sources/growth-curve-t-independent.md

Removing time dependence

It can be difficult to get absolute fitness out of these curves, because when the pool approaches the carrying capacity, all the strains growth rates mutually affect each other.

However, if we're only interested in the relative fitness of multiplexed strains relative to a reference (e.g. wild-type) strain, then we can make this simplification:

$$\frac{d{n}_i}{dt}\mathrm{/}\frac{d{n}_{wt}}{dt}=\frac{d{n}_i}{d{n}_{wt}}=\frac{w_i{n}_i}{w_{wt}{n}_{wt}}\backslash frac\{dn\_\{i\}\}\{dt\}\; /\; \backslash frac\{dn\_\{wt\}\}\{dt\}\; =\; \backslash frac\{dn\_\{i\}\}\{dn\_\{wt\}\}\; =\; \backslash frac\{w\_\{i\}\; n\_i\}\{w\_\{wt\}\; n\_\{wt\}\}$$

The interdependency term cancels out, and time is removed, with the reference strain's growth acting as the clock. Unlike the time-dependent Lotka-Volterra equations, this has a closed-form integral:

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

So now the log of the number of cells of a mutant at any moment ($n_1(t)$) is dependent only on its inoculum ($n_1(0)$), how much the reference strain has grown (i.e. fold-expansion, $\frac{n_{wt}(t)}{n_{wt}(0)}$), and the ratio of fitness between the mutant and the reference ($\frac{n_{wt}(t)}{n_{wt}(0)}$).

Plotting the number of cells of a mutant against the reference fold-expansion gives a straight line (on log scales), with intercept being the mutant inoculum and gradient being its relative fitness with respect to the reference strain's fitness.