# Poisson Nachman Crowell

Even aside from Basener and Sanford, others including Nobel Prize winner Hermann Muller pointed out the human race cannot tolerate very many mutations per individual per generation. The number Muller arrived at was about 1 bad mutation per generation per individual as the limit the human genome can tolerate.

Additionally, so what if an individual has a good mutation if he has 10 bad to go with it. This is like have a slight increase in intelligence while having 10 heritable diseases to go with it. You go one step forward and ten steps back.

Can natural selection arrest the problem? Only if there are enough reproductive resources relative to the number of offspring per couple.

For human populations there was something published by Nachman and Crowell and Eyre-Walker and Keightley using a Poisson distribution as reasonable model for the probability of a eugenically clean individual appearing in the face of various mutation rates.

If it is improbable that an eugenically clean kid can be reproduced by a couple, this makes it hard to weed out the bad. So this is an alternative way to arrive at Muller’s conclusions, which are also Sanford and Basener’s conclusions, and really everyone else’s conclusions as summarized by Dan Gruar: “If ENCODE is right, evolution is wrong.”

This is a simpler argument than the one Basener and Sanford put forward, but to Sanford’s credit, he’s also put the simpler version in his book Genetic Entropy, although the following derivation isn’t in his book, it’s something I ginned up myself. 🙂

So how can we estimate the probability a kid can be born with no defective mutations?

The following derivation was confirmed in Kimrua’s paper (see eqn. 1.4)

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1211299/pdf/1337.pdf

which Nachman and Crowell, and Eyre-Walker and Keightley reference as well.

So now the details:

let U = mutation rate (per individual per generation)
P(0,U) = probability of individual having no mutation under a mutation rate U (eugenically the best)
P(1,U) = probability of individual having 1 mutation under a mutation rate U
P(2,U) = probability of individual having 2 mutations under a mutation rate U
etc.

The wiki definition of Poisson distribution is:

$\huge f(k,\lambda ) = e^{-\lambda }\frac{\lambda^k }{k!}$

to conform the wiki formula with evolutionary literature let

$\lambda = U$

and

$f = P$

Because P(0,U) = probability of individual having no mutation under a mutation rate U (eugenically the best), we can find the probability the eugenically best individual emerges by letting:

$k = 0$

which yields

$\large \large P(k,U) = P(0,U) = \frac{U^0 e^{-U }}{0!} = e^{-U}$

Given the Poisson distribution is a discrete probability distribution, the following idealization must hold:

$\large \sum_{n}P_n =\sum_{i=0}^{\infty}P(i,U) = 1$

thus

$\large \large P(0,U) + \sum_{i=1}^{\infty}P(i,U) = 1$

thus subtracting P(0,U) from both sides

$\large \large P(0,U) + \sum_{i=1}^{\infty}P(i,U) -P(0,U) = 1 - P(0,U)$

thus simplifying

$\large \sum_{i=1}^{\infty}P(i,U) = 1 - P(0,U)$

On inspection, the left hand side of the above equation must be the percent of offspring that have at least 1 new mutation. Noting gain that $P(0,U) = e^{-U}$, the above equation reduces to the following:

$\sum_{i=1}^{\infty}P(i,U) = 1 - P(0,U) = 1- e^{-U}$

which is in full agreement with Nachman and Crowell’s equation in the very last paragraph and in full agreement with an article in Nature: High genomic deleterious mutation rates in homonids by Eyre-Walker and Keightley, paragraph 2.

http://www.lifesci.sussex.ac.uk/CSE/members/aeyrewalker/pdfs/EWNature99.pdf

The simplicity and elegance of the final result is astonishing, and simplicity and elegance lend force to arguments.

So what does this mean? If the bad mutation rate is 6 per individual per generation (more conservative than Gruar’s estimate if ENCODE is right), using that formula, the chances that a eugenically “ideal” offspring will emerge is:

$\large \large P(0,6) = e^{-6} = 0.25\%$

This would imply each parent needs to procreate the following number of kids on average just to get 1 eugenically fit kid:

$\frac{1}{e^{-U}} = \frac{1}{e^{-6}} = 403.42$

Or equivalently each couple needs to procreate the following number of kids on average just to get 1 eugenically fit kid:

$\large \large 2 * \frac{1}{e^{-U}} = 2 * \frac{1}{e^{-6}} \approx 807$

For humanity to survive, even after each couple has 807 kids on average, we still have to make the further utterly unrealistic assumption that the eugenically “ideal” offspring are the only survivors of a selective process.

Hence, it is absurd to think humanity can purge the bad out of its populations — the bad just keeps getting worse.

In truth, since most mutations are of nearly neutral effect, most of the damaged offspring will reproduce, and the probability of a eugenically ideal line of offspring approaches zero over time.

Muller’s number of only 1 new bad mutations per generation per individual. So if anything I understated my case.

There are some “fixes” to the problem suggested by Crow and Kondrashov. I suggested my fix. But the bottom line is to look at what is actually happening to the human genome over time. Are we getting dumber and sicker? I think so. It’s sad.

We can test Basener and Sanford’s prediction by observing whether human heritable diseases continue to increase with each generation. Whether their derivation is right or not, some of their conclusions are observationally and experimentally testable.

On some level, I suppose even Basener and Sanford wished it were not so because it is a tragic conclusion.