What is fitness and how is it maximized?

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I’ve committed to spending the next month trying to review and understand the meaning of the concept of “fitness” in evolutionary biology and population genetics.

I’ve been re-reading sections of Joe Felsenstein’s book on population genetics which he generously provided free-of-charge here:

http://evolution.genetics.washington.edu/pgbook/pgbook.html

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Because of his generosity, I was quite happy to purchase his more famous book:

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Even as a card-carrying creationist, I’ll give my hearty endorsement for both books by Dr. Felsenstein.

In this thread I’ll post my best understanding of how fitness is defined, computed as well as how it might be maximized in theoretical and real populations. This will be an on-going project for the next couple of weeks as I hope to prepare teaching materials on the concept of fitness.

Thanks in advance to those who participate in this discussion.

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Allen Orr provided an article to try to explain the population genetic notions of fitness. He points out the various notions of fitness which I will contrast so as to set the stage for discussing the population genetic definition of fitness.

There are fit traits, fit individuals and fit populations.

The notion of fitness in the medical sense is not exactly fitness in the sense of evolutionary biology.

For example, there are conditions like sickle-cell anemia (the heterozygous expression) that might be considered an unfit trait in the medical sense but reproductively advantageous in a tropical environment. Thus it is seen as “fit” in the evolutionary sense, and unfit in the medical sense.

The evolutionary sense of a fit trait has at least two definitions. Orr says:

Without differences in fitness natural selection cannot act and adaptation cannot occur. Given its central role in evolutionary biology, one might expect the idea of fitness to be both straightforward and widely understood among geneticists. Unfortunately, this may not be the case; although evolutionary biologists have a clear understanding of fitness, the idea is sometimes misunderstood among general geneticists.

:open_mouth:

Fitness and its role in evolutionary genetics - PMC

So there is an idea, like say the color of a peppered moth, of a trait being more fit than another trait.

In the formal population genetic definition the absolute fitness of a trait is measured in terms of the number of offspring that will possess the trait in the next generation. It’s usually stated in terms of a coefficient like W, where W is the number of offspring that proceed from one individual which will have that trait.

Finally, the computer science world in attempting to model and utilize Darwinian ideas has it’s own approaches for defining fitness in evolutionary algorithms that seek to find optimal solutions to problems by searching for the most fit solution per generation. The most famous example is probably Dawkins weasel.

So there are 4 notions of fit for a trait:

  1. Medical notions of fitness
  2. informal notions of reproductively advantageous traits like a certain color of moth
  3. informal notions of fit in computer evolutionary algorithms
  4. formal definition of fit in terms of number of offspring

As of this writing, the notion of fit for individuals that possess many traits, and the notion of fit in terms how to characterize a population as fit or unfit is not yet entirely clear.

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I find that extremely confused. Medical fitness is irrelevant to evolution, notions in computer algorithms are irrelevant (though probably identical to other definitions), and the informal notion of advantageous traits is the same as the definition regarding numbers of offspring. You don’t tell us what Orr thinks are the two definitions, which I suspect are absolute and relative fitness. A hot mess.

Also, “fit” and “unfit” are not how fitness works. It’s a quantitative measure, not a qualitative one.

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Thank you for your comment and criticism, John, I’ll amend my teaching materials accordingly in light of what you said.

Now that is out of the way, I’ll focus on the Pop Gen notion of fitness in the next comment.

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To quote the conclusion of Orr’s paper:

Although Darwinian evolution is founded on the idea that some genotypes have higher fitness than others, the idea of fitness itself is fairly subtle. Fortunately, some, though not all, of the confusion can be cleared by distinguishing between fitness as a phenotype assigned to individuals and fitness as a summary statistic, e.g., absolute fitness, relative fitness, mean fitness, geometric mean fitness, etc. Fitness as a phenotype is generally unproblematic and confusion typically arises only when attempting to determine which summary statistic is most appropriate given a certain evolutionary scenario.

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A given gene can have various forms in a population, these various forms are called alleles.

For example, there are 3 major alleles of the galactosyltransferase gene that eventually determine the human ABO bloodtype.

Unlike humans, some organisms that are asexual. A population of an asexual species can be categorized by what allele for a given gene they carry.

Taking an simplified model for fitness from Dr. Felsenstein’s book;

Suppose we have an asexual population that has individuals carrying an allele we call A and another allele we call a, and the numbers of the individuals at any given time are:

N_A(t) = number of individuals at time t with allele A
N_a(t) = number of individuals at time t with allele a

The number of offspring that each set of individuals will yield in the next generation is W where

W_A = number of offspring each individual with allele A will have
W_a = number of offspring each individual with allele a will have

W is the absolute Darwinian fitness of each of the individuals.

Obviously there are some simplifications in this model, but that illustrates the definition of absolute Darwinian fitness in this most simple model.

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What was that intended to be? If you’re preparing teaching materials, why reinvent the wheel? Any evolutionary biology text should do.

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He isn’t preparing teaching materials. He’s preparing “teaching” materials - scientific material he can cherry pick and spin beyond recognition to push his YEC position. He’s made that abundantly clear over the years.

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The materials are for non-biologist creationists wanting a quick primer. There are lots of IT/Engineer type creationists for example that could benefit from this among people I know anyway.

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Timothy_Horton that was a personal attack on me in what should be a scholarly discussion about fitness. I’m putting you on my ignore list through the preferences feature.

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It would seem that if

W_A > W_a

then direction of evolution should be clear enough as far as where the population will go. But as models became more sophisticated, we see problems such as highlighted in this thread:

Progress in population genetics theory, it then seemed, would come from the application of this method to the study of systems with several genes. This ambition came to a halt in the context of the influential objections made by the Australian mathematician Patrick Moran in 1963. These objections triggered a controversy between mathematically- and biologically-inclined geneticists, with affected both the formal standards and the aims of population genetics as a science. Over the course of the 1960s, the mean fitness method withered with the ambition of developing the deterministic theory.

In addition to preparing teaching materials, I was interested why the more sophisticated models didn’t come with such clear conclusions as in the simple case above. And it seemed to take several decades for the rise and fall of fitness maximization methods.

To that end, it seems important to review the principles of fitness from the ground up to grasp the problems described in the paper in that thread.

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Sal, I really don’t think this is something you are equipped to teach, and no amount of chat on a web forum is going to help with that.

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If in the simple example,

W_A = W_a

there is no natural selection. For there to be natural selection this condition has to be violated. When this condition is violated it becomes meaningful to relate the absolute fitnesses in terms of their ratios with one another, and hence the concept of relative fitness emerges.

Since this is an asexual example there is no real distinction between allele and what is known as a genotype, but I will defer dealing with the nuance of sexually reproducing species for later.

Using Joe’s numbers from page 55-57, suppose

W_A = 101

and

W_a = 100

If we use the uppercase W to signify and absolute fitness we can use the lower case w to signify a relative fitness.

We can define a relative fitness w for each of the alleles A and a scaling down the absolute fitnesses through a process of dividing each of the absolute fitnessess by an arbitrarily chosen absolute fitnesses of one of the alleles.

To illustrate how to do this in this example, let’s arbitrarily choose to divide by W_a to scale the absolute fitnesses into relative ones:

\LARGE w_a = \frac{W_a}{W_a} = 1

\LARGE w_A = \frac{W_A}{W_a} = 1.01

we can further define a quantity s, where s is called the selection co-efficient as:

s \equiv w_A - w_a
this implies

s = w_A - 1

which implies

w_A = s + 1

provided we are scaling the absolute fitness by dividing by W_a. By doing this we are saying the s-coefficient favors A .

We can additionally represent the relationships of the relative fitnesses with this symbolic notation

w_A:w_a

where the colon means we are considering the relative fitness of the alleles A and a

but this notation implies (by substituting the values of w_A and w_a)

1 + s : 1

which is reflected in Joe’s book page 55-57, but I derived it in a slightly different way above.

BUT Joe points out a complication, suppose we make W_A as the value by which to scale the absolute fitnesses down to relative fitnesses?

then,

\LARGE w_a = \frac{W_a}{W_A} = \frac{100}{101}=0.99009900\bar{9}\bar{9}\bar{0}\bar{0}

\LARGE w_A = \frac{W_A}{W_A} = 1

following convention of defining s as:

s \equiv w_A - w_a

which implies
w_A - \frac{100}{101}=1 - 0.99009900\bar{9}\bar{9}\bar{0}\bar{0} = 0.00990099\bar{0}\bar{0}\bar{9}\bar{9}

One immediately sees that the value of s is dependent on which allele is used as a reference! In this case we say the s-coefficient is against a. This can be a source of confusion, obviously!

Furthermore, if allele A is used as a reference, and s=0.00990099\bar{0}\bar{0}\bar{9}\bar{9} the conventional symbolic representation for

w_A : w_a

must be

1 : 1 - s

which implies

1 : 1- 0.00990099\bar{0}\bar{0}\bar{9}\bar{9}

which implies

1 : 0.99009900\bar{9}\bar{9}\bar{0}\bar{0}

To restate where care is in order, note that this form of stating relative fitnesses of w_A:w_a by

1 + s: 1

is saying the selection coefficient, the s-coefficient is in favor of A

Whereas,

1 : 1- s

is saying the s-coefficient s is against a.

And again, s has a different meaning depending on which convention is adopted!

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Interesting to see all those quotes and paraphrases from my book. There are 500 pages to go – this is going to be a long thread … As for the alternative definitions of fitness, no, people who say “he’s always exercising, he’s very fit” aren’t talking about the same thing as the fitness used in evolutionary biology. The fact that users of genetic algorithms in engineering often call something other than the expected number of offspring the “fitness” is their problem, not ours. And no, the limitations on using maximization of mean fitness to predict where a phenotype will change is not directly relevant. When Sal gets further in chapter two, he will actually see there the equations for changes of gene frequency, which use the fitnesses of the individual genotypes. To get the changes of gene frequency, we don’t just wave our hands and say that the gene frequency will go to the value that gives the highest fitness.

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As some-one who has written GAs, I’d like to point out that there isn’t a problem. Words have different meanings in different contexts. Especially when using words in technical contexts.

If a GA writer uses “fitness” to mean “the output from a population member evaluation function” and not “the expected number of offspring” there is no more a problem than there is because you are using “call” to mean “give something a name” rather than “summon into the courtroom”, “communicate with via telephone”, “sing in order to attract potential mates” or “invoke a subroutine”.

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When Sal gets further in chapter two, he will actually see there the equations for changes of gene frequency, which use the fitnesses of the individual genotypes.

Yes, thank you. And speaking of chapter 2, this raises the question I’ve seen in literature, the “correct” formulation of Fisher’s Fundamental Theorem of Natural Selection.

My understanding from Crow and Kimura, 1970, An Introduction to Population Genetics Theory, page 270, is that this is an approximation of a special case of Fisher’s Fundamental Theorem, (or should I say not-so-fundamental Fundamental Theorem of Natural selection, NSF-FTNS):

kimura_crow_1970_NSFFTNS
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which I presume is the special case of single-locus 2-allele discrete time non-continuous population case. Is that right? This looks like this forumula in chapter 2 of Joe’s book:

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Joe, said this formula was Fisher-like a few years ago:
http://theskepticalzone.com/wp/absolute-fitness-in-theoretical-evolutionary-genetics/#comment-99127

Sal, this is a Fisher-like formula, a simplified version of one originally due to Sewall Wright.

I did not discuss the not-so-fundamental Fundamental Theorem of Natural Selection in the book, but I should put in a mention and citation. It turns out to be very hard to find a theorem that comes close to the FNTS and is also provable. There has been a big literature on this, with the most useful recent work by Anthony Edwards and Warren Ewens. You have to make a rather weird theorem to get one that is rigorously true.

The discussion that you found in Chapter II is the place in the book where I cover selection and mean fitness in haploids. See also section II.8 where I cover it for diploids.

That exchange was in December 2015, before Joe’s latest version of his book in 2017 that now mentions the NSF-FTNS.

That said, some authors cite this special case as THE formulation the NSF-FTNS. I presume they do this erroneously? For example Queller 2017:

fundamental_theorems_colored

or this website from Cold Spring Harbor Press:

http://evolution-textbook.org/content/free/notes/ch17_WN17B.html

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But then, I find a re-birth of Price’s characterization in Bill Basene’s paper that presents the following formulation which I can barely make heads or tails of and looks almost nothing like the above!!!

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Which I presume comes from Price’s 1972 paper, “Fisher’s Fundamental Theorem made clear”.

A copy of Price’s 1972 book from which Basener draws upon is here:
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.366.4070&rep=rep1&type=pdf

I presume the more complex looking version is for multiple loci? Is that for diploids too? Is it discrete time non-continuous populations? It was a little harder to discern what it meant than the other simpler formulations… I admit that.

Lessard’s 1997 notation was also a little hard to uncoil. I’m not even touching that yet.

So… what is the correct formulation of Fisher’s theorem? That is the question.

Thanks in advance.

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It’s going to be a long thread.

The discrete-generations formulas for change of mean fitness are for a different model of reproduction than Fisher’s “Fundamental Theorem”. What Fisher’s result assumes has been the subject of much discussion in the FTNS literature. Fisher used a verbal argument and never clarified it by writing a paper with the full mathematics in it. The Wright formula(s) are not exactly the same as the FTNS, having denominators and extra terms in them.

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MANY THANKS! That’s the way I sort of understood what’s happening. I just wanted to make sure I’m understanding why I can’t seem to pin down a stable definition of Fisher’s theorem!

Thank you again. That was immensely helpful.

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In layman terminology, improvements in evolutionary fitness come from any trait in a population that improves the number of fertile offspring over a specified time frame.

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Thanks to Joe, I looked up Ewens work on the matter. His 1989 paper is not immediately available at my university library, however his 2015 paper is.

It will take me at least a week to slug through it, probably more, but the math does look beautiful and interesting. The abstract:

On the interpretation and relevance of the Fundamental Theorem of
Natural Selection

The attempt to understand the statement, and then to find the interpretation, of Fisher’s ‘‘Fundamental Theorem of Natural Selection’’ caused problems for generations of population geneticists. Price’s (1972) paper was the first to lead to an understanding of the statement of the theorem. The theorem shows (in the discrete-time case) that the so-called ‘‘partial change’’ in mean fitness of a population between a parental generation and an offspring generation is the parental generation additive genetic variance in fitness divided by the parental generation mean fitness. In the continuous-time case the partial rate of change in
mean fitness is equal to the parental generation additive genetic variance in fitness with no division by the mean fitness. This ‘‘partial change’’ has been interpreted by some as the change in mean fitness due to changes in gene frequency, and by others as the change in mean fitness due to natural selection. (Fisher variously used both interpretations.) In this paper we discuss these interpretations of the theorem. We indicate why we are unhappy with both. We also discuss the long-term relevance of the Fundamental Theorem of Natural Selection, again reaching a negative assessment. We introduce and discuss the concept of genic evolutionary potential. We finally review an optimizing theorem that involves changes in gene frequency, the additive genetic variance in fitness and the mean fitness itself, all of which are involved in the Fundamental Theorem of Natural Selection, and which is free of the difficulties in interpretation of the Fundamental Theorem of Natural Selection.

On the interpretation and relevance of the Fundamental Theorem of Natural Selection - PubMed