Question

Consider a protein with a surface-exposed histidine residue in a \(\mathrm{pH} 4\) solution. What is the fraction of proteir molecules in which this histidine residue is charged? (Assume that the \(\mathrm{p} K_{a}\) is \(6.0\).)

Short answer

The fraction of protein molecules with charged histidine is approximately 0.01.

Step-by-step solution

Understand the Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is used to relate the pH of a solution to the pKa and the ratio of the concentration of the deprotonated and protonated forms of an acid. The equation is given by:\[ ext{pH} = ext{pKa} + ext{log} \left(\frac{[ ext{A}^-]}{[ ext{HA}]}\right)\]where \( [\text{A}^-] \) is the concentration of the deprotonated form (charged) and \( [\text{HA]} \) is the concentration of the protonated form (uncharged).

Plug Values into the Equation

Given the pH of the solution (4.0) and the pKa of histidine (6.0), substitute these values into the Henderson-Hasselbalch equation:\[4.0 = 6.0 + \log \left(\frac{[\text{A}^-]}{[\text{HA}]}\right)\]

Key concepts

Protein Chemistry

Proteins are large, complex molecules critical to many functions in living organisms. **They are composed of a long chain of amino acids** connected through peptide bonds. Each amino acid has a unique side chain that affects the protein's structure and function. For instance, the side chain of histidine, which contains an imidazole group, can carry a charge depending on the pH of the surroundings.
Histidine is special in proteins because its side chain can act as both a proton donor and acceptor. This allows it to play a key role in enzyme activity and binding reactions. Understanding how these side chains interact with the environment, and how they can be charged, is crucial in fields like enzyme design and drug development.

pKa Values

The pKa value of an acid provides insight into its strength and how it will behave in a given pH environment. **For histidine's side chain, the imidazole ring, the pKa is typically around 6.0.**
This tells us that at a pH lower than 6, like pH 4 for the given problem, the imidazole group is more likely to be protonated and positively charged. Conversely, at pH levels above 6, deprotonation will occur, leaving the imidazole group neutral.
Understanding the pKa is essential for predicting the charge state of a molecule's functional groups. This information is crucial for understanding how proteins will behave and interact in different environments. Such knowledge can aid in designing better pharmaceuticals by predicting protein binding and activity.

Acid-Base Equilibrium

Acid-base equilibrium refers to the balance between protonated and deprotonated forms of a molecule in solution. **In this context, the Henderson-Hasselbalch equation helps determine this balance for the histidine residue in proteins.**
By substituting the known pH and pKa values into the equation:\[4.0 = 6.0 + \log \left(\frac{[\text{A}^-]}{[\text{HA}]}\right)\]
We can compare the concentration of the deprotonated \( [\text{A}^-] \) and protonated \( [\text{HA}] \) forms. The equation indicates that at pH 4, the histidine is likely to be in its protonated form, and thus positively charged.
The ratio \(\frac{[\text{A}^-]}{[\text{HA}]}\) can be found as follows:\[10^{-2} = \frac{[\text{A}^-]}{[\text{HA}]}\]
This shows there is 100 times more protonated histidine than deprotonated, confirming that the predominant form in this scenario is charged. This concept helps in understanding the behavior of proteins under various pH conditions, crucial for applications like drug interactions and enzyme reactions.