Relative Fitness

My intention over the next several weeks is to delve into the five forces of evolution – selection, mutation, migration, drift, and environmental change – in more detail. I will deal with the forces separately, but all of the forces interact.  For example drift is not particularly exciting unless there is selection to magnify its effects.  Similarly, there are (as yet unpublished – I’m working on it) examples of selection having drift like effects on genetic variance.  Finally, in Wade and Goodnight (1991. Science 253: 1015 – 1018) we imposed selection by differential migration.  Thus, it should be clear that the various forces are not independent.  It is not so clear, and honestly I am not certain, whether the forces really are logically distinct, or arbitrary boundaries dictated by the human need to categorize things.

(Please excuse some of the unfortunate formatting:  I don’t seem to be much of a mathematical type setter)

As I pointed out last time, even if they may be fuzzy around the edges these forces logically divide the world:

  • Selection – deterministic change
  • Drift – dispersive change
  • Mutation – random change
  • Migration – change in elements
  • Environmental change – change due to external causes.

I will start with selection, and todays lecture, er, I mean post, will be on relative fitness, and unfortunately boring.  One of the things that I have discovered as I have been doing this blog is that I have these very nice compact concepts, such as how selection works.  When I start to unpack these concepts I discover they are big and messy, and that there are a lot of important details that cannot be just brushed under the carpet, even if I think they are obvious.  Relative fitness is one of those concepts.

It is easiest to introduce relative fitness using a purely genetic system (is it any wonder that the gene centered view is so common place, given that concepts are often much easier to explain!).  Fortunately, J.B.S. Haldane (http://en.wikipedia.org/wiki/J._B._S._Haldane) provided a very easy way of modeling selection in a one locus two allele system.  We start with standard HWC proportions, but add to that a new entry, fitness.  Notice that, in Haldane’s very gene centric view, phenotype is simply skipped.

Table 1

Here S is the strength of selection, which is unfortunate, since in my next post S will mean something else.  Sigh, different modeling traditions lead to confusing symbol choices.  This is an example of “additive gene action”.  Every A1 allele confers a fitness of ½, and every A2 allele confers a fitness of (1-S)/2.  Thus an A1A1 genotype has a fitness of ½ + ½ = 1 etc.  In future posts I will talk about non-additive systems.  In non-additive systems genes interact so that the phenotype is not just the sum of the underlying genes.

The fitnesses are “normalized” fitnesses.  That is, they are numbers that are arbitrarily scaled.  Absolute fitness is the actual value of the fitness trait.  In this example it may be that A1A1 genotypes lay 1000 eggs, A1A2 lay 750 eggs, and A2A2 lay 500 eggs, or perhaps they are probabilities of survival of 100%, 95% and 90%.  In the first example the absolute fitnesses are 1000, 750 and 500, and the second are 100, 95 and 90.  Normalizing is done by dividing by the absolute fitness of the best genotype so the best genotype has a fitness of 1, and S = 0.5 in the first case and 0.05 in the second case.  The problem with normalized fitness is that it is not very useful for predicting the outcome of selection.   It should be obvious that A1A1 has the highest fitness, so the population will eventually fix for the A1 allele (and A2 will be eliminated), but from the normalized fitnesses alone we can’t say anything about the rate at which evolution will proceed.

To determine the change in gene frequency as a result of selection we need to know the “relative” fitness.  To get relative fitness the first step is to calculate the mean absolute fitness in the population.  I am going to do this using mathematical notation, as this will allow it to be generalized beyond the simple one-locus two-allele case:

Equation 1

here ̅w is the mean normalized fitness, Pg is the frequency of the gth genotype (p2, 2pq or q2) and wg is the genotypes normalized fitness.  The equation is general, regardless of the number of genotypes or their frequency, but the result (1-sq) is specific to this simple additive system.  I mention that only because 1-sq is one of those meaningless numbers that is burned irrevocably into my skull.  Don’t let that happen to you!

With the mean normalized fitness we can get the relative fitness by dividing each normalized fitness by the population mean normalized fitness.  The frequency after selection is simply the frequency before selection times the relative fitness.  Thus:

Table 2

Now, we can calculate the rate of change due to selection.  Remembering the frequency of the A1 allele can be calculated as the sum of the homozygotes plus half the heterozygotes:

Equation 2

which is only important because it shows that we can in fact talk about the rate at which selection works.  The relative fitness highlights a couple of important points, however.  Most importantly the relative fitnesses have the term (1-sq), that is there is a gene frequency in them.  Thus, unlike the normalized fitnesses the relative fitnesses are a function not only of the individual, but also of the population in which they are measured, and that the relative fitness will change as the gene frequencies in the population change.  The other thing to notice is that the mean relative fitness is always one (should be obvious), and that those genotypes that will increase in frequency have a relative fitness greater than one, and those that will decrease in frequency have a relative fitness less than one.

 

The other thing to notice is that other than being indexed by genotype the relative fitnesses have no mention of the genes in them.  Thus, we could easily re-write the mean fitness of the population indexing it by individuals: Symbol 1 .  In the example developed above it is simply a different means of indexing by individuals rather than genotypes, so it should give the same mean fitness, although the calculations would not be as pretty.  The relative fitness of the ith individual then becomes Symbol 2 ( Relative fitness symbol– an italic w with a tilde over it — is the universal symbol for relative fitness.).

 

Relative fitness is used very frequently in talking about natural selection, so think of this as a tutorial rather than a real blog entry!  By the way, in your meanderings through population genetics you will discover that frequently people will call fitnesses that are normalized so that the most fit genotype has a fitness of 1 as being relative fitness.  This is a holdover from Haldane’s models, which as you saw above involved dividing by the absolute fitness of the most fit genotype.  This normalized fitness has almost no utility outside of Haldane’s models, so I strongly urge you to refer to what Haldane did as “normalization” of fitness and what I showed you here to be relative fitness.

 

2 Responses to “Relative Fitness”

  1. Jim;

    Density effects are of course important. Basically they are a topic that I hope to talk about when I start talking about population structure. Essentially, I would think about population density in a contextual analysis framework, and consider it to be a contextual trait like any other. It potentially can have effects on individual selection in the form of a change in environment – I don’t think it can be heritable at the individual level, or it can be a heritable selectable trait at the group or higher level. Before I can get to that I need to cover the five basic evolutionary forces, and then start delving into how population structure works. One thing I am finding is that the things I want to talk about are deeper than I thought, and since I am aiming for a 1000 word essay every week, this is taking a long time. (I can’t believe I spent 1000 words just covering relative fitness for example.) The bottom line is, yes, I think population density is important, and no, I don’t see myself getting to it for a couple of months. I hope you can be patient!

  2. james mallet says:

    What about population density?

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