274 Chapter 8. Chemical forces and self-assembly[[Student version, January 17, 2003]]
coding for that protein. The interactions of the residues with one another and with water determine
how the protein folds; the structure of the folded protein determines its function.
In short, proteins are horrendously complicated. How can we say anything simple about such a
complex system?
Some amino acids, for example aspartic or glutamic acid, liberate H+from their own carboxyl
groups, like any organic acid. Others, including lysine and arginine, pull H+out of solution onto
their basic side chains. The corresponding dissociation reactions thus involve transfer of a proton:
Acidic side chain: –COOH –COO−+H+
Basic side chain: –NH 3 + –NH 2 +H+. (8.27)The species on the left are theprotonatedforms; those on the right aredeprotonated.
Each residue of typeαhas a characteristic equilibrium constantKeq,αfor its deprotonation
reaction. We find these values tabulated in books. The range of actual values is about 10−^4.^4 for the
most acidic (aspartic acid) to about 10−^12 for the most basic (arginine). The actual probability that
aresidue of typeαwill be protonated will then depend onKeq,α,and on the pH of the surrounding
fluid. Denoting this probability byPα,wehavefor examplePα=[–COOH]/([–COOH]+[–COO−]).
Combining this definition with the equilibrium condition, [–COO−][H+]/[–COOH]=Keq,αgives
Pα=^1
1+Keq,α/[H+]=^1
1+Keq,α 10 pH.
It’s convenient to rephrase this result using Equation 8.10 as
Pα=(1+10xα)−^1 ,wherexα=pH−pKα. probability of protonation (8.28)The average charge on an acidic residue in solution will then be (−e)(1−Pα). Similarly, the average
charge on a basic residue will beePα.Inboth cases the average charge goes down as pH goes up,
as indeed we can see directly from Reactions 8.27.
Actually, in a protein uncharged and charged residues will affecteach otherinstead of all be-
having independently. Hence Equation 8.28 says that the degree of dissociation of a residue is a
universal function of the pH in its protein environment, shifted by the pKof that residue. Equa-
tion 8.28 shows that a residue is protonated half the time when the ambient pH just equals its
dissociation pK.
Though we don’t know the local pH at a residue, we can guess that it will go up as that of the
surrounding solution goes up. For example, we cantitrateasolution of protein, gradually dripping
in a base (starting from a strongly acidic solution). Initially [H+]ishigh and most residues are
protonated; the acidic ones are then neutral, while the basic ones are positively charged, so the
protein is strongly positively charged. Increasing pH decreases [H+], driving each of Reactions 8.27
to the right and decreasing the net charge of the protein. But at first only the most strongly acidic
residues (those with lowest pK)respond. That’s because the universal function (1 + 10x)−^1 is
roughly a constant except nearx=0,where it rapidly switches from 1 to 0. Thus only those bases
with pKclose to the pH respond when pH is changed slightly; the basic residues remain completely
protonated as we raise the pH from very acidic to somewhat acidic.
Continuing to titrate, each type of residue pops over in turn from protonated to deprotonated,
until under very basic conditions the last holdouts, the strongly basic residues, finally surrender
their protons. By this time the protein has completely reversed its net charge, orvalence;now