Question

Saturation of myoglobin. Suppose that \(\mathrm{O}_{2}\) molecules bind to myoglobin with association constant \(K=2 \mathrm{Torr}^{-1}\) at \(25^{\circ} \mathrm{C}\) and \(\mathrm{pH}\) 7.4. (a) Show a table of the fractional saturation of myoglobin for pressures of \(1,2,4,8\), and 16 Torr \(\mathrm{O}_{2}\). (b) Does the fractional saturation double for each doubling of the pressure?

Short answer

The fractional saturation does not double for each doubling of the pressure. The calculated values show incremental increases but not a doubling.

Step-by-step solution

- Understand the given parameters

The association constant (K) is given as 2 Torr^{-1} at 25°C and pH 7.4. This means that the binding affinity of O2 to myoglobin is known. The formula for the fractional saturation of myoglobin (\theta) can be expressed as \[\theta = \frac{K \times p(O_2)}{1 + K \times p(O_2)} \]

- Calculate the fractional saturation for each pressure

Substitute the given pressures into the formula to calculate the fractional saturation (\theta) for each pressure.For 1 Torr: \[\theta = \frac{2 \times 1}{1 + 2 \times 1} = \frac{2}{3} \approx 0.67 \]For 2 Torr: \[\theta = \frac{2 \times 2}{1 + 2 \times 2} = \frac{4}{5} = 0.80 \]For 4 Torr: \[\theta = \frac{2 \times 4}{1 + 2 \times 4} = \frac{8}{9} \approx 0.89 \]For 8 Torr: \[\theta = \frac{2 \times 8}{1 + 2 \times 8} = \frac{16}{17} \approx 0.94 \]For 16 Torr: \[\theta = \frac{2 \times 16}{1 + 2 \times 16} = \frac{32}{33} \approx 0.97 \]

- Tabulate the results

Create a table displaying the calculated fractional saturations for each pressure.\[\begin{array}{|c|c|}\hline\text{Pressure (Torr)} & \text{Fractional Saturation} \ \hline1 & 0.67 \ \hline2 & 0.80 \ \hline4 & 0.89 \ \hline8 & 0.94 \ \hline16 & 0.97 \ \hline\end{array}\]

- Analyze saturation doubling

Evaluate if the fractional saturation doubles with the doubling of pressure:Compare the values between consecutive pressures from the table:For example,From 1 to 2 Torr: 0.67 to 0.80 (not doubled)From 2 to 4 Torr: 0.80 to 0.89 (not doubled)Continue for other values (8, 16 Torr).It's evident from the results that fractional saturation does not double with each doubling of pressure.

Key concepts

Association Constant

When studying biochemical reactions, an association constant (K) is a crucial parameter. It measures the affinity between two binding partners—in this case, myoglobin and oxygen (O₂). The association constant helps us understand how readily myoglobin binds to oxygen, represented by the equation: \[ K = \frac{[MbO_2]}{[Mb][O_2]} \]
Here, \[ [MbO_2] \] is the concentration of oxygen-bound myoglobin, \[ [Mb] \] is the concentration of free myoglobin, and \[ [O_2] \] is the concentration of free oxygen. A higher K value indicates a higher affinity, meaning myoglobin will bind oxygen more readily. In our problem, K is given as 2 Torr⁻¹, showing a relatively high affinity.

Fractional Saturation

Fractional saturation (θ) tells us the proportion of myoglobin molecules bound to oxygen at any given pressure. Essentially, it's a measure of how 'saturated' myoglobin is with oxygen. The mathematical relationship is:

\[ \theta = \frac{K \times p(O_2)}{1 + K \times p(O_2)} \]
Here, \[ p(O_2) \] is the partial pressure of oxygen. This equation allows us to calculate \[ \theta \] at various oxygen pressures. Using pressures like 1, 2, 4, 8, and 16 Torr, we can observe that the fractional saturation increases but does not double as the pressure doubles. For example, at 1 Torr the saturation is approximately 0.67, whereas, at 2 Torr, it is around 0.80, not quite doubled. This illustrates the non-linear relationship due to the saturation dynamics.

Oxygen Binding

Oxygen binding to myoglobin is an essential biological process, significant in muscle tissue where myoglobin resides. Myoglobin acts as an oxygen reservoir and assists in the transportation of oxygen within muscles. The binding can be influenced by factors such as pressure, temperature, and pH.

As demonstrated in the exercise, the binding affinity remains constant under given conditions (25°C and pH 7.4), and the oxygen binding behavior can be depicted through the association constant (\[ K=2 \text{Torr}^{-1} \]). When oxygen binds to myoglobin, it forms oxymyoglobin (MbO₂). The efficiency of this process is quantified through fractional saturation, which shows the proportion of myoglobin molecules that have oxygen bound to them at different pressures.

Thermodynamics

Thermodynamics plays a vital role in understanding the binding and saturation of myoglobin with oxygen. Thermodynamic principles explain how energy changes during the binding process.

The Gibbs free energy (\[ \bigtriangleup G \]) determines the spontaneity of the binding reaction. The relationship between the association constant and Gibbs free energy is given by:

\[ \bigtriangleup G = -RT \times \text{ln}(K) \]

Where \[ R \] is the gas constant and \[ T \] is the temperature in Kelvin. For our problem, knowing \[ K \] allows us to understand the energy changes that occur when myoglobin binds to oxygen. A large \[ K \] indicates a more negative \[ \bigtriangleup G \], signifying a spontaneous and energetically favorable binding process. Understanding these principles helps unravel the bioenergetics of oxygen transport and storage in muscle tissues.