Natural Selection's Speed Limit and Complexity Bound

Followup to:  An Alien God, The Wonder of Evolution, Evolutions Are Stupid

Yesterday, I wrote:

Humans can do things that evolutions probably can't do period over the expected lifetime of the universe.  As the eminent biologist Cynthia Kenyon once put it at a dinner I had the honor of attending, "One grad student can do things in an hour that evolution could not do in a billion years."  According to biologists' best current knowledge, evolutions have invented a fully rotating wheel on a grand total of three occasions.

But then, natural selection has not been running for a mere million years.  It's been running for 3.85 billion years.   That's enough to do something natural selection "could not do in a billion years" three times.  Surely the cumulative power of natural selection is beyond human intelligence?

Not necessarily.  There's a limit on how much complexity an evolution can support against the degenerative pressure of copying errors.

(Warning:  A simulation I wrote to verify the following arguments did not return the expected results.  See addendum and comments.)

The vast majority of mutations are either neutral or detrimental; here we are focusing on detrimental mutations.  At equilibrium, the rate at which a detrimental mutation is introduced by copying errors, will equal the rate at which it is eliminated by selection.

A copying error introduces a single instantiation of the mutated gene.  A death eliminates a single instantiation of the mutated gene. (We'll ignore the possibility that it's a homozygote, etc; a failure to mate also works, etc.)  If the mutation is severely detrimental, it will be eliminated very quickly - the embryo might just fail to develop.  But if the mutation only leads to a 0.01% probability of dying, it might spread to 10,000 people before one of them died.  On average, one detrimental mutation leads to one death; the weaker the selection pressure against it, the more likely it is to spread.  Again, at equilibrium, copying errors will introduce mutations at the same rate that selection eliminates them. One mutation, one death.

This means that you need the same amount of selection pressure to keep a gene intact, whether it's a relatively important gene or a relatively unimportant one.  The more genes are around, the more selection pressure required.  Under too much selection pressure - too many children eliminated in each generation - a species will die out.

We can quantify selection pressure as follows:  Suppose that 2 parents give birth to an average of 16 children.  On average all but 2 children must either die or fail to reproduce.  Otherwise the species population very quickly goes to zero or infinity.  From 16 possibilities, all but 2 are eliminated - we can call this 3 bits of selection pressure.  Not bits like bytes on a hard drive, but mathematician's bits, information-theoretical bits; one bit is the ability to eliminate half the possibilities.  This is the speed limit on evolution.

Among mammals, it's safe to say that the selection pressure per generation is on the rough order of 1 bit.  Yes, many mammals give birth to more than 4 children, but neither does selection perfectly eliminate all but the most fit organisms.  The speed limit on evolution is an upper bound, not an average.

This 1 bit per generation has to be divided up among all the genetic variants being selected on, for the whole population.  It's not 1 bit per organism per generation, it's 1 bit per gene pool per generation.  Suppose there's some amazingly beneficial mutation making the rounds, so that organisms with the mutation have 50% more offspring.  And suppose there's another less beneficial mutation, that only contributes 1% to fitness.  Very often, an organism that lacks the 1% mutation, but has the 50% mutation, will outreproduce another who has the 1% mutation but not the 50% mutation.

There are limiting forces on variance; going from 10 to 20 children is harder than going from 1 to 2 children.  There's only so much selection to go around, and beneficial mutations compete to be promoted by it (metaphorically speaking).  There's an upper bound, a speed limit to evolution:  If Nature kills off a grand total of half the children, then the gene pool of the next generation can acquire a grand total of 1 bit of information.

I am informed that this speed limit holds even with semi-isolated breeding subpopulations, sexual reproduction, chromosomal linkages, and other complications.

Let's repeat that.  It's worth repeating.  A mammalian gene pool can acquire at most 1 bit of information per generation.

Among mammals, the rate of DNA copying errors is roughly 10^-8 per base per generation.  Copy a hundred million DNA bases, and on average, one will copy incorrectly.  One mutation, one death; each non-junk base of DNA soaks up the same amount of selection pressure to counter the degenerative pressure of copying errors.  It's a truism among biologists that most selection pressure goes toward maintaining existing genetic information, rather than promoting new mutations.

Natural selection probably hit its complexity bound no more than a hundred million generations after multicellular organisms got started.  Since then, over the last 600 million years, evolutions have substituted new complexity for lost complexity, rather than accumulating adaptations.  Anyone who doubts this should read George Williams's classic "Adaptation and Natural Selection", which treats the point at much greater length.

In material terms, a Homo sapiens genome contains roughly 3 billion bases.  We can see, however, that mammalian selection pressures aren't going to support 3 billion bases of useful information.  This was realized on purely mathematical grounds before "junk DNA" was discovered, before the Genome Project announced that humans probably had only 20-25,000 protein-coding genes.  Yes, there's genetic information that doesn't code for proteins - all sorts of regulatory regions and such.  But it is an excellent bet that nearly all the DNA which appears to be junk, really is junk.  Because, roughly speaking, an evolution isn't going to support more than 10^8 meaningful bases with 1 bit of selection pressure and a 10^-8 error rate.

Each base is 2 bits.  A byte is 8 bits.  So the meaningful DNA specifying a human must fit into at most 25 megabytes.

(Pause.)

Yes.  Really.

And the Human Genome Project gave the final confirmation.  25,000 genes plus regulatory regions will fit in 100,000,000 bases with lots of room to spare.

Amazing, isn't it?

Addendum:  genetics.py, a simple Python program that simulates mutation and selection in a sexually reproducing population, is failing to match the result described above.  Sexual recombination is random, each pair of parents have 4 children, and the top half of the population is selected each time.  Wei Dai rewrote the program in C++ and reports that the supportable amount of genetic information increases as the inverse square of the mutation rate(?!) which if generally true would make it possible for the entire human genome to be meaningful.

In the above post,  George Williams's arguments date back to 1966, and the result that the human genome contains <25,000 protein-coding regions comes from the Genome Project.  The argument that 2 parents having 16 children with 2 surviving implies a speed limit of 3 bits per generation was found here, and I understand that it dates back to Kimura's work in the 1950s.  However, the attempt to calculate a specific bound of 25 megabytes was my own.

It's possible that the simulation contains a bug, or that I used unrealistic assumptions.  If the entire human genome of 3 billion DNA bases could be meaningful, it's not clear why it would contain <25,000 genes.  Empirically, an average of O(1) bits of genetic information per generation seems to square well with observed evolutionary times; we don't actually see species gaining thousands of bits per generation.  There is also no reason to believe that a dog has greater morphological or biochemical complexity than a dinosaur.  In short, only the math I tried to calculate myself should be regarded as having failed, not the beliefs that are wider currency in evolutionary biology.  But until I understand what's going on, I would suggest citing only George Williams's arguments and the Genome Project result, not the specific mathematical calculation shown above.

Comments

[Caledonian]

A lot of our DNA was acquired in the days when our ancestors were not yet mammals.

[Tom McCabe]

"Surely the cumulative power of natural selection is beyond human intelligence?"

Even if it was, why would you want to use it? Evolution has thoroughly screwed over more human beings than every brutal dictator who ever lived, and that's just *humans*, never mind the several billion extinct species which litter our planet's history.

[Recovering irrationalist]

So the meaningful DNA specifying a human must fit into at most 25 megabytes.

And that's before compression :-)

[Tom McCabe]

"So the meaningful DNA specifying a human must fit into at most 25 megabytes."

These are bits of entropy, not bits on a hard drive. It's mathematically impossible to compress bits of entropy.

[Scott Aaronson]

Eliezer, your argument seems to confuse two different senses of information. You first define "bit" as "the ability to eliminate half the possibilities" -- in which case, yes, if every organism has O(1) children then the logical "speed limit on evolution" is O(1) bits per generation.

But you then conclude that "the meaningful DNA specifying a human must fit into at most 25 megabytes" -- and more concretely, that "it is an excellent bet that nearly all the DNA which appears to be junk, really is junk." I don't think that follows at all.

The underlying question here seems to be this: suppose you're writing a software application, and as you proceed, many bits of code are generated at random, many bits are logically determined by previous bits (albeit in a more-or-less "mindless" way), and at most K times you have the chance to fix a bit as you wish. (Bits can also be deleted as you go.) Should we then say that whatever application you end up with can have at most K bits of "meaningful information"?

Arguably from some God's-eye view. But any mortal examining the code could see far more than K of the bits fulfilling a "functional role" -- indeed, possibly even all of them. The reason is that the web of logical dependencies, by which the K "chosen" bits interacted with the random bits to produce the code we see, could in general be too complicated ever to work out within the lifetime of the universe. And crucially, when biologists talk about how many base pairs are "coding" and how many are "non-coding", it's clearly the pragmatic sense of "meaningful information" they have in mind rather than the Platonic one.

Indeed, it's not even clear that God could produce a ~K-bit string from which the final application could be reliably reconstructed. The reason is that the application also depends on random bits, of which there are many more than K. Without assuming some conjecture about pseudorandom number generators, it seems the most God could do would be to give us a function mapping the random bits to K bits, such that by applying that function we'd end up most of the time with an application that did more-or-less the same thing. (This actually leads to some interesting CS questions, but I'll spare you for now! :) )

To say something more concrete, without knowing much more than I do about biology, I wouldn't venture a guess as to how much of the "junk DNA" is really junk. The analogy I prefer is the following: if I printed out the MS Word executable file, almost all of it would look like garbage to me, with only a few "coding regions" here and there ("It looks like you're writing a letter. Would you like help?"). But while the remaining bits might indeed be garbage in some sense, they're clearly not in the sense a biologist would mean.

[Caledonian]

Excluding the complex and subtle regulatory functions that non-coding DNA can possess strikes me as being extremely unwise.

There is no DNA in the maize genome that codes for striped kernels, because that color pattern is the result of transposons modulating gene expression. The behavior of one transposon is intricately linked to the total behavior of all transposons, and the genetic shifts they result in defy the simple mathematical rules of Mendelian inheritance. But more importantly, the behavior of transposons is deeply linked to the physical structure of the encoding regions they're associated with.

Roughly half the genome of corn is made up of transposons. Is this 'junk' or not?

[Eliezer Yudkowsky]

Aaronson, McCabe:

Actually, these mathematician's bits are very close to bits on a hard drive. Genomes, so far as I know, have no ability to determine what the next base ought logically to be; there is no logical processing in a ribosome. Selection pressure has to support each physical DNA base against the degenerative pressure of copying errors. Unless changing the DNA base has no effect on the organism's fitness (a neutral mutation), the "one mutation, one death" rule comes into play.

Now certainly, once the brain is constructed and patterned, there are billions of neurons, all of them playing a functional role, and once these neurons are exposed to the environment, the algorithmic complexity will begin to actually increase. But the core learning algorithms must still in principle be specifiable in 25 megabytes. There may not be junk neurons, but there is surely junk DNA.

Now, even junk DNA may help, in a certain sense, because the metabolic load of DNA is tiny, and the more junk DNA you have, the more crossover you can do with a smaller probability of swapping in the middle of a coding gene. This "function" of junk DNA does not depend on its information content, so it doesn't have to be supported against the degenerative pressure of a per-base probability of copying error.

To sum up: The mathematician's bits here are very close to bits on a hard drive, because every DNA base that matters has to be supported by "one mutation, one death" to overcome per-base copying errors.

[J Thomas]

However, mutation rates vary and can be selected. They aren't simply a constraint.

Also, it's been a long time since I've thought about this and I may be wrong, but aren't you talking about 1 bit per linkage group and not one bit per genome? (And the size of linkage groups also varies and can be selected.)

Some viruse genomes face severe constraints on size -- they have a container they must fit into -- say an icosahedral shape -- and it would be a big step to increase that size. And some of those make proteins off both strands of DNA and sometimes in more than one reading frame. 3 proteins from the same DNA sequence. Presumably each protein is less efficient than it might be if the DNA evolved to make it alone, but they do an adequate job of reproducing the virus.

Probably size constraints can usually be fudged better than that.

You can make mathematical theories about evolution, but they're highly sensitive to their beginning assumptions. It's too soon to say how far evolution has gone to produce genetic mechanisms that let evolution proceed more efficiently.

[Scott Aaronson]

Eliezer, so long an organism's fitness depends on interactions between many different base pairs, the effect can be as if some of the base pairs are logically determined by others.

Also, unless I'm mistaken there are some logical operations that the genome can perform: copying, transpositions, reversals...

To illustrate, suppose (as apparently happens) a particular DNA stretch occurs over and over with variations: sometimes forwards and sometimes backwards, sometimes with 10% of the base pairs changed, sometimes chopped in half and sometimes appended to another stretch, etc. Should we then count all but one of these occurrences as "junk"? Of course, we could measure the number of bits using our knowledge of how the sequence actually arose ("first stretch X was copied Y times, then the copies were modified as follows..."). But the more such knowledge we bring in, the further we get from the biologist's concept of information and the closer to the Platonic mathematical concept.

[Recovering irrationalist]

Tom:These are bits of entropy...mathematically impossible to compress

My bad, was thinking of the meaningful base pairs. Thanks for correcting me.

[Dog of Justice]

I interpret Eliezer to be saying that the Kolmogorov complexity of the human genome is roughly 25MB -- the absolute smallest computer program that could output a viable human genome would be about that size. But this minimal program would use a ridiculous number of esoteric tricks to make itself that small. You'd have to multiply that number by a large factor (representing how compressible, in principle, modern applications are) to make a comparison to hard drive bits as they are actually used.

[Recovering irrationalist]

Eek I just noticed an unfortunate way that last comment could be read. I meant I was thinking of material bits of information when I should have thought of information-theoretical bits. I in no way interpret your "bits of entropy" to mean physical, non-meaningful base pairs!

[Scott Aaronson]

OK, I came up with a concrete problem whose solution would (I think) tell us something about whether Eliezer's argument can actually work, assuming a certain stylized set of primitive operations available to DNA: insertion, deletion, copying, and reversal. See here if you're interested.

[Wiseman]

Eliezer, I see two potential flaws in your argument, let me try and explain:

1.) The copy error rate can't directly translate mathematically into how often individuals in a species die out due to the copy error rate. We simply can't know how often a mutation is neutral, good, or detrimental, in part because that depends on the specific genome involved. I imagine some genomes are simply more robust than others. But I believe the prevailing wisdom is that most mutations are neutral, simply because proteins are too physically big to be effected by small changes. Either way, I can't see how anyone knows enough about this to be confident in coming up with specific mathematically calculated numbers.

2.) One bad mutation does NOT equal one death, as far as I see it. Greater intelligence leads to greater capability to cope with detrimental circumstances. Sickle-Cell-Anemia is detrimental, but people live and reproduce with it, and have for generations. But it's almost entirely detrimental, especially if your risk of Malaria is low. It's true, organisms with non-detrimental versions of the genes will gradually take over, but that doesn't mean the detrimental versions can't survive on their own and with just a lower population cap.

And not referring to you in saying this Eliezer, but this whole “Most of the DNA is junk” mantra reeks of conventionalist thinking, a classic form of bias, and has always annoyed me when I saw it in science programs and news articles. Current scientific knowledge knows more about proteins than any other aspect of the function of DNA, so it follows that people will focus on this and gloss over the importance of the other functions of DNA. If you know something very concrete about DNA: proteins, that are amazing enough in themselves, it's very easy to justify the case that the rest is simply junk DNA. I doubt that, I think we just do know what it does yet on a mechanical level.

[Eliezer Yudkowsky]

Scott, the mechanisms you've described indeed allow us to end up with more meaningful physical DNA than the amount of information in it. To give a concrete example, a protein-coding gene is copied, then mutates, and then there's two highly structured proteins performing different chemical functions, which because of their similarity, evolved faster than counting the bases separately would seem to allow.

So the 1 bit/generation speed limit on evolution is not a speed limit on altered DNA bases - definitely not!

The problem is that these meaningful bases also have 10^-8 copying errors per base per generation, so the above does not bypass the total complexity bound on evolution. The total complexity bound on evolution is not some hyper-compressed Kolmogorov algorithmic complexity, it's counting meaningful bases (bases such that if they are changed they degrade the fitness of the organism).

So: Mammalian evolutions can create 1 bit of algorithmic complexity per generation, possibly highly compressed; but the total number of meaningful DNA bases is limited to 100,000,000 bases or less, without compression in the reproductively transmitted genetic material, but potentially with all kinds of unpacking over the lifetime of a particular organism.

[Caledonian]

Quite a lot of mutations are so lethal that they abort embryonic development, yes. This is a severe problem with organisms drawn from a narrow gene pool, like humans and corn, and less so with others. It's worth noting that, if we consider these mutations in the argument, we have to consider not only the children who are born and are weeded out, but all of the embryos conceived and lost as well.

Given how few conceptions actually make it to birth, and how many infants died young before the advent of modern medicine, humans didn't lose two out of four, they lost more like two out of eight-to-twelve.

[A. Coward]

Eliezer, I'm a little skeptical of your statement that sexual reproduction/recombination won't add information...

1. Single base pairs don't even code for amino acids, much less proteins. 2. If we're looking at how a mutation affects an organism's ability to reproduce, we want to consider at least an entire protein, not just an amino acid. 3. There can be multiple genes that are neutral on their own, yet in combination are either very harmful or very beneficial.

Can you provide an argument as to why none of this affects the "speed limit" (not even by a constant factor?)

[Tom McCabe]

"To sum up: The mathematician's bits here are very close to bits on a hard drive, because every DNA base that matters has to be supported by "one mutation, one death" to overcome per-base copying errors."

There are only twenty amino acids plus a stop code for each codon, so the theoretical information bound is 4.4 bits/codon, not 6 bits, even for coding DNA. A common amino acid, such as leucine, only requires two base pairs to specify; the third base pair can freely mutate without any phenotypic effects at all.

"Can you provide an argument as to why none of this affects the "speed limit" (not even by a constant factor?)"

For a full explanation, see an evolutionary biology textbook. But basically, the 1 bit/generation bound is information-theoretic; it applies, not just to any species, but to *any* self-reproducing organism, even one based on RNA or silicon. The specifics of how information is utilized, in our case DNA -> mRNA -> protein, don't matter.

[Caledonian]

Even in the argument, it applies to organisms that lose half of their offspring to selection. It's different for those that lose more, or less.

[billswift]

__Among mammals, it's safe to say that the selection pressure per generation is on the rough order of 1 bit. Yes, many mammals give birth to more than 4 children, but neither does selection perfectly eliminate all but the most fit organisms. The speed limit on evolution is an upper bound, not an average.__

One bit per generation equates to a selection pressure which kills half of each generation before they reproduce according to the first part of your post. Then you say 1 bit per generation is the most mammalian reproduction can sustain. But, more than half of mammals (in many, perhaps most, species) die without reproducing. Wouldn't this result in a higher rate of selection and, therefore, more functional DNA?

[Tom McCabe]

"But, more than half of mammals (in many, perhaps most, species) die without reproducing. Wouldn't this result in a higher rate of selection and, therefore, more functional DNA?"

"Yes, many mammals give birth to more than 4 children, but neither does selection perfectly eliminate all but the most fit organisms. The speed limit on evolution is an upper bound, not an average."

[Caledonian]

But mammals have many ways of weeding out harmful variations, from antler fights to spermatozoa competition. And that's just if they have the four children. The provided 1 bit/generation figure isn't an upper bound, either.

Life spends a lot of time in non-equilibrium states as well, and those are the states in which evolution can operate most quickly.

[A. Coward]

"But basically, the 1 bit/generation bound is information-theoretic; it applies, not just to any species, but to *any* self-reproducing organism, even one based on RNA or silicon. The specifics of how information is utilized, in our case DNA -> mRNA -> protein, don't matter."

OK, and I'm familiar with information theory (less so with evolutionary biology, but I understand the basics) but I'm thinking that the 1 bit/generation bound is -- pardon the pun -- a bit misleading, since:

1. A lot -- I mean a _lot_ -- of crazy assumptions are made without any hard evidence to back them up. (E.g., the "mammals produce on average ~4 offspring, and when they produce more, it's compensated for by selection's inefficiencies.")

2. I'm still not convinced that we're measuring in the right units. Some mutations do absolutely nothing (for example, if a segment of DNA translating to a UAU codon mutated into one translating to UAC), and some make a ridiculously huge difference. This kind of redundancy, along with many other factors, makes me wonder if we need to change the 1 bit by some scaling factor...

[Cyan]

David MacKay did a paper on this. Here's a quote from the abstract:

If variation is produced by mutation alone, then the entire population gains up to roughly 1 bit per generation. If variation is created by recombination, the population can gain O(G^0.5) bits per generation.

G is the size of the genome in bits.

[Fly]

I've been enjoying your evolution posts and wanted to toss in my own thoughts and see what I can learn.

"Our first lemma is a rule sometimes paraphrased as "one mutation, one death"."

Imagine that having a working copy of gene "E" is essential. Now suppose a mutation creates a broken gene "Ex". Animals that are heterozygous with "E" and "Ex" are fine and pass on their genes. Only homozygous "Ex" "Ex" result in a "death" that removes 2 mutations.

Now imagine that a duplication event gives four copies of "E". In this example an animal would only need one working gene out of the four possible copies. When the rare "Ex" "Ex" "Ex" "Ex" combination arises then the resulting "death" removes four mutations.

In fruit fly knock-out experiments, breaking one development gene often had no visible affect. Backup genes worked well enough. The backup gene could have multiple roles: First, it has a special function that improves the animal fitness. Second, it works as a backup when the primary gene is disabled. The resulting system is robust since the animal can thrive with many broken copies and evolution is efficient since a single "death" can remove four harmful mutations.

I've focussed on protein-coding genes, but this concept also applies to short DNA segments that code for elements such as miRNA's. Imagine that the DNA segment is duplicated. Being short, it is rarely deactivated by a mutation. Over time a genome may acquire many working copies that code for that miRNA. Rarely an animal would inherit no working copies and so a "death" would remove multiple chromosomes that "lacked" that DNA segment. On the other hand, too many copies might also be fatal. Chromosomes with too few or too many active copies would suffer a fitness penalty.

On a different note, imagine two stags. The first stag has lucked-out and inherited many alleles that improve its fitness. The second stag wasn't so lucky and inherited many bad alleles. The first stag successfully mates and the second doesn't. One "death" removed many inferior alleles.

Animals may have evolved sexual attraction based on traits that depend on the proper combined functioning of many genes. An unattractive mate might have many slightly harmful mutations. Thus one "death" based on sexual selection might remove many harmful mutations.

Evolution might be a little better than the "one mutation, one death" lemma implies. (I agree that evolution is an inefficient process.)

"This 1 bit per generation has to be divided up among all the genetic variants being selected on, for the whole population. It's not 1 bit per organism per generation, it's 1 bit per gene pool per generation."

Suppose new allele "A" has fitness advantage 1.03 compared to the wild allele "a" and that another allele "B" on the same type chromosome has fitness advantage 1.02. Eventually the "A" and "B" alleles will be sufficiently common that a crossover creating a new chromosome "AB" with "A" and "B" alleles is likely (This crossover probability depends on the population sizes of "Ab" and "aB" chromosomes and the distance between the alleles). The new chromosome "AB" should have a fitness of 1.05 compared to the chromosome "ab". Both "A" and "B" should then see an accelerated spread until the "ab" chromosomes are largely displaced. The rate would then diminish as "AB" displaced "Ab" and "aB" chromosomes. Thus multiple beneficial mutations of the same type chromosome should spread faster than the "single mutation" formula would indicate.

Due to crossover, good "bits" would tend to accumulate on good chromosomes thereby increasing the fitness of the entire chromosome as described above. The highly fit good chromosome thus displaces chromosome with many bad "bits". The good "bits" are no longer inherited independently and each "death" can now select multiple information "bits".

We seem to view evolution from a similar perspective.

Information requires selection in order to be preserved. The DNA information in an animal genome could be ranked in "fitness" value and the resulting graph would likely follow a power law. I.e., some DNA information is extremely important and likely to be preserved while most of the DNA is relatively free to drift. In a species such as fruit flies with many offspring selection can drive the species high up a local fitness peak. Much of the animal genome will be optimized. In a species such as humans with few offspring there is much less selection pressure and the specie gene pool wanders further from local peaks. More of the human genome drifts. (E.g., human regulatory elements are less conserved than rodent regulatory elements.)

[Tom McCabe]

"But mammals have many ways of weeding out harmful variations, from antler fights to spermatozoa competition. And that's just if they have the four children. The provided 1 bit/generation figure isn't an upper bound, either."

Read a biology textbook, darn it. The DNA contents of a sperm have negligible impact on the sperm's ability to penetrate the egg. As for antler fights, it doesn't *matter* how individuals are removed from the gene pool. They can only be removed at a certain rate or else the species population goes to zero. Note than nonreproduction = death as far as evolution is concerned.

"Life spends a lot of time in non-equilibrium states as well, and those are the states in which evolution can operate most quickly."

Yes, but they must be balanced by states where it operates more slowly. You can certainly have a situation where 1.5 bits are added in odd years and .5 bits in even years, but it's a wash: you still get 1 bit/year long term.

"1. A lot -- I mean a _lot_ -- of crazy assumptions are made without any hard evidence to back them up. (E.g., the "mammals produce on average ~4 offspring, and when they produce more, it's compensated for by selection's inefficiencies.")"

The bit rate is O(log(offspring)), not O(offspring), so even if you produced 16 offspring, that's only three bits/generation. How many offspring do you think we have? 8,589,934,592? (= 32 bits/generation)? Selection *will* have inefficiencies, so these are upper bounds.

"This kind of redundancy, along with many other factors, makes me wonder if we need to change the 1 bit by some scaling factor..."

The factor due to redundant coding sequences is 1.36 (1.4 bits/base instead of 2.0). This *does* increase the amount of storable information, because it makes the degenerative pressure (mutation) work less efficiently. Then again, it's only a factor of 35%, so the conclusion is still basically the same.

[Scott Aaronson]

OK, I posted the following update to my blog entry:

Rereading the last few paragraphs of Eliezer's post, I see that he actually argues for his central claim -- that the human genome can’t contain more than 25MB of "meaningful DNA" -- on different (and much stronger) grounds than I thought! My apologies for not reading more carefully.

In particular, the argument has nothing to do with the number of generations since the dawn of time, and instead deals with the maximum number of DNA bases that can be simultaneously protected, in steady state, against copying errors. According to Eliezer, copying DNA sequence involves a ~10^-8 probability of error per base pair, which — because only O(1) errors per generation can be corrected by natural selection — yields an upper bound of ~10^8 on the number of "meaningful" base pairs in a given genome.

However, while this argument is much better than my straw-man based on the number of generations, there's still an interesting loophole. Even with a 10^-8 chance of copying errors, one could imagine a genome reliably encoding far more than 10^8 bits (in fact, arbitrarily many bits) by using an error-correcting code. I'm not talking about the "local" error-correction mechanisms that we know DNA has, but about something more global -- by which, say, copying errors in any small set of genes could be completely compensated by other genes. The interesting question is whether natural selection could read the syndrome of such a code, and then correct it, using O(1) randomly-chosen insertions, deletions, transpositions, and reversals. I admit that this seems unlikely, and that even if it's possible in principle, it's probably irrelevant to real biology. For apparently there are examples where changing even a single base pair leads to horrible mutations. And on top of that, we can't have the error-correcting code be too good, since otherwise we'll suppress beneficial mutations!

Incidentally, Eliezer's argument makes the falsifiable prediction that we shouldn't find any organism, anywhere in nature, with more than 25MB of functional DNA. Does anyone know of a candidate counterexample? (I know there are organisms with far more than humans' 3 billion base pairs, but I have no idea how many of the base pairs are functional.)

Lastly, in spite of everything above, I'd still like a solution to my "pseudorandom DNA sequence" problem. For if the answer were negative -- if given any DNA sequence, one could efficiently reconstruct a nearly-optimal sequence of insertions, transpositions, etc. producing it -- then even my original straw-man misconstrual of Eliezer's argument could put up a decent fight! :-)

[Caledonian]

Read a biology textbook, darn it. The DNA contents of a sperm have negligible impact on the sperm's ability to penetrate the egg.

Defective sperm - which are more-than-normally likely to be carry screwed-up DNA - is far less likely to reach the egg, and far less likely to penetrate it before a fully functional spermatozoan does. It's a weeding-out process.

As for antler fights, it doesn't *matter* how individuals are removed from the gene pool.

Of course it does! Just not to the maximum-bit-rate argument.

Yes, but they must be balanced by states where it operates more slowly.

No, they mustn't. They can theoretically be kept in a constant non-equilibrium.

[Scott Aaronson]

Eliezer, sorry for spamming, but I think I finally understand what you were getting at.

Von Neumann showed in the 50's that there's no in-principle limit to how big a computer one can build: even if some fraction p of the bits get corrupted at every time step, as long as p is smaller than some threshold one can use a hierarchical error-correcting code to correct the errors faster than they happen. Today we know that the same is true even for quantum computers.

What you're saying -- correct me if I'm wrong -- is that biological evolution never discovered this fact.

If true, this is a beautiful argument for one of two conclusions: either that (1) digital computers shouldn't have as hard a time as one might think surpassing billions of years of evolution, or (2) 25MB is enough for pretty much anything!

[Gray Area]

Scott said: "25MB is enough for pretty much anything!"

Have people tried to measure the complexity of the 'interpreter' for the 25MB of 'tape' of DNA? Replication machinery is pretty complicated, possibly much more so than any genome.

[Wiseman]

Actually, Scott Aaronson, something you said in your second to last post made me think of another reason why the axiom "one mutation, one death" may not be true. Actually, it's just an elaberation of the point I made earlier but I thought I'd flesh it out a bit more.

The idea is that the more physically and mentally complex, and physically larger, a species gets, the more capable is it is of coping with detrimental genes and still surviving to reproduce. When you're physically bigger, and smarter, there's more 'surplus' resources to draw upon to help in survivial. Example: There is a rare genetic disorder that causes some people to have no finger prints. This mean's that their manual dexterity is greatly reduced because of lack of friction in the fingers. And while detrimental, this is a historicaly prevelant case that has not gone away just because it's bad for an individual. You can learn to avoid situations where failure in manual dexterity could be fatal, etc.

I also believe it's possible for long standing sections of DNA to evolve and become more robust to mutation once they have "proven themselves". Meaning if a certain series of genes/DNA that serve a benificial function are around long enough, they will become more refined and effective, and especially robust. However this is accomplished specifically, which of course I don't know, I don't see why it's mechanically impossible. Thus, large sections of DNA could essentially be "subtracted" from amount of DNA to be mutated per generation.

Any flaws in this logic?

[Tom McCabe]

"Defective sperm - which are more-than-normally likely to be carry screwed-up DNA - is far less likely to reach the egg,"

Then the DNA never gets collected by researchers and included in the 10^-8 mutations/generation/base pair figure. If the actual rate of mutations are higher, but the non-detected mutations are weeded out, you still get the exact same result as if the rate of mutations is lower with no weeding-out.

"Of course it does! Just not to the maximum-bit-rate argument."

True.

"No, they mustn't. They can theoretically be kept in a constant non-equilibrium."

Yes, they can be- it doesn't change the bit rate. Non-equilibria where the population is shrinking must be balanced by non-equilibria where the population is growing, or the population will go to zero or infinity.

[Mike Johnson]

A mammalian gene pool can acquire at most 1 bit of information per generation.

Eliezer,

That's a very provocative, interestingly empirical, yet troublingly ambiguous statement. :)

I think it's important to note that evolution is very effective (within certain constraints) in figuring out ways to optimize not only genes but also genomes-- it seems probable that a large amount of said "bits" have been on the level of structural or mechanical optimizations.

These structural/mechanical optimizations might in turn involve mechanisms by which to use existing non-coding, "junk" DNA in various ways (which might, in some sense, effectively increase the "bit size" of a single adaption into the megabytes).

It may be telling that we haven't seen, in three billion years and given all the other genetic complexity out there, any organisms evolve a mechanism to clean the junk out of its DNA.

At any rate, I think your argument is interesting, and the topic is simply fascinating, but I take your numbers with a grain of salt. No offense.

[Scott Aaronson]

Wiseman, if it's true that (1) copying DNA inherently incurs a 10^-8 probability of error per base pair, and that (2) evolution hasn't invented any von-Neumann-type error-correction mechanism, then all the objections raised by you and others (and by me, earlier!) are irrelevant.

In particular, it doesn't matter how capable a species is of coping with a few detrimental mutations. For if the mutation rate is higher than what natural selection can correct, the species will just keep on mutating, from one generation to the next, until the mutations finally do become detrimental.

Also, your idea that sections of DNA can become more robust once they've "proven themselves" violates the 10^-8 assumption -- which I'd imagine (someone correct me if I'm wrong) comes from physics and chemistry, not because it's favored by natural selection.

[Eliezer Yudkowsky]

Aaronson: What you're saying -- correct me if I'm wrong -- is that biological evolution never discovered this fact [error-correcting codes].

You're not wrong. As you point out, it would halt all beneficial mutations as well. Plus there'd be some difficulty in crossover. Can evolution ever invent something like this? Maybe, or maybe it could just invent more efficient copying methods with 10^-10 error rate. And then a billion years later, much more complex organisms would abound. All of this is irrelevant, given that DNA is on the way out in much less than a million years, but it makes for fun speculation...

Aaronson: If true, this is a beautiful argument for one of two conclusions: either that (1) digital computers shouldn't have as hard a time as one might think surpassing billions of years of evolution, or (2) 25MB is enough for pretty much anything!

In an amazing and completely unrelated coincidence, I work in the field of Artificial Intelligence. Did I mention DNA is on the way out in much less than a million years?

Cyan, MacKay's paper talks about gaining bits as in bits on a hard drive, which goes as O(N^1/2) because of Price's Equation and because variance in the sum of bits goes as the square root of the number of bits. I spent some time scribbling and didn't prove that this never gains more than one bit of information relative to a fitness function, but I don't see how eliminating half the organisms in a randomly mixed gene pool can make it gain more than one bit of information in the allele frequencies.

[michael vassar]

I think that a subset of Fly's objections may be valid, especially the ones about sexual selection concentrating harmful mutations in a small subset of the population. This could plausibly increase the number of bits by a significant factor. OTOH, 25M is an upper bound, so the actual number of bits could easily still be less.

Great point about evolution not discovering hierarchical error-correcting code Scott A. Chris Phoenix frequently makes similar points about molecular nanotechnology in response to its critics.

Regarding the earlier posts point about evolution not being able to extrapolate the fitness benefits of a gene and save time in it's penetration into a population, I will point out that in financial markets this ability is what creates bubbles. In evolutionary dynamics it can potentially produce runaway sexual selection, which is sort of like a bubble that never resolves itself in a price correction.

[Wiseman]

Scott A. I wasn't suggesting DNA would magically *not* mutate after it had evolved towards sophistication, only that the system of genes/DNA that govern a system would become robust enough so it would be immune to the effects of the mutations.

Anway, evolution does not have to "correct" these mutations, as long as the organism can survive with them, they have as much a chance of mutating to a neutral, positive, or other equally detremental state as it has of becoming worse. As a genome becomes larger and larger, it can cope with the same ratio of mutations it always has. The effects of the mutations don't "add up" as is assumed by Eliezer, they effect the local region of DNA and its related function, and that's it. If an organism happens to have a synergetically enhanced group of detrimental mutations, then yes that one will die, but showing empirically that that would happen more often than not, I thin, would be very difficult.

In any case, I still don't see where the ~25 megabyte number comes from. Wouldn't you need to know precisely how many mutations were detrimental to work that number out? And I'm assuming it's reasonable to say we don't have that information?

[Scott Aaronson]

Wiseman, let M be the number of "functional" base pairs that get mutated per generation, and let C be the number of those base pairs that natural selection can affect per generation. Then if M>>C, the problem is that the base pairs will become mostly (though not entirely) random junk, regardless of what natural selection does. This is a point about random walks that has nothing to do with biology.

To illustrate, suppose we have an n-bit string. At every time step, we can change one of the bits to anything we want, but then two bits get chosen at random and inverted. Question: in "steady state," how many of the bits can we ensure are 0?

I claim the answer is only 3n/4. For suppose pn of the bits are 1, and that we always pick a '1' bit and change it to 0. Then the expected change in the number of bits after a single time step is

D(p) = (1/4) [p^2 (-3) + 2p(1-p) (-1) + (1-p)^2 (1)].

Setting D(p)=0 and solving yields p=1/4.

So we get an either/or behavior: either the mutation rate is small enough that we can keep the functional DNA pretty much exactly as it is, or else a huge fraction of the "functional" DNA becomes randomized. In other words, there's no intermediate regime where the functional DNA keeps mutating around within some "interesting" region of configuration space, without spilling out into a huge region of configuration space.

Note that for the above argument, I've assumed that the "interesting" regions of DNA configuration space are necessarily small -- and in particular, that they can't correspond to Hamming balls of size c*n where c is a constant. This assumption is basically a restatement of our earlier observation that natural selection hasn't discovered error-correcting codes. As such, it seems to me to be pretty secure biologically. But if this assumption fails then so does my argument.

[A. Coward]

"'Life spends a lot of time in non-equilibrium states as well, and those are the states in which evolution can operate most quickly.'

Yes, but they must be balanced by states where it operates more slowly. You can certainly have a situation where 1.5 bits are added in odd years and .5 bits in even years, but it's a wash: you still get 1 bit/year long term."

This seems to contradict your earlier assertion that the 1 bit/generation rate is "an upper bound, not an average." It seems to me to be more analogous to a roulette wheel or the Second Law of Thermodynamics (relax! I'm not about to make a creationist argument just 'cause I said that!), so a gene pool can certainly acquire more than 1.36 bits (or whatever the actual figure is) in some generations, but in the long run "the house always wins."

"The factor due to redundant coding sequences is 1.36 (1.4 bits/base instead of 2.0). This *does* increase the amount of storable information, because it makes the degenerative pressure (mutation) work less efficiently. Then again, it's only a factor of 35%, so the conclusion is still basically the same."

Thank you. As long as everyone's clear that the speed limit is O(1) bits/generation (over long stretches? on average?) and not necessarily precisely 1 bit no matter what, I'm happy.

Scott: "What you're saying -- correct me if I'm wrong -- is that biological evolution never discovered [error-correcting codes]...[O]n top of that, we can't have the error-correcting code be too good, since otherwise we'll suppress beneficial mutations!"

Whoa -- that's really helpful. Scott, as usual, you've broken through all the troubling and confusing jargon ("equilibrium? durr...") so that some poor schmuck like me can actually see the main point. Thanks. =)

But of course, evolution itself is a sort of crude error-correcting code -- and one that discriminates between beneficial mutations and detrimental ones! So here's my question: Can you actually do asymptotically better than natural natural selection by applying an error-correcting code that doesn't hamper beneficial mutations? Or is natural selection (plus local error-correction of the form existing in DNA) optimal?

[Caledonian]

With sufficiently large selection populations, it's not clear to me how anything could be better than natural selection, since natural selection is what the system is trying to beat. Any model of natural selection will necessarily contain inaccuracies.

[Scott Aaronson]

So here's my question: Can you actually do asymptotically better than natural natural selection by applying an error-correcting code that doesn't hamper beneficial mutations?

In principle, yes. In a given generation, all we want is a mutation rate that's nonzero, but below the rate that natural selection can correct. That way we can maintain a steady state indefinitely (if we're indeed at a local optimum), but still give beneficial mutations a chance to take over.

Now with DNA, the mutation rate is fixed at ~10^-8. Since we need to be able to weed out bad mutations, this imposes an upper bound of ~10^8 on the number of functional base pairs. But there's nothing special mathematically about the constant 10^-8 -- that (unless I'm mistaken) is just an unwelcome intruder from physics and chemistry. So by using an error-correcting code, could we make the "effective mutation rate" nonzero, but as far below 10^-8 as we wanted?

Indeed we could! Here's my redesigned, biology-beating DNA that achieves this. Suppose we want to simulate a mutation rate ε<<10^-8, allowing us to maintain ~1/ε functional base pairs in a steady state. Then we simply stick those 1/ε base pairs (in unencoded form) into our DNA strand, and also stick in "parity-check pairs" from a good error-correcting code. These parity-check pairs let us correct as many mutations as we want, with only a tiny probability of failure.

Next we let the physics and chemistry of DNA do their work, and corrupt a 10^-8 fraction of the base pairs. And then, using exotic cellular machinery whose existence we get to assume, we read the error syndrome off the parity-check pairs, and use it to undo all but one mutation in the unencoded, functional pairs. But how do we decide which mutation gets left around for evolution's sake? We just pick it at random! (If we need random bits, we can just extract them from the error syndrome -- the cosmic rays or whatever it is that cause the physical mutations kindly provide us with a source of entropy.)

[Eliezer Yudkowsky]

Fly: Imagine that having a working copy of gene "E" is essential. Now suppose a mutation creates a broken gene "Ex". Animals that are heterozygous with "E" and "Ex" are fine and pass on their genes. Only homozygous "Ex" "Ex" result in a "death" that removes 2 mutations.

Now imagine that a duplication event gives four copies of "E". In this example an animal would only need one working gene out of the four possible copies. When the rare "Ex" "Ex" "Ex" "Ex" combination arises then the resulting "death" removes four mutations.

Fly, you've just postulated four copies of the same gene, so that one death will remove four mutations. But these four copies will suffer mutations four times as often. Unless I'm missing something, this doesn't increase the bound on how much non-redundant information can be supported by one death. :)

McCabe: The factor due to redundant coding sequences is 1.36 (1.4 bits/base instead of 2.0). This *does* increase the amount of storable information, because it makes the degenerative pressure (mutation) work less efficiently. Then again, it's only a factor of 35%, so the conclusion is still basically the same.

This increases the potential number of semi-meaningful bases (bases such that some mutations have no effect but other mutations have detrimental effect) but cancels out the ability to store any increased information in such bases.

[J Thomas]

Eliezer, could you provide a link to this result? Something looks wrong about it.

Fisher's fundamental theorem of natural selection says the rate of natural selection is directly proportional to the variance in additive fitness in the population. At first sight that looks incompatible with your result.

You mention a site with selection at 0.01%. This would take a very long time for selection to act, and it would require that there not be stronger selection on any nearby linked site. It seems implausible that this site would have been selected before, with the result that it should be a 50:50 chance whether each change is a small favorable or unfavorable one. Tiny selective effects are neutral for all practical purposes. But tiny unfavorable changes have only a tiny chance to spread in the same way that you have little chance to win big at the casino when you make a very long series of small bets with the odds a little bit against you. Tiny favorable changes have only a very small chance to spread because they're usually lost by accident before they get a large enough stake.

Your numbers are clearly correct when all mutations are dominant lethal ones. When the mutation rate is high enough that half the offspring get a dominant lethal mutation, and there are only twice as many offspring as parents, then the population can barely survive those mutations and any higher mutation rate would drive it extinct.

I'm not sure that reasoning applies to mutations that affect relative reproduction rather than absolute, though. When a mutation lets its bearer survive better than other individuals when competing with them, but they survive just fine when it isn't around, that could be a different story.

Clearly there are limits to the rate of natural selection. It's proportional to the variance in fitness, so anything that limits the variation in fitness limits the rate of evolution. Mutation and recombination create variation in fitness, and there's some limit on the mutation rate because of mutations that reduce the absolute level of functioning of the organism. But the reasoning expressed in the post doesn't look convincing to me.

[J Thomas]

"Now with DNA, the mutation rate is fixed at ~10^-8."

Well no, it isn't. Not to get too complicated, usually the mutation rate is lower than that, but occasionally things happen that bring the mutation rate rather higher. We have things like DNA repair mechanisms that are mutagenic and others that are less so, and when the former get turned on we get a burst of mutations.

"Since we need to be able to weed out bad mutations, this imposes an upper bound of ~10^8 on the number of functional base pairs."

Definitely no more than 10^8 sites that would mutate into dominant lethals. For lesser deleterious mutations it gets murkier.

_But there's nothing special mathematically about the constant 10^-8 -- that (unless I'm mistaken) is just an unwelcome intruder from physics and chemistry. So by using an error-correcting code, could we make the "effective mutation rate" nonzero, but as far below 10^-8 as we wanted?_

Yes, and it happens some.

_Indeed we could! Here's my redesigned, biology-beating DNA that achieves this. Suppose we want to simulate a mutation rate ε<<10^-8, allowing us to maintain ~1/ε functional base pairs in a steady state. Then we simply stick those 1/ε base pairs (in unencoded form) into our DNA strand, and also stick in "parity-check pairs" from a good error-correcting code. These parity-check pairs let us correct as many mutations as we want, with only a tiny probability of failure._

It's been years since I've looked at this. I may have some of it wrong and it might have changed while I wasn't looking. But one way we used to handle that was to keep track of which strand of DNA is the old known strand and which is the new one. Then if there's a mismatch, you repair the new one instead of the old one.

If you have two copies of the DNA sequence and one of them is being replicated while the other waits, and there's an error, you can copy DNA from the reserve copy and splice it into one or both of the new ones.

Since each DNA repair system might possibly do misrepair under some circumstance, and since they are potentially disruptive, it makes some sense that they would only be activated when needed.

[Wiseman]

If a species can deal with detrimental mutations for several generations, then that simply means that the species has more time to weed out those really bad mutations, making the "one mutation, one death" equation inadequate to describe the die off rate based purely on the mutation rate. Yes, new mutations pop up all the time, but unless those mutations directly add on to the detrimental effects of previous mutations, the species still will survive another generation.

To add on to my other argument that we "know too little" to make hard mathamatical calculations on how big a functional genome can be, we also shouldn't work under the assumption that mutation rates are static. Wikipedia's "Mutation rate" article states the rate varies from species to species, and there is even some disagreement as to what the human rate is. There is NO REASON why a species can't evolve redudent, error correction copy mechanisms so the mutation rate is right at the sweetspot, providing variation but not so much as to cause extinction.

AGAIN, I still advocate that the original point Eliezer made can't be proven untill we know exactly how many mutations are detrimental. As a neutral mutation simply doesn't count, no matter how many generations you look forward, and benificial mutations can counter detrimental ones.

[Eliezer Yudkowsky]

A comment from Shtetl-Optimized discussion:

It’s actually a common misconception that biological systems should have mechanisms that allow a certain number of mutations for the purpose of accruing beneficial adaptations. From an evolutionary perspective, all genes should favor the highest possible fidelity copies. Any genes that have any lower copying fidelity will necessarily have fewer copies in future generations and thus lose the evolutionary game to a gene with higher copying fidelity.

Remember, folks, evolution doesn't work for the good of the species, and there's no Evolution Fairy who tries to ensure its own continued freedom of action. It's just a statistical property of some genes winning out over others.

[Nick Tarleton]

From an evolutionary perspective, all genes should favor the highest possible fidelity copies.

Hmm... Suppose there are two separated populations, identical except that one has a gene that makes the mutation rate negligibly low. Naturally the mutating population will acquire greater variation over time. If the environment shifts, the homogeneous population may be wiped out but part of the diverse population may survive. So in this case, lower-fidelity copying is more fit in the long run. This is highly contrived, of course.

[Wiseman]

Disagree. Any genome that has lower copy fidelity will only be removed from the gene pool if the errors in copy actually make the resultant organism unable to survive and reproduce, otherwise it's irrelevant how similar the copied genese are to the original. If the copy error rate produces detrimental genes at a rate that will not cause the species to go extinct, it will allow for any benificial mutations to arise and spread themselves throughout the gene pool at 'leisure'. As long as those positive genese are attached to a genome structure which produces mutations at a specific rate, that mutation rate genome will continue to exist because it's 'carried' by an otherwise healthy genome.

Sexual reproduction supports this concept very well. Fathers share only a portion of their actual genome with their offspring, (effectively a very low copy fidelity from parent to offspring.) And yet this is the most powerful type of reproduction because it allows for rapid adaptation to changing enviroments. However it arose, it's here to stay.

[Scott Aaronson]

Remember, folks, evolution doesn't work for the good of the species, and there's no Evolution Fairy who tries to ensure its own continued freedom of action. It's just a statistical property of some genes winning out over others.

Right, but if the mutation rate for a given biochemistry is itself relatively immutable, then this might be a case where group selection actually works. In other words, one can imagine RNA, DNA, and other replicators fighting it out in the primordial soup, with the winning replicator being the one with the best mutation properties.

[J Thomas]

"It’s actually a common misconception that biological systems should have mechanisms that allow a certain number of mutations for the purpose of accruing beneficial adaptations. From an evolutionary perspective, all genes should favor the highest possible fidelity copies. Any genes that have any lower copying fidelity will necessarily have fewer copies in future generations and thus lose the evolutionary game to a gene with higher copying fidelity."

I sympathise with the sentiment, but the example is wrong.

A sequence that affects mutation rates within its own linkage group will be selected along with the mutations it allows.

Since on average there are more deleterious mutations than favorable ones, on average mutations will have a net negative effect at first. However, the negative effects on average increase logarithmicly over time (because the worst of them have their effects up front and the minor ones keep costing longer), while the positive effects on average increase logisticly. That is, at first they increase exponentially but that's reduced for two reasons -- they will tend to saturate the population and toward the end they will face competition from other advantageous mutations.

So the mean time between mutations ought to be arranged by the lnked mutation-rate gene to maximise the average benefit. Too many mutations too soon result in too big an upfront cost. Too few mutations and something else outcompetes.

When a linkage group can't compete with a mutant that has better survival, it dies out completely. If it mutates at the proper rate it has a *chance* to survive past the time it would have been outcompeted. Or something descended from it has that chance. Each gene sequence surviving today is descended from one that didn't mutate so fast that it failed and also didn't mutate so slow that it was replaced by something better. It's no good doing perfect copying if the result is extinction.

So a gene that affects mutation rate ought to optimise the mutation rate on its own linkage group. But it gains nothing by increasing the mutation rate on other linkage groups. If it's with them for one generation then it shares in the bulk of the unfavorable selection. But then it gets separated from the mutations it caused and it does not share in the long-term favorable effects.