The following amino acid is called lysine. I was asked to calculate its isoelectric point, with the given $mathrm pK_mathrm a$ values.

I’ve searched a lot, and the most helpful post that I found was How do I calculate the isoelectric point of amino acids with more than two pKa’s?
According to orthocresol’s answer:

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.

Let’s call the ends $e_1, e_2$ and $e_3$ (from left to right).

Approach $#1$

• deprotonate $e_3$ (i.e., carboxyl group)
• deprotonate $e_1$ or $e_2$ [neutral point]

So, $mathrm pK_mathrm a$‘s of $e_1$ and $e_2$ are relevant.

$$Rightarrow mathrm{pI} = frac{10.53 + 8.95}{2} = 9.74$$

But, is there some limit to number of protonations/deprotonations or some procedures to follow?

For instance,

Approach $#2$

• deprotonate $e_1$
• deprotonate $e_3$ [neutral point]
• deprotonate $e_2$ and protonate $e_1$ [neutral point]

This time, $mathrm pK_mathrm a$‘s of $e_3$ and $e_1$ are relevant. But, the calculated $mathrm{pI}$ isn’t correct.

So, how can I validate the approaches?

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Problem source: FIITJEE study material

$mathrm pK_mathrm a$ and $mathrm{pI}$ values table for amino acids: https://www.anaspec.com/html/pK_n_pl_Values_of_AminoAcids.html