Kekkei Genkai are a recessive or dominant trait? It is possible that someone born from two parents without Kekkei Genkai in a clan that have one, inherit the characteristic?
Kekkei Genkai Genetics
Good question. Unfortunately, I did not study biology, but I did do some research (asking some biology-savvy people, digging out my 10th-12th grade biology book). Still, take everything here with a barrel of salt unless you see this post getting peer-reviewed.
Generally speaking, yes, I think that Kekkei Genkai can be recessive traits, resulting in "hereditary carriers" (people with a gene that is not expressed). However, this does not apply to all Kekkei Genkai, and technically, you can also end up a hereditary carrier of a dominant gene.
(I hope you don't mind me explaining irl biology first; if you do, skip the first half of this post until you reach the Sharingan section)
Recessive and Dominant Inheritance Patterns
Recessive or dominant inheritance occurs because genes always come in double packs - one half of a child's chromosomes originates from the mother and the other half originates from the father. However, those two gene variants (also referred to as "alleles") do not always contain the same genetic information. This is when a child will end up with a "heterozygous genotype".
As Anon previously mentioned, one of the two heterozygous alleles will usually play a recessive role in relation to a dominant allele, the latter of which will "win" against the other and determine the phenotype.
For communication purposes, we will always assign each allele a letter. Dominant alleles use upper case letters (eg. "B") and recessive alleles use lower case letters (eg. "b").
To visualize this, if one parent has brown eyes and contributes a dominant (B) gene while the other parent has blue eyes and contributes a recessive (b) gene, the child will end up with a heterozygous genotype (Bb), which contains both genes.
Since the allele for brown eyes (B) dominates the blue eyes allele (b), the child will have brown eyes as a phenotype. The child remains a carrier of the recessive blue eyes allele (b), however. The recessive allele is still there, you simply cannot see it.
If a child instead inherits two blue eye alleles (b) from their parents, they will be born with a homozygous genotype (bb), translating into blue eyes. This is the only circumstance under which an otherwise recessive gene won't be hindered by a dominant one.
Additionally, a genetic trait itself is never inherently recessive or dominant. "Recessive" and "dominant" merely describe the relationship between two alleles. A gene that is recessive toward one gene can still be dominant in relation to another (eg. brown eyes > green eyes > blue eyes).
TL;DR Dominant traits (B) are always expressed. Recessive genes (b) are only expressed if no dominant gene is present. Dominant (B) + Recessive (b) -> Dominant Inheritance (Bb) Dominant (B) + Dominant (B) -> Dominant Inheritance (BB) Recessive (b) + Recessive (b) -> Recessive Inheritance (bb) "Dominant" and "recessive" are not absolute, as they don't refer to the genes themselves but their interaction with another gene instead. A trait (g) dominant to a recessive trait (b) can still be recessive to a different trait (B). b < g < B
The Limits of Dominance and Recessiveness
The problem is, those previously established rules surrounding recessiveness and dominance do not always apply. Two well-known examples are co-dominance (eg. blood types) and incomplete dominance (eg. red + white = pink flowers) in which neither allele in a heterozygous genotype is recessive and instead results in an entirely new phenotype.
Not just that, dominant/recessive gene inheritance patterns are not the be-all and end-all of gene inheritance and are sometimes mixed with and influenced by polygenic, epistatic, or gonosomal inheritance patterns.
Polygenic Inheritance
In the real world, polygenic inheritance is common in traits like skin color and height. In the world of Naruto, it might come into play for traits like nature affinity or chakra potency.
If a polygenic inheritance pattern is present, the trait will be determined not by a yes/no question but instead by the cumulative effort of multiple genes. It will create a spectrum of phenotypes, depending on both the number and the exact genes that have been inherited.
To visualize the concept, imagine dropping food dye into a glass of water. The more drops of food dye and the bigger the drops of food dye are, the stronger the color inside the glass becomes.
In reference to Anon's ask: If a Kekkei Genkai were to be polygenic, parents might pass on a Kekkei Genkai to a child in spite of not having access to the ability themselves, simply just because their child reached a certain "threshold" whereas the parents did not (eg. your water nature affinity is not strong enough for you to acquire ice release). This can be entirely independent of "gene dominance".
Epistasis
Epistasis, similar to polygenic inheritance (as well as co-dominance and incomplete dominance), describes the interaction of two or more non-allelic genes. Unlike polygenic inheritance, epistasis produces an entirely new phenotype rather than a spectrum of phenotypes.
Epistasis comes in various forms, hence coming up with just one analogy wouldn't be doing it any justice. Sometimes, one gene masks the effect of another (eg. a person without hair cannot be blonde), and sometimes, genes instead require one another for a phenotype to go into effect (eg. sweet pea flowers will be purple but only if two specific mutations interact - otherwise, they will be white). One famous real-world example is mice and their coat colours, but as for Naruto, epistasis may play a role in the inheritance of the Rinnegan, among others.
In reference to Anon's ask: Just like before, this can easily create a situation similar to the traditional hereditary carrier. A parent may carry a gene that they themselves have no access to, as it either requires the presence or absence of yet another gene.
Gonosomal Inheritance
Additionally, as opposed to being located on one of the 22 different autosomal chromosome pairs, genes can also be passed on through gonosomes (X and Y chromosomes) and therefore rely on the gender of the child.
Traits passed down via the father's Y Chromosome will always be present in his sons, never in his daughters. If it's via the father's X Chromosome, it will never be inherited by his sons, only his daughters. If it's on the mother's X Chromosome, sons have a 50% chance of inheriting the gene. For girls, this number is entirely reliant on recessiveness/dominance.
In reference to Anon's ask: If a Kekkei Genkai were to be gonosomal, the father could never be a "mere" carrier and would typically always have access to the ability.
In-Universe Examples
Now, there are multiple different Kekkei Genkai in the world of Naruto, all of which are likely determined by different, non-allelic genes entirely. This means that we cannot make blanket judgments about all Kekkei Genkai when it comes to their respective inheritance patterns - We can only make case studies.
However, any resemblance to the previously mentioned inheritance patterns might just be coincidental after all. Kishimoto's writing itself doesn't give me much of a basis when trying to gauge the extent of his expertise in genetics. Honestly, that's just how wild genetics can be, giving a biological basis even to the random whims of an author.
Regardless, I'd like to go through a couple examples.
Example: The Sharingan
Since I usually discuss Uchihas, let's use the Sharingan as our first example and apply a purely dominant/recessive inheritance pattern both for simplicity's sake as well as to better accommodate Anon's ask.
For communication purposes: E = Dominant Sharingan Gene e = Recessive Sharingan Gene N = Dominant Non-Sharingan Gene n = Recessive Non-Sharingan Gene
Sasuke and Sakura have a daughter, Sarada, who has inherited the Sharingan. Sakure doesn't carry the genetic information for the Sharingan on either allele, meaning that Sasuke is the only parent whose genes contributed to the manifestation of Sarada's Sharingan. This outcome can only arise under the condition that at least one of Sasuke's alleles is dominant (En or EE) in relation to Sakura's. So, depending on whether Sasuke's genotype is homo- or heterozygous, these are the two most probable possibilities:
Option 1: Sasuke's genotype is homozygous and dominant (EE)
Option 2: Sasuke's genotype is heterozygous and has a non-Sharingan gene (En)
But all of this only applies under the condition that Sakura's alleles are homozygous.
As a hypothetical, there might be multiple different "non-Sharingan" genes in the Naruto-verse. Some of these non-Sharingan genes may be dominant (N) and some others might be recessive (n) in relation to the Sharingan gene (E). Under this condition, both NE genotypes (the Sharingan is not expressed), as well as En genotypes (the Sharingan is expressed), may occur.
Option 3: Sakura's genotype is heterozygous (Nn)
In yet another scenario, Sakura may have unexpected Uchiha ancestry and secretly carry the Sharingan gene. This may hypothetically be the case if one (or more) of Sakura's ancestors had been married to or had had an affair with an Uchiha. Since Sakura herself never awakened the Sharingan, we'd have to assume the gene that was contributed by Sakura's Uchiha ancestor would be recessive (e).
Option 4: Sakura's genotype is heterozygous and has a Sharingan gene (Ne)
And, of course, there are multiple combinations and variations of all of these previously outlined options. I won't explain them all here, but you get the gist. That being said, the Sharingan being a "generally more dominant" trait remains the more probable option here.
Next to Sarada's heterozygous (En) type, we know for a fact that Indra Otsutsuki's genotype was heterozygous (En) as well. Since Indra's mother couldn't possibly have possessed the Sharingan, she must have contributed a recessive gene. So, if nothing else, the only confirmed non-Sharingan gene in circulation is a recessive gene.
While the Sharingan itself is still supposedly rare within the Uchiha Clan, this is easily explained with "low penetrance", as @the-real-sasuke-uchiha suggested. The obvious threshold here is the Sharingan's awakening condition in the form of psychological trauma and/or turmoil, constituting an environmental factor.
Example: The Byakugan
I won't go as far into the details as last time, but the previously illustrated Sharingan logic also applies to the Byakugan.
In the case of the Byakugan (b or B), we know that only one of Hinata's children inherited the Dojutsu (assuming this is not the result of low penetrance instead). Boruto got a recessive allele from Hinata (b or n), whereas Himawari received a dominant allele. For this, either Naruto and/or Hinata must have a heterozygous genotype.
Hierarchy: N > b > n
Hinata is homozygous (bb): Boruto (Nb) = Hinata (bb) + Naruto (Nn) Himawari (bn) = Hinata (bb) + Naruto (Nn) Hinata is heterozygous (bn): Boruto (Nb, nn, or Nn) = Hinata (bn) + Naruto (Nn or nn) Himawari (bn) = Hinata (bn) + Naruto (Nn or nn)
And of course, then there's also the possibility that Naruto's and Hinata's genes are co-dominant on the "Byakugan allele". This might have expressed itself through Himawari's blue eyes, which only turn white whenever the Byakugan is activated. Alternatively, this is just a regular eye pigmentation gene, unrelated to the Byakugan.
Example: Nature Combinations
While Earth and Water Release are the basis of Wood Release, in the manga, Kakashi points out that their simultaneous activation is what defines the Kekkei Genkai, meaning that the presence of two basic chakra nature affinities is insufficient. As a result, the inheritance of nature combinations must be influenced by two factors:
A polygenic collection of genes determining your individual affinities for the five basic chakra natures (Earth, Water, Fire, Wind, Lightning), which are the foundation of the Kekkei Genkai.
An additional gene that determines your ability to activate two or more chakra natures simultaneously and combine them.
So, more likely than not, we are talking about epistasis. Without affinity for two or more basic chakra natures as a foundation, the Kekkei Genkai gene itself will be masked (read: useless).
Of course, this doesn't mean that dominant and recessive inheritance patterns don't play a role. Both the nature affinity as well as the Kekkei Genkai genes are likely subject to recessive/dominant inheritance patterns.
If we apply this to Haku and his Kekkei Genkai, we can assume that he inherited a dominant Ice Nature Combination gene (I) from his mother (compare with Chapter #29). Unless Haku's father had secret ancestry from the Yuki Clan, he should've naturally lacked the combinator gene.
Haku (In) = Mother (II or In) + Father (nn)
But it is also entirely possible that Haku's mother could not use Ice Release herself, due to lacking Wind (A) and/or Water (B) Release (or both). That would make Haku's mother more of a "carrier" of the gene rather than a user, meaning that her husband would have contributed the genes for Haku's nature affinities.
Now, unlike Ice Release, Hashirama's Wood Release might be recessive instead as none of his relatives have ever been said to possess the Kekkei Genkai and there is no known association with the Senju Clan as a whole.
Back to Anon
Now, this was a whole lot of needless fantasy science, but it was fun regardless. I hope I could give you an approximate idea of just how complicated genetics are how inheritance patterns might affect the distribution of Kekkei Genkai within a blood lineage.
Regarding your initial question, Anon, there are no definitive answers, unfortunately, for multiple reasons:
"Recessive" and "Dominant" are not absolute and vary from gene interaction to gene interaction. Heck, there might even be different versions of the same Kekkei Genkai gene if we are really unlucky.
Different Kekkei Genkai may not (and most likely don't) follow the same inheritance patterns as other Kekkei Genkai.
While we can make educated guesses, we don't know the definitive inheritance patterns (polygenic, epistatic, gonosomal, etc.) of any Kekkei Genkai. From an irl POV, we don't even know whether Kishimoto had any thoughts on the matter.
But once again, recessive/dominant inheritance patterns are not the only setup that can lead to surprise Kekkei Genkai users born to parents without Kekkei Genkai abilities.
Nature Combinations almost certainly do lead to such as they most likely rely on epistasis (unless, I don't know, the Ice Release gene always comes in a triple pack with Water and Wind release).
The Byakugan might just meet the criteria for a recessive inheritance pattern (unless Hinata's genotype is heterozygous), and even if it isn't, it could be of low penetrance instead.
The Sharingan appears to be mostly dominant.
