Question

Mutations of bacteria in our gut (a) The populations of the \(E\). coli in the guts of a collection of humans can be large enough that multiple mutations can occur simultaneously in one bacterium. Suppose that a very particular combination of \(k\) point mutations is required for a pathogenic strain to emerge and that these must all arise in one cell division (as could be the case if the subsets of these mutations are deleterious). With the point mutation rate per base pair per cell division of \(\mu,\) what is the probability \(m_{k}\) that this occurs in a single cell division? The simplest assumption is that the probabilities of the different mutations are independent. (b) In a human large intestine, the density of bacteria is estimated to be about \(10^{11.5}\) per milliliter, of which a fraction of about \(10^{-4}\) are \(E\) coll. Estimate how many \(E\) coli per person this implies. In a population of \(N\) humans, with \(n\) \(E\) coli in each of their guts, in \(T\) generations of the \(E\). coli estimate the total probability \(P_{k}\) that the particular combination of \(k\) mutations occurs at least once. (c) With the population of Silicon Valley over one year, what are the chances this occurs for \(k=2 ?\) For \(k=3 ?\) Some crucial factors in your estimate are \(\mu \approx 10^{-10}-10^{-9}\) mutations per base pair per cell division and the generation time of \(\bar{E}\). colt. the standard lab result is that \(E\). coll divide every 20 minutes. A low-end estimate for the division rate of \(E\). coli in human guts is about once every few days. Why is this more realistic? Given these and other uncertainties, how big are the uncertainties in your estimates of \(P_{2}\) and \(P_{3} ?\) (Problem courtesy of Daniel Fisher.)

Short answer

Probability depends on the number of simultaneous mutations, population size of bacteria, and the number of generations. The uncertainties in this calculation could be due to variable mutation rates, fluctuating E.coli population sizes, and assuming mutations are independent.

Step-by-step solution

Determine the Probability for Single Cell Division

We are trying to find the probability of 'k' mutations happening in one cell division. Given the point mutation rate per base pair per cell division as '\(\mu\)' the probability '\(m_{k}\)' should be calculated as: \(\mu^{k}\), where '\(\mu\)' is multiplied by itself 'k' times. This because it is a combination of 'k' independent mutations.

Estimation of E.coli Population

With bacterial density to be about \(10^{11.5}\) per milliliter and only a fraction of about \(10^{-4}\) are E.coli. Therefore, the total population of E.coli would be: \(10^{11.5} \times 10^{-4} = 10^{7.5}\) E.coli per milliliter.

Estimating the Probability of k Mutations

The total number of 'n' E.coli in each of the 'N' human guts in 'T' generations would be 'n' times 'N' times 'T'. Now, the total probability \(P_{k}\) that the particular combination of 'k' mutations occurs at least once can be calculated as: \(P_{k} = 1 - (1 - \mu^{k})^{n \times N \times T}\).

Calculating Chances for a Given Population and k=2 and k=3

Substitute given values and calculate the possibility for \(k=2\) and \(k=3\) given the factors of mutation and division rate.

Calculating Uncertainties

With these estimations there could be many sources of uncertainties. This could be related to mutation rates, under or over estimation of E.coli population, assuming independence of mutations, etc. These uncertainties would create a range on the probability values calculated.

Key concepts

E. coli Population

E. coli bacteria are a significant part of the microbial life within the human gut. Though they are a fraction of the total bacteria present, they can still reach significant numbers.
The human large intestine is estimated to host about \(10^{11.5}\) bacteria per milliliter. Since only a fraction of these, roughly \(10^{-4}\), are E. coli, this translates to approximately \(10^{7.5}\) E. coli bacteria per milliliter.
This estimate helps scientists understand and predict the behaviors and developments, such as mutations, that occur in the E. coli population within a human host.

Mutation Probability

Calculating the probability of mutations is crucial for understanding bacterial evolution and the development of drug resistance.
A point mutation refers to a single base change in the DNA sequence, and its occurrence is a random event with a very low probability per base pair per cell division, denoted as \(\mu\).
When estimating the likelihood of 'k' mutations happening simultaneously in a single cell division, the formula used is \(\mu^{k}\), where \(k\) is the number of specific mutations required. This represents independent and rare mutation events occurring together during one division cycle.

Independent Mutations

The assumption of independent mutations simplifies the calculation of complex genetic events like multiple point mutations required for a mutation to be significant.
Each mutation occurs independently of others, meaning the probability of multiple mutations is the product of their individual probabilities.
This can be mathematically represented as \(\mu^{k}\), emphasizing that each of the \(k\) mutations has an equal chance of occurring without any influence from the occurrence of another mutation.
This independence is a theoretical assumption but provides a basis for calculating and predicting mutation probabilities in populations.

Generation Time of E. coli

Generation time is a measure of how quickly a population of E. coli can double in size. In laboratory settings, E. coli typically divides every 20 minutes, making it one of the fastest replicating organisms.
However, in the human gut, this rate is much slower, with estimates suggesting division occurs once every few days. This is more realistic due to less optimal growth conditions, competition for resources, and the presence of other microbial populations.
The slower generation time in vivo impacts the overall chances of mutations building up, thereby influencing genetic diversity and evolution in natural settings.