Question

The protein hemoglobin (Hb) transports \(\mathrm{O}_{2}\) in mammalian blood. Each 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 \(P 50\) 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 \(2.53 \mathrm{kPa},\) and adult hemoglobin has a P50 value of \(3.57 \mathrm{kPa}\). Use these data to estimate how much larger \(K_{c}\) is for the aqueous reaction \(4 \mathrm{O}_{2}(g)+\mathrm{Hb}(a q) \rightleftharpoons\left[\mathrm{Hb}\left(\mathrm{O}_{2}\right)_{4}(a q)\right]\) in a fetus, compared to \(K_{c}\) for the same reaction in an adult.

Short answer

The Kc value for the aqueous reaction of O2 binding to Hb in a fetus is approximately 2.977 times larger than the Kc value for the same reaction in an adult. This indicates that fetal hemoglobin has a higher affinity for oxygen compared to adult hemoglobin.

Step-by-step solution

Write the reaction equation

The reaction of O2 binding to Hb can be written as: \[ 4\, O_2(g) + Hb(aq) \rightleftharpoons [Hb(O_2)_4](aq) \]

Express the reaction in terms of equilibrium constant Kc

The Kc of this reaction can be written as: \[ K_c = \frac{[Hb(O_2)_4]}{[Hb] [O_2]^4} \]

Utilize the P50 values at 50% saturation

At 50% saturation, the concentrations of Hb and [Hb(O2)4] are equal. Therefore, we can write the Kc expressions for fetal and adult hemoglobin as: - Fetal hemoglobin: \[ K_{c,f} = \frac{[Hb(O_2)_4]}{[Hb] \times (2.53\,kPa)^4} \] - Adult hemoglobin: \[ K_{c,a} = \frac{[Hb(O_2)_4]}{[Hb] \times (3.57\,kPa)^4} \]

Compare Kc values for fetal and adult hemoglobin

To compare the Kc values for fetal and adult hemoglobin, we will take the ratio of the two Kc expressions. Cancel out equal terms (in this case [Hb(O2)4]) in both numerator and denominator, and then simplify: \[ \frac{K_{c,f}}{K_{c,a}} = \frac{[Hb] \times (3.57\,kPa)^4}{[Hb] \times (2.53\,kPa)^4} \] Since the [Hb] terms are the same in both expressions, they cancel out, and the ratio simplifies to: \[ \frac{K_{c,f}}{K_{c,a}} = \frac{(3.57\,kPa)^4}{(2.53\,kPa)^4} \]

Calculate the numerical value of the Kc ratio

Evaluate the expression to determine the numerical value of the Kc ratio: \[ \frac{K_{c,f}}{K_{c,a}} = \frac{(3.57)^4}{(2.53)^4} \approx 2.977 \] #Conclusion# The Kc value for the aqueous reaction of O2 binding to Hb in a fetus is approximately 2.977 times larger than the Kc value for the same reaction in an adult. This indicates that fetal hemoglobin has a higher affinity for oxygen compared to adult hemoglobin.

Key concepts

Protein Oxygen-Binding Capacity

Understanding the protein oxygen-binding capacity is key to grasping how hemoglobin functions in the body. Hemoglobin (Hb) is a protein in red blood cells tasked with transporting oxygen from the lungs to various tissues throughout the body. Each hemoglobin molecule can bind to four oxygen molecules, making it highly efficient in oxygen transport. This binding is represented by an equilibrium constant, denoted as \( K_c \), which reflects how easily hemoglobin can pick up and release oxygen under specific conditions.

The measurement frequently used by biochemists to evaluate protein oxygen-binding capability is the \( P_{50} \) value. This value defines the partial pressure of oxygen when 50% of hemoglobin is saturated. A lower \( P_{50} \) value means hemoglobin has a higher affinity for oxygen, as it requires a lower oxygen pressure to achieve the same level of saturation. Conversely, a higher \( P_{50} \) value implies that the protein holds oxygen less tightly, indicative of a lower affinity.

In biological scenarios, understanding the balance of oxygen binding and release is crucial for ensuring that tissues receive an adequate supply of oxygen, which is essential for metabolism and energy production.

Fetal Hemoglobin

Fetal hemoglobin (HbF) plays a vital role during pregnancy, ensuring that the developing fetus can efficiently draw oxygen from the mother's blood. Unlike adult hemoglobin, fetal hemoglobin has a different structure that gives it a higher affinity for oxygen. This heightened affinity is reflected in its lower \( P_{50} \) value of 2.53 kPa, compared to adult hemoglobin. This means that fetal hemoglobin can become more saturated with oxygen at lower oxygen pressures found in the placenta.

The enhanced oxygen-binding capacity of fetal hemoglobin facilitates oxygen transfer from the mother to the fetus, effectively catering to the metabolic demands of the rapidly growing fetus. In the biochemical reaction \( 4 \, O_2(g) + Hb(aq) \rightleftharpoons [Hb(O_2)_4](aq) \), the fetal hemoglobin shows a larger equilibrium constant \( K_{c,f} \) than that of adult hemoglobin. This higher \( K_{c} \) is quantitatively supported by the calculated ratio of 2.977, signifying that fetal hemoglobin binds oxygen approximately three times more effectively than its adult counterpart.

Adult Hemoglobin

Adult hemoglobin (HbA) is the predominant form of hemoglobin postnatally. It has a specific role in oxygen transport throughout the human body. Unlike fetal hemoglobin, adult hemoglobin has a higher \( P_{50} \) value of 3.57 kPa. This indicates a moderate oxygen affinity, designed to efficiently release oxygen to the tissues where it is most needed.

The interaction of oxygen with adult hemoglobin is a carefully balanced process. Due to this balance, hemoglobin efficiently picks up oxygen in high concentrations (as found in the lungs) and releases it in lower concentrations where it is needed (such as in peripheral tissues). In scenarios where more oxygen is required – such as during exercise or in high altitudes – the affinity adjustments in adult hemoglobin allow for effective distribution of oxygen. This adaptability makes adult hemoglobin crucial for responding to various physiological demands.

The lower equilibrium constant \( K_{c,a} \) of adult hemoglobin compared to fetal hemoglobin ensures that once a person is born, the hemoglobin can efficiently deliver oxygen, ensuring all body systems work optimally. By comparing the equilibrium constants, it's clear that adult hemoglobin's lower affinity allows better oxygen release, meeting adult metabolic needs.