Building blocks of protein structures: Physics meets biology

Tatjana Škrbić, Amos Maritan, Achille Giacometti, George D. Rose, and Jayanth R. Banavar

Phys. Rev. E 104, 014402 – Published 8 July 2021

Abstract

The native state structures of globular proteins are stable and well packed indicating that self-interactions are favored over protein-solvent interactions under folding conditions. We use this as a guiding principle to derive the geometry of the building blocks of protein structures—α helices and strands assembled into $β$ sheets—with no adjustable parameters, no amino acid sequence information, and no chemistry. There is an almost perfect fit between the dictates of mathematics and physics and the rules of quantum chemistry. Protein evolution is facilitated by sequence-independent platforms, which can elaborate sequence-dependent functional diversity. Our work highlights the vital role of discreteness in life and may have implications for the creation of artificial life and on the nature of life elsewhere in the cosmos.

Received 21 January 2021
Accepted 22 March 2021
Corrected 16 September 2021

DOI:https://doi.org/10.1103/PhysRevE.104.014402

Physics Subject Headings (PhySH)

Proteins

Physics of Living Systems

Corrections

16 September 2021

Correction: The previously published Figure 5 contained an error in the angle given in panel (h) and has been replaced. Corresponding changes have been made to the caption of Figure 5, Table 1 and its caption, and the caption of Figure 2.

Authors & Affiliations

Tatjana Škrbić ^1,2, Amos Maritan ³, Achille Giacometti ^2,4, George D. Rose ⁵, and Jayanth R. Banavar ^1,*

¹Department of Physics and Institute for Fundamental Science, University of Oregon, Eugene, Oregon 97403, USA
²Dipartimento di Scienze Molecolari e Nanosistemi, Università Ca' Foscari Venezia, Campus Scientifico, Edificio Alfa, via Torino 155, 30170 Venezia Mestre, Italy
³Dipartimento di Fisica e Astronomia, Università di Padova and INFN, via Marzolo 8, 35131 Padova, Italy
⁴European Center for Living Technologies (ECLT), Ca' Bottacin, Dorsoduro 3911, Calle Crosera, 30123 Venezia, Italy
⁵T. C. Jenkins Department of Biophysics, Johns Hopkins University, 3400 North Charles Street, Baltimore, Maryland 21218-2683, USA

^*banavar@uoregon.edu

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Vol. 104, Iss. 1 — July 2021

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Images

Figure 1
Optimal geometry of space-filling helix. (a) A segment of ten residues of a helix from phage T4 lysozyme protein 1L56 (residues 61–70). The ribbon represents the helical trace formed by the $C_{α}$ atoms, the spheres denote the heavy backbone and side chain atoms in the helix, and the transparent tube is a guide to the eye. (b) and (c) Top views of two continuum helices, both with a helix pitch $P$ to helix radius $R$ ratio $η = (P / 2 π R) \sim 0.4$ and a local radius of curvature of the helix, $R_{local} = R (1 + η^{2}) \sim 1.16 R$ . The tube radii Δ in the two cases are different: $Δ / R_{local} = 1 / 2$ and 1, respectively. (b) When $Δ$ is less than $R_{local}$ , there is empty space in the interior. When $Δ$ is bigger than $R_{local}$ , the turn is too tight leading to a kink, as is sometimes observed in a garden hose (not shown). (c) The sweet spot occurs when $Δ = R_{local}$ , leading to maximization of the local self-interaction. (d) and (e) Side views of two helices with $η$ values of 0.8 and $\sim 0.4$ , respectively. In both cases, $Δ$ has been chosen to be the local radius of curvature of the latter helix ∼ $1.16 R$ . (d) When $η$ is larger than $\sim 0.4$ , there is empty space between successive turns and the nonlocal self-interaction is not maximized. In the other limit of small $η$ (not shown), successive turns of the tube overlap and this is forbidden sterically. (e) A Goldilocks situation here is when $η$ is tuned just right to $\sim 0.4$ yielding $(Δ / R) \sim 1.16$ for a continuum space-filling helix maximizing both local and nonlocal self-interaction. The top and side views of the optimal continuum helix are shown in (c) and (e) respectively. Panels (f) and (g) show how these results can be captured analytically (see text) for a continuum and a discrete tube, respectively.
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Figure 2
Coordinate system at a discrete location $i$ along the tube axis. The bond length $b$ , assumed here to be a constant, is the distance between successive points. The angle $θ_{i}$ is the angle subtended at $i$ by points $(i - 1)$ and $(i + 1)$ along the tube axis. $θ_{i}$ was taken to be the magnitude of the angle between neighboring bonds $(i - 1, i)$ and $(i, i + 1)$ along the chain. $μ_{i}$ is the dihedral angle between the planes $π_{1}$ and $π_{2}$ formed by [ $(i - 2), (i - 1), i$ ] and [ $(i - 1), i, (i + 1)$ ], respectively, or equivalently the angle between the binormals in a Frenet reference frame at points $(i - 1)$ and $i$ . Knowledge of the coordinates of the previous three points $(i - 2, i - 1, i)$ and the variables $(θ_{i}, μ_{i})$ are sufficient to uniquely specify the coordinates of the point $(i + 1)$ .
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Figure 3
Optimal packing of strands. (a) A single two-dimensional zigzag strand (with a rotation angle of $180^{\circ}$ ) lying in the plane of the paper. This planarity can only occur for a discrete tube and is forbidden for a tube in the continuum. Alternate points along a strand are colored red and blue (right and left points). There are two equivalent choices for a straight tube axis, one lying along the line of blue (left) points (blue or the left axis) or the line of red points (red or the right axis). Two distinct space-filling arrangements for strand packing are shown corresponding to (b) red axis–red axis (right axis–right axis) tubes [or equivalently blue axis–blue axis (left axis–left axis) tubes (not shown)] and (c) red axis–blue axis (right axis–left axis) (not shown). The two cases correspond to antiparallel and parallel β sheets with distinct distance constraints. The yellow point $M_{j}$ lies midway between the blue (left) points $j - 1$ and $j + 1$ . The maximization of self-interaction dictates that the distances $(i, j)$ in (b) and $(i, M_{j})$ in (c) ought to be $2 Δ \sim 5.26 Å$ to ensure space filling. (d) and (e) show the histograms of the distances $(i, j)$ and $(i, M_{j})$ in the interior of antiparallel and parallel $β$ sheets in protein structures. The black vertical lines show the theoretical prediction of $2 Δ \sim 5.26 Å$ . The mean values of both histograms are the same as the theoretical prediction (see Table 1).
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Figure 4
Two views of the local structure representation of proteins. (a) ( $θ, μ$ ) plot of the PDB data set (see Table S1 of the Supplemental Material) comprising 4416 proteins and 972 519 residues. Here, the local conformations of residues are shown in the ( $θ, μ$ ) plane. For strands, a $μ$ value that deviates from $\sim 180^{\circ}$ is the signature of a twisted strand, which is still locally planar. The plot shows chiral symmetry breaking; i.e., the points are not symmetrically placed around $μ = 180^{\circ}$ . Our simplified analysis does not attempt to account for this. (b) ( $θ, μ$ ) coordinates of random samples of 12 000 points each from the interior of α helices (orange); antiparallel (green) and parallel (red) $β$ sheets; and $β$ turns [the two interior sites of $(i, i + 3)$ hydrogen-bonded residues with no helical residues] (blue). The tight turns have $θ$ values similar to those of helices. Unlike for helices and turns, the $θ$ values of strands are not constrained. The black $X$ in both panels shows our prediction of the geometry of the space-filling helix.
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Figure 5
Distribution of α helix characteristics. (a) Distribution of the experimentally determined bond lengths (consecutive $C_{α} - C_{α}$ distances). The bond length in the theory was chosen to be the mean bond length of 3.81 Å and sets the characteristic length scale. The other panels show the distributions of (b) the rotation angle, (c) the rise per residue, (d) the helix radius, (e) $θ$ , (f) $μ$ , (g) the local radius of curvature, and (h) $ψ = 180^{\circ} - ∠ [π (i - 1, i, i + 3), π (i - 1, i + 2, i + 3)]$ , where the angle $∠ [π (i - 1, i, i + 3), π (i - 1, i + 2, i + 3)]$ is the dihedral angle between the planes defined by the points $(i - 1, i, i + 3)$ and $(i - 1, i + 2, i + 3)$ in Fig. 1. The triangles formed by the two triplets ought to be congruent but they are not coplanar. The black line in each of the panels (except the first) shows the zero parameter theoretical prediction. Overall, there is excellent accord between theory and observations from protein structures.
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Figure 6
Consilience between mathematics and biochemistry. The figure shows three views each of two short proteins. (a)–(c) is the 56-residue long protein 3GB1 comprising four strands assembled into sheets along with a single helix. (d)–(f) is a protein of the same length, 2KDL, comprised of a three-helix bundle. Each panel shows a uniform tube, with the theoretically predicted radius of 2.6 Å, whose axis passes through the $C_{α}$ atoms. The sole exception is the $β$ sheet for which hydrogen bonding was identified using dssp [11]), where every other $C_{α}$ atom is considered [as explained in Figs. 3 and 3]. The tube color varies continuously from red to blue (via gray) as its axis moves from the N terminal to the C terminal (in the black and white image, the tube is darkest at the two ends). The heavy atoms of the side chains sticking outside the tube are shown. The maximization of the self-interaction through space-filling is evident.
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