Doujutsu refers to techniques of inherited advanced bloodlines expressed via the eyes, such as the sharingan and byakugan. This article will address the possible ways the sharingan of the Uchiha clan and the byakugan of the Hyuuga clan may be inherited.
If we assume that the genes for the sharingan and byakugan follow simple genetic rules, such as dominant and recessive genes, we can map the probabilty of the offspring of the clan having these powers. The mother (M1,M2) and father (F1,F2) each have two genes that can be inherited. In the following examples, the parents' genes are depicted as the row and column headers while the genes inherited by the offspring are depicted by the resultant cells. In actuality, there are three genes involved with the eye color trait rather than just two.
Genotype refers to the actual genes inherited, while phenotype is the trait that is observed.
The expression of the sharingan is likely to be a recessive trait as it does not appear in all members of the Uchiha clan. If we assume that the gene for the sharingan is recessive (s) and the normal dark colored eye of the Uchiha (B) is dominant, the following probabilities of the offspring having the sharingan can be mapped as follows:
B = normal (black) eye
BB x BB
ss x ss
In the case where the parents both have the sharingan gene and both express the sharingan, all children will have ss genotype and s phenotype (i.e. all offspring will be able to activate the sharingan).
BB x Bs
In the case where one parent does not have the sharingan gene, and the other parent has one sharingan gene (a carrier of one gene does not express the sharingan), half will have BB genotype and half Bs genotype. All will have B phenotype (i.e. all offspring will not be able to activate the sharingan).
BB x ss
In the case where one parent expresses the sharingan and the other does not carry the sharingan gene, all children will have Bs genotype and B phenotype (i.e. all offspring will not be able to activate the sharingan).
Bs x ss
In the case where one parent expresses the sharingan and the other is a carrier, 50% will have Bs genotype and 50% will have ss genotype. 50% will have B phenotype and 50% s phenotype (i.e. 50% will be able to activate the sharingan).
Bs x Bs
In the case where both parents are carriers (neither can activate the sharingan), 25% will have BB genotype, 50% Bs genotype, 25% ss genotype. 75% will have B phenotype and 25% s phenotype (i.e. 25% of offspring will be able to activate the sharingan).
The byakugan is assumed to be a dominant trait since all the members of the Hyuuga clan have white eyes that are presumably capable of doujutsu. However, it may be that although all the Hyuga have white eyes, only those with the byakugan (B) gene have special abilities while those with the regular recessive white eye (w) genes do not.
B = byakugan
w = normal white eyes
BB x BB
In the case where both parents have byakugan genes, all children will have BB genotype and B phenotype (i.e. all offspring will have the byakugan ability).
ww x ww
In the case where neither parent has the byakugan gene, all children will have ww genotype and w phenotype (i.e. no offspring will have the byakugan ability).
BB x Bw
In the case where one parent has two byakugan genes, and the other parent has one byakugan gene (both express the byakugan), half will have BB genotype and half Bw genotype. All will have B phenotype (i.e. all offspring will be able to activate the byakugan).
BB x ww
In the case where one parent has two byakugan genes and the other does not carry the byakugan gene, all children will have Bw genotype and B phenotype (i.e. all offspring will be able to activate the byakugan).
Bw x ww
In the case where one parent has one byakugan gene and the other has none, 50% will have Bw genotype and 50% will have ww genotype. 50% will have B phenotype and 50% w phenotype (i.e. 50% will be able to activate the byakugan).
Bw x Bw
In the case where both parents have one byakugan gene (both can activate the byakugan), 25% will have BB genotype, 50% Bw genotype, 25% ww genotype. 75% will have B phenotype and 25% w phenotype (i.e. 75% of offspring will be able to activate the byakugan).
Some diseases like hemophilia are linked to genes on the female X chromosome. Most hemophiliacs are male because the Y chromosome does not contribute to traits other than gender determination and the gene for hemophilia is recessive. If the advanced bloodline is inherited along the maternal line then the following probabilities occur:
XX = female
XY = male
X* = sharingan gene on the X chromosome
XX* x XY
In the case where the mother is a carrier (neither parent can activate the sharingan), 50% of the offspring (male and female) will carry the sharingan gene, but only the males (X*Y) can activate the sharingan. This is the most common way exhibited hemophilia is inherited.
XX x X*Y
In the case where the father has the sharingan ability and the mother is not a carrier, then no offspring will be able to activate the sharingan. In this scenario, the male offspring do not inherit the sharingan gene. The female offspring are carriers only since the gene is recessive. (Hemophilia tends to skip a generation.)
XX* x X*Y
In the case where the father has the sharingan ability and the mother is a carrier, then half the male offspring can activate the sharingan. All the females have the sharingan gene but only half can activate the sharingan.
X*X* x X*Y
In the case where both parents have the ability to activate the sharingan, all offspring will also be able to activate the sharingan.
X*X* x XY
In the case where only the mother can activate the sharingan, all the females will be carriers (cannot activate the sharingan) while all the males will be able to activate the sharingan.
In addition to the basic statistics of recessive and dominant inheritance, the gene population and gene frequency should also be considered. For example, even though blood type O is recessive and types A and B are dominant, O is the most common blood type, especially in Asian countries. Given the basis of survival of the fittest, shinobi with the sharingan or byakugan are more likely to survive, are more desirable mates, and are expected to procreate to a greater extent. This means those with the advanced blood limit are more likely to perpetuate the clan and thus the gene population becomes purer or more concentrated. However, contrarily, those with the advanced blood limit are more likely to become shinobi and take risks, and are more likely to die, and will thus decrease the gene frequency.
The sharingan is said to have evolved from the byakugan. Mutations are the natural way to evolve. Mutations tend to be recessive so in order for the first sharingan user to be born, both parents would have to have at least one sharingan mutation, or the mutations could have occurred only in the offspring's genes, but simultaneously.
Another possible way the sharingan developed is via genetic engineering. The Uchiha may have been bred with the intention of creating the superior shinobi warrior. The Hyuuga clan are considered nobility and their lives have to be protected. Thus the Uchiha were created from their genes in order to protect Konoha (this theory ties into the "cursed" sharingan bloodline - the Uchiha clan was created to fight and die for their village). Of course there are other theories more supernatural in nature such as a mating between a tengu (type of demon) and a Hyuuga.
Haku's advanced bloodline of being able to use ice jutsu was inherited from his mother, and was probably a dominant trait since his father was a normal civilian. His advanced bloodline may not be a doujutsu but may still follow the same genetics of the byakugan (dominant) rather than the sharingan (recessive).
Itachi claims that the mangekyou sharingan can only be activated after killing your best friend. It is unlikely to be the only condition. More likely is the death of any close loved one for which the sharingan user bears responsibility. The guilt under those conditions can release a number of stress hormones which may be the basis of activating the genes for the mangekyou sharingan.
There are known cases of stress activated diseases. Besides the common headache, ulcer, and heart disease, certain genes and viruses may be activated by stress hormones and neurotransmitters. For example, shingles, which occurs from the chicken pox virus, can be activated by stress. Stress weakens the immune system. After infection with chickenpox, the virus resides in a non-active state in the nerve tracts that emerge from the spine. When it is re-activated, it spreads along the nerve tract, causing pain or a burning sensation.
If the sharingan originated as a mutation, such as from the incorporation of a viral strand of RNA, then other genes on the viral RNA can activate under specific circumstances. For example, HIV belongs to a class of viruses called retroviruses, which have genes composed of RNA. The genes of humans and almost all other organisms are made of DNA. Like all viruses, HIV replicates inside cells by commandeering the cell's machinery to reproduce. But a person may remain free of HIV-related symptoms for years.