Histidine vs Lysine

Among the amino acids, histidine and lysine, which is more basic?
And why?
I tried protonating and checking electronic effects but couldn't reach a satisfactory conclusion.




As their name imply, each amino acid has at least one amine and one acid functional group, but in most amino acids, the basicity of the amine is offset by the carboxylic acid group at physiological $\mathrm{pH}$, and hence considered neutral.


Amines are basic because the nitrogen has an unshared electron pair that can accept an $\ce{H+}$ more readily than that water does. They will even, to some extent, take an $\ce{H+}$ from water, leaving a hydroxide ($\ce{OH-}$) ion. There are three amino acids that have basic side chains at neutral $\mathrm{pH}$, which are named arginine (Arg), lysine (Lys), and histidine (His).


Arginine is the most basic among them because it contains guanidine side group, $\ce{-(CH2)4NHC(=NH)NH2}$, which is basic.


Lysine has two amine groups, which makes it overall basic because of the second isolated amine group $(\ce{-(CH2)4NH2})$.


Histidine, on the other hand, contains imidazole group, which is also basic.


The $\mathrm{p}K\_\mathrm{a}$'s of all three side chains are high enough that they tend to bind protons, gaining a positive charge in the process (See below chart for $\mathrm{p}K\_\mathrm{a}$ values). The $\mathrm{p}K\_\mathrm{a}$ values are in order of


$$\ce{-(CH2)4NHC(=NH)NH2} > \ce{-(CH2)4NH2} > \ce{-(CH2)-(\text{4-imidazole})}$$


(see the chart), thus, lysine is considered to be more basic than histidine in physiological conditions. Please also note that, in the following chart: $\mathrm{p}K\_\mathrm{a1} = \alpha$-carboxyl group, $\mathrm{p}K\_\mathrm{a2} = \alpha$-amine group, $\mathrm{p}K\_\mathrm{a3} =$ side chain group, and $\mathrm{pI} =$ the isoelectronic point.


$$
\begin{array}{lccccc}
\hline
\textrm{Amino Acid} & \mathrm{p}K\_\mathrm{a1} & \mathrm{p}K\_\mathrm{a2} & \mathrm{p}K\_\mathrm{a3} & \mathrm{pI} & \textrm{acid/base} \\
\hline
\textrm{Aspartic acid} & 1.88 & 9.60 & 3.65 & 2.77 & \textrm{acidic} \\
\textrm{Glutamic acid} & 2.19 & 9.67 & 4.25 & 3.22 & \textrm{acidic} \\
\textrm{Arginine} & 2.17 & 9. 04 & 12.48 & 10.76 & \textrm{basic} \\
\textrm{Lysine} & 2.18 & 8.95 & 10.53 & 9.74 & \textrm{basic} \\
\textrm{Histidine} & 1.88 & 9.17 & 6.00 & 7.59 & \textrm{basic} \\
\hline
\end{array}
$$


I thank [University of Calgary, Canada](http://www.chem.ucalgary.ca/courses/351/Carey5th/Ch27/ch27-1-4-2.html) and [University of Wisconsin, Madison, WI](https://www.chem.wisc.edu/deptfiles/genchem/netorial/modules/biomolecules/modules/protein1/prot14.htm) for relevant data.


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