# Is this formula describing the loss of unfixed genes from a population accurate?

In the 1954 science fiction novel Search the Sky by Frederick Pohl and Cyril M. Kornbluth, a formula is featured that "quantitatively describes the loss of unfixed genes from a population".

$$L_T=L_0 e^{-T/2N}$$

"A final word. There has been much loose talk among the troops about the slogan of the Joneses, which goes $L_T=L_0 e^{-T/2N}$. Some uninformed people actually believe it is an invocation which gives the Joneses supernatural power and invulnerability. It is not. It is merely an ancient and well-known formula in genetics which quantitatively describes the loss of unfixed genes from a population. By mouthing this formula, the Joneses are simply expressing in a compact way their ruthless determination that all genes except theirs shall disappear from the planet and the Joneses alone survive. In the formula $L_T$ means the number of genes after the lapse of $T$ years, $L_0$ means the original number of genes, $e$ means the base of the natural system of logarithms and $N$ means number of generations."

Is this formula indeed an accurate description?

Or was it made up by the authors? I find it suspicious that it contains two different measures of time, years and number of generations, while they seem related to me.

An earlier version of this question omitted the minus sign in the equation. I noticed it when finding it again on Google Books.

• Honestly, you should really look at some scientific literature in this area. Do not judge it by speculation. If you can not find anything, then assume no. – Adam Radek Martinez Jan 20 '17 at 18:34
• @RadekMartinez - I assume that the point is to ask the experts, who might already know the right equation. – Obie 2.0 Jan 20 '17 at 18:42
• I also find this formula odd. Isn't the length of a generation generally fixed? And if so, shouldn't the number of generations in a given time be proportional to time, rendering T/N independent of time? – Obie 2.0 Jan 20 '17 at 18:44
• Number of generations would be proportional to number of years, but every species would have a different ratio. Since P & K were talking about only humans, that would be a constant. But it appears to be a more general formula with generation length independent of number of years. But you're right, It seems odd. – jlawler Jan 20 '17 at 22:02
• I have edited your title so readers have an idea what the formula relates to without having to read the question. Helps them and you. (Gentle hint for any future questions you ask.) – David Jan 21 '17 at 22:18

## 2 Answers

The expression is based on a real one, but missing a minus sign (EDIT: not anymore! See comment below), and the explanation is garbled. For the actual equation, see "eigenvalue effective population size" in, e.g., WJ Ewens, Mathematical Population Genetics. There should be a negative sign in the exponent: $L_T = L_0 e^{-T/2N}$. Note that you need this minus sign so that "genes" (actually heterozygosity) are lost with time rather than growing exponentially in number!

As for the errors in the explanation: $T$ should be measured in generations rather than years, and $N$ should be the number of individuals (or rather the "eigenvalue effective number") instead of the number of generations.

• Oh yes, you are probably right! +1. So, the definition of the symbols were wrong (generation vs years), the meaning of the equation was wrong (loss of genes vs loss of heterozygosity) and the equation was wrong (minus sign missing). Wow, that's quite a distance from the original formulation! – Remi.b Jun 13 '17 at 13:56
• Actually, [the minus sign is there](books.google.com/books?id=FCd6CwAAQBAJ&pg=PA132&lpg=PA132&dq="There+has+been+much+loose+talk+among+the+troops+about+the+slogan+of+the+Joneses"). Either I overlooked it or the previous scan I saw omitted it. I'll update my question, if you don't mind. – SQB Jun 13 '17 at 13:57

I keep my answer for information purpose but the right answer is @DanielWeissman answer. It is made very clear once the OP noticed the missing minus sign.

I think this formula comes from nowhere. The supposed meaning of this formula is very unclear. As you said, it makes little sense to have two measures of time (in years and in generations) in the same equation.

Fixation of new alleles

I am not sure what is meant by gene loss. I would suppose, it refers to fixation of new alleles. Under the assumption that all mutations are neutral, then the probability of fixation of new alleles is $\frac{1}{2N}$, where $N$ is here the population size. If the neutral mutation rate is $\mu$, then the number of new mutations per generation is $2N\mu$ and therefore the rate of fixation of new alleles is $2N\mu\frac{1}{2N} = \mu$. Per consequence, after $t$ generations the number of new alleles $L_t$ is $L_t = L_0 + t \mu$ (assuming it makes sense to talk about the number of newly fixed allele at time $t=0$).

Loss of genes

Muller's ratchet (typically on the Y chromosome in mammals) does lead to systematic loss of genes. There are here a number of models and a final expression for the rate of loss of genes is not that easy.