# A Study on the Covalent Geometry of Proteins and Its Impact on Distance Geometry

Simon B Hengeveld, Mathieu Merabti, Fabien Pascale, Thérèse E Malliavin

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Simon B Hengeveld, Mathieu Merabti, Fabien Pascale, Thérèse E Malliavin. A Study on the Covalent Geometry of Proteins and Its Impact on Distance Geometry. 6th International Conference on Geometric Science of Information (GSI'23), Aug 2023, Saint Malo, France. pp.520-530, 10.1007/978-3-031-38299-4_54. hal-04183511

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
A study on the covalent geometry of proteins and
its impact on distance geometry *

Simon B. Hengeveld¹, Mathieu Merabti², Fabien Pascale², and Thérèse E
Malliavin²[0000-0002-3276-3366]

¹ IRISA, CNRS UMR 6074, University of Rennes 1, Rennes, France.
simon.hengeveld@irisa.fr

² Laboratoire de Physique et Chimie Théoriques (LPCT), CNRS UMR 7019,
University of Lorraine, Vandoeuvre-lès-Nancy, France
therese.malliavin@univ-lorraine.fr

**Abstract.** The Distance Geometry Problem (DGP) involves determin-
ing the positions of a set of points in space based on the distances be-
tween them. Due to the importance of measuring spatial proximity's
between atoms in structural biology, Distance Geometry finds a natural
application to protein structure determination. When an instance of the
DGP is discretizable, the search tree can be explored using the (interval)
branch-and-prune algorithm (iBP). A statistical study on a database of
protein structures shows that slight variations are observed in the cova-
lent geometry of the molecules, depending on the Ramachandran region
of the protein residues. These variations are in agreement with observa-
tions made on high resolution X-ray crystallographic structures. These
slight variations may cause atomic clashes when using iBP algorithms
to calculate a folded protein structure. In this contribution, we compare
two software implementations (MDJEEP and IB-PNG) that attempt to
address this issue.

**Keywords:** Distance Geometry Problem · interval Branch-and-Prune · local optimisation · protein structure determination · covalent geometry

# 1 Introduction

The Distance Geometry Problem (DGP) consists of finding the realization of
a given set of points in a *K*-dimensional Euclidean space, where the distances
between (some of) the pairs of points are known [20]. Formally, we can define the
DGP using a simple weighted undirected graph *G* = (*V*, *E*, *d*) [14, 19], where the
vertices *V* correspond to the points that we want to realize and the existence of
an edge (*u*, *v*) ∈ *E* implies that we know the distance from *u* to *v*. The weighting
function *d* assigns each edge to a real interval δ(*u*, *v*). This interval may be
degenerate.

<footnote>* CNRS, Lorraine University, University of Rennes</footnote>
**Definition 1.** Given a simple weighted undirected graph $G = (V, E, d)$ and an integer $K \in \mathbb{Z}_+$ the (assigned) DGP asks whether a realization

$$x : v \in V \rightarrow x_v \in \mathbb{R}^K \qquad (1)$$

exists, such that

$$\forall \{u, v\} \in E, \quad ||x_u - x_v|| \in \delta(u, v) \qquad (2)$$

where $|| \cdot ||$ is the Euclidean norm.

An important application of the DGP is found in the context of structural biology. In this case, the dimension $K$ is equal to 3, the vertices $V$ of $G$ represent atoms, and the edges $E$ (and their weights $d$) represent distance constraints between these atoms. Generally, these constraints can be divided into four sets. The first set corresponds to holonomic constraints [4], derived from the covalent geometry of molecules (using bonds and bond angles). The second set of distances arise from the knowledge of torsion angles within the molecules [10, 15]. These torsion angles, combined with the bonds and bond angles lets us compute distances between some of the atoms in the backbone. The third set of distances are measured between atoms not related by a torsion angle and can be obtained using NMR experiments [9] or other structural biology techniques. The fourth set of restraints is imposed by the impossibility of inter-penetrating the van der Waals radii of the atoms. These three last sets of constraints are mostly defined by intervals, except if a torsion angle is considered to be exactly defined, as for the case of the torsion angle $\omega$ defining the peptide plane. As the distance constraints may contain a lot of uncertainty, the constraints (2) leave space for the weight $d(u, v)$ to map to a degenerate interval $\delta(u, v)$ (an exact distance):

$$||x_u - x_v|| = \delta(u, v), \qquad (3)$$

or to match real a interval $[\delta(u, v), \bar{\delta}(u, v)]$:

$$\underline{\delta}(u, v) \le ||x_u - x_v|| \le \bar{\delta}(u, v). \qquad (4)$$

In the following, we consider the covalent geometry to include, in addition to the holonomic constraints, the torsion $\omega$ angle.

The instances of the DGP in the context of protein structure determination have been shown to belong to the DDGP subclass [10, 13, 15] and may be solved within the branch-and-prune (BP) framework [15], which explores the search tree generated by a discretization process. Early BP algorithms only worked for instances of the DDGP that only contain exact distances. Later, a BP algorithm which could handle both intervals as well as exact distances was proposed, which was referred to as the interval branch-and-prune (iBP) algorithm [13]. This opened the way for calculating protein conformations. Two main implementations of iBP have been introduced over the years: a recursive implementation illustrated by the open-source software MDJEEP [22] and an iterative implementation illustrated by the software IBP-NG [24]. The iBP approach was shown to be effective in various previous works [3, 16, 18] where NMR data were simulated
from known PDB structures [2] or were obtained from experimental restraint files [10].

In the standard iBP approach, as the holonomic distances and torsion angles $\omega$ are considered as exact, the covalent geometry is assumed to be static. However, analyses of high resolution X-ray structures have shown [1] that bond angles as well as the torsion angle $\omega$ of the peptide plane vary along the Ramachandran regions. Section 2 describes a similar analysis, showing observed variations in the covalent geometry. Studying these tiny variations is of crucial importance to describe the architecture of protein structures. In principle, the variations of the covalent geometry could be tackled in the iBP approach by adding an additional level of branching including discrete variations of the bond angles. From a practical point of view, this would increase the algorithm complexity. To avoid this, one straightforward step is to add a different set of bond length and bond angle values for each residue to the input of the algorithm. In this work, we conducted experiments with two iBP implementations. In Section 3 we present more details on the branch-and-prune framework, and describe the two different BP implementations. Next, in Section 4 we present experiments with the two implementations to see how they deal with the variations of the covalent geometry of the proteins. Finally, in Section 5 we draw conclusions and describe perspectives for the future.

## 2 Variations in covalent geometry

To show that there are in fact variations in the covalent geometry of proteins, we analysed a data set of PDB structures obtained by X-ray crystallography. The data-set was prepared using the PISCES server [23]. The criteria for choosing the structures were: an identity smaller than 20% between protein sequences, an X-ray crystallographic resolution better than 1.6 Å and a *R-factor* better than 0.25. The *R-factor* is a measure of how well the refined structure predicts the observed data. Protein chains smaller than 100 residues and displaying no *cis* peptide bonds were selected, producing a data set of 391 structures.

To analyze the covalent geometry, we computed the average values for the bond angles between the backbone heavy atoms (N, C$\alpha$, C, O) as well as for the torsion angle $\omega$ of the peptide plane, using the MDAnalysis package [8]. The results of this analysis are displayed in Figure 1. Uniform averages (left column) are observed along the amino acid types, except for the residues G, P and L. Next, average values have been calculated on various Ramachandran regions, using the definitions determined from high resolution crystallographic structures [11] (right column). These averages display more variations. In addition, one should notice that the torsion angle $\omega$ displays much larger variations than the bond angles, with standard deviations around 15° for $\omega$ and around 2° for the bond angles.
<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Average values of the bond angles between the heavy backbone atoms (N, Cα, C, O) as well as on the torsion angle ω of the peptide plane. Np denotes the nitrogen atom N of the next residue, Cm denotes the carboxyl carbon C of the previous residue, and CA denotes the Cα atom. On the left we see the averages according to the amino-acid types, while on the right we see them according to the Ramachandran region of the residue. The horizontal dashed lines correspond to the angles values from Ref. [5]</figcaption></figure>

3 Branch-and-prune implementations

By making certain assumptions on the vertex order, the search space of DGP
instance may be discretized [17]. DGP instances related to protein structure
determination meet these assumptions [13], making them instances of the Dis-
cretizable Distance Geometry Problem (DDGP) [17]. The discrete search space
has the structure of a tree, where the nodes of the tree represent a feasible po-
sition of an atom v ∈ V. Because the search space is a tree, we can potentially
enumerate the complete set of solutions. The search tree is explored by BP al-
gorithms [15]. In the “branching” phase, the BP algorithms will construct the
search tree, while in a “pruning” phase branches will be removed if they con-
tain solutions that are infeasible given the input distance constraints. The BP
algorithms that can handle the interval distances present in the protein DDGP
instances are referred to as interval BP (iBP) algorithms. However, to ensure
that the DGP instance relating to proteins can be discretized, some distances
must be exact [17]. The inter-atomic distances of which we are most certain
are the bond and bond angle related distances, so in the BP-framework these
distances are regarded as exact. However, the study in Section 2 shows that the covalent geometry contains some variations.

Two error-tolerant IBP-implementations are capable of handling these variations: IBP-NG [24] and MDJEEP [22].

## 3.1 IBP-NG

IBP-NG<fnref n="3" /> makes use of both breadth-first and depth-first iterative tree searches based on a multidimensional index data structure [24]. The iterative order of the tree traversal is based on embedding equations derived from Clifford algebra [12], which makes it ideal for instances with highly repetitive vertex orders. When one converts the torsion backbone angles to a distance, their sign is lost. However, as shown in previous works, the sign of the torsion angles may be very useful during the pruning phase [21]. The aforementioned iterative order lets IBP-NG exploit this sign in the *branching phase* so that it avoids creating unnecessary branches to begin with.

Intervals are handled in a straightforward way. IBP-NG has an input variable *B*, which is referred to as the *branching factor*. In case an interval is encountered, the associated arc is cut into *B* pieces uniformly. For each of these pieces, a new branch is created. This means that IBP-NG branches at most *2B* times for each vertex. In case we have information about the sign of the torsion angle involved in this distance, only *B* branches are created. A drawback of this approach is that IBP-NG may not be able to identify a solution, even if the instance is feasible.

## 3.2 MDJEEP

MDJEEP<fnref n="4" /> is a recursive implementation of the BP framework [22].

It relies on a tree search to compute coordinates for each of the vertices. For the coordinate generation step, an efficient method using a rotation matrix is employed [7]. To allow for error in the input distances, MDJEEP makes use of a coarse-grained representation. This allows it to deal with the continuous feasible subsets of potential atomic positions while preserving the general tree structure [18]. A three-dimensional box is assigned to each vertex *v*, which we suppose contains the *true* position of the atom. After computing an initial position for *v* within its box, it is then allowed to move in a refinement step. This refinement includes local optimization using a spectral projected gradient method. These boxes also come into play when MDJEEP is handling interval distances. The advantage of this refinement step is that it allows MDJEEP to identify solutions where IBP-NG may not be able to.

The published *C*-version of MDJEEP only allows for pure instances that only contain distances. To allow for a proper comparison with IBP-NG, the algorithm was reimplemented into Java and extended so that it is capable of using the sign of the dihedral angles to prune. The Java language was chosen because new

<footnote n="3">³ https://github.com/geekysuavo/ibp-ng</footnote>

<footnote n="4">⁴ https://github.com/mucherino/mdjeep</footnote>
functionalities may more easily be included in the software tool in the future when a higher-level programming language is used. In the experiments presented below, the Java implementation is used.

# 4 Computational experiments

We ran several experiments using both IBP-NG and MDJEEP in order to compare how the iBP-solvers deal with the variations described in Section 2.

## 4.1 Input

The experiments were conducted using a subset of proteins from the PISCES database described in Section 2. Then, for each of these proteins we generated distances and angles to create the DDGP instances to be solved by the two iBP methods. The input consists of the following three types of distances.

The first type are holonomic distances, uniform or non-uniform along the sequence. Normally, these are taken from force field parameter files. In these experiments, we only used the bond lengths from the force field PARALLHDG (version 5.3) [5]. For the bond-angles we take a more targeted approach. Usually, in the input of iBP-algorithms, the bond-angle of a triplet atoms is also taken from the force-field parameters, and thus solely depends on the three atom types of the triplet. In this experiment, we vary the bond angles (and thus their associated distances) depending on which residue in the protein sequence the atoms belong to. The bond angles of each residue were taken from the initial PDB structure.

The second type used are distances derived from the backbone dihedral angles $\phi$ and $\psi$ and $\omega$. The $\phi$ and $\psi$ angles were taken from the PDB structures, and uncertainty was introduced by creating 10 degrees intervals around them. For the $\omega$ angles, the procedure differed between the two implementations. The exact $\omega$ values of the initial PDB structure were used in the IBP-NG runs, while for MDJEEP 10 degrees intervals were used. In order to prune using the torsion angle values directly, we also included the angle intervals directly in the input.

Finally, we have distances arising from van der Waals restraints. The van der Waals radii were scaled by 0.1 in order to allow for some degree of overlap between pairs of van der Waals spheres, as we can observe in real conformations.

Aside from backbone dihedral angles, we also prune using the sign of the improper torsion angles, which defines the stereo-chemistry of L-amino acids. These improper angles were included directly in the input and taken from force-field parameter files.

## 4.2 Results

The two approaches, IBP-NG and MDJEEP, were applied to the data set of protein structures. Among the 391 structures smaller than 100 residues present in the data set, 32 produced no solution for IBP-NG. MDJEEP produced solutions for all
<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Distributions of the root-mean-square deviations (RMSD, Å) of atomic coordinates between the initial PDB structure and the structure recalculated by IBP-NG (left) and MDJEEP (right). Upper panel: effect of the percentage of secondary structures: α helices, and β strands, determined by STRIDE [6], on the RMSD distributions. Lower panel: effect of the number of ω outliers outside of the ranges [175,180] or [-180,-175].</figcaption></figure>

structures in data-set. The quality of the computed structures can be verified by comparing the solutions to the initial PDB structures. In practice, this is done by calculating the root-mean-square deviations (RMSD, Å) between the atomic coordinates. For IBP-NG, for each protein, multiple structures were computed, and the best solution (with the lowest RMSD value) was kept. For MDJEEP, execution was cut after the first found solution, potentially leading to lower quality solutions. This choice was made, because MDJEEP took too long to explore the full search tree.

The distribution of RMSD values (Figure 2) are different for the two approaches, IBP-NG producing smaller range (0-6 Å) than MDJEEP (0-13 Å). Figure 2 shows the distributions of the RMSD values for IBP-NG (left) and for MDJEEP (right). The RMSD values were computed using the MDAnalysis package [8]. The RMSD values (Figure 2, upper panels) are also analyzed according to the percentages of α helix and β strand present in the structures, and calculated using STRIDE [6]. The proteins displaying percentage of α helix larger than 70% of the sequence display RMSD distributions shifted towards smaller values (blue curves). At the contrary, the percentages of β strand larger than 50% induce a shift of distributions towards larger values (green curves). Besides, the presence of a larger number of outliers for ω values (Figure 2, lower panels) induces also a shift of the distributions towards larger RMSD values.
<table><caption>**Table 1.** Averaged differences of bond and torsion angles (°) between the conformations obtained during the runs MDJEEP and IBP-NG, and the PDB structures. The absolute values (above) is the average of the absolute differences. The signed values (below) instead is the average of the signed differences.</caption><thead><tr><th>Experiment</th><th>Averaging</th><th>CmNCA</th><th>NCAC</th><th>CACNp</th><th>CACO</th><th>OCNp</th><th>φ</th><th>ψ</th><th>ω</th></tr></thead><tbody><tr><td rowspan="2">MDJEEP</td><td>absolute</td><td>1.336</td><td>1.296</td><td>1.399</td><td>0.087</td><td>0.071</td><td>1.692</td><td>1.701</td><td>4.165</td></tr><tr><td>signed</td><td>-0.217</td><td>-0.264</td><td>-0.429</td><td>0.016</td><td>-0.054</td><td>1.655</td><td>1.566</td><td>4.023</td></tr><tr><td rowspan="2">IBP-NG</td><td>absolute</td><td>1.400</td><td>1.253</td><td>0.025</td><td>0.056</td><td>0.029</td><td>0.035</td><td>0.036</td><td>0.024</td></tr><tr><td>signed</td><td>-0.162</td><td>0.165</td><td>-2.05e-4</td><td>0.0436</td><td>-1.94e-4</td><td>2.83e-4</td><td>0.66e-4</td><td>-0.19e-4</td></tr></tbody></table>

In general, IBP-NG produces solutions that are closer to the X-ray crystallography structures. This difference between the two approaches can be explained by calculating the averaged differences of bond and torsion angles between the obtained conformations and the initial PDB conformations (Table 1). All angles display differences smaller than 5°, which justifies the two strategies used for avoiding atomic clashes: the local optimization approach or the residue-dependent variation of the covalent geometry. Nevertheless, smaller differences for torsion angles φ, ψ and ω are observed for IBP-NG compared to MDJEEP. By contrast, the differences between the bond angles are similar for the two runs. For all parameters, the signed difference display smaller values than the absolute ones, which agrees with differences displaying alternating positive and negative signs.

Overall, the analysis of RMSD values (Figure 2) as well as of the averaged differences of bond and torsion angles, reveals a stronger influence of φ, ψ and ω values to the variation of protein conformations. The influence of the ω torsion angle can be put in parallel with the larger variations of this angle among the Ramachandran regions located in the loop and β strand regions (Figure 2). This supports the conclusion that the conformational drift of conformations is directly related to the variation of the covalent geometry along the protein sequence. The RMSD shifts observed for proteins containing mostly α or β secondary structures can be put in parallel with the standard deviations of the ω angle in the Ramachandran regions corresponding to the α helix and β strand regions (labels A and B respectively in Figure 1).

Comparing the two implementations, IBP-NG seems to be able to compute structures that are closer to the original X-ray crystallography structures. However, bear in mind that for MDJEEP, only the first found solution for each protein was kept, while for IBP-NG multiple solutions were considered. Furthermore, there was more uncertainty in the input instances of MDJEEP because intervals were used on the ω angles, while for IBP-NG exact values were used. Because MDJEEP takes too long to explore the entire tree, we cannot be sure whether it can potentially give similar quality solutions as ibp-ng. Part of the reason why MDJEEP is slower may be due to the local optimization refinement step. There are many distances in the input, which makes computing the gradient a time-consuming procedure. However, on the other hand, MDJEEP was able to identify solutions for all proteins in the instance because of the local optimization, where as IBP-NG was not.
# 5 Discussion

The work presented here has confirmed the variability of bond angles and ω tor-
sion angle according to the location of the amino acid residues in the Ramachan-
dran maps. This complicates distance-based approaches, when attempting to
determine protein structures in the context of the branch-and-prune framework.
We presented two alternatives which may deal with this problem, encapsulated
by the MDJEEP and IBP-BG software packages. Although both methods permits
to determine solution with bond and torsion angles close to the target values,
they did not permit to systematically reach small coordinate deviations. Never-
theless, the drift underlying the calculations may come from patterns of bond
and torsion angles along successive residues. This could be investigated by ana-
lyzing triplets of amino acids successive in the primary sequence, in place of the
analysis of single amino acids performed here.

The approaches presented here to facilitate the generation of protein struc-
tures can constitute a first step for the application of the iBP algorithm to the
reconstruction of missing regions encountered in X-ray crystallographic or cryo-
EM structures. As many of these regions correspond to loop secondary struc-
tures, the variability of bond and torsion angles specifically observed for these
structures, will be essential to realize an efficient reconstruction. To improve the
results in future work, the strong suits of the two implementations should be put
together into one.

**Acknowledgements** CNRS, Lorraine University and University of Rennes are acknowledged for funding. ANR PRCI “Multi-scale and multi-resolution bio-molecular structure determination by geometric approaches - multiBioStruct” (ANR-19-CE45-0019) was supporting this work.

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