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For most amino acids, the $\mathrm{pI}$ is simply the mean of the amino and carboxyl $\mathrm pK_\mathrm a$'s. However, for tyrosine and cysteine, which have more than one $\mathrm pK_\mathrm a$ value, this rule of thumb doesn't apply.

I see that for tyrosine, it's the $\mathrm pK_\mathrm a$'s of the carboxyl and amino groups that are averaged, but for cysteine it's those of the carboxyl group and the side chain.

I haven't been able to find an explanation of why this is the case, or what the reasoning behind the calculations is. If anyone could help me out here I'd really appreciate it.

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Since the $\mathrm{pI}$ is the $\mathrm{pH}$ at which the amino acid has no overall net charge, you need to average the $\mathrm pK_\mathrm a$ values relevant to the protonation/deprotonation of the form with no net charge. Here are the acid-base equilibria for tyrosine:

Tyr

The form with no net charge is in red (+1 and -1 cancel out to give no net charge). It is the $\mathrm pK_\mathrm a$ values on either side of this form (in blue) that matter, hence the $\mathrm{pI}$ of tyrosine is $5.66$ (the average of $2.20$ and $9.11$).

It just so happens that $2.20$ is the carboxyl $\mathrm pK_\mathrm a$ and $9.11$ is the amino $\mathrm pK_\mathrm a$. If the side chain $\mathrm pK_\mathrm a$ were lower than $9.11$, then you should average the carboxyl and side chain $\mathrm pK_\mathrm a$'s instead.

The same logic applies to cysteine (look up the $\mathrm pK_\mathrm a$ values and draw out the differently protonated forms). You'll find that since the side chain has a lower $\mathrm pK_\mathrm a$ than the amino group, you average the carboxyl and the side chain $\mathrm pK_\mathrm a$'s.

This procedure can of course be extended to the amino acids with acidic side chains (aspartic acid; glutamic acid) and those with basic side chains (lysine; arginine; histidine).

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