Cryotosporidium pervum sporozoites are prolate spheres approximately 4 um long and 1 um wide. The average density of the cell is 1.28 g/mL. Approximately 7% of the wet weight of the organism is soluble protein. Estimate the number of molecules of a particular enzyme per cell if the enzyme has a mW of 80,000 Daltons and represents 0.3% of the total soluble protein.
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To estimate the number of molecules of a particular enzyme per cell, we need to follow a step-by-step calculation:
1. Calculate the volume of an individual Cryotosporidium pervum sporozoite cell:
- The cell shape is described as a prolate sphere, so we can approximate it as an ellipsoid.
- The length of the cell is given as 4 um, and the width is given as 1 um.
- The volume (V) of an ellipsoid can be calculated using the formula:
V = (4/3) * π * a * b^2,
where a and b are the semi-major and semi-minor axes respectively.
- For the given prolate sphere, a = 2 um (half the length) and b = 0.5 um (half the width).
- Substituting the values into the formula:
V = (4/3) * π * 2 * 0.5^2 = (4/3) * π * 2 * 0.25 = (4/3) * 0.5 * π um^3.
2. Convert the volume of the cell to mass using the given density:
- The average density of the cell is given as 1.28 g/mL.
- As volume (V) is in cubic micrometers (um^3), we need to convert it to milliliters (mL):
1 mL = 1,000,000 um^3.
- So, V = (4/3) * 0.5 * π * (1/1,000,000) mL.
- Multiply the volume by the density to obtain the mass:
Mass = V * Density = [(4/3) * 0.5 * π * (1/1,000,000)] * 1.28 g.
3. Calculate the wet weight of the entire organism:
- The wet weight of the organism is not given directly, but it is mentioned that 7% of the wet weight is soluble protein.
- Let WW be the wet weight of the organism in grams.
- Soluble protein weight (SPW) = 0.07 * WW grams.
4. Calculate the weight of the specific enzyme:
- The enzyme represents 0.3% of the total soluble protein.
- Let EW be the weight (mass) of the enzyme in grams.
- EW = 0.003 * SPW grams.
5. Convert the enzyme mass to the number of molecules:
- The molecular weight (mW) of the enzyme is given as 80,000 Daltons.
- The number of molecules (N) can be calculated by dividing the enzyme mass by the molecular weight:
N = EW / mW.
By following these calculations, you should be able to estimate the number of molecules of the particular enzyme per cell.