What is the pI of lysine tripeptide?

What the the $$\mathrm{pI}$$ of a tripeptide of lysine?

$$\begin{array}{l|ccc} \hline \text{Group} & \ce{Lys{-}α{-}NH3+} & \ce{Lys{-}ε{-}NH3+} & \ce{Lys{-}α{-}COOH} \\ \hline \mathrm{p}K_\mathrm{a} & 8.0 & 10.5 & 2.2 \\ \hline \end{array}$$

I have written out the tripeptide, and I know when the hydrogens would come off of it depending on the $$\mathrm{pH}.$$ I know it has to do with the $$\ce{α{-}NH3+}$$ and the $$\ce{ε{-}NH3+}$$:

\begin{align} \mathrm{pI} &= \frac{\mathrm{p}K_\mathrm{a}(\ce{Lys{-}α{-}NH3+}) + \mathrm{p}K_\mathrm{a}(\ce{Lys{-}ε{-}NH3+})}{2} \\ &= \frac{8.0 + 10.5}{2} \\ &= 9.25, \end{align} but it was incorrect. Can anyone please help and explain the process to me?

Swiss-Prot — Compute pI/Mw calculator gives $$\mathrm{pI} = 10.30$$ for KKK sequence in single letter code.

• The tripeptide has 1+3 NH2 and 1 COOH. Calculate the average partial charge for each of them as function of [H+] and of respective acidity constant. And then calculate [H+] resp. pH when their charges are balanced with zero net charge. Jun 30, 2021 at 7:12
• Another thing is acidity constants for alpha NH2 and COOH will be different, compared to a single lysine, as the peptidic bond has different induction effect than amino or carboxy group. But the task may ignore this. Jun 30, 2021 at 9:15

$$\def\VEQ{{\scriptsize-\ce{H+}}{\Large\downharpoonleft\!\!\upharpoonright}\scriptsize{+\ce{H+}}} \begin{array}{rcc} \mathrm{pH} < 2.2 &\quad &\ce{H3\overset{+}{N}-\overset{+}{Lys}-\overset{+}{Lys}-\overset{+}{Lys}-COOH} \\ & & \VEQ \\ 2.2 < \mathrm{pH} < 8.0 &\quad &\ce{H3\overset{+}{N}-\overset{+}{Lys}-\overset{+}{Lys}-\overset{+}{Lys}-COO-} \\ & & \VEQ \\ 8.0 < \mathrm{pH} < 10.5 &\quad &\ce{H2N-\overset{+}{Lys}-\overset{+}{Lys}-\overset{+}{Lys}-COO-}\\ & & \VEQ \\ 10.5 < \mathrm{pH} = \mathrm{pI} &\quad &\ce{H2N-\overset{+}{[Lys-Lys-Lys]}-COO-}\\ \end{array}$$
As you can see, the formal net zero charge doesn't occur between any given $$\mathrm{p}K_\mathrm{a}$$s. Rather, at the highest $$\mathrm{p}K_\mathrm{a}(\ce{Lys{-}ε{-}NH3+}) = 10.5$$ the tripeptide is still going to have $$0.5 + 0.5 + 0.5 -1 = +0.5$$ formal charge considering $$50\,\%$$ deprotonation of residue amine groups at $$\mathrm{pH} = \mathrm{p}K_\mathrm{a}$$ and $$-1$$ charge at the $$\ce{-COO^-}$$ group.
This implies that $$\mathrm{pI}$$ of trilysine must be slightly higher, about $$10.8:$$
For polylysines $$\mathrm{pI}$$ should increase with the chain length. ThermoFisher's Peptide Analyzing Tool suggests theoretical $$\mathrm{pI}$$ values of $$11.0,$$ $$11.2$$ and $$11.3$$ for tetra-, penta- and heptalysine, respectively. This tool, however, refuses to calculate isoelectric points for tripeptides (Input sequence is too short). This can be an indication of insufficient quality of applied algorithm resulting in apparently wrong data suggested by Swiss-Prot in your question.
Note that we didn't corrected $$\mathrm{p}K_\mathrm{a}$$ values for the tripeptide. I wouldn't assume they change drastically, but it's not the identical values either.