Question

The R group of lysine has an amino group that can be positively charged or lose a proton to become neutral. The \(p K_{a}\) of the amino group is \(10.8 .\) Determine the fraction of amino group that is protonated at \(\mathrm{pH}=9.8\) and at \(\mathrm{pH}=11.8\).

Short answer

90.9% protonated at pH 9.8; 9.1% protonated at pH 11.8.

Step-by-step solution

Understand the Problem

We need to determine the fraction of the amino group that remains protonated. This involves applying the Henderson-Hasselbalch equation to calculate the ratio of protonation at given pH values around the known \(pK_a\) value.

Henderson-Hasselbalch Equation Familiarization

The Henderson-Hasselbalch equation is given by: \[ \text{pH} = pK_a + \log \left( \frac{[A^-]}{[HA]} \right) \] where \([A^-]\) is the deprotonated form (neutral) and \([HA]\) is the protonated form (positively charged). We'll rearrange this equation to calculate the fraction of the amino group that is protonated.

Rearrange for Protonation Fraction

Rearrange the equation to find \([HA]/([HA] + [A^-])\): \[ \frac{[HA]}{[HA] + [A^-]} = \frac{1}{1 + 10^{\text{pH} - pK_a}} \] This expression gives the fraction of the amino group that is protonated at a given \( \text{pH} \).

Calculate Fraction at pH 9.8

Substitute \( \text{pH} = 9.8 \) and \( pK_a = 10.8 \) into the rearranged equation: \[ \frac{[HA]}{[HA] + [A^-]} = \frac{1}{1 + 10^{9.8 - 10.8}} = \frac{1}{1 + 10^{-1}} = \frac{1}{1 + 0.1} = \frac{1}{1.1} \approx 0.909 \] This means approximately 90.9% of the amino groups are protonated at pH 9.8.

Calculate Fraction at pH 11.8

Substitute \( \text{pH} = 11.8 \) and \( pK_a = 10.8 \) into the rearranged equation: \[ \frac{[HA]}{[HA] + [A^-]} = \frac{1}{1 + 10^{11.8 - 10.8}} = \frac{1}{1 + 10^{1}} = \frac{1}{1 + 10} = \frac{1}{11} \approx 0.091 \] This means approximately 9.1% of the amino groups are protonated at pH 11.8.

Key concepts

Lysine

Lysine is one of the essential amino acids, which means our bodies cannot synthesize it, so it must be obtained through diet. This amino acid features a unique side chain which contains an additional amino group. This side chain often plays a significant role in biochemical functions such as enzyme activity and protein interaction.

The side chain in lysine is quite flexible in its charge state, owing to the presence of the amino group. This group can carry a positive charge or become neutral based on the environment's pH level.

Specifically, lysine's side chain is particularly interesting because it is protonatable, meaning it can gain or lose hydrogen ions (protons) depending on the acidity or basicity of the surrounding solution. This ability to change charge is crucial in biological systems, where lysine plays roles in structural stabilization and interaction with negatively charged molecules like DNA.

Protonation

Protonation is a chemical process in which a proton (hydrogen ion) is added to a molecule. For amino acids like lysine, protonation mostly affects the side chain and the amino and carboxyl groups connected to the central carbon. Understanding protonation is key for predicting how lysine behaves in different environments.

When lysine's amino group is protonated, it holds a positive charge. This occurs at lower pH levels, where the environment is more acidic. Conversely, when the environment is more basic, at higher pH levels, the amino group tends to lose its proton, becoming neutral.

• Protonation increases the positive charge on the molecule.
• De-protonation (loss of proton) reduces the positive charge, potentially neutralizing the molecule.

Understanding this chemical transformation helps predict lysine's behavior in different biological settings, influencing how proteins fold or bind to other molecules.

pH and pKa Relationship

The relationship between pH and pKa is crucial in determining the protonation state of molecules like lysine. pH measures the acidity or basicity of a solution, while pKa is the specific pH at which a group is 50% protonated and 50% deprotonated. Each acidic or basic group on a molecule has its specific pKa value.

In lysine, the pKa of the amino group is about 10.8. Using the Henderson-Hasselbalch equation, we can calculate how much of this group is protonated at any given pH level. This is achieved by knowing the balance between protonated and deprotonated forms of the group based on the pH:
\[ \text{pH} = pK_a + \log \left( \frac{[A^-]}{[HA]} \right) \]

When the pH is lower than the pKa, the protonated form dominates. Conversely, when the pH is higher than the pKa, the deprotonated form is more prevalent. This dynamic relationship is key to understanding the chemical behavior and interaction of amino acids in various environments.

Amino Acids

Amino acids are the building blocks of proteins, which are essential for many biological functions in our bodies. Each amino acid has a central carbon atom bonded to four different groups: an amino group (-NH₂), a carboxyl group (-COOH), a hydrogen atom, and a variable side chain (R group) which is different for each amino acid.

The side chain determines the unique properties of each amino acid. Lysine, for example, has a side chain with an additional amino group that contributes to its basic nature. These side chains can also interact with each other, influencing protein structure and function.

Amino acids can exist in various ionization states depending on the pH of the environment, which affects their behavior and function. At different pH levels, proteins can change shape, activate or deactivate, all based on how their amino acids react to the environment. Understanding these properties helps us predict how proteins behave in different contexts, and how changes in pH affect their structure and activity.