In the genetic code of living systems, information is coded in an ordered string (polymer) of four bases known as DNA. Each polymer is an ordered combination of four bases: adenine (A), thymine (T), guanine (G) and cytosine (C). These bases can pair up as in A to T and G to C such that each string of DNA can pair up with its complement. Each string of DNA encodes for the production of an ordered string of amino acids, in particular a peptide or protein. Each amino acid is encoded by a particular ordered sequence of three bases (triplet) in the DNA string.
An amino acid can be represented by the general chemical formula:
in which R represents a general group. Each molecule of an amino acid has a tetrahedral shape. The central carbon atom (not shown) is bonded to four different atoms or groups: a hydrogen atom, an amino group (NH2), a carboxyl group (COOH) and a group R (other than glycine, where R is itself a hydrogen atom). For example, when R represents a methyl group the amino acid is alanine.
Amino acids can form chains in which the carboxyl group of one amino acid forms a peptide bond (CONH) with an amino group of an adjacent amino acid. In this way amino acids can form peptides and proteins, which have many different functions in living organisms.
The amino acids making up the body are chiral.
The word “chiral” comes from the Greek word for hand. Amino acids are chiral molecules. Four different objects can be arranged in 3D space in two mirror-opposite ways, like the wrist, thumb, first finger and second finger of the hands. Chirality is a symmetry that distinguishes the left from the right hands. The thumb, first finger and second finger can be ordered either clockwise or anticlockwise relative to the wrist.
In an amino acid, the four different objects are the hydrogen atom, R group, amino group and carboxyl group bonded to a central carbon atom. They point towards the corners of a tetrahedron. All of the amino acids in the human body are of one handedness.
Chemists use a technique known as the Fischer projection to distinguish between the two forms of a chiral tetrahedral molecule. A Fischer projection sets out the four components of a tetrahedral molecule as if they lie at the ends of a cross. The two components positioned horizontally are deemed to project towards (+) the viewer and those positioned vertically are deemed to project away (-) from the viewer. By convention, the carboxyl group of an amino acid is positioned above and the R group below.
When the amino group is positioned on the left and the hydrogen is positioned on the right, the amino acid is said to be in the L configuration. When the amino group is positioned on the right and the hydrogen is positioned on the left, the amino acid is said to be in the D configuration. All amino acids in the human body are in the L configuration.
Instead of viewing an amino acid with both the amino group and hydrogen atom projecting towards (+) the viewer, we can also look at the amino acid with just the hydrogen atom projecting towards (+) the viewer and the other three groups projecting away (-). The order of the R group, amino group and carboxyl group is clockwise for the L amino acid and anticlockwise for the D amino acid. However, if the D amino acid is viewed from the opposite side (in effect reversing all of the signs) then the order of the R group, amino group and carboxyl group is the same as that for the L amino acid viewed from the opposite side.
The significance of the signs and their utility in the coding of relationships will become clear below.
The two different forms of an amino acid, L and D, are called enantiomers. It is not possible to superimpose the four groups of one enantiomeric form on those of the other, no matter how you rotate the molecule. They are mutually exclusive. This can be visualised by taking a disc having a black side and a white side and marked to match up with three of the four objects of one enantiomer on one side and three of the four objects of the other enantiomer on the reverse side.
The two enantiomers are mutually exclusive (XOR/XNOR). The eight objects of the two enantiomers if taken together (interpenetrating) would point to the corners of a cube.
Each face of a cube corresponds with a Fischer projection of a chiral tetrahedral molecule: i.e. with two groups projecting towards (+) and two away (-) from the viewer. There are 24 such projections: four for each of the six faces, and they correspond with the 24 different ways in which four different objects can be permuted (4 x 3 x 2 x 1). With every change from one Fischer projection to another, two of the signs flip and two stay the same.
The drawings of Fischer projections are adapted from a paper by S. Capozziello and A. Lattanzi: “Chiral Tetrahedrons as Unitary Quaternions: Molecules and Particles under the Same Standard”: International Journal of Quantum Chemistry, Vol. 104, 885-839 (2005). The authors conclude in their paper that Fischer projections “constitute a fundamental structure to achieve a quantum chiral algebra, which relates properties of tetrahedral molecules to those of spinor particles.”
Francisco M. Fernández: “On the algebraic structure of central molecular chirality”: J. Math Chem (2016) 54: 552-558 has disclosed that the 24 4×4 permutation matrices (ci) of Capozziello and Lattanzi can be reduced to 24 3×3 matrices. The 3×3 matrices are composed of three rows of two 0s and one +1 or -1. According to Fernández the 4×4 matrices (ci) can be transformed into the 3×3 matrices by means of the following equation:
When the 24 4×4 permutation matrices of Capozziello and Lattanzi are converted to 24 3×4 Rxtci matrices using the equation of Fernandez, these 3×4 matrices constitute ordered triplets of chiralkine numbers. All of the matrices coding for the (+) enantiomer are ordered yellow, blue, green and all the permutation matrices for the (-) enantiomer are ordered blue, yellow, green. All 24 matrices recovered from the equation of Fernandez are coded in XNOR (+ means same and – means different).
Each must have a mirror opposite coded in XOR (- means same and + means different), where the colour order is conserved. (A tetrahedron has a total of 24 symmetry elements and a cube has a total of 48, twice 24). The two mirror-opposite sides are coded in the same chirality.