Red, theory; black, fact.
5-26-2018: Influential biologist Richard Dawkins wrote in "The God Delusion" that a genuine case of irreducible complexity will never be found in biology. A case of irreducible complexity would be some adaptation that would require an intelligent designer because it could never evolve one mutation at a time, and Dawkins believes there is no such intelligent designer in biology.
In classic natural selection, each mutation must be individually beneficial to its possessor in order for selection to increase its prevalence in the population to the point where the next incremental, one-mutation improvement becomes statistically possible. In this way, all manner of wondrous things are supposed to evolve bit by tiny bit.
However, I am seeing irreducible complexity all over the place these days. For example, your upper-jaw dentition must mesh pretty accurately with that of your lower jaw or you can't eat. Thus, the process of evolutionary foreshortening of the muzzle of the great apes to the flat human face could never have happened, assuming that a single mutation affects only the upper or lower jaw. But it did. (Let us gloss over the fact that that is an assumption, because the contrary seems to require non-local rules in development.)
Furthermore, how can any instinctive signaling system evolve one mutation at a time? At a minimum, you always need both the transmitter adaptation and the receiver adaptation, not to mention further mutations to connect the receiver circuit to something useful. The evolution of altruism presents a similar problem. The lonely first altruist in the population is always at a disadvantage in competition with the more selfish non-mutants unless it also has a signaling system that lets it recognize fellow altruists (initially, close relatives) and a further mutation that places the altruistic behavior under the control of the receiver part of this system. Thus, altruists would only be altruistic to their own kind, the requirement for altruism to be selected in the presence of selfishness. Finally, the various parts of this system must be indissolubly linked in a way that the non-altruists cannot fake.
My solution is to label the crossing-over events that occur during meiosis as "tetra-mutations." In crossing over, two homologous chromosomes pair up along their length and swap a long segment of DNA, a process requiring four double chain breaks and their corresponding repairs. Because of the presence of single-nucleotide polymorphisms, the homologous chromosomes are not exactly the same, so that each of the upstream sides of the four chain breaks ends up in a subtly different genetic environment. If the break point falls between a cis-acting regulatory element and the corresponding structural gene, for instance, the former may now control the expression of a slightly different protein. Thus, there could be as many as four distinct functional consequences of one crossing-over event. Why not call that a tetra-mutation?
In this way, a concerted change affecting four distinct sites becomes possible. The two ends of the recombinant segment can in principle be functionally unrelated initially. They become related if both are affected by the same tetra-mutation and the entire change increases fitness and is thus selected.
A single tetra-mutation could in principle produce viable altruism at one stroke because of the number of simultaneous changes involved.
The probability of a combination of simultaneous local changes being beneficial to the organism is much smaller on mathematical grounds than is the probability of a given single-nucleotide change being beneficial. However, these unfavorable statistics are at least partly offset by the existence of a dedicated system for producing tetra-mutations in large numbers, namely meiosis, part of the process of maturation of egg cells and sperm cells.
In the big picture, tetra-mutations provide a way for a species to discontinuously jump into new niches as they open up, possibly explaining how a capacity for this kind of mutation could spread and become characteristic of surviving species over time. This idea also provides a ready explanation for the lack of transitional forms in the fossil record.
5-30-2018: Here is the search description again, in case you missed it or could not see all of it: Sexual reproduction may allow the evolution of irreducible complexity by increasing the intrinsic complexity of the basic building block of change, the mutation.
6-12-2018: Upon further reflection, it seems that the tetramutation construct described above lacks validity because during gamete maturation it falls apart into two bi-mutations, both of which cannot contribute to the same zygote. The bi-mutation is stable, however, because of the intervening translocated DNA segment. It is harder to see how a complex adaptation like altruism could evolve out of nothing but mono-mutations and bi-mutations, but that does not mean the theory put forward in this post is necessarily wrong. One must not argue from lack of imagination. It is an interesting question, actually, what is the minimum set of all mutation types necessary to account for all known adaptations.
8-27-2019: In my ignorance, I have undersold the bi-mutation idea. A very far-reaching change to the genetic information can occur during crossing-over that is not at all subtle and is termed unequal crossing-over. This form of the process arises because of inaccuracies, sometimes major, in the initial alignment of the homologous chromosomes prior to crossing-over. When the process is finished, one chromosome has been shortened and the other has been lengthened, with gene duplication. This is the major source of gene duplication, which, in turn, is a major source of junk DNA, the part that is classified as broken genes. Two questions come to mind. The first is, are anatomical features such as jaw length and axon targets somehow controlled by variations in gene dose? The second, which is a tangent, is, are broken genes really broken or just temporarily switched off by genetic drift at some mutational hot spot in the recognition site of some transcription factor? The analogy here is to a generator in a power plant that has been placed in stand-down mode because of a temporary decrease in the demand for electrical power.
