Problem 5. Protein Concentration in Cells

Question

Consider a growing bacterial cell that is producing a protein involved in binary fission. Suppose that, at a particular point in time, a bacterial cell has a volume of 1000 μm3 (microns cubed) and is growing at a rate of 5 μm3/min. Furthermore, suppose that there are 1000000 molecule of the protein in the cell, and the protein is accumulating at a rate of 10000 molecules/min.
Find the concentration of the protein in the cell and the rate at which the con- centration of protein is changing within the cell (include units). Is the concentra- tion increasing or decreasing? Thoroughly justify your answer with the appropriate derivative rule.

I think that we have to find the derivative of the exponential growth equation A=Pe^(rt), but i am not sure please help?

Answer 1

There is no exponential growth here. The population and volume are not growing by a fixed percentage, but just as a linear amount.

the concentration is measured in molecules/μm^3, so at t minutes later,

c = (10^6 + 10^4t)/(1000+5t)
= 2000(100+t)/(200+t)
= 2000(1 - 100/(200+t))

so, find dc/dt; you will see that the concentration is increasing