Our project used a method called MSS-Maximum Likelihood Estimator Method (MSS-MLE) to calculate the mutation rate in cells, as implemented by FALCOR.

This method starts with the idea that the log of m, the mutation rate, is normally distributed. Using that logic, by guessing a value m, the pdf of exactly r mutants is

p_r = {
  r = 0 -> e^(-m)
  r > 0 -> m / r * sum(i, 0, r - 1, p_i / (r - i + 1))
}

Then, the likelihood of the observed results can be calculated by finding the product of each plate's probability to have that many mutants. By selecting different values of m, and searching until finding the one that gives the highest likelihood of observation, the most likely value of m is found.

Since the values of ln(m) are normally distributed, it can be statistically treated like any other normally distributed random variable, so it can be used in t-tests and the like

In our tests, we had a low plating efficiency, meaning that we did not plate all the cells we grew to check if they were mutants on a single plate. To account for that, given that z is the proportion of the culture plated, we say that the actual value of m is

 m * (z - 1) / (z * ln(z)) 

due how the proportion of mutants is not normally distributed, so splitting up the cells does not linearly correlate.

Calculation Results

We had 4 or so runs of the flux analysis protocol.

The first one was more or less successful for one biobrick and gave us data for IPTG+GFP+EmrR using the falcor calculator (MSS-MLE method):
For EmrR+IPTG+GFP we got 6.723 mutational events (corrected from 1.172 which reflects 10% of the culture); dividing through by the average cells per culture ascertained by the test plate counts 595000000 gave us a mutation rate of 1.13E-08 mutational events/per cell per culture. Doing the same for the pure wt control we got 5.617 mutational events (from 1.437) and 2.84E-09 mutation rate (dividing through by 1975000000).

Note: the (statistically significant) difference in true cell counts might prevent us from comparing the two mutation rates as cell density can effect endogenous mutation rate (some quorum sensing thing - http://www.nature.com/ncomms/2014/140429/ncomms4742/full/ncomms4742.html). also the control should have contained the iptg +gfp plasmid.

The next 2 attempts were characterized by failure (1 of 2: contamination, 2 of 2: I left and there was misreading of protocol). The 4th attempt resulted in not great data as well. Although there was some usable readings for the selective plates Ryan S plated the wrong dilution on the test plates (10^2 instead of 10^6) so getting an accurate cell count from them was not possible. We could use the OD600 reading taken prior to plating, but that also likely does not reflect the accurate amount of cells per culture (OD600 readings give total biomass and online calculators are likely in tune with healthy wild-type e coli growth/death rates). The only strains that ryan collected usable selective plate data/ an OD600 reading for were 2 IPTG+GFP+mutH strains. Using falcor and rough estimates for cell count we get mutation rates of 4.53E-09 and 3.30E-09 for each strain.

EmrR was inconclusive.