Ramachandran plot Phi(ϕ) Psi(ψ) dihedral angle Convention for Zero, Positive and Negative value- old and new

[OP] On today's convention; how we define the zero value, positive value and negative values of phi and psi dihedral angle?

In general, you have four atoms (1, 2, 3, 4) in a chain to define a torsion angle. You rotate around the bond connecting atom 2 with atom 3 to change the torsion angle (bond angles and lengths stay constant). The conformation where 1 and 4 are closest is called zero degrees (cisoid). The conformation where 1 and 4 are furthest apart (transoid, extended conformation) is called 180 degrees. In both cases, all four atoms are in a single plane.

To measure the torsion angles of other conformations, you consider the plane through atoms 1,2,3 compared to the plane through atoms 2,3,4. The angle between these planes is the torsion angle. For a given angle, there are two conformations related by mirror symmetry. The right-handed one is given the positive torsion angle while the left-handed one is given the negative one (see figure in OP's question).

For a mathematical treatment with figures, see https://www.math.fsu.edu/~quine/MB_10/6_torsion.pdf

For the protein main chain, there are multiple ways to define a torsion, e.g. N-CA-C-N or N-CA-C-O. Switching the definition will lead to a shift of torsion angles by 180 degrees. The current convention, however, is to use N-CA-C-N and C-N-CA-C, and this is reflected in the current version of the Ramachandran plot.

The authoratative reference for the definition of torsion angles in general is the IUPAC gold book, citing DOI:10.1351/pac199668122193.

The authoratative reference for which atoms are used to define phi and psi seems to be Richardson, J.S. (1981). "Anatomy and Taxonomy of Protein Structures". Advances in Protein Chemistry. 34: 167–339.

Evidently the old $\phi$ is measured from an angle oriented $180^\circ$ relative to the new convention. This implies that the amide H ($H_N$) rather than carbonyl carbon was used as the reference fourth atom used to define the dihedral, since these atoms are related by a $180^\circ$ rotation. The old choice of atom is opposite on the chain to the modern convention of using the heavy backbone atom (carbonyl carbon). You can obtain angles in the new convention by subtracting $180^\circ$ ie $\phi \rightarrow \phi - 180^\circ$.

The meaning of angles with value of zero for $\phi$' aka $\psi$ is the same. However the values between $180^\circ$ and $360^\circ$ in the original Ramachandran plot can be shifted below zero, as $180^\circ\rightarrow -180^\circ$ and $360^\circ\rightarrow 0^\circ$, or more generally $\psi \rightarrow \psi - 360^\circ$.

By the way inspection shows that this is the correct interpretation. See for instance the following diagram. The extended (beta/turn/pII) region is usually larger than the alpha region. The beta region sits at large values (below $180^\circ$) of $\psi$, whereas the narrower alpha region is close to $0^\circ$ (at negative angles):

In Ref. 1 Ramakrishnan and Ramachandran explain the differences in the conventions:

At a recent conference of some of the representative workers in this field held in Bethesda, it was decided to denote the two dihedral angles about the bonds N-Ca and C.-C' by $\phi$ and $\psi$ respectively, the sense of rotation being the same as that adopted here. The fully extended chain, with N-H and C'=O trans with respect to one another, is to be taken as the standard conformation with $\phi$=$\psi$=0.

It is readily seen that the new $\phi$ is the same as the old $\phi$, but that $\psi$=$180^\circ$+$\phi'$ (and $\phi'$=$180^\circ$+$\psi$). All the data reported here are thus readily converted into the ($\phi$,$\psi$) coordinates. The diagrams in Figs. 2, 3, and 6 have to be shifted up by $180^\circ$, or half the total length, along the vertical direction. In particular, the right-and left-handed alpha helices will have ($\phi$,$\psi$) equal to ($133^\circ$,$123^\circ$) and ($227^\circ$,$123^\circ$). As before, a helix with ($-\phi$,$-\psi$) will be inverse to one with ($\phi$,$\psi$); i.e., it will be of opposite sense, but having the same number of turns per unit.

This paper was finalized well before this meeting, and so the older conventions are adopted here. However, it is proposed to use the new notation in the following papers in this series.

References

1.Ramakrishnan C, Ramachandran GN. Stereochemical criteria for polypeptide and protein chain conformations II. Allowed conformations for a pair of peptide units. Biophysical Journal, 01 Nov 1965, 5(6):909-933. DOI: 10.1016/s0006-3495(65)86759-5.