Kondrashov's Paradox: Why We Haven't Died 100 Times Over

Conversation
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The Many Faces of Genetic Entropy

Since John Sanford published Genetic Entropy & the Mystery of the Genome in 2005, genetic entropy (GE hereafter) has become a staple creationist argument. In his book, he defines GE thusly:

I am using the term entropy as it is most commonly used, i.e., the universal tendency for things to run down or degrade apart from intelligent intervention. Genetic entropy specifically means entropy as it applies to the genome. It reflects the inherent tendency for genomes to degenerate over time apart from intelligent intervention.

(Emphasis his.) Since Sanford likens genetic entropy to physical entropy, and by using the word “universal,” a plain reading suggests that Sanford is contending that GE applies to all life, not just humans or large-bodied vertebrates. Indeed, the only empirical paper ever purported to demonstrate the reality of GE was Carter & Sanford (2012) on mutation accumulation in the H1N1 virus - obviously not a large vertebrate. If viruses can degrade via GE, surely bacteria can as well.

Second, it has been understood that GE, as formulated by Sanford, relies on the existence of a “no selection zone” - a region of the distribution of fitness effects (DFE) in which selection does not operate at all. Sanford describes it like so:

Essentially every beneficial mutation must fall within Kimura’s “no selection zone”. All such mutations can never be selected for.

He emphasizes the word “never,” and throughout uses “no selection.” Again, a plain reading of this text implies that Sanford contends that nearly neutral mutations cannot be selected for or against.

Since 2005, both of these appear to have been walked-back. Carter (2012) has conceded that bacteria at least could survive GE, and Paul Price (@UncensoredPilgrims) now refers to “large multicellular eukaryotes” (LMEs) as those destined for extinction:

Given that most living things are not, in fact, LMEs, GE has fallen far from a universal principle akin to entropy.

It has also been recognized by Paul Price and Sanford himself that effectively neutral mutations are, in fact, selectable. Sanford wrote in 2020:

The nature of near-neutral mutations is such that they are not only un-selectable due to environmental noise, but they are also un-selectable because they are ‘noise’ to each other

And then a few sentences down wrote:

If an individual carries just one near-neutral mutation, it might be very weakly selectable

Hence, un-selectable does not really mean “un-selectable.” Paul effectively said the same thing in my recent debate with him, starting around minute 24:30.

Needless to say, Sanford has changed his mind about what appear to be otherwise core ideas of GE or wrote Genetic Entropy in such a way as to obscure what he really meant.

Creationist supporters of GE appear just as confused as we are. In my recent debate with Paul, he directly equates GE with the stochastic load paradox (Kondrashov, 1995):

Because the stochastic mutation load paradox [genetic entropy] appears real, it requires a resolution.

Above, Paul is quoting Kondrashov but adds the bracketed genetic entropy. He is even clearer in the comments, where he writes:

He said he had not resolved Kondrashov’s paradox (which is, in fact, GE!), which means he now effectively disavows his own published work.

Emphasis mine. As with Sanford, a plain reading of this suggests Kondrashov’s paradox = GE. However, Paul has since walked this back when asked to clarify in the forums:

Perhaps the most confusing part for all of us is the “and then some.” This is truly where the mystery lies. Indeed, we may never know.

Kondrashov’s Paradox Made Clear

Despite the confusion of what the core arguments of GE are and what sorts of organisms they are relevant to, both Sanford and Paul are clear that Kondrashov’s paradox is “at the heart of GE.” While this paradox is often invoked, I’ve never seen a creationist derive the paradox or talk through the assumptions involved in classic genetic load theory. If Kondrashov’s paradox is central to GE, then each assumption involved should be acceptable to creationists and comport to biological reality sufficiently. Thus, let’s derive the stochastic mutation load and chat through how we arrive at the paradox.

Firstly, some background on the genetic load. Unlike GE, the genetic load is an important and well-established concept in population genetics. Broadly, it refers to the reduction in fitness of a population due to recurrent mutation. No population is free of deleterious mutations; even when selection is highly efficient, new harmful mutations continue to appear such that some proportion of the population possesses them. Most eukaryotes possess a large genetic load; humans, for example, likely carry several hundred to even thousands of deleterious mutations per individual. The genetic load is very real.

What is the impact of all these harmful mutations? Let’s derive the classic mutation load from Haldane (1937) to get a taste of load theory. For a single locus, we have three possible genotypes, AA, Aa, and aa. There is recurrent mutation from A > a, which occurs at rate \mu; a is harmful with effect -s, and recessive to A. Let p be the frequency of A and q the frequency of a. Fitness of each genotype is then:

p^2:1, 2pq:1, q^2:1-s

Mean population fitness is thus: \bar{W}=1-sq^2. At equilibrium between selection and mutation, \mu p = sq^2. Since q is deleterious, it’s assumed that its frequency is likely very small. At equilibrium it should be q \approx \sqrt{\mu/s}, and so it’s assumed that p \approx 1. Thus, \mu \approx sq^2. Plugging \mu in for sq^2, we see mean fitness is now: \bar{W}=1-\mu. Thus, the genetic load is the difference in mean fitness between a population with no mutations (W_{max}=1) and those with, \bar{W}:

L=\frac{W_{max}-\bar{W}}{W_{max}}

In the single-locus recessive case, L=\mu (it is 2\mu in the additive case).

For the multi-locus case, the classic load assumes that mutational fitness effects at each locus combine multiplicatively. Thus, the multi-locus load becomes:

\displaystyle \bar{W}=\prod_{i=1}^{n} (1-\mu_i)\approx \prod_{i=1}^{n} e^{-\mu}=e^{-U}

where n is the number of loci and U is the total number of deleterious mutations expected per individual each generation. Notice now that W_{max} is no longer the mean fitness of a population with no mutation at a single locus, but now a population with no mutations at all. This is because the reduction in fitness per locus is 1-\mu, such that the total load becomes L=1-e^{-U}. This is known as the mutation load.

Before we chat through the implications of this, I want to show how Kondrashov’s stochastic load is built using the same machinery as the classic mutation load. In the classic load, the expected frequency of q was \approx \mu / s. This assumes an infinite population in which genetic drift plays no role. But if drift is also important, then N_e will play a part in equilibrium frequencies. Kondrashov showed that, in a single-locus case, the expected frequency is:

\text{E}[q] \approx \frac{\mu/v}{\mu/v + e^{4N_es}}

where v is the rate of back-mutation from a > A. Next, let f be the fraction of nearly neutral sites within the genome:

f=\int_{0}^{1/4N_e} q(s)ds

The average selective effect against each of these sites is:

\bar{s}=f^{-1}\int_{0}^{1/4N_e}sq(s)ds

As with the classic load, if we assume all sites are independent with their own unique fitness effects, then the number of expected harmful near neutral mutations (equivalent to U in the classic load) becomes:

U=2G \int_{0}^{1/4N_e}\text{E}[q]sq(s)ds \approx Gf\bar{s}

where G is the size of the genome. Thus, defining the genetic load as we did above, we arrive at the stochastic load: L=1-e^{-Gf\bar{s}}. As with the mutation load, notice that this load is defined relative to a population with no nearly neutral mutations at all.

What is the Paradox?

Nothing presented thus far is obviously paradoxical. The paradox emerges when we start to plug in numbers. In humans, U \approx 2.2 (Keightley, 2012) - that is, each individual suffers around 2.2 new deleterious mutations per generation. The load is then L=1-e^{-2.2}=0.89. That is, the average fitness of the human population is only 11% of what it would be if there were no harmful mutations. Another way of stating this is that 89% of the population each generation suffer genetic or selective death.

This sounds like a lot of carnage. Some have argued that such a load is intolerable (Kondrashov & Crow, 1993; Reed & Aquadro, 2006). If selection acts on offspring viability, then each human female would need to have 2\frac{1}{1-L}=18.2 children on average to maintain the population size. While this is technically within a human female’s reproductive ability, it’s far above the average. A similar issue emerges in the stochastic load, except that instead of solely relying on the deleterious mutation rate, the dangerous zone is when s falls within 1/G \leq s \leq 1/4N_e.

In this view of the genetic load, there is an apparent paradox: the deleterious mutation rate is too high for population’s to persist, and yet here we are. As Kondrashov asks, “Why have we not died 100 times over?” Creationists interpret this paradox to mean “the mutation rate is too high, thus we couldn’t have evolved.” Either way, the paradox requires a resolution.

Why We Haven’t Died 100 Times Over

A Mutation-free Individual Doesn’t Exist

As the derivations above make clear, the load is defined relative to an individual with no mutations at all. The genetic load imagines taking the average Joe - one of us with all our deleterious mutations - and plopping us into a population of unloaded individuals. In such a context, we might, indeed, do quite poorly. Agrawal & Whitlock (2012) write:

In principle, a load of any magnitude is compatible with species persistence because heavily loaded individuals can have high absolute fitness when competing against one another, even though each would have negligible fitness if forced to compete in a population of unloaded individuals.

Galeota-Sprung et al. (2020) argue that in large multicellular eukaryotes, such an individual is exceedingly unlikely to exist. Sewall Wright (1977) argued the same thing, writing:

…if many loci are involved, the genotype that combines the [optimal] genotypes at all loci is in general so rare theoretically that neither it nor anything approaching it exists in a finite population.

Dobzhansky (1957) wrote concerning the genetic load:

What would a Drosophila and a man be like if they did not carry such recessive mutations? Perhaps they would be a superfly and a superman, but the fact is that such prodigies have never existed on earth. The species Drosophila pseudoobscura and Homo sapiens have been molded in the process of evolution as Mendelian populations which carry mutational loads.

Now, in the side-comments, Paul stated that:

Hopefully, the derivations above show why this is false. Both the mutation and stochastic load assume W_{max}=1; i.e., an individual is totally mutation free.

Galeota-Sprung et al. (2020) calculate the fitness of the most fit individual that might actually exist in a finite population. They find that it is e^{5\sqrt{n\mu s}}, where n is the number of sites capable of suffering deleterious mutations. Let \bar{W}=1, then, assuming the entire genome is functional and capable of suffering deleterious mutations of \bar{s}=0.0001, then L=1.22-1=0.22. This is obviously much lower than a load of nearly 90%. In the context of the mutation and stochastic load paradoxes, far fewer genetic deaths occur because the difference in fitness between the max and average are much smaller. Individuals in real populations aren’t that different.

Relative vs. Absolute Fitness

In both the mutation and stochastic loads, the mean fitness of individuals is relative. However, population persistence and growth rates depend on absolute fitness - the expected number of offspring an individual with a given genotype will have. While the genetic load may reduce a loaded individual’s ability to compete against an unloaded one, how that influences demography is opaque.

Agrawal & Whitlock (2012) derive a few simple expressions to try and connect the genetic load to ecology. They define L as the number of selective deaths that occur, \beta as the amount of precious resources an individual destined to die due to the genetic load consumes before dying, b is the birth rate and d the death rate, and I is the rate of resource conversion to reproduction. The carrying-capacity of the population is then:

\hat{N}=\frac{bI}{d}\frac{1-L}{(1-L(1-\beta))}

Notice that even if L is very large but \beta is small, the genetic load has a tiny impact on population size. For example, if the first term is 10,000 and the load is 0.89, if juveniles only utilize 1% of resources before they die then the equilibrium population size is 9,251 - about 92% of what it would be with no load at all. In other words, the earlier in life (gametes, zygotes, young juveniles) selection acts, the smaller the ecological impact of the load. Importantly, the load is still there, it just has a small demographic impact. Highly loaded populations can persist forever.

Stabilizing Selection

Another key assumption you might’ve noticed is that the mutation and stochastic load assume that selection acts on each mutation independently. This presumes that each mutation has an effect independent of the body that carries it, or the genomic context in which it finds itself.

In reality, most traits are polygenic (e.g., Boyle et al. (2017), Mathieson (2021)). When traits are determined by many genes, mutations don’t have independent fitness effects. Instead, mutations alter a trait value in some direction, the fitness effect of which is determined by the suite of other alleles an individual has. I described this in the debate starting at 38:22.

Stabilizing selection acts to maintain a trait around an optimum, with mutation and drift pulling it away. Charlesworth (2013) showed that the load for a polygenic trait is then a function of the genetic variance: L \approx Sna^2\pi where S is the intensity of stabilizing selection, n is the number of sites, a is the effect size of the mutation, and \pi is the genetic variance. Using data from Ward & Hillis (2012) and Kong et al. (2012), he estimated for humans that Sa^2=5e-7 given N_e=20,000, and with \pi = 0.001, the genetic load is a measly 0.05. That is, 5% of individuals suffer genetic death due to the load in humans. Again, this load is a far cry from 89%, and easily resolves Kondrashov’s paradox.

Nick Barton (2022) argued the same thing with respect to growing evidence that complex traits have a highly polygenic (what is called omnigenic) architecture. He writes:

Lynch (21) has applied this concept to argue that molecular adaptations that are under weak selection cannot be established or maintained in (relatively) smaller populations, imposing a “drift barrier” to adaptation. Along the same lines, Kondrashov (55) has argued that deleterious mutations with Nes ≈ 1 will accumulate, steadily degrading the population. Both ideas seem problematic if we view adaptation as due to optimization of polygenic traits: Organisms can be well adapted even if drift dominates selection on individual alleles, and, under a model of stabilizing selection on very many traits, any change that degrades fitness can be compensated.

The above is the result of the infinitesimal model of quantitative genetics, which I describe here. My opening in the debate was built around this specific resolution of Kondrashov’s paradox, though I’m not sure Paul really understood it.

Soft Selection (Density-dependence)

The final resolution I’ll discuss here (though by no means the last) is the specific mode of selection assumed under paradox models. All of the above discussion assumed that selection is hard, acting on absolute fitness independent of other genotypes. For example, a population of rabbits suffers dramatically cold weather due to a polar vortex and only those with thick enough fur survive.

However, a great deal of selection in nature is not like this at all. Bruce Wallace (1975) coined the term soft selection to describe the situation in which absolute fitness is a function of the availability of some resource and thus the density of superior competitors for that resource. In a species in which males compete for access to females, fitness variance, and hence selection, acts on male competitive ability. Importantly, all females will mate with someone even if the ideal male doesn’t exist.

Haldane (1956) argued that whenever a population is well-adapted to its environment and at or near carrying-capacity, selection is density-dependent (“soft”), operating on competition for space, resources, mates, etc. Near range edges, when populations have colonized a new area, or during dramatic environmental shifts, selection becomes density-independent (“hard”), as individuals struggle to gain a foothold.

As with stabilizing selection, under soft selection the genetic load is determined by variance in fitness between the best and worst competitors. Charlesworth (2013) argued this could be determined using the standard deviation of the natural logarithm for fitness. Assuming \pi = 0.001 and N_e=20,000, as above, with 5\times 10^8 sites under purifying selection, the load is only 0.0088 in humans. If we use the load calculation from the Stabilizing Selection section above, the coefficient of variation (another proxy for fitness variance) was 0.074. While higher than 5%, it’s still an easily tolerable load.

Charlesworth (2013) writes:

In general, if we treat fitness as a function of the probability of success in obtaining access to a limiting resource, the mean fitness of the population relative to the fitness of a hypothetical optimal genotype that has a very low chance of being present in the population is essentially irrelevant.

Summary

In conclusion, I’ve discussed the derivation and implication of the genetic load as it pertains to GE. If it’s true that the stochastic load is central to GE, as Paul claims, it does not appear theoretically to pose any serious issue to evolution. Note that I have refrained from discussing any empirical implications of the stochastic load (e.g., natural populations maintain high fitness despite relatively low population numbers). To summarize, I’ve made the following claims:

  1. Both the mutation and stochastic load are measures of the difference in fitness between an idealized, mutation-free population and a genetically loaded one.
  2. A mutation-free individual never exists in eukaryotes, implying the experienced load is much less because individuals are competing against equally loaded individuals.
  3. Genetic load is a population genetic measure, not a demographic one. To link the load to demography requires knowing when selection acts. If it acts early, its impacts are much diminished.
  4. Most traits are polygenic and hence don’t experience selection on individual alleles, but on the sum of their effects. In such a case, the genetic load is a function of the genetic variance, which is very small.
  5. For populations at or near carrying-capacity, most fitness variance is competitive and thus determined by density-dependent processes. The load is thus a measure of fitness variance, which is tiny in natural populations.
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Never say never! Here I am again, because the truth is I won’t sleep at night if I allow Hancock to make these claims unopposed. That was the whole purpose of the debate I did with him, but most of my time was spent on refuting Hancock version 1.0, which has now morphed into Hancock version 2.0 (with a completely different argument). At least this one more time, I am being forced to post a response again.

These cannot be honestly described as anything other than a strawman argument and quotemining. Hancock claims familiarity with Sanford’s work, and claims to have read Sanford’s book. How can that be?

On pg. 74 (3rd Ed), Sanford writes,

…one prominant geneticist [wrote] a paper entitled, “Why have we not died 100 times over?” (Kondrashov, 1995). The problem of the unselectability of near-neutrals is very real.

A large homogeneous population in a homogenous environment (for example, a typical bacterial culture) is more resistant to genetic drift because it sees much less noise and experiences much more efficient selection. Such populations usually have simpler genomes, fewer mutations per genome, and far fewer inter-genic interactions. Furthermore, they exist in large numbers and have very high rates of reproduction. Most importantly, every cell is subject to selection, independently, at every cell division. Selection in such systems is more effective, more precise, and can have much higher resolution. This means that in bacteria, a much smaller proportion of the genome is near neutral and unselectable. This is why theorists [like Hancock!] typically prefer to use microbial samples.

So here we see that Sanford does not consider bacteria to be endangered by GE in the same way as LMEs, right out of his book.

What we also see is that Sanford is directly referencing Kondrashov. So any reading of what Sanford means when he says “unselectable” that would put him in conflict with Kondrashov is entirely false. As I have been forced to point out multiple times, Sanford wrote his book to a lay audience. Saying “unselectable” is entirely accurate in the sense that drift overwhelms selection for effectively neutral mutations. It does not imply that there is a total lack of selection, or a selection coefficient of 0 (which would be strict neutrality, not near neutrality). All throughout his book, Sanford refers to these as near neutrals, not strict neutrals. Hancock is doing far worse here than being uncharitable. He’s totally misleading his audience about Genetic Entropy, and about Sanford’s views.

I must confess that I originally didn’t understand what Hancock meant by “a mutation free individual”. It turns out that all he is saying is that the genetic load model depends upon the assumption of a theoretical possibility of no mutations (which is entirely reasonable). Whether such a theoretical possibility was ever a reality entirely depends upon your worldview, but it is by no means the case that you must start with zero mutations segregating in order to model GE. Far from it. You can start with whatever number of segregating mutations you like, and fitness will still decline!

This is sleight of hand. Hancock has now subtly changed the conversation away from fitness effects (which he now denies exist on a per-mutation basis!) over to this amorphous concept of “traits”. And of course, now Hancock gets to assume that “traits” have selectable differences. No more problem folks! Just change the subject and the problem seems to vanish. But the problem is that this is not an actual resolution of the paradox at all. It’s pretending it doesn’t exist by pretending that most mutations are not, in fact, deleterious at all. It’s a rejection of neutral theory and a rejection of the basic population genetics that drove Kimura, Kondrashov, Lynch (and so many others’) work.

It doesn’t matter if selection is hard or soft. If the mutation is below the drift barrier, the effect of selection is negligible and is overwhelmed by drift. We see this happen in real time when we model it in SLiM, as I did.

Hancock has spent years making videos attacking John Sanford and Genetic Entropy, and now is acting like he has had no idea until now that Kondrashov’s stochastic mutation load paradox has anything to do with GE. Will he go take down all his videos on YouTube where he attacked John Sanford without even having a basic understanding of his work? Let’s see.

Hancock, like Charlesworth, is simply ignoring the problem and declaring it solved when it isn’t. I think it will be an interesting test to see if @sfmatheson chooses to take Hancock to task for this, or remains silent, given that his son works with the Masel Lab where they are still actively attempting to resolve Kondrashov’s paradox, and on whose website they state with reference to it, “We still don’t have an answer”. Clearly Charlesworth’s handwaving wasn’t satisfactory in their eyes, at least!

Yes, this is exactly like saying that a half-empty gas tank is a measure of fullness with respect to an idealized full gas tank.

This statement simply amounts to an a priori assertion that the evolutionary worldview is true.

No it doesn’t. It requires knowing, or inferring, a selection coefficient. Lethal selection events that occur in the womb, etc. won’t even show up in the measured DFE in the first place, so they are entirely outside of consideration.

The fact that traits are polygenic is a problem, not a help, to evolution. It means we need multiple genes to be working functionally together at the same time in concert, yet we have no mechanism other than randomness to produce that effect. It also reinforces the impossibility of selection acting to create such a state in the first place.

No, the load is determined by the sum effects of the segregating mutations on the average fitness in the population. That’s what Hancock tried to model in SLiM.

As exasperating as this game of whack-a-mole is, I pray my efforts here actually benefit someone.

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Before this topic turns into a game of Brockian Ultracricket, I’m going to treat this thread like an “Office Hours” thread for @talkpopgen and @UncensoredPilgrims. (I am assuming their mutual permissions at this point, please let me know if you object.

Others can comment too, but please keep your comments brief and to the point. Also consider if someone before you have already made the same point. Please use Side Comments to hash out what should be asked and how to ask it, keeping chatter in the main thread to a minimum.

I will move comments to/from Side Comments as needed, but maybe not quickly.

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So, let’s take this pile one steaming lump at a time.

First, here is what Zach actually wrote:

Go ahead and read it. I’ll wait.

And here is what Joanna Masel and her colleagues write on the lab website, quotemined by our apologist:

In the title of a 1995 paper, Alexey Kondrashov posed the puzzle “Why have we not died 100 times over?”. We still don’t have an answer.

Literate adults can see the difference, but I’ll spell it out for others: Zach claims that load (as described within GE) is not a “serious issue to evolution” (@talkpopgen please clarify or correct as needed), while Masel et al. are saying (at the beginning of a paragraph that poses a question; see below), that we don’t yet understand why populations persist in the face of load. Masel et al. do NOT say that this is an issue for evolution. If there is disagreement between Masel et al. and Zach, it is about the extent to which the load paradox has been solved. Zach discusses several important factors, which he seems to view as “resolutions,” but he does not claim that the paradox is “solved.” I’d love to hear from him about how he views the status of the paradox today. Here, in the meantime, is what the Masel group writes on their website, in full:

In the title of a 1995 paper, Alexey Kondrashov posed the puzzle “Why have we not died 100 times over?”. We still don’t have an answer. It should take one selective death to purge one new deleterious mutation, but the mean number of new deleterious mutations per human seems to be more than 2, exceeding the number of selective deaths available. Most work on mutation load periodically renormalizes relative fitness to deal with the fact that fitness keeps declining no matter what, or else treats only a subset of the genome. Either way, this is dodging the problem, not tackling its fundamentals, which are not specific to human mutation rates or to small population sizes. One historical solution to the mutation load paradox was synergistic epistasis, often modeled in extreme forms such as truncation selection. Since then, data has come out showing that the mean effects of two deleterious mutations are extraordinarily close to multiplicative. We are exploring models by which very weak epistasis (compatible with data) might be sufficient and/or a “ratchet” occurs in which many small-effect deleterious fixations are counterbalanced by a much smaller number of large-effect beneficial fixations.

What we can all see, then, is that the apologist either doesn’t understand what he’s reading, or has chosen to quote it in ways that obscure what both Zach and the Masel lab think and do.

Now about my son and his work in the Masel lab. (He was a grad student with Joanna and finished 3 years ago.) I dare the apologist to read the most recent preprint from my kid and Joanna (and their colleagues) and then write a summary here on the forum. The preprint is here, and I’ve mentioned it on the forum before. It needn’t take more than a half hour to read the Discussion while looking at their Figure 6.

I think they might. I don’t know if the forum is read regularly by people who – like me, twelve years ago – are descontructing their Christian faith. But those people might find your writings helpful, for reasons that I will leave unwritten.

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Please can you explain that comment.

If genetic entropy is true then it will be the case that there are no mutation-free individuals and that individuals are competing against equally loaded individuals. And clearly that should reduce the effect of the genetic load in respect to within-species competition. Indeed, if that were not the case there would be selection reducing the genetic load.

So why should it amount to an “a priori assertion that the evolutionary worldview is true” ? Did you mean to refer to something else?

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Where is the quotemine? I sure didn’t see one. If we don’t know how populations can persist in the face of a load then

  1. The paradox is not solved and

  2. That’s a problem for evolution, since evolution requires that to happen.

He certainly does. If there is no serious problem for evolution, then the paradox must be solved! Kondrashov himself clearly considered it a serious problem. But wait, there’s more!

Turn to the debate, at timestamp 50:51, where I directly ask Zach if he has solved Kondrashov’s Paradox. He answered,

“No, but it has been solved, and I’m happy to talk you through it if you like.”

By what you have written, it would seem that you disagree with him, am I correct?

I understand what they wrote quite clearly. It is in direct opposition to Hancock’s claim that Kondrashov’s Paradox has been solved. I have not quotemined them in any way. You’re going to have to pick sides here: the Masel Lab or Hancock?

I have already read through it. It contradicts Hancock’s statements and published model, and they calculated an effective population size even smaller than the one I cited in the debate (they calculated it to be 7500). Which is making Kondrashov’s Paradox worse, not better.

The simulation they did bears little resemblance to anything realistic with respect to humans, and the numbers of generations they went through in the sim would be sufficient to kill off the species (200,000 generations!). That’s 100x as many generations as it took to kill off the species in SLiM using accurate human parameters (and over-generous fitness scaling logic).

That’s very saddening. The new faith you have adopted, faith in evolution, will not be able to save you or provide satisfying answers to the questions of life. I pray you will reconsider.

Side Comments on Kondrashov’s Paradox
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Careful with simplifying assumptions, such as pretending that traits, such as size and proportion, are not commonly polygenic and regulatory. Being larger or smaller overall or in terms of proportion can have deleterious or advantageous tradeoffs, depending on environment or competition. Such a selectable trait is the result of many genes, which do not necessarily all pull in the same direction.

Come on. Making an if…then argument does not imply any sort of novel realization of the premise. Did you miss that or are just deflecting?

That the universe including earth is ancient and the general framework of evolution is true, is not an a apriori assumption, but a necessary conclusion of manifold observations and clear thinking readily accessible to any layperson. I do not need to question that the earth is round and I need not revisit that it is old to satisfy the dogma of others.

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That’s correct, and that is in fact the problem. Since most individuals are nearly equally loaded with mutations, there’s no real mutation differential on which selection can act to purify the deleterious load.

When I said it was an a priori assertion, I was referring to Hancock’s claim that a mutation-free individual never exists. On a biblical worldview, all of the originally-created animals and humans would have been mutation-free. But as I also stated previously, one does not need to assume a mutation-free starting point in order to see genetic entropy happen. You can start with any amount of load you want, it will only affect the end result of how long it takes you to reach extinction.

That amounts to an equivocation. Now you’re using “genetic load” to mean something totally different than what Kondrashov and Sanford meant by it. Genetic load is not relative, it’s absolute. How many mutations are segregating in the population, what is the mean deleterious allele frequency, and what is the average fitness?

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I have sent several comments in the queue back for revisions. Please makes sure the post you are responding to has not changed, since you started writing, and update accordingly if it has.

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This despicable comment is a brilliant encapsulation of why the apologist’s writing here can help those who are finding their way out of Christianity and into freedom.

The apologist deserves no further response from me.

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I might be talking out of my arse here since I am in no way a specialist in this field, just trying to make sense of all the things I’ve read on the topic, but I can’t help wondering if some of the people working in the field don’t interact enough with fields like microbiology and molecular biology? Since they’re proposing some sort of paradox with what seems to me an already-known solution.

The solution to this putative paradox seems so simple to me, it just requires a bit of biochemistry to see how it works (it’s the physical/biochemical cause of diminishing returns epistasis). The data seems to have come from microbiology and biochemistry experiments.

Since mutations affect fitness through their physical/biochemical effects, and genetic components function ultimately through affinity-based effects, deleterious mutations are those that push the affinity too far in one direction (too strong or too weak, too specific or too promiscuous). Biochemically, if the affinity has degraded to many small effect mutations, a single mutation that creates a strong bond can compensate for many bonds that have become weakly degraded.

When enough deleterious mutations have accumulated in any given gene or locus that have degraded it’s function, a single large-effect compensatory one can restore it. This just IS the solution to the paradox. It’s true in both prokaryotes and eukaryotes (completely regardless of their size or the number of cells) because the underlying biochemical principles that govern the molecular functions of life are the same.

What is interesting to my view here is that it seems to me Fisher predicted the phenomenon with his geometric model, apparently entirely from rather abstract mathematical principles rather than any consideration of biochemistry. It is fundamentally correct.

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So in other words your objection was essentially pointless. There is nothing wrong with describing the situation according to the conclusions of evolutionary science - indeed that is exactly what he should be doing. As you admit it made no difference to the argument - but if it had you would still have been wrong to object as you did, since it would show a problem in your argument.

I don’t think so. Fitness is always dependent on environment, and for within-species competition the relative fitness of the other competitors is surely relevant. I don’t think that this aspect can simply be defined out of existence - and I see no reason to think that Kondrashov would disagree. How then is it taken into account?

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Do tell. Dr. Matheson and Dr. Joanna Masel can close up shop, problem solved!

This is a hopeless oversimplification. DNA is not merely about “bonds”, it is about information which encodes for functional complexity. This fact is attested to all throughout the literature.

You are also mistaken when you think this is a novel suggestion you’re making. It boils down to Kimura’s original proposed solution: mega-beneficial mutations. It has been addressed in Sanford’s book.

Kondrashov himself specifically rejected this solution:

“According to Kimura (1983: 248) VSDMs do not
cause any problem because (i) they accumulate very
slowly and (ii) their impact can be easily counter-
balanced by rare fixations of beneficial alleles. I do not
think that this is correct.”
(1995 paper)

If you get 5 geneticists in the room, and ask what the solution is, you’ll get 10 responses at least, many of them mutually incompatible.

This is, by the way, not the response given by Hancock as to how this paradox was resolved.

I used accurate parameters in my simulation, based on the literature, and it clearly showed fitness decline. Your claim that rare mega-beneficial mutations can come in and compensate for the load is simply not supported by the evidence, and is illogical on the face of it. I explained that in my original talk on genetic entropy quite clearly. It’s analogous to adding new functional parts to a car that is uniformly rusting out in all areas.

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I think this is what Matheson et al. are getting at in this paragraph of their Discussion:

Our simulations confirm that the solution to population persistence is a pattern of many small mutations, each of which cannot be effectively cleared, being counteracted by compensatory mutations with global effects. This pattern is part of drift barrier theory 23,24,80–87.

Their paper doesn’t emphasize the biochemical/molecular underpinnings of this, but I can assure you that the authors are not unaware of it and are well connected to biochemistry and microbiology.

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A select coefficient describes the relative fitness of a particular variant relative to other variants within a given population. The term does not apply on an individual basis.

They can be “working functionally together” AND also provide robustness against mutation in the form of redundancy. For example, 5 genes work together to produce phenotype X. But phenotype X is maintained if only 1 gene is mutated, and phenotype X is only marginally compromised if only 2 genes are mutated.

This is exactly why most phenotypes are quantitative, a product of many small discrete and often interdependent additive effects.

Such robustness is also why evolution is definitely possible.

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Genetic Load Clarifications

Given Paul’s response, I think I was unclear about the assumptions of the genetic load model so I want to begin by spelling them out explicitly:

  1. Mutations are assumed to impact viability. In this context, selection = death of the individual. If s = -0.01, then 1 out of every 100 individuals die without reproducing.
  2. Selection is assumed to act on each mutation independently.
  3. Each mutation has its own effect independent of the genomic context it is within – i.e., no epistasis and traits are relatively simple such that selection can “see” each individually.
  4. 1-3 is also true in all the SLiM models, both mine and Paul’s.
  5. The genetic load is a measure of the relative fitness difference between a population with no mutations and one with mutations. For example, a load of 50% means that the loaded population has 50% the number of offspring as a hypothetical population with no mutations. (Ask yourself: how many is that, exactly?)

Response

Now, to the specific content of Paul’s response.

What Sanford wrote:

…the universal tendency for things to run down or degrade apart from intelligent intervention. Genetic entropy specifically means entropy as it applies to the genome. It reflects the inherent tendency for genomes to degenerate over time apart from intelligent intervention.

Essentially every beneficial mutation must fall within Kimura’s “no selection zone”. All such mutations can never be selected for.

Notice the use of the words universal, specifically, and his emphasis on the word never. Furthermore, note that the first quote is the only time Sanford gives a definition of GE in the book. When you define a concept, it should be exact. And yet, Paul claims:

Paul seems to be arguing that when Sanford uses the word universal or never, so long as he uses different verbiage later on, then he has not in fact defined GE in such strong terms. The fact that you can find instances in which Sanford clearly does not mean that GE is universal or that un-selectable always means un-selectable doesn’t mean that I quote-mined him – it means that he contradicted himself. He could have said generally or rarely. Words have meaning.

The prefix un- means “not” or “the opposite of.” Hence, unselectable literally means “not selectable,” which implies a total lack of selection. Again, Sanford’s own words were “…all such mutations can never be selected for.”

If we’re more generous, then “unselectable” is at best misleading. Consider the case in which Ns = 0.5 (beneficial, near neutral) and Ns = -0.5 (deleterious, near neutral). The former is 7.38 times more likely to fix than the latter, and 2.31 times more likely to fix than a strictly neutral mutation. This seems nothing like “unselectable”, in any sense of the word.

To clarify, it’s that the load itself depends on it. That is, the load is relative to a hypothetical, mutation-free individual. For example, for U = 2.2, then L = 0.89, so fitness of loaded individuals are only 11% of unloaded ones. But that means nothing demographically. If we imagine a mutation-free superhuman, how many children would they have on average? Maybe they’d have 50 children! But a typical loaded human, with only 11% their fitness, can have a max of 5.5.

The point here is that to translate the load into the question “will this population persist?” requires additional assumptions about how and when selection acts as Agrawal & Whitlock (2012) show.

It does not, it’s a mathematical certainty given our genome size. For such an individual to exist at equilibrium – which is what the stochastic load is measuring, to be clear – would require population sizes so large as to ensure that an individual could be born without any mutations. This assumption is reasonable in bacteria (Galeota-Sprung et al. (2020)), but certainly not humans.

Consider a mutation that kills you as a zygote with a probability of 0.0001. That means that 1 out of every 10,000 fertilization events will result in a selective death. This is, quite literally, a selection coefficient of 0.0001, but occurs in the womb. These mutations can still be inferred in the DFE because they will be slightly less represented in adults relative to their rate of occurrence.

Now consider a mutation that doesn’t kill you until you’re 15 – old enough to have used-up a lot of resources but not yet old enough for reproduction. This mutation – despite having the same selective coefficient – is demographically more harmful. A parent has wasted a great deal of reproductive potential funneling resources into an individual destined for death as opposed to a virtually unnoticed zygotic death. This is quite literally what the equation from Agrawal & Whitlock (2012) demonstrates that I showed in the OP.

You did not model soft selection in SLiM – you modelled hard selection. Both our simulations are irrelevant to soft selection. This is because each mutation had an effect on fitness irrespective of any other member of the population (i.e., density-independent). Ecological theory (e.g., Haldane (1956); Wallace (1975)) can’t just be thrown out. As any ecologist will tell you, most selection is density-dependent, driven by competition for space, resources, and mates.

As I explained, when selection is soft, the load is greatly reduced because it is drive solely by fitness variance, which is small in natural populations.

This is surprising. The entire agricultural enterprise is built on the fact that “traits” are selectable. The fact that dog breeds exist are because “traits” have selectable differences. This is a curious statement to me. And yet, underlying the vast majority of those traits are thousands of genes each of which have extremely small effects. This is what I explained in my opening in the debate. It also forms the theoretical foundation of quantitative genetics, going back to Fisher (1918). “Traits” are not amorphous – I gave several examples in the debate: corn oil production, beak depth, human height.

But we can go further. Imagine an enzyme that has an optimal expression level of 1000 transcripts per cell. If the initial expression level is 200, any mutation that increases expression is beneficial. This can occur via gene duplications, promoter recruitment from transposons, etc. Once at 1000 transcripts, any additional duplications are now deleterious, as are any deletions. The point of quantitative genetics is that mutational fitness effects are contextual.

I demonstrated mathematically that it is (see Charlesworth 2013), even at an optimum where all mutations would be harmful. My entire argument during the debate hinged on this fact. As Hledick et al. (2022) demonstrate, stabilizing selection on a great many alleles of extremely small effects maintained information with greater efficiency than strong selection. If you think this is incorrect, you need to demonstrate: 1) traits are not as polygenic as we think; 2) mutational effects are independent of genomic context; and 3) mutations have larger effects than we think they do.

I don’t think you realize that what you’re saying is “quantitative genetics is a rejection of basic population genetics.” This is especially odd given that both Kimura and Lynch started their careers in quantitative genetics. What is important to realize is that the degree to which classic population genetics is applicable vs. quantitative genetics is an empirical question. If we assume fairly simple traits – melanism in peppered moths, sickle-cell anemia, etc. – then classic population genetics works fine. But when traits are controlled by a very large number of genes, the underlying assumptions of classic population genetics fails. The degree to which these models fail is an area of active debate that can only be resolved with empirical data (e.g., how polygenic are traits? How do genes interact? What’s the relative importance of epistatic vs. additive genetic variance?).

I think this is a good time to reference Stephen’s response. My purpose in the OP (and the debate) is demonstrating we have resolutions to the paradox. Paul is correct that I said “we have resolved” in the debate – this is a incorrect and I appreciate Paul pointing it out. What I should have said is that we have resolutions to the paradox. Joanna would say the same thing, as noted in Matheson et al. (2025), which offers one such resolution.

Paul is flustered that there are so many possible responses to Kondrashov’s paradox, but the fact is that there simply a multitude of possible resolutions. I listed a few in the OP, Joanna has others, and Kondrashov himself has his own (he suggests synergistic epistasis). The lingering question is which is correct – that is, which of these resolves the paradox? Likely it is a mix. It is “unresolved” insofar as we don’t yet know which of the many resolutions is actually the solution. This is an empirical question, and empirical work always lags behind theory.

This leads me to this question. I would argue (to reveal my own bias) that it’s the reverse. These are theoretical questions that can be resolved with molecular biology, but the questions came first and as the data rolls in, we try to incorporate them into our models. Too often it seems like molecular biologists (e.g., the ENCODE consortium) make sweeping claims about evolution without reference to theory. Undoubtedly, we should all interact more often. I think a great example of this is Palazzo & Keijou (2022).

Under the infinitesimal model, Fisher’s geometric model of adaptation works beautifully. When traits are not polygenic or have large effect sizes, Wright’s shifting-balance theory works better (e.g., Lande & Arnold 1983).

I suppose this is good company to be in. Brian Charlesworth is perhaps the most famous living population geneticist (Joe, who I know is on this forum, is definitely in the running) with over 74,000 citations. His doctoral advisor was John Maynard Smith, who was the student of Haldane himself. Seems like it’d be worth it to read his work a little more carefully.

Lastly, as I mentioned above, Paul is flummoxed that more than one argument exists. This was particularly evident in the debate, where it seemed like he expected the debate to hinge on mine and Dan’s 2024 paper responding to Basener & Sanford (2018). However, the debate topic was “Are mutational effects a problem for evolution?” When he contacted me to request the debate, he did not say “I’d like to debate your paper” or “can you defend your paper”, but requested a broad debate about genetic entropy. In addition, he requested to be in the affirmative. When you are in the affirmative, you build a case – the negative deconstructs your case. Paul spent a great deal of his affirmative deconstructing things I had not said during the debate.

The argument I presented during the debate is what I consider the strongest argument against GE – that is, that most complex traits are highly polygenic, and thus selection is exceptionally effective despite being weak on each individual allele (e.g., Barton 2022). I chose this argument because it is much broader than Hancock & Cardinale (2024), is not restricted to a response to a single paper (Basener & Sanford 2018), and does not suffer from modelling assumptions (which we extensively discuss in the paper!). Furthermore, the infinitesimal predicts real, biological data, and has immense practical importance in agriculture.

Again, if Paul wished to debate specifically our paper, he should have said so and placed me in the affirmative (e.g., “Did Hancock & Cardinale 2024 disprove genetic entropy?”).

Paul’s Behavior

I want to conclude by commenting on Paul’s behavior here. Speaking to Paul directly, I understand that you feel as though you’re Daniel in the lion’s den in this forum, and that is enough to make anyone prickly. But by referencing Stephen and his son, attempting to pit them either against one another or me, is like walking up to the lion and slapping it in the face. Then, to say the following:

reveals the depth of your disrespect. You could have cited the paper or the Masel lab if you’d like, without specifically tagging a father in reference to his son. If you don’t understand why that is reprehensible, then spend some time away from the internet for a while. Touch some grass. Make some human connections. At the very least, keep your insults directed at me and don’t drag in bystanders.

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Side Comments on Kondrashov’s Paradox
Side Comments on Kondrashov’s Paradox
Side Comments on Kondrashov’s Paradox
Side Comments on Kondrashov’s Paradox
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2 posts were merged into an existing topic: Side Comments on Kondrashov’s Paradox

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Since a large fraction of individuals aren’t viable enough to be born, that’s trivially false.

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I’m sorry but I don’t think this paragraph implies what you present it as. You present this in response to Zach’s claim, that Sanford originally described GE as universally affecting all life.

But the paragraph you quote does not successfully constitute a refutation of what Zach wrote. That Sanford gives a number of reasons why bacteria would be less affected by GE does not mean he thinks it would be completely unaffected by GE.
At no point in what you quote there does Sanford write that bacteria are completely excluded from or immune to GE.

In fact as this .pdf written by Christopher Rupe and John Sanford (in 2015) proves, Sanford absolutely thinks bacteria such as the E coli in the LTEE (and the H1N1 virus) are subject to GE:

They write(my bold):

In summation, the most famous evolution experiment ever conducted (LTEE) that is being proclaimed to the world as a dramatic proof of “observable evolution”, is ironically one of the most powerful demonstrations of genetic entropy and de-evolution. This is consistent with the Biblical view of origins. The Bible teaches that because of Adam’s sin (see LogosRa.org article for genetic evidence for a literal Adam and Eve ancestry) we live in a fallen creation that is subject to the “bondage of decay” (Romans 8:21). Ever since the fall, the genomes of all living creatures have been degenerating due to the accumulation of mutations – this includes the populations of E. coli described in this article.

There is no strawmanning of Sanford going on here and you’re reading something into that vague paragraph it doesn’t say.

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I’m sorry but no, it’s entirely about bonds. Chemical bonds, and the physics that apply to them. It’s how the information comes to express itself, through chemical interactions. The information is an abstraction that only matters in so far as it takes some sort of physical effect. A naked DNA molecule left to itself in solution, no matter how much information you somehow compressed into it’s sequence, would accomplish nothing and would not be meaningfully different in terms of what it does from a DNA molecule of the same length with a completely random and scrambled sequence.

A protein coding gene, say, is only protein coding in virtue of a co-existing and already expressed physical translation system, where the bonds that form between the DNA and the transcriptional and translational RNAs and proteins can decode that gene. Without transcriptional and translation system components, the information would not get expressed, and thus have no physical effect. It would just be a DNA molecule that was relatively chemically inert.

In that sense it is only information in the context of someone (or something, some system) who/which can transcribe and translate it. The sequence information in genetic polymers is, like the fitness effects of mutations, entirely contextual and not intrinsic. An arbitrary mutation A->C is not intrinsically beneficial or deleterious. That effect arises only out of an already existing genetic and environmental context. The word ‘horse’ does not intrinsically mean horse, as any imaginable combination of letters and sounds could in principle take it’s place, and thus it’s meaning only exists in a context where communicators have been taught to interpret it as such.

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