sources/read-counts-expansion.md

Accounting for sequencing subsampling per sample

Each sequencing sample $s$ could be over- or under-sampling the population relative to the first timepoint by some factor $\phi_s$.

$$\log \frac{c_i(t)}{c_i(0)}=\log {\varphi }_s\frac{n_i(t)}{n_i(0)}=\log {\varphi }_s+\frac{w_i}{w_{wt}}\log \frac{n_{wt}(t)}{n_{wt}(0)}\backslash log\; \backslash frac\{c\_i(t)\}\{c\_i(0)\}\; =\; \backslash log\; \backslash phi\_s\backslash frac\{n\_i(t)\}\{n\_i(0)\}\; =\; \backslash log\; \backslash phi\_s\; +\; \backslash frac\{w\_i\}\{w\_\{wt\}\}\; \backslash log\; \backslash frac\{n\_\{wt\}(t)\}\{n\_\{wt\}(0)\}$$

Variables:

• $c_i(t)$: Read (or UMI) count of strain $i$ at time $t$
• $\phi_s$: The ratio of sampling depth at time $t$ to that at time $0$ for sample $s$

The factor $\phi_s$ is the ratio of the ratio of read counts between samples and the ratio of cell counts between samples for any strain (assuming each strain is sampled without bias):

$$\log {\varphi }_s=\log \frac{c_i(t)}{c_i(0)}-\log \frac{n_i(0)}{n_i(0)}\backslash log\; \backslash phi\_s\; =\; \backslash log\; \backslash frac\{c\_i(t)\}\{c\_i(0)\}\; -\; \backslash log\; \backslash frac\{n\_i(0)\}\{n\_i(0)\}$$

We can get rid of the nuisance parameter $\phi_s$ (which is difficult to measure becuase we don't know the true number of cells for each strain and sample) using the following trick.

We have the equation for read counts for mutant $i$ (same as above):

$$\log \frac{c_i(t)}{c_i(0)}=\log {\varphi }_s+\frac{w_i}{w_{wt}}\log \frac{n_{wt}(t)}{n_{wt}(0)}\backslash log\; \backslash frac\{c\_i(t)\}\{c\_i(0)\}\; =\; \backslash log\; \backslash phi\_s\; +\; \backslash frac\{w\_i\}\{w\_\{wt\}\}\; \backslash log\; \backslash frac\{n\_\{wt\}(t)\}\{n\_\{wt\}(0)\}$$

And for the reference strain (relative fitness is 1):

$$\log \frac{c_{wt}(t)}{c_{wt}(0)}=\log {\varphi }_s+\log \frac{n_{wt}(t)}{n_{wt}(0)}\backslash log\; \backslash frac\{c\_\{wt\}(t)\}\{c\_\{wt\}(0)\}\; =\; \backslash log\; \backslash phi\_s\; +\; \backslash log\; \backslash frac\{n\_\{wt\}(t)\}\{n\_\{wt\}(0)\}$$

We can make $\phi_s$ disappear by taking the difference:

$$\log \frac{c_i(t)}{c_i(0)}-\log \frac{c_{wt}(t)}{c_{wt}(0)}=\frac{w_i}{w_{wt}}\log \frac{n_{wt}(t)}{n_{wt}(0)}-\log \frac{n_{wt}(t)}{n_{wt}(0)}\backslash log\; \backslash frac\{c\_i(t)\}\{c\_i(0)\}\; -\; \backslash log\; \backslash frac\{c\_\{wt\}(t)\}\{c\_\{wt\}(0)\}\; =\; \backslash frac\{w\_i\}\{w\_\{wt\}\}\; \backslash log\; \backslash frac\{n\_\{wt\}(t)\}\{n\_\{wt\}(0)\}\; -\; \backslash log\; \backslash frac\{n\_\{wt\}(t)\}\{n\_\{wt\}(0)\}$$

This is equivalent to:

$$\log \left(\frac{c_i(t)}{c_{wt}(t)}\frac{c_{wt}(0)}{c_i(0)}\right)=(\frac{w_i}{w_{wt}}-1)\log \frac{n_{wt}(t)}{n_{wt}(0)}\backslash log\; \backslash left(\; \backslash frac\{c\_i(t)\}\{c\_\{wt\}(t)\}\backslash frac\{c\_\{wt\}(0)\}\{c\_i(0)\}\; \backslash right)\; =\; \backslash left(\backslash frac\{w\_i\}\{w\_\{wt\}\}\; -\; 1\; \backslash right)\; \backslash log\; \backslash frac\{n\_\{wt\}(t)\}\{n\_\{wt\}(0)\}$$

So the ratio of the count ratio of a strain to the reference strain at time t to the count ratio of a strain to the reference strain at time 0 is dependent only on the relative fitness and the true fold-expansion of the reference strain.

Plotting the ratio of the count ratio of a strain to the reference strain at time t to the count ratio of a strain to the reference strain at time 0 should give a straight line (on a log-log) plot, with intercept 0 and gradient equal to the relative fitness minus 1.