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Problem session – T. Hwa
1. Simultaneous and hierarchical usage of carbon sources.
E. coli cells grow with growth rate 𝜆1 in a medium with saturating concentration of the
substrate C1 as the sole carbon source, and with growth rate 𝜆2 in the same medium
except that C2 is the sole carbon source (and in saturating concentration).
a. Suppose the uptake of each substrate C1 and C2 is regulated only by “catabolite
repression” via the cAMP signaling system described in the lecture. Can you write down
a formula for the growth rate 𝜆12 that the cells can attain in a medium with saturating
concentrations of both substrates C1 and C2?
[Recall that for an enzyme E such as transporter under the control of “catabolite
repression”, its concentration [E] is given by [𝐸] = 𝐸0 ∙ (𝜆 − 𝜆0 ), where 𝜆 is the growth
rate, 𝐸0 is an enzyme specific constant, 𝜆0 is a strain-specific constant. Also, given
saturating concentration of the substrate, the carbon influx carried by E is proportional to
[E]. You may also assume that the carbon influx is proportional to the growth rate 𝜆.]
b. It is desired to have cells consume only the carbon substrate supporting faster growth
when multiple carbon sources are present. Try to come up with a regulatory strategy to
implement this behavior.
2. Density-dependent motility.
A mutant strain of E. coli executes a random walk by swimming and tumbling as wildtype E. coli cells do, but only when density is below a threshold, 𝜌𝑐 . Above this density,
the cells hardly swims and can be regarded as immotile. Cell density is signaled by a
small molecule (generically called a “quorum sensor”), which is secreted by the cell, has
a short life time, and can reenter the cell at sufficiently high density (to turn off motility).
One drop of such cells is spotted in the middle of a soft agar plate rich in nutrient.
a. What do you expect to happen qualitatively to the population of these cells?
[You may assume the initial density to be much below 𝜌𝑐 and the cells can swim happily
in the soft agar.]
b. Write down a minimum set of differential equations to describe the dynamics of this
system.