Charge of an amino acid sequence

Question

I have the following problem:

A polypeptide has the following sequence (amino end on the left) $$\mathbf{lysine - aspartic~acid-histidine-arginine-serine}$$

Using the $pK_a$ information that follows, what would be the approximate charge on this molecule at pH = 3.3?

$pK_a$ values for the various groups in the molecule are as follows:

\begin{array}{|cc|}\hline \rm Alpha~carboxyl & 2.2 \\ \rm Alpha~amino & 9.5 \\ \rm R-group~amino & 11.7 \\ \rm R-group~carboxyl & 4.5 \\\hline\end{array}

After my calculations I got +2 and my answer was wrong. How to solve it correctly?

My effort:

$\ce{NH3}$ has $+1$ when on and $0$ when off. $\ce{NH3}$ are Lysine, Histadine, and arginine. $\ce{COOH}$ has $0$ when on and $-1$ when off. $\ce{COOH}$ are Serine and aspartic acid. Based on the pH and $pK_a$ values the answer should be 2. Lysine=+1 Aps=0 His=+1 Arg=+1 Serine=-1

Answer 1

When the pH is lower than the pKa, the group will be protonated, and when the pH is higher than the pKa, the group will be deprotonated. Your explanation is correct that basic groups will be positively charged when protonated and neutral when deprotonated and that acidic groups will be neutral when protonated and negatively charged when deprotonated. Shown below is the compound in it's fully protonated state with the given pKa's.*

The given pH is 3.3 which means that:

• N-terminus: pH < pKa, protonated basic group, +1
• lys-sidechain: pH < pKa, protonated basic group, +1
• asp-sidechain: pH < pKa, protonated acidic group, 0
• his-sidechain: pH < pKa, protonated basic group, +1
• arg-sidechain: pH < pKa, protonated basic group, +1
• C-terminus: pH > pKa, deprotonated acidic group, -1

The overall charge is +3.

*An aside, the pKa's for the side chains are a gross approximation here. It turns out they are sufficient to answer the question, but given a higher pH, the answer wouldn't be correct.