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Some readers will be unfamiliar with the genetic terms and concepts necessary to follow this essay, and will need a crash course, below. If on the other hand you already know what Hox genes do and how dominance and recession works, skip ahead to the main essay.
Conversely, those who would like to see a full glossary of genetic terminology, including concepts which are outwith the scope of this particular essay, can find one here.
To begin with, the characteristics and appearance of a living organism (its "phenotype") are controlled by a combination of its genetic coding ("genotype") inherited from its parents, and the environment in which it developed. Genetic coding is carried by a special, complex molecule called DNA (deoxyribonucleic acid), which is made up of chains of smaller molecules called nucleotide bases, which function like little letters which can be put together in different sequences to form words. Particular sequences of these bases tell the organism how to manufacture the different chemicals which assemble the structures which go to make up a physical body.
Very long strings of DNA are coiled and folded up to form tiny sausage-shaped structures called chromosomes. In all but the most primitve organisms the chromosomes are found at the centre of each of the organism's cells, contained in a little membrane bag called the nucleus.
Many bacteria have just one, circular chromosome, but any reasonably advanced species of organism will have its own unique pattern ("karyotype") of many different chromosomes, all with characteristic sizes and shapes. In organisms which reproduce sexually there will be a matched pair of each type of chromosome, one from each parent, in each cell.
There are two exceptions to this. One is red blood cells, which have no nucleus. The other is the gametes, the egg and sperm cells or their plant, algal or fungal equivalents, which have one of each type of chromosome, not two - although those single chromosomes each contain a random mixture of elements from the two chromosomes of that type which the organism inherited from its parents, a different mix in every individual gamete. When an egg and a sperm cell fuse they create a new organism which has two of each type of chromosome, one from the sperm cell and one from the egg, each containing a mixture of material derived from the parent's parents.
The sections of these DNA supercoils which code for the production of particular chemicals, and therefore control particular characteristics in the organism, are called genes. The position on a given chromosome at which a given gene is found is called the locus of the gene. Because chromosomes come in pairs (except in gametes) there are two genes at each locus, one on each chromosome, and therefore two genes influencing each characteristic for which the genes code.
The exception is the sex chromosomes. In placental and marsupial mammals females have a matching pair of so-called X-chromosomes (although only one of these will be active in any given cell), one from each parent, and males have one X-chromosome from the mother and one Y-chromosome from the father. Because the Y-chromosome is smaller than the X there are some genes on the X-chromosome which do not have a matching, paired gene in males, and there are also a few genes which exist only on the Y-chromosome.
The exact nucleotide-base sequence of a gene, and therefore the precise structure of the chemical for whose production the gene codes, may change over time, due to transcription errors when dividing a cell to make two (or, in the case of sex cells, four) new ones, or to damage caused by exposure to harmful chemicals or to e.g. too much ultraviolet radiation. These changes are called mutations, and if they occur in cells in the testicles or ovaries they can be passed on to the organism's offspring and alter its development. If they occur in a non-sexual cell which cannot give rise to a gamete they do not influence the organism's offspring, and are called "somatic" mutations.
There are also "chromosomal mutations" which occur when whole or partial chromosomes are wrongly sorted into a new gamete cell, resulting in duplications or omissions which lead to major developmental changes such as Down's Syndrome, XXY males and the like, and there are also whole-genotype mixups called chimaeras and mosaics. Chimaeras form when two fraternal twin embryos fuse, creating an organism who is a mixture of two genetic types: if the original embryos were different sexes this can have some very strange results, and even if they were the same gender you can get oddities such as thighbones whose bone marrow comes from different embryos, one on each side, and creates two simultaneous, warring blood groups, or women whose children test genetically as not being theirs because their ovaries have a different genetic make-up from their skin and blood. Mosaics occur when a somatic mutation - that is, a transcription error in a cell which is not a gamete or about to generate a gamete - early in embryonic development results in blocks of cells with slightly different genes, which then go on to produce a patchwork organism - albeit one in which the conflicting cell-lines differ less than they do in a chimaera, and are unlikely to be of different genders unless the mutation was the loss of the Y-chromosome. These large-scale mutations - chromosome duplication or loss and the formation of mosaics and chimaeras - are unlikely to be relevant to the inheritance of magic, however.
Ordinarily genetic, non-chromosomal mutations which generate new genes occur only rarely, but occasionally you get a chromosome with a weak fracture-point at a particular locus, resulting in frequent mutation at that point, especially in older individuals or in males who have done a lot of sunbathing (exposing the testicles to excessive levels of U.V. over their lifetime) and then pass mutated sperm to their offspring. Achondroplasia, for example, a type of dwarfism, usually arises by spontaneous mutation in this way, which is why achondroplastics often do not have a dwarf parent, even though the gene is dominant.
Mutation generally results in there being several variant versions of the gene found at any given locus. These groups of variant genes associated with a particular locus are called alleles. If an organism has two matching alleles/genes at a given locus it is said to be homozygous at that locus; if it has two different alleles it is heterozygous.
Generally speaking, if you take two different alleles at a given locus one allele will be dominant and the other recessive. What this means is that if you have a copy of the dominant gene on one chromosome, and a copy of the recessive gene at the same locus on the matching, paired chromosome, the effect on the organism's development will be the same as if it had two copies of the dominant gene. The recessive gene is "carried", that is, its presence is masked by the dominant gene, and it may be passed down through several generations before making its presence felt. Only if an organism inherits two copies of the recessive gene, one from each parent, or one copy of that gene and one of one which is even more recessive, will that gene affect how the organism looks and develops.
You can have chains of dominance and recession at the same locus, so for example at the Colour locus the gene for full (dark) normal hair or fur colour is dominant to all other variants, red is recessive to full-colour but dominant to blonde, and blonde is recessive to both. If you have one or two full-colour genes your hair or fur will be full colour; if you have two red genes or one red and one blonde your hair or fur will be red; and only if you have two blonde genes will your hair or fur be blonde.
The normal mechanism of dominance is that the dominant gene codes for the production of some particular chemical which contributes to the phenotype of the organism, and the recessive gene either doesn't produce that chemical, or produces an altered version which has a different or lesser effect. If one copy of the dominant gene/allele is sufficient to produce enough of the "normal" chemical to produce the same effect as having two of the dominant gene, then that gene is fully dominant. If the effect of having just one of the dominant gene and one recessive is to produce a rather patchy, weakened version of what you would expect to see if there were two copies of the dominant gene, that gene is said to be "incompletely dominant". If having one of each allele results in an evenly intermediate state, neither allele is dominant to the other and they are said to be "co-dominant".
There is also a phenomenon called "epistasis", which means that a gene at one locus masks the effect of a gene at another locus. An individual who is homozygous for the albinism gene, for example, will have no colour except for the pink tinting caused by seeing blood under the skin and within the eyes, and although they may have genes at other loci which would normally give them stripes or yellow eyes those charactacteristics will be rendered invisible by the albinism gene, which is therefore said to be epistatic to the genes for stripes or eye-colour.
Finally, to understand this essay you will need to know what a Hox (homoeotic complex) gene is. Hox genes are able to switch other genes on and off: as such they control the head-to-tail order in which various structures appear during embryological development, initiating e.g. a whole sequence of genes for forming a limb, in the manner of a program calling sub-routines.
Mutations of Hox genes in insects result in strange aberrations such as legs growing where antennae should be. Vertebrates with Hox gene mutations normally die before birth, but it has been suggested that mutations in Hox genes are involved in major changes such as the loss of the hind-limbs in whales, and that, very early on in animal evolution, Hox genes were responsible for the formation of different phyla. [A phylum is a group of animals with a body-plan very different from that of any other group of animals, such as molluscs, vertebrates etc.: the equivalent groups in plants are called divisions.]
It is thought that Hox mutations may sometimes result in the reappearance of ancestral features which had been switched off - such as a tail occurring in a normally tailless species. The characteristic was suppressed in the first place by a Hox mutation which switched off the sequence of genes which coded for it, but that code is still present. This code is no longer being subjected to selection pressure to keep it accurate, and over the very long term it will become corrupted by random mutations - but it will probably remain readable, and capable of being reactivated by a reverse Hox mutation, for many millions of years. Mice with a Hoxa2 mutation, for example, have extra, non-functional jaw-bones - of a reptilian pattern which hasn't been seen in mammals probably for hundreds of millions of years.
Note - in case it turns out to be relevant to the transferrence of magic - that in addition to their own DNA and their own genes, the cells of multi-celled organisms also contain tiny symbiotic bacteria called mitochondria, which enable the cell to process oxygen, and plants also have symbiotic bacteria called chloroplasts, which carry out photosynthesis. These tiny symbiotic bacteria are passed on from the mother, in the egg cell, and have their own separate DNA. It is also occasionally possible for viruses to insert snippets of new DNA into the host's chromosomes, and if this happens in the ovaries or testicles the new DNA may be passed on to any offspring as if it were a regular gene. For some reason the "blue" fur colouration (really a smokey mid-grey) often crops up in hitherto blueless furry mammals by this route.
It is firmly established in Potterverse canon that "Muggle-born" witches and wizards (or "wizwitches" as they are called in the excellent Mansion of E web-comic) occur quite frequently in non-magical, Muggle families. On the face of it it would be possible for magic to be a dominant gene which arises by a spontaneous mutation at a fragile fracture point, like achondroplasia, but JK Rowling has stated that all Muggle-born wizwitches have a wizwitch somewhere in their ancestry. This means that magic very rarely arises by spontanous mutation (although it must have done so at least once, somewhere, to exist at all) and is sited at a stable locus; instead it is controlled by a rare recessive gene which is carried by Muggles for many generations until they happen to pair with another Muggle who carries the same recessive gene, at which point there will be a one in four chance that any child they may have will be a wizwitch.
So far, so simple. It's normal to name a locus after the most dominant allele at that locus, but that would mean calling the relevant locus "Muggle", which would be confusing when what we're discussing is the inheritance of magic. Therefore, I'm going to call it "Mage". So, we have a dominant gene M at the Mage locus, a single copy of which is enough to produce a Muggle, and a recessive gene m two of which produce a wizwitch.
Before we go any further, Red Hen has raised the question of whether there is one gene for magic, or many. There are certainly many different kinds of magic and many different talents for magic, at least two of which - being a Parselmouth or a Metamorphmagus - are innate.
I do not think it's likely that being a wizwitch is in itself controlled by multiple genes at multiple loci (although there may be two, for reasons explained below), as we do not hear of people who have only one strong magical skill and no others. At least, I suppose divination might be controlled by a separate gene, as you can get clairvoyant Muggles and wizwitches with zero divinatory ability, and a case could be made for Muggle legends about shapeshifters deriving from people who were Animagi or Metamorphmagi without being full wizwitches, but actual spell-casting, wand-mediated magic of the kind which gets you a place at Hogwarts seems to be one thing. Anybody who can cast any spell, can probably cast any other spell at least a little bit.
If we assume that innate skills such as being a Metamorphmagus only occur in people who also possess wand-type magic, it may be that these characteristics are controlled by genes which do one thing in a Muggle, and another in a wizwitch - that they are genes whose manifestation is altered by the presence of homozygous m at the Mage locus.
On the other hand, it may be that there is an entire suite of genes for different types of magic, and that the genes at the Mage locus are Hox genes which switch the whole block on or off. For present purposes it doesn't really matter whether m is a simple gene which in itself conveys magical power which then manifests differently according to what other genes may be present at other loci, or whether it is a ruling Hox gene which switches on a group of many other genes all of which convey different kinds of magic - but if it is the latter then there are going to be all sorts of side issues about how the differing powers are passed on. Just bear in mind that whenever I refer to a gene at the Mage locus, it could be a Hox gene.
The problem is, we also know that, as Ron says in CoS, "A Squib is someone who was born into a wizarding family but hasn't got any magic powers. Kind of the opposite of Muggle-born wizards, but Squibs are quite unusual." But a wizwitch cannot carry the Muggle gene M at the Mage locus - it cannot be passed down through the generations to reoccur unexpectedly - because we've already established that M is the dominant gene at that locus. If you have even one of it, you're a Muggle.
It's also not clear whether Squibs are functionally the same as Muggles or not. JKR has stated that they can't see Dementors (so Mrs Figg was lying about that to protect Harry). Filch can see Hogwarts, but we don't know whether that's because he has higher magical abilities than Muggles, or just because he's been let through the wards, as the Grangers were let into Diagon Alley. He is able to use some magical implements - but we don't know that Muggles can't. Filch and Mrs Figg both have intense relationships with their cats (including in Mrs Figg's case a half-Kneazle) who act like their familiars. The fact that Neville Longbottom's family suspected he was a Squib, even though he clearly had some magic, since he was accepted into Hogwarts, suggests either that there are degrees of Squibness, or that all Squibs have at least a little residual magic.
The following possibilities exist, and there may be more I haven't though of yet.
1: The Squib gene is an allele ms on the Mage locus, recessive to m. Depending on whether or not Squibs do have some residual magic, it either switches magic off again totally, producing an effect similar to M by a different chemical route, or it switches magic on, like m, but to a much lesser degree. It could be that m is incompletely dominant to ms, so that possession of one magic and one Squib gene results in an individual with weak magic (in which case both parents of a Squib would have weakened magic, even if they wouldn't admit to it), or it could be that Mage is a Hox gene locus and people with weak magic have some problem within the suite of magic genes regulated by the Hox gene.
2: The Squib gene is the regular, dominant Muggle gene M recurring by spontaneous mutation at a fracture point. If it is the case that Squibs still do have some residual magic, more than Muggles do, and that Muggles - even ones who are carrying the m gene - have no magic at all, then this option can be ruled out: otherwise it is possible but mildly unlikely. Dominant genes usually perform some chemical process "correctly" while recessive ones represent some sort of failure in a process, so for M to recur spontaneously a mutation would have to be putting something right, repairing a damage, which would mean it was to some extent going against entropy. It is conceivable however that magic itself recognises the m gene as some form of damage in the gametes and tends towards putting it right, eliminating itself in the process, or that some medical process which is being used to treat another condition causes this repair as a side-effect. However, it is still somewhat unlikely, because Rowling's statement that all Muggle-borns have a magical ancestor suggests that the locus is stable.
3: The Squib gene is a dominant (because one copy of it is enough) epistatic gene at a locus other than Mage, occurring by spontaneous mutation at a fracture point and masking or partially masking the magic-enabling effect of m (this is Red Hen's suggestion).
4: The Squib gene is a rare recessive epistatic gene at a locus other than Mage, carried invisibly by generations of wizwitches until two carriers come together to have a child who is homzygous for the recessive gene, which then masks or partially masks the magic-enabling effect of m. It is possible that excessively mundane people such as Vernon are homozygous for both the Muggle and Squib genes.
5: The Squib gene is either a dominant (or incompletely dominant) gene at Mage, or a dominant epistatic gene at a locus other than Mage, being newly introduced into the genotype by a virus. This is mildly unlikely because you would expect that either it would be a much rarer occurrence even than we are told the birth of a Squib is, or if it wasn't as vanishingly rare as all that, somebody would have noticed that it followed on from some particular infection in one of the parents.
6: Daniel Amdurer's suggestion:
M is any allele that doesn’t code for a working mage protein. Mutations can occur, turning m into M.
MM = pure Muggle
Mm = Squib
mm = wizwitch
Many Muggles would be Squibs, but nobody would really know the difference - they’ve got very little magic and so just seem like any other Muggle.
All Muggle-borns are actually Squib-born.
If indeed Squibs do have some magic, less than wizwitches but more than Muggles, then all these options except 2 raise the possibility that Muggle clairvoyants and other psychics are Muggle-born wizwitches whose magic is being choked off at a low level by the Squib gene.
There is another genetic question, which is the issue of why wizwitches are so rare - around one in six thousand compared with the Muggle population. You would have thought that any gene which enhanced the bearer's ability to defend themselves, to keep warm, to recover from injuries, to impress a mate and to find food would be very advantageous and would spread rapidly through the population.
However, in evolutionary terms it doesn't matter how happy and comfortable an individual is but only how many descendants (or e.g. nieces and nephews bearing the same genes, since genes are fungible) they bequeath to future generations. Any gene which leads to lots of future individuals carryng that gene is by definition spread and perpetuated. Being safe and well-fed matter in evolutionary terms only because these things usually enhance reproductive success.
In the real world, my personal experience has been that clairvoyance and similar psychic powers do exist, but that they are not very powerful and they require huge, exhausting amounts of energy to run. They are therefore only advantageous and only selected for in populations such as coastal fishermen, who are exposed to random danger to such an extent that even a small increase in the ability to predict where the next threat is coming from is worth the extra energy expenditure.
In the Potterverse, I'd say there's enough canon evidence to support the idea that although being homozygous for m increases the comfort of the bearer and can greatly enhance life-span in certain individuals, it doesn't lead to a greatly increased average lifespan (a calculation based on all the wizwitches whose age at death we know, other than the Flamels and the casualties of the Final Battle, gives an average lifespan of seventy-three) and it does radically reduce fertility. Unless, for some reason, you're a Weasley.
That I recall, we only hear of five wizarding couples that have more than two children, and the mode seems to be one child. Those more prolific couples are Molly and Arthur Weasley; the parents of Molly Weasley née Prewett and her two brothers; Ginny Potter née Weasley and Harry Potter; the Dumbledores and the parents of the three Black girls. That is, we only hear of two wizarding families who have more than two children, and aren't directly descended from the Prewetts, and only one family - the children of Molly Prewett - with more than three.
On the other hand wizwitches whom we know or strongly suspect to be only children include Harry, Neville, Hermione, Draco, Luna, Severus, Minerva, Tonks, Teddy, Barty Crouch, Tom Riddle, Remus and James, while those whom we know or strongly suspect to be in pairs of just two siblings include Sirius and Regulus, Luna's children, Ron and Hermione's children, the Creeveys, the Patils, the Gaunts, the Carrows, the Lestrange brothers, Umbridge and her (Pottermore canon) Squib brother, and Lily. This means that far from the m gene spreading like wildfire, the wizarding community must struggle to keep its numbers up.
In this case, you would expect that the genes for magic had initially taken hold and spread in hunter-gatherer communities living in such extreme conditions - perhaps during the last Ice Age - that being sub-fertile but having enhanced ways of protecting the few children you managed to produce resulted in more living descendants than having a lot of kids you couldn't feed or protect. Once conditions improved and infant mortality among Muggles began to fall - especially after the introduction of agriculture - magic became progressively less useful, in evolutionary terms.
As for the Squib gene, excessivelyperky has suggested that it may be analogous to the sickle cell gene. This is a co-dominant gene found in some persons whose ancestors came from areas of Africa heavily affected by malaria, and which alters the shape of the red blood cells. Individuals who are hymozygous for sickle cell suffer from a painful and debilitating form of anaemia, but the gene persists in the population because heterozygous individuals have improved resistance to malaria, with few or no unpleasant side-effects.
Similarly, excessivelyperky proposes that having a single copy of the Squib gene conveys some clear benefit and little or no damage, explaining the persistence of the gene in the wizarding population, even though being an actual Squib radically reduces a person's chances of finding a mate within that community. If she's right, I would suggest that what having a heterozygous Squib gene does is to improve the individual's fertility, with little or no effect on their magic.
If the Squib gene merely masks the gene for magic rather than replacing it - basically all the options above except 1) - then it is likely that in many, probably most cases Muggle-borns are descended from Squibs who left the magical world and married into Muggle families.