 Anyone up for tic-tac-toe and genetics? It’s not exactly a game, but grab a cup of coffee and let’s explore the intriguing design in genetic systems.

In parts 1, 2, and 3 of this series, we observed three features of genetic systems that suggest design. In this article we look at isomorphic systems to find yet another analogy in the genetic code.

## Analogy: Genetic Code-Like (GCL) Binary Representation

Recall the genetic code mapping table discussed in part 1 and part 2 of this series. This table describes the mapping between the 64 codons and the 20 amino acids.

Researchers in Venice, Italy, have identified a specific and unique number system and have used it to mathematically model the genetic code. In this work, the 64 codons and the 20 amino acids are assigned to numerical elements within the system, referred to as the genetic code-like (GCL) binary representation. The GCL binary representation is a unique model that holds true to the specific redundancy features in the natural genetic code mapping.1 It is a mathematical model of the underlining physical/chemical processes related to genetic information processing—a so-called structural isomorphism.

An isomorphism is a one-to-one correspondence between the elements of two sets such that the result of an operation on elements of one set corresponds to the result of the analogous operation on their images in the other set. If two sets are isomorphic with respect to certain properties, then those properties that are true of one of the sets must also be true of the other.

For example, a six-sided die and a bag from which a number 1 through 6 is chosen are isomorphic. As another example, tic-tac-toe and the “game of 15” are isomorphic. In the game of 15, players take turns saying a number between 1 and 9. Numbers may not be repeated. Both players aim to say three numbers that add up to 15. Although perhaps not obvious, the defining characteristics of this number game are identical to those of tic-tac-toe. It turns out that both games are based on the well-known (to mathematicians) magic square.

The GCL binary representation and the genetic code are also isomorphic systems (sets). So, characteristics that are true of the GCL binary representation must also be true of the genetic code. (See here and here for more details of isomorphic systems.)

What are the characteristics of the GCL binary representation? The European researchers noted that this mathematical model exhibits:2

• Palindromic symmetry
• Parity symmetry
• Organized redundancy
• A rich mathematical structure

Such elegant symmetry, organization, and structure speak of a code that has been designed for a purpose—no mere afterthought of evolutionary chance events.

Also, the GCL binary representation makes possible the existence of error detection/correction codes that operate along the strands of DNA. A parity code (as discussed in part 3 of this series) is one example of such a technique.

If a parity code or similar technique functions along strands of DNA using the GCL binary representation, then dependence must exist in the genetic data along the strand. In other words, the data must be correlated to some degree.

Assuming the GCL binary representation and using two different robust statistical analysis methods, the research team discovered significant short- and long-range correlation peaks in actual DNA sequences. These results confirm that actual DNA sequences using this specific model satisfy a basic prerequisite for such error-minimizing techniques.

The team noted that “an error-control mechanism implies the organization of the redundancy in a mathematically structured way,” and that “[t]he genetic code exhibits a strong mathematical structure that is difficult to put in relation with biological advantages other than error correction.”

Thus, scientific advance has uncovered a peculiar and unique mathematical model that accounts for the key properties of the genetic code. This model exhibits symmetry, organized redundancy, and a mathematical structure that would be vital for the existence of error-coding techniques operating along the DNA strands. Actual DNA data tested using this model gives a strong hint that such further error-coding techniques may very well exist and provides impetus for future study in this area. Purposeful creation seems a reasonable conclusion.

Part 5 (the last entry) of this series will discuss the impact of these analogies on William Paley’s watchmaker argument.

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##### Endnotes
1. It describes exactly the 1st level of degeneracy (i.e., redundancy) in the natural genetic code (i.e., the number of codons which map to specific amino acids), and gives deep insight into the 2nd level of degeneracy (i.e., the association between specific codons and specific amino acids).

2. A full treatment of how the GCL binary representation displays these features is very technical and difficult to communicate without active use of visual aids. Nevertheless, this note is a brief attempt. See here for the necessary visual aid. Table 3 shows the actual GCL binary representation of the natural genetic code. The numerical elements in the model are associated with biochemical elements in the genetic code. The whole numbers 0–23 along the table edges map to amino acids. The 6-bit binary strings map to codons. The degeneracy number is shown in the center of the table, along with the amino acids coded for. The degeneracy number indicates how many different codons code for each particular amino acid. A careful inspection will show that all 64 codons are present, along with all 20 amino acids. Palindromic symmetry is seen as a reflection through the middle of the table in a left to right fashion. Palindromic amino acids are always associated in pairs. For example, tryptophan (Trp) and methionine (Met) are a pair of palindromic amino acids. Note that if the 6-bit codewords are folded on top of each other through the middle of the table, they form bitwise negated pairs, reflecting a very peculiar mathematical structure for palindromic symmetry. Also note the symmetry evident in the parity, again through the middle of the table. The light entries are odd parity (odd number of 1’s) and the dark entries are even parity (even number of 1’s). The interested reader is referred to the journal paper for further details.