One thing that often used to happen, perhaps not so much any more, is that people will say that we don’t need to worry about levels of selection because all selection can be reduced to selection acting directly on genes. George Williams perhaps put this view best, first with his principle of parsimony, which argues that reductionism is the right perspective:

“In explaining adaptation, one should assume the adequacy of the simplest form of natural selection, that of alternative alleles in Mendelian populations, unless the evidence clearly shows that this theory does not suffice”

and in the same book, and more explicitly, which says that reductionism is works:

“No matter how functionally dependent a gene may be, and no matter how complicated its interactions with other genes and environmental factors, it must always be true that a given gene substitution will have an arithmetic mean effect on fitness in any population.”

All I can say to this is GAHHHH!

*Merida expresses her opinion on genetic reductionism (taken from http://giphy.com)*

I think a lot of people know that you cannot think of selection as acting on genes, but a lot of people also can’t articulate why it doesn’t work. So, if anybody asks you, the simple answer is that reductionism doesn’t work because of interactions. At the individual level this will primarily be gene interactions of dominance and epistasis.

In a fully additive system there would be no problem, and this IS the problem. Our intuition about genetics was developed using simple additive models. In an additive system, knowing at what level selection was acting would be nice information, but the fitness of the phenotype can always be algebraically reduced to fitness effects on individual loci. In other words, in additive systems, how the genes are packaged really doesn’t affect the effect of genes on the phenotype. To see this consider a phenotype affected by a single locus additive trait:

Genotype | A_{1}A_{1} |
A_{1}A_{2} |
A_{2}A_{2} |

Frequency | p^{2} |
2pq | q^{2} |

Fitness | 1 | 1-Z/2 | 1-Z |

(I use Z to emphasize that we are not talking about fitness. Selection will be affected by the packaging for the simple reason that some of the selection is on heterozygotes). We can calculate the average effect of the A_{1} allele on the phenotype we would discover that it is:

Original genotype | genotype after substitution | probability | change |

A_{1}A_{1} |
A_{1}A_{1} |
p^{2} |
0 |

A_{1}A_{2} |
A_{1}A_{2} |
½ 2pq | 0 |

A_{1}A_{1} |
½ 2pq | Z/2 | |

A_{2}A_{2} |
A_{1}A_{2} |
q^{2} |
Z/2 |

So, the average effect of the A_{1} allele is:

Now consider a haploid system

Genotype | A_{1} |
A_{2} |

Frequency | p | q |

Fitness | 1 | 1-Z/2 |

The average effect with the same phenotypic effects (adjusted for ploidy). Now the local average effect of the A_{1} allele is:

Original genotype | genotype after substitution | probability | change |

A_{1} |
A_{1} |
p | 0 |

A_{2} |
A_{1} |
q | Z/2 |

So, the average effect of the A1 allele is: you guessed it:

The effect of the allele on the phenotype is not affected by the packaging.

Now lets do the same thing with a dominant system:

Genotype | A_{1}A_{1} |
A_{1}A_{2} |
A_{2}A_{2} |

Frequency | p^{2} |
2pq | q^{2} |

Fitness | 1 | 1 | 1-Z |

Now the average effect of the A_{1} allele on the phenotype becomes:

Original genotype | genotype after substitution | probability | change |

A_{1}A_{1} |
A_{1}A_{1} |
p^{2} |
0 |

A_{1}A_{2} |
A_{1}A_{2} |
½ 2pq | 0 |

A_{1}A_{1} |
½ 2pq | 0 | |

A_{2}A_{2} |
A_{1}A_{2} |
q^{2} |
Z |

So, the average effect of the A_{1} allele is:

turning to the haploid system

Genotype | A_{1} |
A_{2} |

Frequency | p | q |

Fitness | 1 | 1-Z/2 |

Now the local average effect of the A_{1} allele is:

Original genotype | genotype after substitution | probability | change |

A_{1} |
A_{1} |
p | 0 |

A_{2} |
A_{1} |
q | Z/2 |

The average effect in the haploid system is now different than in the diploid system,.

In other words, if we add the simplest possible form of nonadditivity the packaging does matter. Trust me it gets worse. I am way to lazy to put up tables for average effects in epistatic systems, but I have talked about this before. It turns out that the variance in local average effects is a measure of how the average effects of alleles are to genetic background. I have talked about these before, but it bears re-posting the relevant figure:

The important point is that the variance in local average effects is zero in additive systems, but non-zero when there are any sort of interactions. This means that the reducability of fitness effects on to genes is a reasonable exercise in additive system, but simply is not meaningful in epistatically interacting systems. To see how bad this can be, consider long-term directional selection in a system with AXA epistasis. Depending on the starting gene frequencies the average effect of an allele can actually reverse signs. For what it is worth, the dashed lines are the local average effects for an additive system, and the solid lines are the local average effects for AXA epistasis. This shows the contrast between additive systems and epistatic systems. For the additive system, if you were to evaluate the fitness effects in generation zero they would provide a pretty good estimate of the fitness at the end (in this deterministic system an exact estimate). On the other hand, for the epistatic system, estimates of allelic effects made in generation zero rapidly become useless, and by the time fixation is reached they are exactly wrong.

In one sense, Williams is absolutely correct. At any given instant it is certainly possible, in principle, to do a least squares regression analysis and assign fitness effects to individual loci. However in an epistatically interacting system those fitness assignments are ONLY good for the moment, or perhaps the generation, in which the assignment is done. Those effects will change as gene frequencies change, and not just gene frequencies at the locus under study, but gene frequencies at any other loci as well. So, my point is not that the assignment cannot be done, but rather that the assignment carries no information that is useful beyond the moment.

Next time I talk about why reductionism does work!

On a Mac: alt/option + o (or O) gives ø (or Ø).

I just think the arithmetic mean is weird to use, because of the way we think about selection coefficients. They are relative increases in fitness, and so work on a multiplicative way. A mutation of s=0.1 increases fitness by 10 percent, and a second mutations with s=0.2 by another 20 percent (total w=1.1*1.2), so it just feels strange to use the arithmetic mean for the total fitness.

But whatever I feel about it, it’s very hard to measure individual fitness components, which you would have to do to test it.

Bjorn;

I put in the arithmetic mean mostly because I didn’t really think it through and have no opinion one way or another. For average effects the arithmetic mean is what Fisher (and Falconer) use. So, I am open to being corrected on that, but you would have to tell me why something else would be a better choice.

By the way, I can’t figure out the keys I need to spell your name correctly.

Also, why arithmetic mean?