Edit: I'll try and post a more simplified version later
Re-Edit: Necessity has required that i repost this before i got around to simplifying it. I also dont have time right now to make the formatting pretty (bold, underlined, quote marks etc). I'll get back to it later.
Blue birds are only blue because their yellow pigment production is faulty, creating what should really be known as a 'parblue' budgie or faulty green essentially. There are four known distinct mutations of the fully functional yellow gene (lets call it G) - gene B1, B2, By2 (YF2) and GF. B1 is the first known faulty mutation, and in the presence of G (Eg G B1 budgie) the fully functional G gene is able to fully compensate for the faulty B1 gene and gives you what we call a green budgie split for blue. The same goes for all parblue genes - in the presence of a single functional gene, there is no phenotypical change as yellow production from the one functional gene is sufficient to fully compensate for the faulty gene.
The alleles of the Yellowface and Blue series can be divided into two groups. One group contains the homoalleles (or alleles with mutation sites in approximately the same position) b2, by2, and bg (which i have called GF above). The other group contains only a single member thus far, the common allele for blue, b1. In the latter, the mutation clearly occurred in a different area of the gene giving it the property of being able to interact in such a way that a series of very active hybrid enzymes is produced.
It is also important to note before going on that:
The budgerigar produces two types of yellow pigment. Each allele or combination of alleles in the series codes for enzymes which synthesize these yellow pigments to varying degrees.
When you get a double factor B1 bird, there is NO ability to produce a working enzyme for yellow pigment. This is the allele that is responsible for creating the first blue birds. At some unknown time, a second distinct mutation occured that phenotypically gave the same result as the faulty B1 allele. This is the B2 allele, and double factor B2s are identical to B1's, however the B1 mutation is located on a different part of the gene, and although faulty, seems more functional than the B2 allele(although not enough to produce a working enzyme on its own).
When paired together, B1 and B2 seem to be able to compensate PARTIALLY for eachothers faults, and TOGETHER they are able to produce a weak enzyme that produces a light yellow pigment in the face (and sometimes this is also found in the wings or tails of these birds). This combination of genes produces what we call the YF1 mutant blue. A 'double factor YF1' is not B1B1, but B2B2, as the B1 allele is on a different part of the gene, allowing it to interact with and produce varying degrees of pigment when paired with B2, By2 and GF. Breeding a B2B2 to a B1B1 bird will produce 100% B1B2 offspring, or 100% 'single factor' YF1 mutants. Technically single factor and double factor YF1 is incorrect, as these birds are actually a combination of two genes, in a similar way that a full body clearwing is a combination of the clearwing and grewying genes which are mutations on different parts of the same gene and are compensatory when paired together.
In an attempt to improve upon the YF1 mutant, the Keston bird farm in Kent, england selectively bred a large number of birds. Although they were unable to improve the colouration of the YF1 mutant, their attempts resulted in the first commercial production and sale of "Rainbow budgies" as well as the appearance of a new mutation, the YF2 mutant (By2).
Keston Farm continued to use selective breeding until it developed the Yellow-Face Budgerigar Type II, which became known as the Golden-Face in many Budgie circles. The primarily difference is that where the Type I is a lemonyellow and the yellow coloring is limited to the head and mask, with the Type II the yellow coloring is a deeper and richer yellow – a golden yellow, that spreads to the body of the bird and often times even gives the bird a yellow tint.
(I'll link the document later).
The By2 gene is a semi-functional gene, therefore it is semi-dominant to blue, or co-dominant. Thus is can truly be called "single factor" or "double factor" as there are differences in the appearance of both. In its single factor form (By2B1), the bird appears as above - A deeper richer yellow than YF1 that spreads into the body and creates a yellow tint (seafoam green as we call it). In its double factor form, the yellow pigment can be slightly stronger but the yellow production seems limited to the face and upper chest of the bird (By2By2) as well as the short lateral tail feathers.
Later still a further and final distinct mutation accured in australia. This is the Goldenface (which i have called the GF, but should really call bg). This is similar in breeding and appearance to the YF2 (By2) allele, but produces a much stronger yellow enzyme.
From what i have read and understand, there remains a great confusion over yellowface breeding due to the variety in offspring that can be hard to understand without realising the number of seperate mutations that have occured in the yellow face or 'parblue' gene. Here is a summary based on what i can find:
B1B1 Produce the defective B1 enzyme
B2B2 Produce the defective B2 enzyme
B1B2 Produce the defective B1 and B2 enzymes, and the semi-functional B1B2 hybrid enzyme creating the classic Blue mutant 1 (YF1)
By2B1 Produces the defective B1 enzyme and the Hyperactive By2B1 enzyme, creating the single factor Blue Mutant 2 (single factor YF2) we know and love which has a yellow wash (eg seafoam green).
I thought it would help to introduce the classic YF2 and then add that YF2 gene breeds exactly like the goldenface gene, although in all combinations produces a lighter yellow than GF. Read below to find out the breeding combinations of GF.
BgfBgf ( Bgf = bg) produces the active enzyme Bgf and creates the classic Double Factor Goldenface, with a strong yellow colour face, and in adults this can sometimes spill into the chest area.
BgfB1 produces the defective B1 enzyme, the active enzyme Bgf and the hyperactive hybrid enzyme BgfB1 giving a bird that is almost identical to the Blue mutant 2 (YF2) except it produces a much brighter yellow in the face and body. In the case of the body, it is a deeper shade of green than By2B1 (YF2 mutant) but still lighter than a normal green.
BgfB2 Produces the defective B2 enzyme, the active enzyme Bgf and the active hybrid enzyme BgfB2 producing a bird paler than a double factor goldenface but brighter than a YF1 mutant. This is the classic Single Factor Goldenface.
BgfBy2 is as follows:
Composites between Mutant 2 Yellowface and Goldenface do exist and likewise produce three versions of the enzyme namely (bg/bg) enzyme, (by2/by2) enzyme, and (bg/by2) hybrid enzyme. All three versions of the enzyme are active but the hybrid enzyme is not overactive. These birds closely resemble the double factor Goldenface.
Here are the links to articles that will elaborate: