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The helix with 3.7 residues per turn and the pleated sheet

by PhilipJ on 7 April 2006

It is, however, not unreasonable to expect that haemoblogin and myoglobin contain chains of the same type, because these proteins appear to be closely related in several ways; and furthermore, the repeat distance, the interchain distance, and the number of residues per repeat, are similar in these two proteins to the corresponding features of α-keratin. It will therefore be assumed as a working hypothesis that the chain configurations in large classes of proteins resemble one another closely, while bearing in mind that this hypothesis is based on slender evidence and may have to be abandoned when further experimental data are available.
– Bragg et al. [1]

Bragg, Kendrew, and Perutz

The above quote shows how little we knew of protein structural components in October, 1950, but it was a promising start. Sir Lawrence Bragg, J. C. Kendrew, and M. F. Perutz were attempting to understand the x-ray crystallography data obtained from protein samples, and had finally concluded that similarities between x-ray diffraction images of different (though related) proteins implied some common structural elements existed between them. They then proceeded to try and enumerate all possible configurations of a folded polypeptide chain which could give rise to the observed x-ray data, after settling on a few assumptions. They assumed that “the chains are held together by hydrogen bonding between >NH and >C0 groups of nearby amino acid residues”, meaning that the the chain is thrown into a series of rings which must be ruptured at the hydrogen bond before unfolding can take place. They also “regard those structures in which all NH and CO groups are hydrogen bonded as inherently more probable, because their free energy is presumably lower.” The understatement of their paper is

It will be evident that the number of configurations formally possible is rather large, and even though many of the formal schemes are found on examination not to be sterically possible it is still true that the number of configurations which should be examined in comprehensive survey is considerable.

To try and narrow the field, they exclude any structures which do not satisfy having a repeat distance of just over 5 angstroms (a characteristic of the x-ray data), and having three amino acid residues per repeat, though four residues per repeat are also considered plausible.

What followed are some 20 pages of helical chain configurations which all looked roughly like this, all of which for plainly stated reasons did not conform to observed x-ray data (particularly in bond lengths), and, as such, were wrong. But helices were on the right track, and this paper spurred other giants of the day into action.

Pauling & Corey’s helices and pleated sheet

Linus Pauling, Robert B. Corey, and visiting physicist H. R. Branson took a different approach at trying to determine the major structural elements of proteins. As arguably the world’s leading structural chemist, Pauling believed that knowing the details of the amino acids and short peptide strands that are the fundamental units of proteins, he could infer the major components of proteins. From their paper [2],

[W]e have been attacking the problem of the structure of proteins in several ways. One of these ways is the complete and accurate determination of the crystal structure of amino acids, peptides, and other simple substances relating to proteins, in order that information about interatomic distances, bond angles, and other configurational parameters might be obtained that would permit the reliable prediction of reasonable configurations for the polypeptide chain.

Pauling believed that the amide group (the backbone connecting the sequence of amino acids together) would be planar because of resonant electrons “between the carbon-oxygen and carbon-nitrogen positions” like so:

Not only was it predicted by Pauling theoretically, it “is now so well grounded and its experimental substantiation so extensive that there can be no doubt whatever about its application to the amide group.” Pauling further restricted his study to finding all hydrogen-bonded structures for a single polypeptide chain in which the carbonyl C=O group of each amino acid residue accepts an N-H hydrogen bond from another residue. They believed this problem was tractable (unlike the huge number of structures predicted by Bragg, Kendrew, and Perutz) because of the planar nature of the peptide group, and adhering to the precise bond lengths and angles determined by previous x-ray scattering experiments.

Indeed, there were “five and only five configurations for the chain that satisfy the conditions” and three of these were unsatisfactory—the length of the hydrogen bonds required would be much longer than the 0.272 nm observed previously. Of the two which remained, one had 3.69 amino acid residues per turn in the helix, and each amide group was hydrogen-bonded to the third amide group beyond it along the chain. The other had 5.13 residues per turn, and each amide group was bonded to the fifth amide group beyond it. They were cleverly called “the helix with 3.7 resides per turn” and “the helix with 5.1 residues per turn”, but have come to be known the familiar α helix and rarely observed γ helices. The 3.7-residue helix as it appeared in the original paper is shown to the right.

The authors then compare their helices to those proposed by Bragg et al. [1]:

None of these authors propose either our 3.7 or 5.1 residue helices. On the other hand, we would eliminate by our basic postulates all of the structures proposed by them. The reason for the difference in results obtained by other investigators and by us through essentially similar arguments is that both Bragg and his collaborators…discussed in detail only helical structures with an integral number of residues per turn, and moreover assume only a rough approximation to the requirements about interatomic distances, bond angles, and planarity of the conjugated amide group, as given by our investigations of simpler substances. We contend that these stereochemical features must be very closely retained in stable configurations of polypeptide chains in proteins, and that there is no special stability associated with an integral number of residues per turn in the helical molecule.

Indeed, had the physicists in the Cavendish walked across the courtyard to chemist Alexander Todd’s lab, history might be different. Todd recalled [4],

despite the proximity, Bragg never, to my knowledge, set foot in the chemical laboratory…until one day…he came to my room in a somewhat agitated state of mind, bearing a bunch of papers in his hand. [Asked whether I preferred the α helix over the helices that Bragg and coworkers invented,] I think, given the evidence, any organic chemist would accept Pauling’s view. Indeed, if at any time since I have been in Cambridge you had come over to the chemical laboratory, I…would have told you that.

A second paper [3] shortly followed, presenting the structure for the now-famous β sheet shown above. This paper outlined a configuration for polypeptide chains that placed the planar peptide backbone in the plane of a sheet, with adjacent sheets of parallel or anti-parallel orientation able to form hydrogen bonds. In the span of two months in 1951, the secondary structure puzzle was completed.

The most humorous detail of the entire story is that Pauling had been aware of the 3.7 turn helix structure for years. As a visiting professor at Oxford in 1948, Pauling fell ill and spent several days in bed. After boring of his detective novels, he instead “took a sheet of paper and sketched the atoms with the bonds between them and then folded the paper to bend one bond at the right angle, what I thought it should be relative to the other, and kept doing this, making a helix, until I could form hydrogen bonds between one turn of the helix and the next turn of the helix, and it only took a few hours of doing that to discover the α helix.” For technical reasons with x-ray data at the time, he held off publishing this result until seeing [1]. He then felt the need to publish his findings because, “I knew that if they could come up with all the wrong helices, they would soon come up with the right one!”


Upon seeing Pauling’s original paper in PNAS, Perutz said,

“I was thunderstruck by Pauling and Corey’s paper. In contrast to Kendrew’s and my helices, theirs were free of strain; all the amide groups were planar and every carbonyl group formed a perfect hydrogen bond with an imino group four residues further along the chain. The structure looked dead right. How could I have missed it? ... I cycled home to lunch and ate in oblivion of my children’s chatter and unresponsive to my wife’s inquiries as to what the matter was with me today.”

Perutz saw that Pauling’s helix was like a spiral staircase with the residues forming the steps, and that each “step” was 0.15 nm in height. Because of this feature, an x-ray diffraction spectrum should show a strong peak 0.15 nm away from the planes perpendicular to the fiber axis.

“In mad excitement, I cycled back to the lab and looked for a horse hair that I had kept tucked away in a drawer… After a couple of hours, I developed the film, my heart in my mouth. As soon as I put the light on I found a strong reflection at 0.15 nm spacing, exactly as demanded by Pauling and Corey’s helix.”

He later scattered x-rays from hemoglobin, a protein we now know is made up of primarily α helices, and saw the 0.15 nm peak. Writing to Pauling, “The fulfillment of this prediction and, finally, the discovery of this reflection in hemoglobin has been the most thrilling discovery of my life.” He and Kendrew went on to win the nobel prize for x-ray analysis of haemoglobin, and interest in protein structure, and the relation to function and dynamics has continued ever since.

[1] Polypeptide Chain Configurations in Crystalline Proteins, Lawrence Bragg, J. C. Kendrew, and M. F. Perutz, Proceedings of the Royal Society of London Series A 203, 321-357 (1950).

[2] The Structure of Proteins: Two Hydrogen-Bonded Helical Configurations of the Polypeptide Chain, Linus Pauling, Robert B. Corey, and H. R. Branson, Proceedings of the National Academy of Sciences USA 37, 205-211 (1951).

[3] The Pleated Sheet, A New Layer Configuration of Polypeptide Chains, Linus Pauling and Robert B. Corey, Proceedings of the National Academy of Sciences USA 37, 251-256 (1951).

[4] The discovery of the α-helix and β-sheet, the principle structural features of proteins, David Eisenberg, Proceedings of the National Academy of Sciences USA 100, 11207-11210 (2003).

  1. Doug    4488 days ago    #

    Is it unreasonable to speculate that since hemoblogin and myoglobin contain helical chains of the same type and are associated with the circulatory system with a heart having electrical pumping properties that these helices may be complex harmonic oscillators?

    If so, then there may be a relationship to the Schrodinger wave equation of quantum mechanics and / or the phasor equation of Steinmetz in electrical engineering.

  2. Andre    4488 days ago    #

    They are harmonic oscillators in the sense that anything sitting in a potential well is a harmonic oscillator for small enough displacements and they are being constantly bombarded by water molecules and so they are oscillating in some “complex” way.

    BUT, the only relationship to Schroedinger’s equation is the obvious one and has nothing to do with the heart’s electrical stimulation: at some lower level the parts of the protein will show their quantum nature and at that scale you would need quantum mechanics to describe their distribution of states.

    Similarly, the elasticity of DNA has nothing to do with its helicity for low forces: it’s well described by models of purely entropic elasticity. That means DNA’s stretching and bending modes don’t play a significant role. See Phil’s post.

    Moral: 1) Small is different. It looks like a spring, but macroscopic analogies to springs won’t get you very far. 2) Keep it simple. The mechanics of molecules can be surprisingly well described by classical models. Don’t go all quantum unless something forces you to.

  3. Doug    4486 days ago    #

    Andre – I understand what you are saying and agree that you are probably correct – although I am wondering if there is an accompanying imaginary unit ‘i“ as used first by Grassmann then Clifford then Steinmetz then Schroedinger.

    Isn’t it ironic that ‘What is Life?’ by Schroedinger inspired Crick, Watson, Wilkins and Franklin in their search for the structure of DNA – which of course is the double helix.

    I am just amazed that the helix is found at so many different physical gauges and involved in apparent electromagnetic energy processes – that I suspect some type of similar but not identical physics relates to this occurrence. I do realize that there may be no relationship at all.

    I also speculate if these helices may be nested as the nested bubbles in the string landscape of Polchinski and Buosso or embedded as the embedded minimal surface [and space dividing] helicoids of MSRI or Weber, Hofman and Wolf of Indiana.

    With Witten and Penrose emphasizing helicity in their respective twistor theories and Borcherds having a string-dimension in his proof of Monster Moonshine there may be some way to find an application for string theory – if the helix is considered a string with energy properties.

    I would find loop quantum gravity more appealing if there was a complex helical loop.

    Thanks for allowing my comments and for your response.

    Have you thought of discussing your interests with microbiologist Carl Woese [UI-UC] since I understand his undergraduate work was in math and phyiscs?

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