sources/read-counts-stationary.md

But we don't actually know the true fold-expansion of the reference strain, since it's not directly observed.

But we do know some things about it. For example, when the pool is far from carrying capacity, the fold-expansion will be close to exponential:

$$\log \frac{n_{wt}(t)}{n_{wt}(0)}=w_{wt}t\backslash log\; \backslash frac\{n\_\{wt\}(t)\}\{n\_\{wt\}(0)\}\; =\; w\_\{wt\}\; t$$

And at the carrying capacity, the fold-expansion stops changing with time, arrested at the carrying capacity minus the other cells in the pool:

$$\log \frac{n_{wt}(t)}{n_{wt}(0)}=\log \frac{K-{\mathrm{\Sigma }}_j{n}_j}{n_{wt}(0)}\backslash log\; \backslash frac\{n\_\{wt\}(t)\}\{n\_\{wt\}(0)\}\; =\; \backslash log\; \backslash frac\{K\; -\; \backslash Sigma\_j\; n\_j\}\{n\_\{wt\}(0)\}$$

So at very early timepoints, long before carrying capacity is reached,

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

or equivalently,

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

So at early timepoints, the log-ratio of a strain's counts to reference counts increases linearly over time with the fitness difference between the strain and the reference. The fitness difference is useful, but the ratio would be better. (Alternatively, we could use a known value of the reference fitness measured separately).

And after carrying capacity is reached,

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

So in this regime, the log-ratio of a strain's counts to reference counts is fixed. Subtracting the log-ratio of a strain's counts to reference counts in the input leaves:

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

But we still can't get the relative finess without dividing by the constant

$$\log \frac{K-{\mathrm{\Sigma }}_j{n}_j}{n_{wt}(0)}\backslash log\; \backslash frac\{K\; -\; \backslash Sigma\_j\; n\_j\}\{n\_\{wt\}(0)\}$$

which we don't know directly. However, the final trick is to use a non-growing control, so that

$$\frac{w_i}{w_{wt}}=0\backslash frac\{w\_i\}\{w\_\{wt\}\}\; =\; 0$$

That means that for this control,

$$\log \frac{c_i(t)}{c_{wt}(t)}-\log \frac{c_i(0)}{c_{wt}(0)}=-\log \frac{K-{\mathrm{\Sigma }}_j{n}_j}{n_{wt}(0)}\backslash log\; \backslash frac\{c\_i(t)\}\{c\_\{wt\}(t)\}\; -\; \backslash log\; \backslash frac\{c\_i(0)\}\{c\_\{wt\}(0)\}\; =\; -\; \backslash log\; \backslash frac\{K\; -\; \backslash Sigma\_j\; n\_j\}\{n\_\{wt\}(0)\}$$

We can then get the relative fitness ratio for the other strains directly.