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What is the charge of the amino acid lysine at $\mathrm{pH}=2$? $\mathrm{p}K_\mathrm{a}$'s are carboxylic: 2.18; amino: 8.95; side chain: 10.79.

The side chain of lysine is an amine group, so I assume that when fully protonated it has positive charge ($\ce{NH3+}$).

At $\mathrm{pH} = 2$, everything will be protonated because their $\mathrm{p}K_\mathrm{a} > \mathrm{pH}$.

This gives a net 0 (carboxy) + 1 (amino) + 1 (side chain) = +2 charge. Why do sites such as this say that at $\mathrm{pH} = 2$, lysine's charge is only +1, not +2?

There's probably something really simple I don't know. If anyone can help point it out, that would be great.

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At $\mathrm{pH}=2$, the charge will be:

(charge near neutral pH, 2 groups positive, one negative) + (the fraction of the pKa = 2.18 group that is protonated)

=1 + (the fraction of the $\mathrm{p}K_\mathrm{a} = 2.18$ group that is protonated)

$=1 + 1/(1 + 10^{\mathrm{pH}-\mathrm{pK}}) =1 + 1/(1 + 10^{2-2.18})=1.60$

So the charge is +1.60, which is closer to 2 than 1, as you are saying.

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    $\begingroup$ Well it worth to pointing that's average charge and modal value is +2 $\endgroup$
    – Mithoron
    Mar 16 '18 at 17:21
  • $\begingroup$ I'll mark this answer as correct in a second, but if +2 is the average charge, then what is 1.6? $\endgroup$ Mar 16 '18 at 17:35
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    $\begingroup$ @LittleDragon 1.60 is the average charge. 1.60 is the charge that you would measure by electrophoresis. $\endgroup$
    – DavePhD
    Mar 16 '18 at 17:39
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[OP] This gives a net 0 (carboxy) + 1 (amino) + 1 (side chain) = +2 charge.

This is approximately correct. See DavePhD's answer for a more accurate treatment.

[OP] Why do sites such as this say that at $\mathrm{pH} = 2$, lysine's charge is only +1, not +2?

The table you cite is for proteins. When lysine gets incorporated into a protein, it forms peptide bonds with its carboxylate and its main chain amino function groups. Therefor, these groups no longer exist and don't get protonated or deprotonated (unless the lysine is the C-terminus or the N-terminus, in which case the table gives the wrong value). In short, the table is about the charge state of the side chain, not the original free amino acid.

[question cited] What is the charge of the amino acid lysine at $\mathrm{pH}=2$? $\mathrm{p}K_\mathrm{a}$'s are carboxylic: 2.18; amino: 8.95; side chain: 10.79.

The $\mathrm{p}K_\mathrm{a}$ values given are correct for lysine in water. In a protein, they might differ depending on the chemical environment of the specific lysine residue you are looking at. They would also depend on ionic strength and on cosolvents, if any.

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