[Big-Daddy]

What is the actual definition of the isoelectric point of an amino acid, peptide, etc.?

Because someone told me that it is "the pH at which the amino acid is neutral". Well there is only one way necessarily that the amino acid can be neutral - let us suggest that we're dealing with an amino acid that protonates once and deprotonates once from its neutral form, so that we have the forms HA, H₂A⁺ and A^- in solution; the true criterion for the amino acid to be "neutral" is that [H₂A⁺]=[A^-]. But obviously, looking at the full charge balance for the solution, there is not much reason to call this an isoelectric point, because it automatically means that [H⁺]=[OH^-].

So that can't be it.

What is the definition then, for a general amino acid/peptide/etc., of its isoelectric point?

[Babcock_Hall]

The isoelectric point means the pH at which the amino acid has no net charge.  The Wikipedia entry is OK, from a quick read through.  Can you think of an experimental way to determine this?  The question of the definition of the isoelectric point can get a little bit subtle, because of a similar quantity called the isoionic point, but I think we can probably leave the latter one out for now.

[Big-Daddy]

The isoelectric point means the pH at which the amino acid has no net charge.

Forget experimental for now, what about theoretical?

Most (simple) amino acids will exist in three forms in solution, in various concentration ratios: the (overall neutral) zwitterion HA, the protonated cation H₂A⁺ and the deprotonated anion A^-.

For "the amino acid to have no net charge", you need [H₂A⁺]=[A^-]; I can think of no other good interpretation. But if this is the case then as I showed in the OP, [H⁺]=[OH^-] for the pure acid in solution and there is self-evidently no point in defining the isoelectric point as a certain pH - it just has to be neutral!

From the Wikipedia article's formula pH=1/2(pKa1+pKa2) I have come up with an alternative definition - the pH at which the concentration of the neutral form HA is at its maximum. This can be proved to follow exactly the formula pH=1/2(pKa1+pKa2), and unlike the other definition is a non-trivial case. But OTOH the solution is not really going to be electrically neutral with respect to amino acid alone unless pKa1=pKa2=(1/2)pKw, which must be exceedingly rare if not impossible. In which case of course we get pH=1/2pKw again - the solution is itself proton-neutral as I postulated before.

[Babcock_Hall]

Hmm... One thing to consider is that amino acids with negatively charged side chains (Asp and Glu) have a very different definition of pI.  It is the average of pK₁ and pK_R.  I don't know the derivation, but I can offer a plausibility argument with respect to this formula.  At the calculated pI, the nitrogen atom has one full positive charge, and the fractional charges for the carboxylate oxygens on the main carboxylate group and the side chain carboxylate group sum to -1.  Try this for glutamic acid, and see what you think.  Of course, amino acids with positively charged side chains have a different formula for calculating pI.

[Big-Daddy]

Ok, thanks for the help. It confirms my suggestion that isoelectric point is probably defined as the point in pH at which the concentration of the neutral form of the amino acid/peptide will be maximal.

[Babcock_Hall]

No, I don't believe that it does, at least not with respect to glutamate or lysine, amino acids with negatively or positively charged side chains, respectively.  Glutamic acid has three pK_a values:
pK₁ = 2.1
pK₂ = 9.5
pK_R = 4.1

The side chain of glutamate is a neutral acid; therefore, we use calculate pI = (pK₁ + pK_R)/2.  The calculated pI for glutamate is 3.1.  This is the pH at which glutamate has no net charge.  Therefore, the positive and negative charges must balance exactly.

At pH 3.1, the nitrogen atom is essentially 100 % ionized, in other words there is a charge of positive one on the nitrogen.  This is true because the nitrogen is essentially 100% in the ammonium (-NH₃⁺) form.  This can be shown with the Henderson-Hasselbalch (H-H) equation.  If you carry out this calculation, you will see that the nitrogen is actually 99.99996% in the ammonium form, which is so close to 100% as to make the difference negligible.  Thus the sum of fractional negative charges on the two carboxyate groups must sum to negative one so as to make the overall charge zero.

Let us use HA and A- to symbolize the carboxyate group next to the nitrogen atom in its conjugate acid and conjugate base form, respectively.  This is the group with pK1 = 2.1.  We will use HB and B- to symbolize the side chain carboxylate group in its conjugate acid and conjugate base forms, respectively.   This is the group with pK_R = 4.1.  Now let us use the H-H equation two separate times to calculate the fractional negative charge on the two carboxylate groups.

For the carboxylate group next to the ammonium group, 3.1 = 2.1 + log{(A-)/(HA)}
At any pH (not just the pI), the fraction of molecules in the conjugate acid form and the fraction of molecules in the conjugate base form must sum to 1.00.  In other words,
1.00 = (HA) + (A-)
Solving two equations in two unknowns, we find HA = 0.0909, and A- = 0.9091
Therefore, there is 0.9091 (90.9%) of a negative charge on this carboxylate group.  We can say that this carboxylate group (the one next to the alpha-carbon) spends 90.9% of its time in its conjugate base (ionized) form.

By the same reasoning, there is 0.0909 (9.1%) of a negative charge on the side chain carboxylate group.  Specifically, 3.1 = 4.1 + log{(B-)/(HB)}, and 1.00 = (HB) + (B-).  Thus the side chain carboxylate group is mainly in the conjugate acid form where it is neutral and only slightly in the conjugate base form where it is negatively charged.  The opposite is true for the carboxylate group next to nitrogen.  It spends most of its time in the negatively charged form.

[Big-Daddy]

Thanks for the information. But I don't see the contradiction with my previous post. If we take glutamic acid, the only difference we have is that now we've got 2 possible acidic protons in the neutral form of the acid, leading to the maximum associated form of the acid having 3 protons: in other words we're dealing with four possible forms each existing in some small quantity at all points, H₃A⁺, H₂A, HA^-, A^2-.

H₃A⁺ and A^2- concentrations are not maximal at a certain concentration of [H⁺], rather only at limits of pH, i.e. asymptotic. But we can say that, if there are a total of 3 acidic dissociations from the most protonated form H₃A⁺ and these are labelled pK_a1, pK_a2 and pK_a3 respectively in order of size, then pH=1/2(pK_a1+pK_a2) generates the maximal concentration of H₂A and pH=1/2(pK_a2+pK_a3) the maximal concentration of HA^-.

Because of your ordering of pKa values, we can safely say that pK_a1 refers to the carboxylic acid next to the NH3 (you symbolized it A), pK_a2 to the carboxylic acid on the far side (you symbolized B) and pK_a3 to the dissociation from ammonium.

So if the isoelectric point refers to the pH where the concentration of the neutral form (here, H₂A) is maximal, that would be defined for this case as pI=1/2(pK_a1+pK_a2), which is identical to what you called pI=1/2(pK_a1+pK_R) since K_a2 is what you called K_R.

Please correct me if I am wrong.

[Babcock_Hall]

The link below may be helpful.  It discusses the difference between theoretical and operation definitions of the isoionic point.  One wishes it had a little bit more to say about isoelectric point.  It is a link to an article by William P. Bryan in the Biochemical Education 6(1) (1978). 
http://onlinelibrary.wiley.com/store/10.1016/0307-4412(78)90164-4/asset/5690060117_ftp.pdf?v=1&t=hols55ui&s=b97772f90d4343bc833594f85ec3e9466bb148f1

[Big-Daddy]

Thanks. I think it is fairy clear on both isoelectric point and isoionic point -

The former is defined as the net charge over all forms of the amino acid being 0 at this point (a condition that, notably, cannot be brought about by dilution unless the isoelectric point is at pH = 7; in which case, the ions added to shift the pH invalidate the possibility of using the charge balance equation for the solution). The latter is defined as the net number of protons lost or gained by all forms of the amino acid being 0 at this point. Unless complexation of amino acid forms is present (at least, it's the only case I can think of) the isoionic and isoelectric point should be equivalent.

Thanks again for the link.