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

lysine-structure-with-pka-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?


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

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    $\begingroup$ The first approach I'd say is the correct one. It's obvious that the isoelectric point will be between e1 and e2 because lysine is a dibasic aminoacid, therefore the two amino groups will have a bigger influence on the pI than the acidic group, thus rendering the pI basic. In these cases, as far as I know, you have to find the average of 2 pKa-values that are going to have the biggest influence on the pI. In this case, since there are 3 pKa-values of which 2 are basic, the pI will be the arithmetic mean of those values. $\endgroup$ – TheRelentlessNucleophile Jul 18 '20 at 14:21
  • $\begingroup$ Related: Various pI calculators for proteins give different results: ExPASy (8.75), isoelectric.org (9.04), and Prot pi (8.475). $\endgroup$ – Peter Mortensen Jul 19 '20 at 12:53
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Isoelectric point of an amino acid is the $\mathrm{pH}$ at which the molecule carries no net charge[1]. It can be calculated by the average of the relevant $\mathrm pK_\mathrm a$ values as you have mentioned.

Your confusion seems to stem from choosing the relevant $\mathrm pK_\mathrm a$ values. For this we should refer to the titration curve of the amino acid.

For a neutral amino acid[2]:

titration curve 1

From the curve we can infer that the $\mathrm{pI}$ is simply the average of the two $\mathrm pK_\mathrm a$ values of the carboxylic acid and the amino group.

For a basic amino acid[2]:

titration curve 2

From the curve we can infer that the $\mathrm{pI}$ is simply the average of the two $\mathrm pK_\mathrm a$ values of the two amino groups. The $\mathrm pK_\mathrm a$ of the carboxylic acid group is not relevant.

For an acidic amino acid[3]:

enter image description here

From the curve we can infer that the $\mathrm{pI}$ is simply the average of the two $\mathrm pK_\mathrm a$ values of the two carboxylic acid groups. The $\mathrm pK_\mathrm a$ of the amino group is not relevant.

Here are examples for all three cases:

enter image description here

References:

  1. Wikipedia

  2. Titration curves for neutral and basic amino acids

  3. Titration curve for acidic amino acids

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