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 oxygen-binding capacity, biochemists use a measure called the P50 value, defined as the partial pressure of oxygen at which \(50 \%\) of the protein is saturated. Fetal hemoglobin has a P50 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 \(4 \mathrm{O}_{2}(g)+\mathrm{Hb}(a q) \longrightarrow\) \(\left[\mathrm{Hb}\left(\mathrm{O}_{2}\right)_{4}(a q)\right] .\)

Short answer

The equilibrium constant \(K_c\) is about \(1.41\) times larger for the aqueous oxygen-binding reaction in fetal hemoglobin compared to adult hemoglobin.

Step-by-step solution

Identify the relationship between the equilibrium constant \(K_c\), P50, and the partial pressure of oxygen \(({\rm P_{O_2}})\), which is as follows: \(50\% = \frac{K_c \cdot ({\rm P_{O_2}})}{1 + K_c \cdot ({\rm P_{O_2}})}\) We will use this formula to calculate the equilibrium constants for both fetal and adult hemoglobin. #Step 2: Calculate the equilibrium constant for fetal hemoglobin#

Substitute the P50 value of fetal hemoglobin (19 torr) into the formula: \(0.5 = \frac{K_{c,f} \cdot 19}{1 + K_{c,f} \cdot 19}\) Solve for \(K_{c,f}\): \(K_{c,f} = \frac{0.5}{19 - 0.5 \cdot 19} = \frac{0.5}{9.5}\) \(K_{c,f} = 0.0526\) #Step 3: Calculate the equilibrium constant for adult hemoglobin#

Substitute the P50 value of adult hemoglobin (\(26.8\) torr) into the formula: \(0.5 = \frac{K_{c,a} \cdot 26.8}{1 + K_{c,a} \cdot 26.8}\) Solve for \(K_{c,a}\): \(K_{c,a} = \frac{0.5}{26.8 - 0.5 \cdot 26.8} = \frac{0.5}{13.4}\) \(K_{c,a} = 0.0373\) #Step 4: Calculate the ratio of equilibrium constants#

Find the ratio between the equilibrium constants of fetal and adult hemoglobin: \(\frac{K_{c,f}}{K_{c,a}} = \frac{0.0526}{0.0373}\) \(\frac{K_{c,f}}{K_{c,a}} \approx 1.41\) So, the equilibrium constant \(K_c\) is about \(1.41\) times larger for the aqueous oxygen-binding reaction in fetal hemoglobin compared to adult hemoglobin.

Key concepts

Equilibrium Constant

The equilibrium constant, symbolized as \(K_c\), is crucial in understanding how easily a reaction occurs at a constant temperature. For hemoglobin, the reaction of interest is its ability to bind oxygen.
In simple terms, \(K_c\) tells us how favorably hemoglobin can bind with oxygen to form oxyhemoglobin. A higher \(K_c\) value signals a greater affinity, meaning hemoglobin binds more readily to oxygen. This is particularly useful when comparing different types of hemoglobin, like fetal and adult hemoglobin.
Here's how the equilibrium constant ties into hemoglobin:

• Hemoglobin's \(K_c\) for oxygen binding demonstrates how effectively it can capture oxygen from the environment.
• When calculating \(K_c\), we use the P50 value, which represents the partial pressure of oxygen at which hemoglobin is 50% saturated.
• Fetal hemoglobin has a higher \(K_c\) compared to adult hemoglobin, illustrating its stronger ability to bind oxygen under the same conditions.

Understanding \(K_c\) provides a deeper insight into hemoglobin's performance in oxygen transport.

Oxygen-Binding Capacity

Oxygen-binding capacity refers to the amount of oxygen hemoglobin can hold. This is a vital feature, especially since hemoglobin's main role is to transport oxygen in the blood.
The ability of hemoglobin to bind oxygen is influenced by several factors, including the structure of the hemoglobin and the surrounding environment. Here’s a closer look:

• Each hemoglobin molecule has four sites where oxygen molecules can attach.
• These binding sites work cooperatively. This means that once one oxygen molecule binds, it becomes easier for the next one to attach.
• Fetal hemoglobin is specifically designed to have a higher oxygen-binding capacity. This ensures that the fetus can effectively capture oxygen from the mother’s blood.

This competitive binding ability makes hemoglobin remarkably efficient at transporting oxygen to where it's needed the most in the body.

P50 Value

The P50 value is a measure used to describe the efficiency of hemoglobin’s ability to bind oxygen. Specifically, it indicates the partial pressure of oxygen at which half of the hemoglobin's capacity is saturated with oxygen.
This metric is incredibly valuable as it provides insight into hemoglobin’s affinity for oxygen under varying conditions.
Let's break it down:

• A lower P50 value means higher affinity, as hemoglobin reaches half-saturation with less oxygen.
• Fetal hemoglobin has a lower P50 value (19 torr) compared to adult hemoglobin (26.8 torr), reflecting its superior ability to capture oxygen from maternal blood.
• The difference in P50 values explains why fetal hemoglobin is more effective in oxygen uptake under the same conditions.

By examining the P50 value, researchers and students gain a better understanding of how hemoglobin behaves in biological and medical contexts.