Sanford's Mendel's Accountant proven rigged

[Roy (34-year veteran)]

@Rumraket

That it was rigged was already known and have been discussed here before. Exactly how it was rigged was only speculated on.

I dug into it a couple of years back:

@Roy

The second parameter γ can be viewed as controlling the fraction of mutations that have a large absolute fitness effect. Instead of specifying γ directly, we select two quantities that are more intuitive and together define γ. The first is θ, a threshold value that defines a “high-impact mutation.” The second is q, the fraction of mutations that exceed this threshold in their effect. For example, a user can first define a high-impact mutation as one that results in 10% or more change in fitness (θ = 0.1) relative to the scale factor and then specify that 0.001 of all mutations (q = 0.001) be in this category.

So q defines the fraction of “high-impact” mutations, and θ defines how high that impact is. But θ isn’t an absolute change in fitness, it’s a change “relative to the scale factor”.

What’s the scale factor?

It’s defined on the previous page:

Our function, expressed by Equation (1), maps a random number x, drawn from a set of uniformly distributed random numbers, to a fitness effect d(x) for a given random mutation as follows:

d(x) = dsf exp(-axγ), 0 ≤ x ≤ 1 (1)

Here dsf is the scale factor which is equal to the extreme value which d(x) assumes when x = 0. We allow this scale factor to have two separate values, one for deleterious mutations and the other for favorable ones. These scale factors are defined relative to the initial fitness value assumed for the population before we introduce new mutations. In Mendel, we assume this initial fitness value to be 1.0. For deleterious mutations, since lethal mutations exist, we choose dsf_del = −1. For favorable mutations, we allow the user to specify the (positive) scale factor dsf_fav. Normally, this would be a small value (for example, 0.001 to 0.1), since it is only in very special situations that a single beneficial mutation would have a very large effect.

So even though you can set a minimum threshold θ for some fraction of beneficial mutations, that threshold value is then scaled down by at least an order of magnitude. And although it’s called a threshold value, the maths given suggests that the fitness change is at the threshold, rather than above it.[1] This threshold value, which apparently can be between 0.001 and 0.1, is applied to a scale factor which is also between 0.001 and 0.1, leading to a beneficial mutation fitness effect range of 0.001×0.001 = 0.000001 to 0.1×0.1 = 0.01 - the range shown in the screenshot above.

Mendel’s accountant uses different fitness effect ranges for deleterious vs beneficial mutations. It does not allow beneficial mutations to have a fitness effect greater than 0.01.

Since deleterious and beneficial mutations have different scaling factors, reversing a deleterious mutation won’t bring fitness back to the previous value.