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

Update: possible duplicate: What is the precise definition of Ramachandran angles?. Question modified.

G.N. Ramachandran et al, in their own work (PMC) (DOI), did not used phi(ϕ) and psi(ψ) as we use today. They used phi(ϕ) and phi-dash(ϕ') defined as follows:

The conventions we have adopted for the standard configuration ϕ = ϕ' = 0°) and for the positive sense of rotation for the two angular parameters ϕ and ϕ' are indicated in Fig. 1.

Where Fig. 1 is: Also they had adapted a very different values, 0 to 360 degrees for ϕ and ϕ'. They provided the Ramachandran plot like this: This looks very different from the Ramachandran plot we see nowadays because:

1. This (Ramachandran's original) graph looks flipped from todays.

2. The zero degrees value of present day is perhaps different from Ramachandran's Zero degrees. Because the current day zero degree correspond to Ramachandran's 180 degree (of original graph).

So my question is basically:

On todays convention; how we define the zero value, positive value and negative values of phi and psi dihedral angle? and how does the new and old convention relate?

Prior research:

I did not find anything on intensive internet search. I got one graphic without any reference I am doubtful about this diagram because its Zero degrees seem to be the zero degree of Ramachandran's original plot.

Update: As the user @Buck Thom said

The meaning of angles with value of zero is the same.

But sill I looked both old and new graph and found what is labelled as 0 degrees in the new graph, has been labelled as 180 degrees in the old graph as follows • @BuckThorn Thank you. I did not mean 360 degrees I mean 180 degrees. Actually I tried to manually reconstruct the Left handed alpha helix, by considering the Web clip (3rd picture). Turned out I made a right handed helix. The confusion gone when I followed Ramachandran's original paper. Seemingly the 2 conventions are very different. Apr 25 '21 at 19:01
• Apr 26 '21 at 3:24

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.

• "The meaning of angles with value of zero is the same" Then why 0 of new graph is corresponding with 180 degrees of old graph? I am not getting this. Apr 25 '21 at 18:31
• It's because in a circle zero degrees is the same as 360 degrees. Or, alternatively, you can move clockwise say and call the angles positive, or counterclockwise, and call the angles negative. Apr 25 '21 at 18:37
• Not 360 its 180 i think I failed to express what I wanted to say. let me update the question Apr 25 '21 at 18:38
• I agree, that is confusing! My current assessment is that they confused the symbols somehow (seems unlikely though). In any case the explanation in my answer shows the difference between the different coordinates, even if that quote from the article is not easy to decipher. Apr 26 '21 at 9:09
• I tried to find out what the Bethesda conference decided on the the 1960s, and whether it is different from the current definition, but did not succeed. So maybe the footnote in the Ramakrishnan and Ramachandran paper is inaccurate, or the community decided to switch the atoms defining phi and psi later on. With the sources at hand, it is impossible to tell. Apr 27 '21 at 13:08

I've attached a figure to show the dihedral angles. Hope this helps. Part of a peptide chain. The four atoms defining the $$\psi$$ (N to N) and $$\phi$$ (CO to CO) torsion angles are shown on the right. As these chains are drawn as if completely extended, the angles are $$180^\text{o}$$. When the main chain atoms are eclipsed, the angles are zero. The 6 atoms in the shaded areas are in the same plane.

Ramachandran plot together with the structure for bacteriorhodopsin. Ramachandran plot of the $$\phi$$ and $$\psi$$ torsion angles in the protein bacteriorhodopsin (pdb 1FBB), which contains extensive $$\alpha$$ - helices. Most angles cluster around the values $$\phi = -56^\text{o}, \psi = -53^\text{o}$$ typical of an $$\alpha$$ - helix. A regular or ideal helix would have angles $$-60^\text{o}$$ and $$-50^\text{o}$$. The area of $$\beta$$-sheet structure, of which there is very little in this protein, is in the top left corner bounded by approximately $$-55$$ and $$+60$$ degrees.

The structure of the protein shows extensive helical structure. The retinal chromophore, which is positioned in the centre of the column of helices, is shown also. • Thanks for your answer but can you please add how do I define phi=zero? on which conformation the phi would be called zero? similarly for psi, and other positive and negative values? Apr 25 '21 at 20:18
• My questiom is even more basic. Its about how we are counting phi and psi; from which frame of reference. Apr 25 '21 at 20:20
• I think this is what you asked, in the top figure 'As these chains are drawn as if completely extended, the angles are 180 degrees. When the main chain atoms are eclipsed, the angles are zero. The 6 atoms in the shaded areas are in the same plane.' Apr 26 '21 at 6:50
• as to positive negative you will just have to try with a structure such as the one above. If you use python I have code that will draw diagrams and can addd maths about dihedral angles if needed. Apr 26 '21 at 7:00
• The protein data bank has a good simulator. Apr 26 '21 at 7:07

This is not really an answer. This is some conclusions or assumptions.

1. The ϕ (old) and the ϕ (new) dihedral angle:

It is quite clear that the graph look is unchanged along horizontal axis. Where ϕ (old) corresponds to ϕ (new) as follows:

ϕ (old) ................. ϕ (new)

0 deg ......... .......... -180 deg

180 deg ....... .......... 0 deg

360 deg .... .......... +180 deg

Conclusion:

ϕ (Old) - 180 degrees = ϕ (new)

or, ϕ (old) = ϕ (new) + 180 degrees.

2. The ϕ' (old) and the ψ (new) dihedral angle

Note that diagram is scroll shifted. Where ϕ' (old) and ψ (new) are related as followes:

ϕ' (old) ..... ..... ψ (new)

0 deg ............ 0 deg

180 deg ........... 180 deg, -180 deg.

360 deg ........... 0 deg

It means we do not have a frame shift between ϕ' (old) and ψ (new) , although there is the difference in direction of counting the angle, from where the negative sign comes.

[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.

• The Cisoid and Transoid is a good way to understand. Apr 26 '21 at 3:36