Peer Review #1 of "An exhaustive survey of regular peptide conformations using a new metric for backbone handedness (h) (v0.1)"

8 The Ramachandran plot is important to structural biology as it describes a peptide backbone in the context of its dominant degrees of freedom – the backbone dihedral angles φ and ψ (Ramachandran et al., 1963). Since its introduction, the Ramachandran plot has been a crucial tool to characterize protein backbone features. However, the conformation or twist of a backbone as a function of φ and ψ has not been completely described for both cis and trans backbones. Additionally, little intuitive understanding is available about a peptide’s conformation simply from knowing the φ and ψ values of a peptide (e.g., is the regular peptide defined by φ = ψ = −100◦ left-handed or right-handed?). This report provides a new metric for backbone handedness (h) based on interpreting a peptide backbone as a helix with axial displacement d and angular displacement θ , both of which are derived from a peptide backbone’s internal coordinates, especially dihedral angles φ , ψ and ω . In particular, h equals sin(θ)d/|d|, with range [−1,1] and negative (or positive) values indicating left(or right)-handedness. The metric h is used to characterize the handedness of every region of the Ramachandran plot for both cis (ω = 0◦) and trans (ω = 180◦) backbones, which provides the first exhaustive survey of twist handedness in Ramachandran (φ ,ψ) space. These maps fill in the ‘dead space’ within the Ramachandran plot, which are regions that are not commonly accessed by structured proteins, but which may be accessible to intrinsically disordered proteins, short peptide fragments, and protein mimics such as peptoids. Finally, building on the work of Zacharias and Knapp (2013), this report presents a new plot based on d and θ that serves as a universal and intuitive alternative to the Ramachandran plot. The universality arises from the fact that the co-inhabitants of such a plot include every possible peptide backbone including cis and trans backbones. The intuitiveness arises from the fact that d and θ provide, at a glance, numerous aspects of the backbone including compactness, handedness, and planarity. 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 INTRODUCTION 30 The backbone of a protein (Fig. 1a) can twist and turn into numerous conformations (folds), in part 31 due to the amino acid sequence that the protein displays. Understanding how a backbone twists is of 32 great importance to the field of biochemistry, since understanding the structure of a protein goes a long 33 way towards understanding how a protein functions (Alberts et al., 2002; Berg et al., 2010). While the 34 conformation of a peptide backbone is dependent on a number of parameters (bond lengths, bond angles, 35 and dihedral angles), Ramachandran et al. (1963) recognized that the twist of a peptide backbone can be 36 described to a great degree by the dihedral angles φ and ψ (Fig. 1a). 37 Today, two-dimensional (φ ,ψ) plots are called Ramachandran plots (or ‘maps’), and are introduced in 38 undergraduate biology textbooks as a guide for understanding a peptide backbone’s general conformational 39 state or ‘twistedness’ at a glance (Bragg et al., 1950; Pauling and Corey, 1951b; Pauling et al., 1951; 40 Linderstrøm-Lang, 1952; Laskowski et al., 1993; Chothia et al., 1997; Hooft et al., 1997; Cooper and 41 Hausman, 2013; Alberts et al., 2002; Laskowski, 2003; Ho et al., 2003; Eisenberg, 2003; Berg et al., 42 2010; Mannige et al., 2016). The Ramachandran plot is especially useful because (stable) proteins are 43 hierarchical in structure (Linderstrøm-Lang, 1952): the final (tertiary) conformation of a structured protein 44 is composed of discrete secondary structures – regular structures – that interact with each other and 45 which are strung together by loops that are less regular (Alberts et al., 2002; Berg et al., 2010). Each 46 PeerJ reviewing PDF | (2017:01:15478:2:1:NEW 15 Apr 2017) Manuscript to be reviewed


INTRODUCTION
The backbone of a protein (Fig. 1a) can twist and turn into numerous conformations (folds), in part 31 due to the amino acid sequence that the protein displays. Understanding how a backbone twists is of 32 great importance to the field of biochemistry, since understanding the structure of a protein goes a long 33 way towards understanding how a protein functions (Alberts et al., 2002;Berg et al., 2010). While the 34 conformation of a peptide backbone is dependent on a number of parameters (bond lengths, bond angles, 35 and dihedral angles), Ramachandran et al. (1963) recognized that the twist of a peptide backbone can be 36 described to a great degree by the dihedral angles φ and ψ (Fig. 1a). 37 Today, two-dimensional (φ , ψ) plots are called Ramachandran plots (or 'maps'), and are introduced in 38 undergraduate biology textbooks as a guide for understanding a peptide backbone's general conformational 39 state or 'twistedness' at a glance (Bragg et al., 1950;Pauling and Corey, 1951b;Pauling et al., 1951; , the -ve diagonal within the Ramachandran plot (dashed line described by φ = −ψ) divides right-handed peptides from left-handed peptides, which leads to the naïve picture of handedness (d). Zacharias and Knapp (2013) showed that this picture is over simplistic, however an in-depth characterization of the backbone in all regions was not performed, and will be done here for both cis (ω = 0) and trans backbones (ω = π). Panel (a) is modified from Mannige et al. (2016). Due to low incidence within the studied database (see Methods), the two left-handed helices in (b) are arbitrarily marked and have no statistical significance. All molecular representations in this text are shown in 'licorice' form, with the colors red, blue and white representing oxygen, nitrogen and carbon atoms. regular peptide structure describes a backbone whose per-residue (φ , ψ) values are generally the same, 47 and therefore their 'locations' on the Ramachandran plot act as structural landmarks (Fig. 1b). 48 So far, our understanding of the Ramachandran plot has been limited mostly to structured proteins that  Interest in how a backbone may be represented as a helix emerged shortly after the first secondary

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Given that there are three backbone atoms associated with a residue (Fig. 1a), d = d n,α + d α,c + d c,n 124 and θ = θ n,α + θ α,c + θ c,n . Here, d i, j and θ i, j respectively refer to the axial and angular displacement 125 between adjacent atoms i and j. Subscripts 'n', 'α', and 'c' respectively refer to the backbone nitrogen, 126 α-carbon and carbonyl carbon atoms (Fig. 1a). The notation used by Shimanouchi and Mizushima (1955) 127 was in terms of matrices, which were then simplified by Miyazawa (1961) into trigonometric terms.

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In particular, Miyazawa (1961) noted that the total residue-residue axial displacement (d) and angular 129 displacement (θ ) may be retrieved using the following two equations. Manuscript to be reviewed d sin The ranges for d and θ , respectively, are [−λ , +λ ] and [0, 2π) (the positive limit λ is defined by allowed 131 values for the various internal coordinates). As above, subscripts 'n', 'α', and 'c' respectively refer to the 132 backbone nitrogen, α-carbon and carbonyl carbon atom types. The dihedral angles φ , ψ, and ω represent 133 the traditional symbols for backbone dihedral angles, which may be otherwise denoted as τ n,α , τ α,c , and 134 τ c,n(+1) , respectively.

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Finally, for any type of atom (say α-carbons), the radius or distance from the helical axis ρ α is defined by (3) Miyazawa (1961) noted that the right-hand side of Eqn. 3 is also the squared distance between adjacent atoms of the same type (denoted here as d 2 α for α-carbons), which allows for a more simplified form The distance between adjacent α-carbons (d α ) is ∼ 3.8Å for trans peptides and ∼ 3Å for cis peptides. Eqns. 1 and 2 may be substantially simplified (Miyazawa, 1961), given that backbone bond lengths and angles are much less 'tunable' when compared to dihedral angles (Ramachandran et This equation is especially relevant to peptides as they occur predominantly in trans conformations (ω = π). However, given the prevalence of cis backbones in peptide mimics such as peptoids (Mirijanian 2 ρ c and ρ n are obtained by the following subscript conversions: (α → c, c → n, n → α) and (α → n, c → α, n → c).
Note that Eqns. 5 through 8 are simplifications of Eqns. 1 and 2, and are therefore prone to some for the purposes of this report, given that this report primarily discusses features within platonic regular 155 backbones.

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On the one-to-one correspondence between ( ( (φ φ φ , , , ψ ψ ψ, , , ω ω ω) ) ) and Given a particular value of ω, every (φ , ψ) pair points to exactly one (d, θ ). However, when using Eqns. 1 and 2, one value of ω can not be replaced with a periodically equivalent version of ω (the same can be said for φ and ψ). For example, using ω = x + 2π instead of ω = x will maintain the magnitude of d and θ , but the signs will not remain conserved. This is because every summand in Eqns. 1 and 2 contains either a sine or cosine of [±φ ± ψ ± ω]/2. The issue arises because of the '2': even though the angle x is considered to be equivalent to the angle x + 2π, and even though cos(x + 2π) equals cos(x) (due to angle periodicity), cos([x ± 2π]/2) = cos(x/2 ± π) = − cos(x/2) (note the negative sign). Similarly, sin([x ± 2π]/2) = − sin(x/2). Therefore, even though the angles ω and ω + 2π may be considered to be equivalent angles, expressions such as cos([x − ω + 2π]/2) and cos([x − ω]/2) are only equal in magnitude and not in sign. I.e., a one-to-one correspondence between (φ , ψ) and (d, θ ) is only possible if one insists on specific values for ωs. For this reason, this report proposes to wrap the value of an amide backbone ω ′ between [∆, ∆ + 360 • ) using where % represents the modulus function, and ∆ describes the start of the range [∆, ∆ + 2π). Choosing 158 ∆ = −90 • would ensure that the distribution of both cis (ω = 0 ± 5 • ) and trans (ω = 180 ± 5 • ) will 159 remain contiguous. Using this system, cis and trans backbones are respectively represented by ω = 0 160 (and not 2π) and ω = π (not −π) for trans backbones. The rest of this report assumes these values of ω 161 for cis and trans backbones.

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These points lead to the conclusion that a strict one to one-to-one correspondence between (φ , ψ, ω) 163 and (d, θ ) does not exist, since multiple sets of the former may be backmapped from the latter (by 164 reconfiguring Eqns. 1 and 2). Yet, a one-to-one correspondence may be ensured by discarding as solutions 165 all but the one set of (φ , ψ, ω), whose φ and ψ lie within a preset range -e.g., [0, 2π) or [−π, π) -and 166 whose ω does not change after being wrapped by Eqn. 9.

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Given these relationships, this paper proposes a new metric for backbone handedness that depends on the sign of d and the value of θ : The range of h is [−1, 1], with negative (or positive) values indicating that the overall twist of the backbone 179 is left(or right)-handed. Also, |h| is proportional to the extent to which the backbone is twisted. Note Two estimates for chirality, χ 1 and χ 2 , used to validate the new measure of handedness h (Eqn. 10), were previously used by Kwiecińska and Cieplak (2005) and Kabsch and Sander (1983), respectively. The equations are: Here, N is the peptide length, i is the peptide residue number and the position of each α-carbon is  Finally, a more backbone-agnostic metric of chirality has been introduced by Solymosi et al. (2002), which is replicated here purely for completeness: that are nonetheless chiral opposites (Testa, 2013). It is for this reason that this report chooses to be 230 careful to not claim that Eqn. 10 is a metric for peptide/backbone chirality, but of peptide backbone twist 231 handedness. However, estimates for backbone chirality (e.g., Eqns. 11 and 12) may be used as surrogates 232 for twist chirality to validate h (Eqn. 10), as both are related but not the same. For the discussions below, since the residue-by-residue behavior of the peptide is of primary relevance, 240 the former definition is chosen as the relevant scope for flatness.

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As described in Fig. 3, the helical parameters d and θ respectively refer to an axial displacement along 242 the helical axis and an angular displacement in a plane perpendicular to the helical axis. For example, 243 d = 0 indicates a helix flattened along its helical axis (Fig. 4, Scenario 1). This means that all regular 244 peptides with d = 0 will be ring-like at some peptide length (shown in a following figure for a range of Scenario 2). Similarly, θ = oπ indicates that every alternate atom (of the same type) along the backbone 253 will be linear, and every adjacent atom will be diametrically opposite to each other (Fig. 4, Scenario 3); 254 i.e., θ = oπ indicates that all atoms of the same type will lie on a plane that is parallel to the helical axis.

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In short, θ = 0 codes for backbones that are linear (optimally extended for a fixed d) and θ = π describe 256 peptides that zig-zag along a plane perpendicular to the helical axis (for a fixed d). Finally, as is evident Manuscript to be reviewed Figure 5. The handedness of an ordered trans peptide within the Ramachandran plot. Panel (a) displays the relationship between backbone parameters (φ , ψ) and the associated helix parameters of curvature sin(θ ) (top; Eqn. 1) and axial displacement d (bottom; Eqn. 2). As shown in Fig. 3, the handedness of a helix is a function of these two variables (h; Eqn. 10). Panel (b) is a map of backbone chirality (h) as a function of φ and ψ. The boundaries, θ = π (' ') and d = 0 (' '), correspond to backbones that are equally flat, but which are respectively optimally extended and curved (see discussion in text). Panel (b) shows that the naïve expectation of handedness in a Ramachandran plot (Fig. 1d) is inaccurate. Interestingly, our naïve expectations would be upheld if one were only to have sampled regions of the Ramachandran plot dominated by known proteins (a; regions enclosed by ' ' indicate 90% occupancy). An example of the behavior of one 'slice' of (b) is shown in (c). Each snapshot represents a peptide backbone that is either in a distinct region of handedness or at a boundary.
in Fig. 4, θ = eπ conformations are not available to peptide backbones. Therefore, θ = oπ (e.g., π or 258 180 • ) will be the most extended type of backbone (for a fixed d). These relationships show how, a priori, 259 the curve of a backbone with particular (d, θ ) may be interpreted. not conserve structure along lines that conserve φ + ψ (and so R only works for trans backbones), yet 269 any two backbones with nearly identical θ 's will also be conserved in structure (see, e.g., Fig. 5a, top).

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This feature of θ will be true irrespective of the nature of the amide dihedral angle ω (Eqn. 1).  Fig. 5c); however, atoms of the same type lie in a single plane that is perpendicular to the helical axis 282 (Fig. 4). In short, within the Ramachandran plot, d = 0 (' ') and h = π (' ') code for flat backbones 283 that are respectively either optimally curved (at a given θ ) or optimally extended (at a given d). A future 284 report will discuss how these simple rules may be combined to make conjectures about novel secondary 285 and tertiary structures.

Manuscript to be reviewed
b.  In the same vein as Fig. 6a, Fig. 6b displays h, χ 1 and χ 2 as a function of φ and ψ for all-cis regular 295 backbones. This appears to be the first complete description of chirality of an all-cis backbone (ω = 0).

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Interestingly, the boundaries for d = 0 and θ = π switch in cis backbones, with the -ve diagonal and 297 curved boundaries being caused by d and θ , respectively. Additionally, Fig. 6a reiterates the idea that b. c. d. e. Figure 7. The landscape of backbone chirality as a function of amide dihedral angle ω. As ω is changed, the features of the landscape smoothly transform from the landscapes of ω = ±π to ω = 0. For all values of ω, it is evident that the naïve view of chirality (Fig. 1d) is wrong: at least four distinct regions of chirality (separated by boundaries d = 0 and θ = π) are evident in each scenario. Although only five snapshots (values of ω) are shown, all integer values of ω were tested, which corroborates the fact that the naïve view of backbone handedness (Fig. 1d) is universally incorrect.
with length two, given that any peptoid of length greater than two would result in overlapping atoms.

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However, such a structure (one with d = 0 and θ = π) is not possible in trans peptides, even in theory, 307 because the boundaries associated with d = 0 and θ = π do not intersect (Fig. 6a); this is also evident in d. in easy to read pictograms. For that reason, switching the map from one range to another means that the 328 two types of scientists -each used to a distinct range -will not be able to converse as seamlessly.

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Therefore, the following question must arise: which range -[−π, π) or [0, 2π) -is able to convey 330 more information with the least amount of effort? Fig. 8 shows the handedness of a trans backbone (a,b) 331 and cis backbone (c,d) in the two frames of reference. From (a) and (b) it is evident that general trends in 332 the map for trans backbones remain the same in both frames of reference: the negative diagonal (θ = π) 333 locally separates right-handed regions from left-handed regions, while the curved line (d = 0) -which 334 also separates handedness -also appears to be in generally the same regions (albeit inverted in curvature).

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The cis backbones, however, look dramatically different in the two frames of reference: the range [−π, π] 336 separates handedness in a more complicated manner (c), while, for the most part, the -ve diagonal appears 337 to meaningfully separate handedness when the plot ranges from 0 to 2π (d). For this reason, purely when 338 looking at handedness, and especially in the case of cis backbones, the Ramachandran plot that ranges 339 between 0 and 2π appears to be more meaningful. features of a peptide backbone just from its (φ , ψ) angles (Fig. 9a). This prompted Zacharias and 344 Knapp (2013) to introduce a new representation for backbone degrees of freedom in the form of a 345 polar graph. In this polar representation, the θ is the angular coordinate (azimuth) and d is the radial 346 coordinate. An example of one such representation is shown in Fig. 9b, with the direction of increasing θ 347 reversed (compared to the cited report) to maintain relative positions of secondary structures within the 348 Ramachandran plot (Fig. 9a). Zacharias  Manuscript to be reviewed of the polar representation (Fig. 9b): θ , which is an angle and therefore periodic, can remain periodic as 350 the angular coordinate in the graph.

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However, the format proposed by Zacharias and Knapp (2013) (Fig. 9b) is incomplete for a few 352 reasons: 1) d < 0 peptides (the bottom-left and top-right regions of Fig. 5a, bottom) will never be observed 353 in this map since only structures with d ≥ 0 are allowed; 2) all peptides with d = 0 (marked by ' ' 354 in every preceding Ramachandran plot) will be compressed into one point at the center, even though 355 Fig. 6c shows a range of legitimate d = 0 conformations; 3) while the graph is θ -periodic, the values for 356 θ in peptides are constrained within one [0, 2π] period (peptides range between θ = π/4 and 2π − π/4; 357 vertical dotted lines in Fig. 4); i.e., periodicity in θ is not required for the faithful representation of 358 peptides. Fortunately, even though this system is not universal (again, since d < 0 structures are not So far, this report has focused on regular or simple backbone conformations, i.e., those that are formed 374 from the same φ and ψ angles repeated along the backbone. This is particularly because a simple and 375 visually intuitive correspondence exists (Figs. 3 and 4) between a regular backbone (described by myriad 376 internal coordinates) and a helix that is described simply by (d, θ ). However, there is a possibility that d 377 and θ are useful even in isolation, when the unreasonable constraint of perfect backbone regularity is 378 lifted. An example of such a departure from regularity follows. twists, but opposite in handedness, which allows for these secondary structures to remain linear, albeit in a 385 meandering way (Mannige et al., 2015). Eqn. 10 also describes these two states as opposite in handedness 386 and similar in twist extent: the h for the two states are −0.34 and 0.51, respectively (the difference in 387 magnitude is within the range of the standard deviation in h [0.391] for the β-sheet). Similarly, the α-sheet 388 proposed by Pauling and Corey (1951a) is constructed by alternating between α (D) and α L backbone 389 states, yet this motif is linear because each state describes equal but opposite handedness h = ±0.41.

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These points raise the possibility that, even in the absence of perfect backbone regularity, the values d, 391 θ , and h may be considered to be residue-specific properties that may be combined to readily provide 392 insights about higher order structures.

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This report introduces a metric for backbone handedness (h) that is based on modeling the backbone as 395 a helix [ Fig. 2; Miyazawa (1961)]. In particular, h, which is a combination of the helical parameters θ 396 (angular displacement) and d (axial displacement), ranges from -1 and 1, and is negative (or positive) 397 when the backbone twist is left(or right)-handed (with larger |h| indicating greater extent of twistedness).

398
This metric (h) was used to characterize every regular backbone's twist within the Ramachandran plot, 399 for both cis and trans peptides. In doing so, this report dispels a naïve view of handedness (Fig. 1d), 400 which states that backbone handedness in the Ramachandran plot is separated by the negative-sloped two common frames of reference -φ , ψ ∈ [−π, π) and [0, 2π) -indicates that the less commonly used 405 frame [0, 2π) may be more appropriate for interpreting cis backbones (Fig. 8). 406 The behavior of a backbone in cis and trans Ramachandran plots look dramatically different (Fig. 6), 407 and so scientists dealing with new structures that have a combination of cis and trans backbones can 408 not use one Ramachandran plot to faithfully describe these structures. Interestingly, the parameters θ 409 and d combine all features (internal coordinates) of a contorting backbone, including the amide dihedral 410 angle ω, which means that (θ , d) can describe any peptide backbone, irrespective of ω. Therefore, the 411 Cartesian plot with θ and d as the x-and y-axis, respectively, serves as a unique plot for any peptide 412 backbone (Fig. 9), with specific values and boundaries containing deep structural meaning (Fig. 4). These