by Joe Felsenstein and Michael Lynch
The blogs of creationists and advocates of ID have been abuzz lately about exciting new work by William Basener and John Sanford. In a peer-reviewed paper at Journal of Mathematical Biology, they have presented a mathematical model of mutation and natural selection in a haploid population, and they find in one realistic case that natural selection is unable to prevent the continual decline of fitness. This is presented as correcting R.A. Fisher’s 1930 “Fundamental Theorem of Natural Selection”, which they argue is the basis for all subsequent theory in population genetics. The blog postings on that will be found here, here, here, here, here, here, and here.
One of us (JF) has argued at The Skeptical Zone that they have misread the literature on population genetics. The theory of mutation and natural selection developed during the 1920s, was relatively fully developed before Fisher’s 1930 book. Fisher’s FTNS has been difficult to understand, and subsequent work has not depended on it. But that still leaves us with the issue of whether the B and S simulations show some startling behavior, with deleterious mutations seemingly unable to be prevented from continually rising in frequency. Let’s take a closer look at their simulations.
Basener and Sanford show equations, mostly mostly taken from a paper by Claus Wilke, for changes in genotype frequencies in a haploid, asexual species experiencing mutation and natural selection. They keep track of the distribution of the values of fitness on a continuous scale time scale. Genotypes at different values of the fitness scale have different birth rates. There is a distribution of fitness effects of mutations, as displacements on the fitness scale. An important detail is that the genotypes are haploid and asexual — they have no recombination, so they do not mate.
After giving the equations for this model, they present runs of a simulation program. In some runs with distributions of mutations that show equal numbers of beneficial and deleterious mutations all goes as expected — the genetic variance in the population rises, and as it does the mean fitness rises more and more. But in their final case, which they argue is more realistic, there are mostly deleterious mutations. The startling outcome in the simulation in that case is there absence of an equilibrium between mutation and selection. Instead the deleterious mutations go to fixation in the population, and the mean fitness of the population steadily declines.
Why does that happen? For deleterious mutations in large populations, we typically see them come to a low equilibrium frequency reflecting a balance between mutation and selection. But they’re not doing that at high mutation rates!
The key is the absence of recombination in these clonally-reproducing haploid organisms. In effect each haploid organism is passed on whole, as if it were a copy of a single gene. So the frequencies of the mutant alleles should reflect the balance between the selection coefficient against the mutant (which is said to be near 0.001 in their simulation) versus the mutation rate. But they have one mutation per generation per haploid individual. Thus the mutation rate is, in effect, 1000 times the selection coefficient against the mutant allele. The selection coefficient of 0.001 means about a 0.1% decline in the frequency of a deleterious allele per generation, which is overwhelmed when one new mutant per individual comes in each generation.
In the usual calculations of the balance between mutation and selection, the mutation rate is smaller than the selection coefficient against the mutant. With (say) 20,000 loci (genes) the mutation rate per locus would be 1/20,000 = 0.00005. That would predict an equilibrium frequency near 0.00005/0.001, or 0.05, at each locus. But if the mutation rate were 1, we predict no equilibrium, but rather that the mutant allele is driven to fixation because the selection is too weak to counteract that large a rate of mutation. So there is really nothing new here. In fact 91 years ago J.B.S. Haldane, in his 1927 paper on the balance between selection and mutation, wrote that “To sum up, if selection acts against mutation, it is ineffective provided that the rate of mutation is greater than the coefficient of selection.”
If Basener and Sanford’s simulation allowed recombination between the genes, the outcome would be very different — there would be an equilibrium gene frequency at each locus, with no tendency of the mutant alleles at the individual loci to rise to fixation.
If selection acted individually at each locus, with growth rates for each haploid genotype being added across loci, a similar result would be expected, even without recombination. But in the Basener/Stanford simulation the fitnesses do not add — instead they generate linkage disequilibrium, in this case negative associations that leave us with selection at the different loci opposing each other. Add in recombination, and there would be a dramatically different, and much more conventional, result.
Technical Oddities
Most readers may want to stop there. We add this section for those more familiar with population genetics theory, simply to point out some mysteries connected with the Basener/Stanford simulations:
1. One odd assumption that they make is that any fitness class that has a frequency below 1 part in a billion gets set to 0. This is not a reasonable way to take genetic drift into account, as all fitness classes are subject to random fluctuations. We imagine such a treatment is a minor issue, relative to the enormous mutation pressure imposed in their study. But someone should check this, which can be done as their Javascript source can be downloaded and then made comprehensible by a Javascript beautifier.
2. The behavior of their iterations in some cases is, well, weird. In the crucial final simulation, the genetic variance of fitness rises, reaches a limit, bounces sharply off it, and from then on decreases. We’re not sure why, and suspect a program bug, which we haven’t noticed. We have found that if we run the simulation for many more generations, such odd bouncings of the mean and variance off of upper and lower limits are ultimately seen. We don’t think that this has much to do with mutation overwhelming selection, though.
3. We note one mistake in the Basener and Sanford work. The organisms’ death rates are 0.1 per time step. That would suggest a generation time of about 10 time steps. But Basener and Stanford take there to be one generation per unit of time. That is incorrect. However the mutation rate and the selection coefficient are still 1 and 0.001 per generation, even if the generations are 10 units of time.
Joe Felsenstein, originally trained as a theoretical population geneticist, is an evolutionary biologist who is Professor Emeritus in the Department of Genome Sciences and the Department of Biology at the University of Washington, Seattle. He is the author of the books “Inferring Phylogenies” and “Theoretical Evolutionary Genetics”. He frequently posts and comments here.
and
Michael Lynch is the director of the Biodesign Center for Mechanisms of Evolution at Arizona State University, and author of “The Origins of Genome Architecture” and, with Bruce Walsh, of “Genetics and Analysis of Quantitative Traits”. Six of his papers are cited in the Basener/Stanford paper.
The smarter ones are off writing books to sell to the chumps.
Glen Davidson
DNA_Jock,
So if I burn a fire and you drive your car the events are dependent because both depend on carbon chemistry?
How are the origin of transcription factors dependent on the origin 3 prime and 5 prime splicing sites?
DNA_Jock,
Sure, and you remove them from the equation.
DNA_Jock,
Not when you make the claim the equation is inherently wrong. The equation may need to be modified based on discussion. We are talking about a 1300 AA protein and we have not even brought a chaperone protein into the discussion.
Yes, they are “not independent”. There’s no need for a causal relationship between the two events, merely correlation — you may be confusing yourself with your elision from “not independent” to “dependent”.
Imagine that there is no carbon chemistry. Has that affected both probabilities?
Your equation is inherently wrong if any two events in it are not independent. Weed out all of the not-independent pairs and get back to us.
Or, you could learn about conditional probabilities.
Rumraket, to Sal:
Rumraket,
You’re asking Sal if he will be honest. I think you already know the answer to that question.
Still too stupid to grasp an iterative process with selection feedback and inheritance takes far less time than pure random trial and error.
But we have. Science has known about it for the better part of a century. You can learn about it in any freshman Biology 101 class.
Just because you personally refuse to understand it doesn’t make the process of evolution not exist.
Faizal Ali:
The smarter creationists (including Basener and Sanford, it seems) know they will have their asses handed to them if they participate here. The dumber creationists forge ahead obliviously.
Adapa,
Really? Do you understand the claim you are making?
DNA_Jock,
I think all the events I mentioned work well in a probability equation with the possible small modification or error in the overall probability due to minor “not independent” factors. At the end of the day we don’t have a viable evolutionary mechanism even when you talk about the evolution of a single protein.
Not one we can dumb down enough for you to understand it anyway. But that’s not science’s problem, it’s all yours.
Adapa:
colewd:
Heh.
Most of them won’t hang out at Sandwalk either
Sal, You are being unclear again. I have no idea what you are getting at. Please explain why fitness variation would pose a problem for the breakdown of LD.
The question regarding Basener and Sanford’s example is, have they discovered some new argument showing that selection is ineffective against deleterious mutation?
The answer is no.
The cases that you are bringing up are known: (1) that if there are too many slightly deleterious mutations with small selection coefficients they can cause fitness to continually decline, but only if as one gets away from high fitness the frequencies of advantageous mutations does not rise. And (2) if there is no recombination one can get Muller’s Ratchet effects that continually reduce fitness.
Those are not new, and they are not exciting.
No, there seemed to be something unexpected in the Basener/Sanford example. It did not have finite populations so Muller’s Ratchet did not apply.
But MIke and I have picked it apart and what do we find? No recombination, and a distribution of net fitnesses of the haploid genomes that implies a seriously unrealistic pattern of interaction between the effects of mutants at different loci. With the result that there is a strong, and bizarre, pattern of linkage disequilibrium — selection at different loci is counterposed so that the net effect is that mutation overwhelms selection.
So it’s back to arguing about the previous effects, which are well-known, not new.
WTF, they are all carbon so they are not independent? Then NOTHING is independent. Name ONE thing that is independent.
This is the most fucked up logic I have ever seen. Is that what they teach in math school?
Or maybe in math school they teach that weather is an example of something that is self-organised and self-sustained.
I guess its sort of like when an objects falls off a table, it sustains falling until it stops falling, and then it sustains not falling until it falls again. And none of the events are independent. Because they all involve gravity!
You object to the concept of a self-sustaining cause, and in it’s place you advocate a self-sustaining cause. And you are not even aware of it.
Sad, ironic, hilarious.
Joe Felsenstein,
How often does true haploidy even occur in Nature?
Individual prokaryote cells can contain linear chromosomes in multiple copies.
https://microbewiki.kenyon.edu/index.php/Chromosomes_in_Bacteria:_Are_they_all_single_and_circular%3F
Not that it matters – we have long known the existence of merodiploids permits true sexual reproduction in the Joe Felsenstein sense of the word if not Allan Miller’s: i.e. recombination with outsourcing.
So, what exactly is the distinction between haploids and diploids, again?
Meanwhile, I am struck by a recent paper explaining the acquisition of multiple copies of genes that promote healthy sperm function and mitigate gene loss via mutation.
I always reckoned the number of palindromic repeats in the Y was to prevent illegitimate COs with the X during Metaphase I.
A recent study suggests otherwise. They recorded a high rate of “gene conversion events” within the palindromic sequences on the Y chromosome – which allows damaged genes to be repaired using an undamaged back-up copy as a template.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5591018/
Doesn’t this also throw a wrench into the analysis?
Joe,
Thanks for your response. First off, I agree from the standpoint of Fisher’s actual theorem, it doesn’t seem to be reflected explicitly in development neo-Darwin/moder synthesis theory.
I read some of the references by Edwards (whom you recommended). Edwards said in:
R.A. Fisher’s gene-centred view of evolution and the Fundamental Theorem of Natural
Selection
But I think a charitable reading of Basener Sanford (2017) would permit the claim Fisher was first to link Darwinism and Mendelism based on Fisher’s 1918 paper:
https://en.wikipedia.org/wiki/The_Correlation_between_Relatives_on_the_Supposition_of_Mendelian_Inheritance
The not-so fundamental Fundamental Theorem came later in the 1930 paper. I scoured the earlier edition your book Theoretical Evolutionary Genetics to actually find a formula stating Fisher’s theorem, and that’s why I noticed it’s complete absence from earlier editions of your book, so I know from that, it is definitely your view Sewall Wright’s formula was foundational, not Fisher’s.
It is understandable one might think Basener Sanford 2017 claim Fisher’s formula was foundational, but I can attest I appraised them in 2016 and thereafter of our discussion in December 2015 where you said Fisher’s theorem was not so fundamental:
So understandably I read the meaning of the paper differently than you would! But their choice of words and the meaning of what they are saying is worth clarifying. I leave that discussion between you Michael Lynch and Bill and John….
But, backing up a bit, my understanding is that according to Queller 2017, there is a move (including Michael Lynch and Walsh) to create the hierarchy depicted below.
Also, while backing up a bit, there are three formulas. One is what I call the Bonkers Formula which Gruar apparently used to argue “If ENCODE is right, evolution is wrong.” This applies to recombining diploid populations like humans.
I derived it here:
http://www.creationevolutionuniversity.com/science/?p=22
The way I interpret the Bonkers Formula is that it is relatively independent of the mutation/selection balance formula. This formula would take precedence over the mutation/selection balance formulas since hypothetically, from a medical standpoint, we can have harmful traits that have neutral to “beneficial” selection coefficients. Thus the ratio of “deleterious” to “beneficial” is moot if the absolute number of deleterious traits is high enough. It doesn’t make sense that a population is getting better if for every increase of IQ points and memory we add a kidney defect…..thus there is the never ending clash of “fitness” in the pop gen sense vs. fitness in the medical sense. So as far as Michael Lynch’s studies on “compensatory mutations”, they are of little comfort to those suffering heritable diseases, since having 13 babies like Octomom doesn’t necessarily translate into more personal well being….
I call the above formula the Bonkers Formula because Graur claimed ENCODE was “bonkers” because of that formula (though I think he punched some numbers in his calculator wrong). It also seems to accord with what you said on page 157-158 of your book for recombining diploid population in relation to ENCODE:
So my reading of what you say is that independent of Fisher’s theorem, there is a point where enough bad mutation will lead to decline. My understanding (which could be wrong) is that mutational load may or may not be independently derivable from Fisher’s theorem? Is that right?
Then there are the mutation selection formulas
https://en.wikipedia.org/wiki/Mutation%E2%80%93selection_balance
haploid:
diploid:
These aren’t derived from Fisher’s formula, as far as I can tell. Aren’t they for the infinite population size case, and aren’t they assuming s is some fixed value rather than s being a mean? Do those formulas apply in the finite population case???
Thanks in advance.
stcordova,
This shows their claim that fitness is generally decreasing in the population based on mutations overwhelming selection. Previously in the paper they show that selection can also reduce genetic variation so there are two counter forces to basic evolutionary theory. I agree with Joe F that it is unlikely these models have not been explored in this way.
I think what might really be new here is someone is looking at the data and the models and questioning if the overall theory is rational. In one case we get genetic break down and in the other we get a lack of genetic variation to generate diversity. What is needed for evolutionists is the Goldilocks theorem.
Speaking of fundamental theorem’s of evolutionary biology, the diagram above place’s Price’s equation as THE most foundational, not Fisher’s theorem.
The irony is that Price became a creationist!
I must give credit to JohnnyB for finding the above gem for me 11 years ago:
colewd,
Thanks for your thoughts. I’m divided on my views of pop gen because the notion of fitness is like quicksand. Relative fitness is treated for the most part like some unchanging constant in a lot of analyses. There are a few papers that uses changing relative fitness, but they are hard mathematically even for “simple” cases.
This doesn’t look right at all! All the math is like a skyscraper built on quicksand. So, I decided to focus more on biochemistry and chromatin and 4D nucleome than pop gen and phylogeny.
A lot of this discussion in this thread is a learning exercise for me. I probably will explore some of this more in my public “notebooks” or personal “journals” at the spinoff of this blog:
TheSkepticalForum:
http://theskepticalforum.org/index.php?board=2.0
No disrespect intended to the field, but population genetics is less a priority for me personally than basics in biology like cellular biology, immunology, neurosicience, anatomy and physiology, virology, microbiology, etc. These are undergrad fields I’m not sufficiently acquainted with but which I think I need to get acquainted with.
Haha, oh man.
No, it’s entirely sensible. What you seem to have realized is that no mutation is unconditionally beneficial in any and all circumstances. We can always imagine some environment or circumstance, or being embedded in a larger genetic background, in which that mutation has a negative effect on reproductive success.
That isn’t a problem with the concept of fitness, rather it shows that the total number of possible environments, and genetic backgrounds in which those mutations occur, is incredibly large, and involves incredibly many different possibilities. Why you think that is a problem with evolution, or population genetics, is far from obvious. I mean aside form your a priori commitment to creationism.
Rumraket,
No it’s not. 🙂
phoodoo:
What do they teach in those mysterious places called “schools”, wonders phoodoo.
Still is!
We were talking about probabilities, and how to calculate them correctly.
Here’s a gravity-driven example, if that would help you understand:
Imagine two meteorites.
P(meteorite A is accelerating towards earth) = 1/1,000,000
P(meteorite B is accelerating towards earth) = 1/1,000,000
Okay so far?
colewd is saying that P(meteorite A AND meteorite B are both accelerating towards earth) = P(A) x P(B) = 1/1,000,000,000,000
And my response is “Not if they are next to each other”
[if they are next to each other, phoodoo, P(A AND B) will be ~= 1/1,000,000 — you guys would be off by six orders of magnitude.]
Adapa used the word ‘stupid’ to describe colewd’s position. Not a word I would generally use on TSZ, but (in a truly delightful coincidence) precisely the word my 18-y-o daughter used to describe his position. She just did Bayes’ theorem in school.
Time for a refresher, perhaps?
P.S. There is a way out of the “everything depends on carbon chemistry!” objection, but when you guys finally figure out what it is, you will also realize that it deep-sixes your calculations. Hint: I just mentioned it.
Never was. 🙂
DNA_Jock,
Hint: Math isn’t the real world Jock, its something we create to try to understand the real world.
Look, look, there is a squirrel Jock!
Now, were you going to name one event, ANY event that is an independent event, according to your rather outlandish distortion of reality?
Roll a pair of fair dice in a fair unbiased manner. The number which appears on one die is independent of the number which appears on the other die.
Not according to Jock.
DNA Jock said no such thing. You need to work on your reading comprehension or your honesty, or both.
He hired you as his spokesman?
Seems an odd choice, but I guess he has his reasons.
That’s right! That’s why it is important to get the math right! Otherwise you will end up mis-understanding things about the real world.
Sure. I’ll play.
A meteor crashes into Alpha Centauri today, and phoodoo types the letter q today.
Also independent events:
BUT
Phoodoo rolls a blue die today and gets a six
Phoodoo rolls a red die today and gets a six
NOT independent events.
I think it is safe to say that phoodoo will never understand the difference.
Have you spotted the hint yet, phoodoo?
I was commenting on your dishonesty, not speaking for him.
Well Jock, both DO involve atoms, sooooo…..I guess not independent.
stcordova,
You left out what happened to George Price after that …
DNA_Jock,
What if they are 50k miles apart? What if they are 500k miles apart?
Would you say your analogy fits well with the conditions necessary to translate a 1300AA protein? Do you think their dependent relationship is like two meteorites probability of being on similar spacetime curvature? Is it possible that each are more like two meteorites 500k miles apart where the “and” probability equation would be a good probabilistic estimation?
You’re not going to get away with that handwavy answer every time someone makes an argument that involves some calculations.
After all their math could corresponds well to what happens in reality. People playing poker at a high level are doing math, because it helps enormously to be able to work out the odds, because that math actually corresponds to what is happening on the poker table.
Reading that story now. Holy hell. Sad how Jesus just can’t seem to give good advice to Price.
Often it describes reality more closely than mere verbal models. I guess some people prefer the narrative, and conveniently forget that that is also just something we create to try and understand the real world.
Great question, Bill! What do you think?
My youngest daughter wants you to know that, in fact, P(A AND B) = P(A) x P(B | A), and she says that if you want to use the approximation P(A) x P(B) instead, then it is incumbent on YOU to DEMONSTRATE that P(B | A) is sufficiently close to P(B). For all pairs in your series.
Seriously, this is High School math.
As to Adapa’s ‘claim’ that
if only someone had written a noddy simulation in Basic that demonstrated this fact. Maybe searching for a Hamlet quote.
Which math corresponds to what happens in reality, all of it? What percent? What is the math on how much the math corresponds to reality?
You are not going to get away with that claim, without showing the math to prove the math is mostly right.
LOL! 😀
Have you ever seen someone try so hard to not understand such a simple concept?
DNA_Jock,
Can’t say as I blame you for bailing on your “well, they are all carbon chemistry so not independent” logic nonsense.
To think you came up with that whopper and it wasn’t even a weekend. The tequila must have been really flowing at your office party, whew!
That is the subject of debate here. But “math isn’t reality” isn’t an argument, it’s a handwave. An empty dismissal.
Those questions are nonsensical.
That also doesn’t make sense. You check whether your calculations correspond to reality by comparing the results of your calculations, to observations.
It is correct that, in infinite-population models, the mutant will rise to a frequency of 1 if the selection coefficient is equal to, or greater than, the mutation rate. So with a selection coefficient of 0.001, a mutation rate as low as 0.001 would be sufficient. (Haldane says that in his 1927 paper and it is easy to show).
To answer your other questions:
(1) The Basener and Sanford model is for an infinite population, even though they seem to think otherwise. It uses the infinite-population deterministic iteration from Claus Wilke.
(2) Yes, with smaller selection coefficients (in the tail of the Gamma distribution that is towards 0), mutation rates that are even smaller might be sufficient. But the mutation rates to the particular alleles that have those small selection coefficients are in fact even smaller than 0.001, and that would have to be taken into account. Because one only mutates to a particular value of the selection coefficient a small fraction of the time.
In any case, the Basener/Sanford model assigns selection coefficients, not to individual mutants at individual loci, but to haploid genomes. And the mutation rate is not per locus but per genome. And the particular distribution used in their ultimate example implies strong interaction between those loci in determining the fitness of the haploid genome.
Also, to take finiteness of the population into account one needs a Markov process, not a feterministic iteration, and one gets out an equilibrium distribution, not a single equilibrium frequency.