Philosophy is written in a great book that is continually open before our eyes (that is the Universe), but it cannot be understood unless one first learns to decipher the language and the characters in which it is written. It is written in a mathematical language, and the characters are triangles, circles, and other geometric figures, without which it is impossible for us to understand a word of it; without them it is a vain wandering through a dark labyrinth.

Galileo Galilei, The Assayer

The geometry of the backbone (or main chain) of a peptide is completely described by a sequence of dihedral angles (also known as torsion angles): the angle Φ is the angle around the chemical bond between the alpha carbon of one amino acid and the azote of the same amino acid; the angle Ψ is the angle around the axis of the bond between the alpha carbon and the other carbon (I call it beta carbon, but this is a personal notation) of the same amino acid. This definition is incomplete, of course, because as we all know, a dihedral angle is defined not only by an axis but also by two planes that intersect in that axis. Not only that, these two angles have a sign, so we must specify a positive arrow of rotation. This can be done in several ways. The problem is that the most efficient way would be by just drawing the planes the torsion angles are defined by. Now, through the years I discovered that this drawing is usually lacking in books, leading to some confusion, particularly in those who study biochemistry without a particular interest in a quantitative description of molecular structures. In Figure 1 you find the graphical description of Φ and Ψ in a well-know book of biochemistry. Do you see the planes the angles are defined by? In Figure 2, two further examples of illustrations that are not suited at completely describing the two dihedral angles, from a two other well-know books, one about bioinformatics, the other one a booklet about protein structures. The second one is the best one, but you still have to mentally merge two drawings (FIG. 5 and FIG. 6) to get the full picture. I hope I won’t be sued…

Figure 1. Dihedral angles, from Principles of Biochemistry, by Voet D, Voet J, and Charlotte WP.

Figure 2. Two illustrations for the description of the torsion angles. The one on the left is from Bioinformatics, Sequence and Genome Analysis, by David W Mount. On the right: The Anatomy and Taxonomy of Protein Structures, by Jane S Richardson.

Now, it is possible that I have just been unlucky with the books of my personal library. Or it may be that the illustration with the three planes we need to correctly define Φ and Ψ (we just need to add a plane to the amide planes), would not be clearly readable. I tried years ago to draw the planes for Ψ (the illustration is in this blog post) and I have now completed it with the other torsion angle, by drawing a tripeptide (Figure 3). It is just a handmade drawing, but I think it serves the scope. This is what I call the lost illustration.

Figure 3. A peptide of three amino acids. The amide planes are shaded grey. A further plane has been added to fully characterize the torsion angles Φ and Ψ. Pencil and pen on paper, by Paolo Maccallini. For the signs of the two angles, a clockwise rotation is considered positive. So, in this case, we have Φ of about -90° and Ψ of about 150°.

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