# evaluate the hemoglobin function

Problem 10.6. Myoglobin and haemoglobin are oxygen carrying molecules in the human body.

Hemoglobin is found inside red blood cells, which flow from the lungs to the muscles through the bloodstream. Myoglobin is found in muscle cells. The function

calculates the fraction of myoglobin saturated with oxygen at a given pressure p torrs. For example, at a pressure of 1 torr, M(1) = 0.5, which means half of the myoglobin (i.e. 50%) is oxygen saturated. (Note: More precisely, you need to use something called the â??partial pressureâ?, but the distinction is not important for this problem.) Likewise, the function

calculates the fraction of hemoglobin saturated with oxygen at a given pressure p.

(a) The graphs of M(p) and H(p) are given below on the domain 0 â?¤ p â?¤ 100; which is which?

(b) If the pressure in the lungs is 100 torrs,what is the level of oxygen saturation of the hemoglobin in the lungs?

(c) The pressure in an active muscle is 20 torrs. What is the level of oxygen saturation of myoglobin in an active muscle? What is the level of hemoglobin in an active muscle?

(d) Define the efficiency of oxygen transport at a given pressure p to be M(p) â?' H(p). What is the oxygen transport efficiency at 20 torrs? At 40 torrs? At 60 torrs? Sketch the graph of M(p) â?' H(p); are there conditions under which transport efficiency is maximized (explain)?

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Problem 10.6. Myoglobin and haemoglobin are oxygen carrying molecules in the human body.

Hemoglobin is found inside red blood cells, which flow from the lungs to the muscles through the bloodstream. Myoglobin is found in muscle cells. The function

calculates the fraction of myoglobin saturated with oxygen at a given pressure p torrs. For example, at a pressure of 1 torr, M(1) = 0.5, which means half of the myoglobin (i.e. 50%) is oxygen saturated. (Note: More precisely, you need to use something called the "partial pressure", but the distinction is not important for this problem.) Likewise, the function

calculates the fraction of hemoglobin saturated with oxygen at a given pressure p.

(a) The graphs of M(p) and H(p) are given below on the domain 0 ≤ p ≤ 100; which is which?

By simply finding the value of each function when p = 20 we can determine the appropriate graph:

Myoglobin:

, while we can probably infer from this that the myoglobin function is represented by the higher graph we should calculate the value for the other function just to make sure:

This confirms our previous assumption.

(b) If the pressure in the lungs is 100 torrs,what is the level of oxygen saturation of the hemoglobin in the lungs?

We must evaluate the hemoglobin function when p = 100:

(c) The pressure in an active muscle is 20 torrs. What is the level of oxygen saturation of myoglobin in an active muscle? What is the level of hemoglobin in an active muscle?

Since we already did these calculations for part (a) this is easy to show again:

(d) Define the efficiency of oxygen transport at a given pressure p to be M(p) − H(p). What is the oxygen transport efficiency at 20 torrs? At 40 torrs? At 60 torrs? Sketch the graph of M(p) − H(p); are there conditions under which transport efficiency is maximized (explain)?

We have already calculated M(20) and H(20) so:

M(20) - H(20) = 0.952 - 0.324 = 0.628

The following table shows the rest of the calculations:

p M(p) H(p) M(p) - H(p)

20 0.952381 0.324182 0.628199219

40 0.97561 0.769625 0.205985253

60 0.983607 0.912256 0.071350774

The graph of the function M(p) - H(p) is:

And the graph with the other two functions included:

From this it can be seen that there is a maximum value of transport efficiency. It appears to occur when p is approximately 8. This is the point at which the difference in the two functions is the greatest. We may be able to find a more exact value but that would require some higher calculus.

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