Problem 100 - Chemistry The Central Science

Question

The protein hemoglobin (Hb) transports \(\mathrm{O}_{2}\) in mammalian blood. Each \(\mathrm{Hb}\) can bind \(4 \mathrm{O}_{2}\) molecules. The equilibrium constant for the \(\mathrm{O}_{2}\) -binding reaction is higher in fetal hemoglobin than in adult hemoglobin. In discussing protein oxygenbinding capacity, biochemists use a measure called the \(P 50\) value, defined as the partial pressure of oxygen at which \(50 \%\) of the protein is saturated. Fetal hemoglobin has a \(\mathrm{P} 50\) value of 19 torr, and adult hemoglobin has a P50 value of 26.8 torr. Use these data to estimate how much larger \(K_{c}\) is for the aqueous reaction \(4 \mathrm{O}_{2}(g)+\mathrm{Hb}(a q) \longrightarrow\left[\mathrm{Hb}\left(\mathrm{O}_{2}\right)_{4}(a q)\right]\) .

Step-by-Step Solution

Answer

The equilibrium constant \(K_c\) for the aqueous reaction is approximately 3.95 times larger for fetal hemoglobin than for adult hemoglobin.

Step 1: Write down the given information

We are given the following information: - Fetal hemoglobin \(P50\) value: 19 torr - Adult hemoglobin \(P50\) value: 26.8 torr

Step 2: Write the reaction equation and expression for the equilibrium constant

The reaction equation is given as \[4 O_2(g) + Hb(aq) \rightleftharpoons [Hb(O_2)_4(aq)]\] The expression for the equilibrium constant \(K_c\) for this reaction is: \[K_c = \frac{[\mathrm{Hb}(O_2)_4]}{[\mathrm{Hb}](\mathrm{P}_{O_2})^4}\]

Step 3: Find the partial pressures at 50% saturation

At 50% saturation, half of the hemoglobin molecules are bound with oxygen, and half are unbound. Therefore, we have: For fetal hemoglobin, \(\frac{1}{2}[\mathrm{Hb}] = [\mathrm{Hb}(O_2)_{4}] \Rightarrow [\mathrm{Hb}] = 2[\mathrm{Hb}(O_2)_{4}]\) \(P_{O_2} = 19~\text{torr}\) For adult hemoglobin, \(\frac{1}{2}[\mathrm{Hb}] = [\mathrm{Hb}(O_2)_{4}] \Rightarrow [\mathrm{Hb}] = 2[\mathrm{Hb}(O_2)_{4}]\) \(P_{O_2} = 26.8~\text{torr}\)

Step 4: Calculate the ratio of \(K_c\) for fetal and adult hemoglobin

We will now express \(K_c\) in terms of the given quantities and then find the ratio of \(K_c\) for fetal and adult hemoglobin. For fetal hemoglobin, \[K_{c_{\text{fetal}}} = \frac{[\mathrm{Hb}(O_2)_4]}{[\mathrm{Hb}](\mathrm{P}_{O_2})^4} = \frac{1}{2} \cdot \frac{1}{(19 \ \text{torr})^4}\] For adult hemoglobin, \[K_{c_{\text{adult}}} = \frac{[\mathrm{Hb}(O_2)_4]}{[\mathrm{Hb}](\mathrm{P}_{O_2})^4} = \frac{1}{2} \cdot \frac{1}{(26.8 \ \text{torr})^4}\] Now, we find the ratio between \(K_{c_{\text{fetal}}}\) and \(K_{c_{\text{adult}}}\): \[\frac{K_{c_{\text{fetal}}}}{K_{c_{\text{adult}}}} = \frac{1/(2 \cdot (19 \ \text{torr})^4)}{1/(2 \cdot (26.8\ \text{torr})^4)} = \left(\frac{26.8}{19}\right)^4\] Using a calculator, we get \[\frac{K_{c_{\text{fetal}}}}{K_{c_{\text{adult}}}} \approx 3.95\] So, the equilibrium constant \(K_c\) for the aqueous reaction is approximately 3.95 times larger for fetal hemoglobin than for adult hemoglobin.

Key Concepts

Oxygen Binding Capacity

Oxygen binding capacity refers to the ability of hemoglobin to bind with oxygen molecules. Hemoglobin, a protein found in the blood, is crucial for transporting oxygen from the lungs to the rest of the body. Each hemoglobin molecule has the capacity to bind up to four oxygen molecules. This binding ability is determined by how tightly hemoglobin holds onto the oxygen.
The measure of this capacity often involves the equilibrium constant, denoted as \(K_c\). A higher \(K_c\) value indicates a stronger binding ability, meaning that hemoglobin holds oxygen more tightly.
Factors such as pH levels, temperature, and the presence of other molecules can influence oxygen binding capacity. For instance, fetal hemoglobin typically has a higher affinity for oxygen than adult hemoglobin. This characteristic ensures that the fetus receives sufficient oxygen supply while in the womb even if the mother’s blood oxygen level is lower. This unique difference is reflected in the equilibrium constant values, where fetal hemoglobin exhibits a higher \(K_c\) compared to adult hemoglobin.

Hemoglobin Saturation

Hemoglobin saturation is a term used to describe the percentage of hemoglobin molecules in the blood that are bound to oxygen at any given time. Typically expressed as a percentage, hemoglobin saturation is an important indicator of how effectively oxygen is being transported in the bloodstream.
A critical concept related to hemoglobin saturation is the \(P_{50}\) value. This value denotes the partial pressure of oxygen at which 50% of hemoglobin is saturated with oxygen. In simple terms, a lower \(P_{50}\) value indicates higher oxygen affinity, meaning the hemoglobin binds oxygen even at lower pressures.
Some factors influencing hemoglobin saturation include:

• Variation in oxygen pressure
• Differences in hemoglobin types (fetal vs. adult hemoglobin)
• Environmental conditions like altitude

Fetal hemoglobin, for example, has a \(P_{50}\) value of 19 torr, which is lower than the 26.8 torr for adult hemoglobin, demonstrating its higher affinity for oxygen.

Protein Chemistry

Protein chemistry is the study of the structure and function of proteins. Proteins are vital macromolecules, consisting of long chains of amino acids. They play critical roles in virtually all biological processes, acting both as building blocks and functional molecules.
Within this realm, hemoglobin is a well-studied protein due to its biological importance in oxygen transport. Its unique ability to reversibly bind oxygen is central to its function and is determined by its quaternary structure, which includes four subunits.
Key aspects of protein chemistry in relation to hemoglobin include:

• Understanding the protein folding and structure-function relationship
• Exploring how mutations and variations (e.g., fetal versus adult hemoglobin) affect function
• Examining the interactions between hemoglobin and small molecules such as oxygen and carbon dioxide

Protein chemistry is instrumental in elucidating how certain modifications and environmental conditions affect hemoglobin's function, offering insights into how variations in structure can lead to different oxygen-binding capacities and physiological effects.