2.1 The helical curve fitting algorithm

A general helical curve in three dimensional (3D) space could be represented as: (1) where r = {x, y, z} (In this paper, bold lower-case letters denote 3D vectors and bold capital letters denote either 3D rotation matrices or 3D curves.) is a point on the curve, r₀ = {x₀, y₀, z₀} its origin, and R the rotation matrix that specifies its helical axis n with respect to a coordinate system. The first three helical parameters, radius (r), pitch (p) and turn angle (t), define a standard helical curve, x = rsint, y = rcost, z = pt with its origin at {1.0,0.0,0.0}, its center at {0.0,0.0,0.0} and its helical axis along the + Z axis. Together with n and r₀ these five parameters completely define a general helical curve. Though r, p, t could be computed directly from the virtual bond length, bond angle and dihedral angle of a quadruple of C_αs [5], no simple analytic expression has been derived for the computation of a helical curve that best fits the coordinates of a quadruple of C_αs, that is, a helical curve that has the minimum RMSD (Δ_i) between the four C_αs and their closest points on the curve. In fact, this minimization (or curve fitting) problem is equivalent to finding the solutions to a high-degree monomial. The complexity of searching for a series of different helical curves that best fit a series of segments of backbone atoms is likely to be the reason why no genuine helical curves have been used to model a biomolecular helix. In the following we describe briefly an efficient algorithmic solution to this minimization problem.

We begin with the computations of r, p and t using previously-derived analytic expressions [5], and denote their values as r_m, p_m and t_m. Then we proceed as follows to search discretely and exhaustively over two intervals, [r_m − δ_r, r_m + δ_r] and [p_m − δ_p, p_m + δ_p], for the r and p values of a helical curve that best fits the coordinates of a quadruple of C_αs of residues i, i + 1, i + 2 and i + 3. Both δ_r and δ_p are user-specified constants.

1. Δ_i = ∞ {the initial RMSD}
2. For each r in [r_m − δ_e, r_m + δ_r]
    For each p in [p_m − δ_p, p_m + δ_p]
     Compute t {the turn angle}
     Generate a helical curve {by Eq 1}
     Best-fit the curve to the four C_αs using singular-value decomposition(SVD) to compute R
     If d_q < Δ_i
      Δ_i = d_q
      r_i = r, p_i = p, t_i = t, R_i = R

where d_q is the RMSD between the quadruple of C_αs and their closest points on the helical curve; r_i, p_i, t_i and R_i are, respectively, the helical parameters and rotation matrix for the helical curve that best-fits the quadruple. Given both r and p and the distance d_{i, i + 1} between two consecutive C_αs, t could be computed as follows: . Singular-value decomposition (SVD) is applied to compute Δ_i and R_i; and from R_i, both n_i and helix center c₀ for this quadruple of C_αs could be calculated. In fact, the SVD step guarantees that the computed helical curve best fits the coordinates of the quadruple of C_αs. The set of five helical parameters and the centers for all the quadruples of consecutive C_αs in a protein chain or Ps in a DNA strand are computed by sliding over its sequence a window of four C_α or P atoms.

2.2 The helix score for a protein helix

Except for the last three residues at the C-terminus of a protein chain, to each residue i of a protein sequence is assigned a helix score h_i. (2) where r_i, p_i, t_i,Δ_i are computed as above using a quadruple of C_αs of residues i, i + 1, i + 2 and i + 3. The constants μ_r, σ_r; μ_p, σ_p; μ_t, σ_t and σ_Δ are respectively the normal distribution parameters for r, p, t,Δ that are determined as follows using the respective data sets for r, p, t,Δ computed over a non-abundant set 𝕊 of 3,287 X-ray structures in the PDB with each of them has a resolution ≤ 2.0Å, a R-factor ≤ 25.0% and at least three helices. We have applied the program dssp [22] to assign a total of 44,456 helices for the protein structures in 𝕊. The three parameters μ_r, μ_p and μ_t define a standard helical curve that represents an average over all the protein helices in 𝕊. For ease of reference, we call it the standard protein helix. The term quantifies the spatial difference between a C_α atom and its closest point on the model while the helix score itself determines the deviation of the model from the standard protein helix: the higher the score the larger deviation from the standard protein helix.

2.3 The helix model

The model is computed through an averaging process that merges into a single curve all the helical curves for the quadruples of backbone atoms obtained by sliding over a protein helix or a DNA strand. Starting with either the N-terminus of a protein helix or 5’-terminus of a DNA strand, the model curve c_{ai ai + 1} between two consecutive atoms, a_i and a_{i + 1}, is computed as follows. (3) where {a₁,a₂,a₃,a₄,a₅,…,a_n} denotes n consecutive backbone atoms along a protein helix or a dsDNA strand, and h_{i, i + 1} is the segment between two consecutive atoms, a_i and a_{i + 1}, of the helical curve H₁₄ that is computed using a quadruple of atoms {a₁,a₂,a₃,a₄}, the segment of H₁₄ between a_{i + 1} and a_{i + 2}, and the segment of H₁₄ between a_i+2 and a_i+3. Please see Fig 1 for an illustration.

3.1 The accuracy of the helical model

The application of the model to a set of 27,105 X-ray protein structures in the PDB with a resolution from 0.46Å to 3.5Å and less than 70% sequence identity confirms the model’s accuracy. Though the RMSDs (Δ_is in Eq 2) between the backbone C_α atoms and their closest points on the model range from 0.0 − 0.3Å for all the 262,266 DSSP-assigned protein helices, more than 95% of the helix residues have their Δ_is less than 0.08Å. As illustrated in Fig 2, the deviations between the experimental C_α positions and the model are barely discernible for the protein structures ranging from ultra-high resolution (pdbid 1EJG, 0.46Å), to medium resolution (pdbid 2RH1, 2.6Å), and to low resolution structures (pdbid 3ZC1, 3.3Å, S1 Fig in the Supporting Information (SI)). In general, the model accuracy is not affected by the residue’s helix score. For DNAs, the spatial difference between a backbone P atom and its closest point on the model could be large with a typical value from 0.0Å to 0.5Å(Fig 3a). Even with the relatively large difference between the model and the P atoms in DNAs, our model is superior to all the previous helix models for DNAs because the generated curves conform to a genuine helical curve much better than a series of splines generated by the previous models do (Please see S4f, S4j and S5 Figs in the SI for examples).

3.2 The helix score and the structural and functional significance of the model

Our model consists of a series of helical curves each one best fits the coordinates of a quadruple of backbone atoms. The individual curves could differ largely from each other depending on the extent of their deviations from a genuine helical curve. The deviation represents the structural difference among different biomolecular helices. They are described in our model by both the helical parameters and the RMSD Δ. Specifically we have defined a helix score for a protein helix residue (Eq 2) that includes both the deviation from the standard protein helix and the minimum RMSD achieved by the curve fitting algorithm. Either the score or the local deviations from a genuine helical curve could be visualized for individual protein residues or DNA nucleotides. A systematic survey over all the protein and DNA structures in the PDB is currently under way for the structural and functional significance of the helix score. As shown in Table 1 a preliminary study on a set of 3,446 x-ray structures with a resolution between 1.0Å–2.0Å and with less than 70% sequence identity has found a correlation between the residue’s helix scores and their locations in the proteins: the higher the score the higher probability of the residue being on a protein surface (Table 1).

The same study also shows that the helix residues in a protein-ligand binding site tend to have higher helix scores than the rest as illustrated in the three figures: Figs 3 and 4 and S1 Fig in the SI. The residues with high helix scores indicated by different colors are concentrated in the ligand binding sites where the ligand could be either a DNA molecule (Fig 3) or a compound (Fig 4) or other protein subunits (S1 Fig). Though it has been documented before [23–25] that the π and left-handed helices in proteins have higher probabilities to be in a ligand binding site, no scores have been proposed previously for a quantitative and consistent description of the distortions in all the three types of protein helices. One key advantage of a consistent helix score is that the scores for different homologs in the same protein family could be used to determine their structural similarity (conservation) and variation (Fig 4). In addition, with our model a polyline could be constructed for each biomolecular helix by connecting together the centers of the individual helical curves along either a protein helix (Fig 4b and S1 Fig) or a DNA strand (S4g and S5 Figs). For ease of reference, we call such a polyline helix center polyline. As illustrated in Fig 4b the abrupt changes (turns) in such a polyline occur often at a protein-ligand interface. Compared with the protein helices it is more tricky to quantify the deviations from a genuine helical curve for DNA helices because of their large structural variations. However our model is still able to provide a qualitative description for local deviations such as the twist away (Fig 3b) from an ideal helix ribbon diagram generated by a genuine helical curve (S4f, S4j and S5 Figs).

3.3 The application to helix visualization

It is a well-known problem in the modeling of a biomolecular helix as a series of low-degree splines that if the curves pass exactly through every backbone atom then the ribbon diagram generated by the model is choppy. On the other hand, if additional steps are applied or more than four backbone atoms are used for spline computation in order to smooth out the choppiness, then the side chains could become detached from the ribbon. The choppiness or detachment is inherent with a helix model that uses a series of splines where either the degrees of the splines are not high enough or the number of splines is not large enough for an accurate representation of a general helical curve that best fits the backbone atoms. Specifically the choppiness is due to nonsmooth changes in curvature along a biomolecular backbone while the detachment results from the spatial differences between the backbone atoms and the spline model. Though efforts have been made in the past [16, 18] to smooth out either the choppiness or detachment, either of them remains for the visualization of biomolecular helices in general and DNA helices in particular [17, 18] (please see S4 Fig). In our model the choppiness and detachment are simultaneously reduced to a great extent because the model itself is composed of a series of helical curves each one best fits a quadruple of backbone atoms. Specifically each helical curve has a unique curvature and the averaging process guarantees that the curvature for the final curve (the model) changes smoothly along a biomolecular backbone, consequently our model is able to almost eliminate the choppiness in protein helices (Figs 2, 4 and 5 and S1 Fig) and to greatly reduce it in DNA helices (S4f, S4g and S5 Figs). Similarly the averaging process keeps the distance between a backbone atom and its closest point on the model close to the minimum achieved by the curve fitting algorithm and thus the detachment problem is greatly alleviated. Fig 6 illustrates the differences between the ribbon diagram generated by our model and the diagrams by two previous programs. Please see the supporting information for the comparisons with four other molecular visualization programs (see S2 Fig). Our helix ribbon diagrams are most similar to the iconic ribbon diagram hand-drawn by Jane Richardson [7] (see S3 Fig). It is also rather similar to the ribbon diagram by UCSF Chimera [16] that uses a series of splines each one fits to five backbone atoms (Fig 6d). In contrast to the ribbon diagram by UCSF Chimera, our model minimizes the distance between a backbone atom and the ribbon diagram.

When four successive backbone P atoms are used to generate a ribbon diagram for a DNA helix, there often exist large variations among the diagrams for different DNAs and obvious deviations from a diagram that is generated using a single genuine helical curve (please see S5a Fig of the SI for an ideal helix ribbon diagram generated by our model using a single genuine helical curve). The variation and deviation are partly due to the small helix turn angle in a typical DNA helix. The turn angle for a typical DNA helix is 34°, about one third of the turn angle for a typical α-helix in proteins. If instead of using all the successive P atoms, when only the first atoms of every triple of consecutive P atoms are used to compute the model, the resulting helix ribbon diagram becomes much similar to both a protein helix diagram (S4h and S4j Fig) and an ideal helix ribbon diagram (S5a Fig). In this case the turn angle per atom is very close to that in a typical protein helix. However, such a DNA model has lower accuracy (S4i and S4j Fig).

3.4 The data set and the molecule visualization program

To evaluate the performance of our algorithm, we have downloaded from the current version of the PDB a set of 27,105 x-ray protein structures that have at most 70% sequence identity and each of them has at least one helix according to the PDB. From them a set of 3,287 high-resolution structures, 𝕊, each of them has at least three helices, a resolution ≤ 2.0Å and a R-factor ≤ 25.0% are selected for the statistical analyses of helical parameters and RMSDs.

We have implemented our helix model and its visualization in C++/Qt/openGL and included them as a module in our structure analysis and visualization program. The default values for the two parameters, δ_r and δ_p, required for the computations of helix parameters r, p are set to 0.25 and the step size for both intervals, [r_m − δ_r, r_m + δ_r] and [p_m − δ_p, p_m + δ_p], is 0.01. The program is written in Qt5.3/openGL4.3/GLSL4.3 and is available upon request.