Suppose there are two kinds of E. coli (bacteria), "red"...

Suppose there are two kinds of E. coli (bacteria), "red" ones and "green" ones. Each reproduces faithfully (no sex) by splitting into half, red $\rightarrow$ red + red or green $\rightarrow$ green + green, with a reproduction time of 1 hour. Other than the markers "red" and "green", there are no differences between them. A colony of $5,000{ }^{\text {"red" }}$ and 5,000 "green" E. coli is allowed to eat and reproduce. In order to keep the colony size down, a predator is introduced which keeps the colony size at 10,000 by eating (at random) bacteria. (a) After a very long time, what is the probability distribution of the number of red bacteria? (b) About how long must one wait for this answer to be true? (c) What would be the effect of a $1 \%$ preference of the predator for eating red bacteria on (a) and (b)?

Suppose there are two kinds of E. coli (bacteria), "red" ones and "green" ones. Each reproduces faithfully (no sex) by splitting into half, red $\rightarrow$ red + red or green $\rightarrow$ green + green, with a reproduction time of 1 hour. Other than the markers "red" and "green", there are no differences between them. A colony of $5,000{ }^{\text {"red" }}$ and 5,000 "green" E. coli is allowed to eat and reproduce. In order to keep the colony size down, a predator is introduced which keeps the colony size at 10,000 by eating (at random) bacteria.
(a) After a very long time, what is the probability distribution of the number of red bacteria?
(b) About how long must one wait for this answer to be true?
(c) What would be the effect of a $1 \%$ preference of the predator for eating red bacteria on (a) and (b)?

Key Concepts

Genetic Drift

Genetic drift refers to the random fluctuations in the frequencies of types (or alleles) within a finite population due to random sampling effects. In processes where reproduction and death occur randomly, small differences can eventually lead to one type disappearing entirely even if there is no selective advantage.

Moran Process

The Moran process is a model used in population genetics to study the stochastic evolution of allele frequencies in a fixed-size population. In this process, one individual reproduces and one (possibly different) individual is randomly removed, keeping the population size constant while allowing chance fluctuations to drive changes in type frequencies.

Fixation

Fixation is the outcome in a finite population when one allele (or type) comes to completely dominate, leading to the extinction of all other competing alleles. Under neutral drift, fixation occurs purely by chance and, given enough time, the probability of eventual fixation is determined by the initial frequency of the allele.

Selection (Biased Predation)

Selection in this context refers to any systematic bias that makes one type more likely to be affected by a random event, such as predation. A preference by the predator for one type introduces non?neutral dynamics, shifting the fixation probabilities and potentially altering the time scale of the process.

Time to Fixation

Time to fixation is a measure of how long it takes for random drift (or drift coupled with selection) to result in the complete dominance of one type in the population. In neutral processes like the Moran model, this time is typically proportional to the population size, with selective biases potentially speeding up the process by disproportionately driving one type to extinction.

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