the
isoelectric point of a protein, which
we will call the pI. This is a measurable property of a protein: it is the pH
14
at which the
protein carries no overall electric charge. This is something that is often used to
characterise and isolate proteins, for example, by performing electrophoresis (moving
particles through a porous substance with an electric current) in a gel with a pH gradient.
Proteins have electric charges because certain kinds of amino acids, together with the
chain termini (the unlinked ends), are capable of accepting or losing a hydrogen ion (H
+
).
The groups that are capable of gaining a hydrogen ion, and thus a positive charge, are
called basic: this includes the residues arginine, lysine and histidine
15
and also the N-
terminus of the protein, the start of the chain where there is a free amine group. The
groups which lose a hydrogen ion gain a negative charge; they are neutral before the loss.
These groups are called acidic and include the residues aspartic acid, glutamic acid,
cysteine, tyrosine and the C-terminus of the protein: the end of the protein chain where
there is a free carboxylic acid group.
In any given situation whether or not these basic and acidic groups carry a charge
depends on the hydrogen ion concentration of the environment: the pH. In a solution with
a low pH the concentration of H
+
is high,
16
and so there are lots of free ions to bind to the
basic groups, giving them a positive charge, and also lots of free ions to bind to the acidic
groups, removing the negative charge and making them neutral. Conversely, with a high
pH, the concentration of H
+
is low, whereupon the ions are lost from the protein; basic
groups become neutral and acidic groups are left with a negative charge.
The different basic and acidic groups do not bind to hydrogen ions equally strongly. For
example, aspartic acid very easily loses H
+
; at neutral pH 7.0 they are almost all lost, but
for tyrosine at a neutral pH hardly any are lost. The strength of any acid or base can be
described by the acid dissociation constant, referred to as the pK
a
. This has a formal
mathematical definition using the concentrations of hydrogen-bound and unbound
components,
17
but is most easily remembered as the pH at which on average half of the
groups will be bound with H
+
. Any one specific group can of course only be bound to a
whole hydrogen ion or no hydrogen ion, so these constants represent the average over time
as H
+
is dynamically lost and gained. The pK
a
value for aspartic acid is 4.4, so at pH 4.4 it
will have H
+
half of the time, and thus its average electric charge will be −0.5: half
negative because the free half is negative. For aspartic acid, as pH
increases it will
become increasingly negatively charged as it will be bound to H
+
less of the time.
Conversely the pK
a
value for lysine is 10.0, thus at pH 10.0 it will be half bound by H
+
,
but because this residue is basic the ions add a positive charge, rather than neutralise a
negative one. So for lysine, as the pH
decreases more H
+
binds and it becomes more
positively charged.