The genetic code uses an alphabet of only 4 letters, namely C, A, G, and T. These represent the four possible bases that make up DNA. The letters are used to form 3-letter words, called codons. There are therefore 43 = 64 different codons, of which 61 code for amino acids. Since there are only 20 amino acids, there are obviously lots of synonyms. The three-letter words very naturally lend themselves to a three-dimensional arrangement:
You can rotate the image by dragging it with the mouse. Double-click to restore the original orientation.
The three dimensions have the following meaning (in the original orientation):
where * is a wildcard, denoting any of the 4 possible bases. In an expression such as G**, the two wildcards are independent, so G** refers to 16 different codons.
Some people have a hard time visualizing things in three dimensions, but for those who can manage it, this is a vastly more faithful representation, compared to the circular “rosetta stone” diagrams or other two-dimensional representations. For starters, you can see there are lots of places where it doesn’t matter if the third base changes. This is the back-to-front direction in the diagram (in its initial orientation).
There are additional places where it doesn’t matter if the third base changes from one purine to another, or if it changes from one pyrimidine to another. This is easy to see, even on the rosetta-stone diagrams. Let r denote either purine (G or A), and y denote either pyrimidine (C or T):
The second base is essentially always important – more important than the first, and much more important than the third. The only exceptions, i.e. the only places where a group of codons crosses a second-base boundary, are the Stop codons (which are very exceptional, because they don’t code for any amino acid), and serine, which is exceptional because when you change the second base you also need to change the first base to make up for it.
The triangular flag in the upper-right corner of each tile has the following meaning:
The synonyms and the proximity relationships are important because of point mutations. The mutation rate is itself under genetic control (because of proofreading). It is smallish but definitely not zero. As the saying goes:
For example, there are four ways of making threonine. A mutation that changes one of those into another is a completely silent mutation, and therefore subject to genetic drift. In the long term you would expect each codon to appear one fourth of the time.
It turns out that two of the threonine codons are adjacent to lysine codons, while the other two are not. Similarly, one of them is adjacent to methionine, while the other three are not, as you can see in the diagram.
Now suppose that due to some new evolutionary pressure it becomes important to evolve from Thr to Met at some point in the molecule. You would expect that 1/4 of the population could make that change with just one additional point mutation. A single mutation is enormously more feasible than anything requiring two mutations in series, especially if the intermediate step would be a deleterious or lethal mutation.
This discussion ignores all sorts of details about diploidy, crossover mutations, et cetera ... but it’s not wrong as far as it goes. It’s part of the story.
For more about what mouse-motions correspond to what sort of rotations, see reference 1.