Here I will treat some basic questions in population genetics. For personal reasons, I tend to include all the algebra.
What is the relationship between per-generation growth rates and the Malthusian parameter, the instantaneous rate of growth?
Let n_{i}(t) be the number of organisms of type i at time t, and let R be the per-capita reproductive rate per generation. If t counts generations, then
Now we wish to move to the case where t is continuous and real-valued.
As before,
where the last simplification follows from L'HÃ´pital's rule. Explicitly, let ε = Δt. Then
The solution to the equation
If two organisms grow at different rates, how do their proportions in the population change over time?
Let r_{1} and r_{2} be the instantaneous rates of increase of type 1 and type 2, respectively. Then
This result says that the proportion of type 1, p, changes most rapidly when p = 0.5 and most slowly when p is very close to 0 or 1.
The logit function , which takes , induces a more natural space for considering changes in frequencies. Rather than tracking the proportion of type 1 or 2, we instead track their log odds. In logit terms, with ,
This differential equation L_{p}'(t) = s has the solution
showing that the log-odds of finding type 1 changes linearly in time, increasing if s > 0 and decreasing if s < 0.
Insert math here.
We have three strains, i, j and r, where r is a reference strain. Strains i and j have fitness and . Define the selection coefficient as usual. We have data consisting of triples (g = number of generations, n_{i} = number of cells of type i, n_{r} = number of cells of type r). We have data consisting of pairs (g = number of generations, p_{ir} = n_{i} / n_{r}) where n_{i}=number of cells of type i and n_{r} = number of cells of type r.
What is the best estimate, and error, on s_{ij}?
Assuming exponential growth, lnp_{ir} =
Let .
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