[dalke]

I'm a bit doubtful about the statistics, and I think I know why. There's a circularity to your use of "deleterious mutations" and "bad mutation."

Is the mutation which causes sickle-cell anaemia a "good mutation" or a "bad mutation"? It increases reproductive fitness in places where malaria is or was common, so it must be good, in an evolutionary sense.

How many deaths has it caused once the population of people carrying the haemoglobin gene mutation migrated to a location without malaria? Is that mutation now "good" or "bad"? How do you incorporate those numbers into your statistics?

Is the loss of eyesight a deleterious mutation? Definitely for a bird of prey, but not so for cave-dwelling creatures living in absolute darkness. For that matter, some people are attracted to people who wear glasses (and wearing zero-prescription glasses is such a turn-off!), so it might increase reproductive fitness.

Evolution doesn't know the future. If a population loses genetic resistance to a disease that's seemingly extinct, is that a "good" or "bad" mutation? How long does it take to judge that? After 1,000 years, should some thawed carcass reintroduce it and the species become extinct, does that count finally as a bad mutation and a single death?

For a real world example, consider the birds of New Zealand. They filled ecological niches which elsewhere were filled by mammals. Were these good mutations or bad ones? And when rats and weasels and cats and more were introduced to New Zealand, helping make many of those species extinct, then did those mutations retrospectively become deleterious?

If a genetic madness affects the leader of the US Strategic Air Command to issue orders which end up nuking a dozen Soviet cities, then what are the other cases which make that average out to one? If the nuking didn't occur, then what would the average have been?

What of a mutation which causes a speciation event? Is that a good mutation or a bad one? It's better for one environment and worse for the other.

There's a 10^-9 chance (1-in-a-million) that a "bad" mutation will mutate again back to the "good" form. With nearly 7 billion people in the world, that almost certainly happens a few thousand times every generation. In a generation we may be able to cure some genetic diseases through genetic engineering, so a "bad" mutation can be fixed.

With all those in mind, I can't figure out a way to get the numbers to come out "1" unless the definition of deleterious is defined to make it come out that way.

[bermanoid]

Yes, the one-mutation-one-death idea is vastly oversimplified when it comes to the real world, so I shouldn't have presented it as being more meaningful than it is. But while it can't be taken as a mathematical truth in the unsimplified real world, the "moral of the story" will holds (that worse mutations can't spread as far as less bad ones).

It's rather simple to prove in the simplified case, it's just a typical steady state assumption. If a population is in an equilibrium state, then the rate at which any mutation is introduced has to be equal to the rate at which it is removed from the population. So if one mutation has a 1% chance to kill its owner each generation, then to maintain equilibrium (in other words, to make sure the prevalence of the mutated gene in the population is stable), every time the mutation shows up anew, it must spread to 100 people, killing one of them. One mutation, one death.

Yes, that's super simplified, it neglects the possibility of multiple mutations, positive or neutral ones, back-mutation, interactions between members of the population, non-equilibrium states, etc. These will change the details of the math, sometimes quite substantially.

But the basic idea, that the worse a mutation is the less prevalent it will be, should hold.

[dalke]

With the steady-state assumption, doesn't every beneficial mutation also lead to a single death?

[bermanoid]

Beneficial and neutral mutations are essentially left out of the equation - they would spread to 100% of the population, so in the steady state, the probability of mutation newly creating a beneficial mutation has to be 0% (since it's already present in every member).

The motivation for ignoring beneficial mutations (and back-mutations to beneficial states) is that they're extremely rare as compared to deleterious ones - most selection pressure in nature is aimed at merely preserving the functionality in the genome, weeding out new deleterious mutations rather than supporting new beneficial ones (though that is a critical role in the very long term, of course).

[dalke]

I still think that your definition of beneficial and deleterious is defined post-hoc as "survives/does not survive long enough to reach the stable state."

In my example earlier regarding extinct bird species on New Zealand, were there ever any beneficial mutations? After all, the genes no longer reproduce.

My point is that there is no stable state, so it's better to have a shorter-term definition of "beneficial" and "deleterious" based on relative reproduction fitness compared to others in the species population over a short time frame.

Additionally, beneficial and neutral mutations do not always spread to 100% of the species population. A Y-linked trait won't spread without a male lineage.

[bermanoid]

I still think that your definition of beneficial and deleterious is defined post-hoc as "survives/does not survive long enough to reach the stable state."

Yup, that's a pretty good description of the assumption/definition, for all the good and bad things it brings with it; I mostly agree with the rest of what you've said.

FWIW, I've mainly seen the one-mutation-one-death rule applied to arguments that attempted to place informational speed or capacity limits on the process of evolution, and in most cases it has turned out that these arguments fail when applied to the real world because of precisely the types of arguments that you've made against this rule (sexual recombination and the fact that evolution is never in a steady state tend to be the biggest problems).

An aside on the topic of defining beneficial/deleterious mutations: the problem is a very difficult one, because in reality the effect of a mutation is at best distributional and not measurable along a single axis of goodness/badness. I don't know much about the state of the art here, but if I was going to sit down and try to figure out a way to try to estimate "benificial-ness" of a mutation (even as a distribution), I'd say you've put your finger on exactly the right question to focus on, that there's a tension between what might be immediately beneficial and what would be beneficial in the longer term. For instance, a gene that made someone have babies like crazy by always sacrificing one's grandchildren for nourishment would be fantastic for 1-generation-out fitness, but terrible for 2-generations-out. Similarly, in an environment where pre-reproductive death was very common, an adaptation that decreased the likelihood to have children by a small factor but significantly increased the ability to keep them alive would be very beneficial at 2-generations but deleterious at 1. So pure-local effects measured 1 generation down the line won't necessarily tell us enough (though in the vast majority of cases, it's probably good enough).

On the other hand, a purely-global view is not right either, because as you've mentioned, environmental changes or genetic shifts within the species can turn previously helpful traits into "bad" ones. So measuring regret after the fact will not suffice, and further, it misses the fact that at the moment the mutation happens, there is some distribution that describes the likely outcomes of that mutation given the current state of the world, using no information from the future (even if we don't know it or can't feasibly calculate it). The extinct bird species definitely had beneficial mutations even though some freak event wiped them out later, and we need to account for that. So something more subtle is required.

In order to do better we'd probably have to make some assumptions to eventually cut off the familial dependencies, like perhaps that the presence or absence of a gene in an N-th grandparent can have no direct causal effect on the survival of the animal in question (i.e. direct nurture effects are limited in time - even this assumption is questionable in the face of things like family wealth). We'd also have to assume that we could quantify the expected changes in the environment into some sort of distribution, including distributional assumptions for expected changes in the rest of the population (luckily these changes should be rather slow, when measured in generations). Then we could in theory compare the expected size of a family tree branching from a member that possesses a new mutation after that N-th generation, to the base case, the family tree without that mutation. Of course, the "expected size" is a trivialization of the real distribution, which would better describe the possible effects of a mutation (and the details of that distribution might strongly effect the population dynamics).

It quickly gets messy, and requires a lot of assumptions. And even after all that, we wouldn't have a very clean number to work with, i.e. we couldn't easily continuous-ize the situation and write down an ODE that took the "beneficial-ness" of a mutation and showed us what would happen, because other details of the distribution would be important, as well.