Abstract
Selfassembling processes are ubiquitous phenomena that drive the organization and the hierarchical formation of complex molecular systems. The investigation of assembling dynamics, emerging from the interactions among biomolecules like aminoacids and polypeptides, is fundamental to determine how a mixture of simple objects can yield a complex structure at the nanoscale level. In this paper we present HyperBeta, a novel opensource software that exploits an innovative algorithm based on hypergraphs to efficiently identify and graphically represent the dynamics of \(\beta\)sheets formation. Differently from the existing tools, HyperBeta directly manipulates data generated by means of coarsegrained molecular dynamics simulation tools (GROMACS), performed using the MARTINI force field. Coarsegrained molecular structures are visualized using HyperBeta ’s proprietary realtime highquality 3D engine, which provides a plethora of analysis tools and statistical information, controlled by means of an intuitive eventbased graphical user interface. The highquality renderer relies on a variety of visual cues to improve the readability and interpretability of distance and depth relationships between peptides. We show that HyperBeta is able to track the \(\beta\)sheets formation in coarsegrained molecular dynamics simulations, and provides a completely new and efficient mean for the investigation of the kinetics of these nanostructures. HyperBeta will therefore facilitate biotechnological and medical research where these structural elements play a crucial role, such as the development of novel highperformance biomaterials in tissue engineering, or a better comprehension of the molecular mechanisms at the basis of complex pathologies like Alzheimer’s disease.
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Introduction
Supramolecular selfassembly arises from the interplay of noncovalent intermolecular and intramolecular interactions, ruling the autonomous organization of molecules into ordered patterns upon exposure to specific environmental conditions or external stimuli. Unlike covalent bonds, noncovalent interactions (electrostatic interactions, \(\pi\)effects, van der Waals forces, hydrophobic effects) do not involve sharing of electron pairs between atoms. Hence, they are characterized by low energies, poor directionality, and reversibility^{1}. These atomictonanoscale features give rise to different macroscale properties such as selfhealing or recovery of the original shape, usually not available to covalently bonded structures^{2,3}. Most of the processes in biological systems—such as the formation of organelles, DNA replication or protein folding—emerge from spontaneous biomacromolecular selfassembly^{3}. Inspired by these mechanisms, researchers have developed different classes of selfassembling molecules suitable for application in several fields, such as electronics^{4}, material science^{5}, and regenerative medicine^{3}.
In the last decade, the rapid progresses in this field have been supported by the improvement of experimental and, in particular, computational methods. Among them, molecular dynamics (MD) played a major role in the investigation of supramolecular selfassembly^{6}. Conventional allatom (AA) MD simulations usually consider atoms as the interaction sites, that is, the points where the potential energy functions of each interaction are calculated. However, due to their exceptional computational requirements, AA models can be inadequate for the investigation of complex systems along timescales that are comparable to reality. In order to overcome this intrinsic limitation of AA modeling, coarsegrained (CG) models have been developed^{7}.
CG modeling consists in grouping multiple atoms as individual interaction sites, named grains or beads, thus reducing the overall computational effort. This strategy was widely applied to investigate various selfassembling systems, like peptides or lipids^{8,9}. Two different approaches to CG modeling for biological systems have been proposed: (1) shapebased CG methods, where a small number of CG beads—typically, 10–50 beads with 200–500 atoms per bead— mimic the overall macromolecule shape; (2) residuebased CG, where several atoms—typically, 10–20 atoms per bead—are grouped into a single CG interaction site, which usually represents a single residue, a sidechain, or a group of backbone atoms^{9,10}. MARTINI is the bestknown and widespread residuebased CG force field for the simulation of biomolecular systems^{9}. MARTINI CGMD simulations are largely used in the field of supramolecular chemistry and structural biochemistry. Despite their unquestionable advantages, i.e. reduced computational costs and accurate description of molecular movements, MARTINI CGMD simulations suffer from a huge limitation in tracking the structural changes involved in protein folding. Indeed, these simulations do not allow to monitor the noncovalent interactions leading to the formation of secondary structures. In addition, in MARTINI CGMD simulations the secondary structure arrangements are limited by imposing of harmonic potentials among backbone grains. This means that MARTINI CGMD simulations are not suitable for the study of transitions among different secondary structures. Instead, MARTINI CGMD simulations find applications for the study of the arrangements of protein structures within supramolecular aggregates, such as selfassembling peptides (SAPs) nanofibrils. Due to the lack of information concerning noncovalent interactions, the main limitation of MARTINI CG simulation is that analytic tools for the quantitative tracking and visualization of selfassembling patterns are still lacking. The development of such tools is mandatory to achieve a deeper understanding of selfassembling phenomena^{10,11}.
In supramolecular chemistry, SAPs were widely used as models for the investigation of Alzheimer’s disease, and in the latter years they found several applications also in the field of tissue engineering^{12}. SAPs selfassemble into \(\alpha\)helix or \(\beta\)sheet secondary structure patterns. The amount of \(\beta\)sheet content in SAPs supramolecular structures usually wellcorrelates to their mechanical properties at the macroscale, which, in turn, can have significant effects on either attached or encapsulated cells in tissue engineering applications^{11}. In this work, we present HyperBeta , a novel tool that fills the gap in stateoftheart methods for the analysis of CG molecular structures. HyperBeta was specifically developed for the analysis of CGMD simulations of SAPs systems, and allows the automatic identification and realtime rendering of \(\beta\)sheets in MARTINI CGMD dynamics. Differently from Morphoscanner^{13}, a tool for CGMD simulations of SAPs systems, HyperBeta relies on an approach based on hypergraphs and additional geometric constraints. Moreover, HyperBeta calculates several statistics about the composition of \(\beta\)sheets, and embeds a highquality 3D engine that exploits sophisticated visual cues to simplify the interpretation of distance and depth relationships among the grains and the peptides. The methodology presented in this paper was validated on multiple proteinaceous structures, showing that it can be successfully exploited to obtain relevant details on SAP processes and kinetics.
Results
In the MD simulation analysis, the secondary structure assignment relies on the recognition of the hydrogen bond pattern or on the equivalent threedimensional topological pattern of backbone atom groups. The features of MARTINI forcefield hamper the analysis of CGMD simulations with software like DSSP^{14} or STRIDE^{15}: on the one hand, DSSP algorithm assigns secondary structures to single amino acid by identifying hydrogen bonds, which are not defined in MARTINI CG model; on the other hand, STRIDE assigns amino acid secondary structures by using hydrogen bonds and intrachain dihedral angle potentials. The HyperBeta analysis workflow does not use the information inherent to molecular connectivity and intrachain dihedral angles, whereas it relies on the CG beads Cartesian coordinates.
HyperBeta processes GROMACS files^{16} exported using the Gromos87 format, representing a single structure or multiple frames of a MD run, along with the number of aminoacid residues per peptide (group length, GL). The input file contains the MARTINI backbone grains corresponding to a single structure or multiple frames (here named “snapshots”) of a MD run. The user can easily introduce both information using HyperBeta ’s GUI (see Supplementary File #1). Once the GROMACS files are processed, HyperBeta ’s visualization tool visualizes the whole animation of the peptides selfassembly. As shown in Fig. 1, HyperBeta ’s rendering allows to track different peptides using different colors. When a grain or a peptide is selected, simulated fogging and depthoffield^{17} provide visual cues of distance relationships between the objects. In addition, HyperBeta highlights the key statistics such as the number of components and their relative compositions in order to have a quantitative description of \(\beta\)sheets. Then, HyperBeta summarizes other useful statistics such as the fraction of grains belonging to \(\beta\)sheets in a translucent panel placed in the topleft corner of the screen.
In order to validate HyperBeta, we processed four protein structures with known characteristics downloaded from the Protein Data Bank (PDB)^{18}. Then, the proteins were CGmapped according to the MARTINI model and subsequently analyzed using HyperBeta. The tested structures were:

the lamininglike module (PDB ID 1d2s), a highmolecular weight protein belonging to the extracellular matrix and constituting the biologically active part of the basal lamina (\(GL=10\))^{19};

the \(A \beta (142)\) fibrils (PDB ID 2mxu), the initial and predominant constituents of the amyloid plaques that characterize Alzheimer’s disease (\(GL=32\))^{20};

an engineered Boriella OspA structure (PDB ID 2fkg) consisting of \(\beta\)hairpin repeats connected by turn motifs (\(GL=9\))^{21};

the Escherichia coli \(\beta\)clamp (PDB ID 3bep), a subunit of the DNA polymerase III holoenzime, characterized by antiparallel \(\beta\)sheet structures (\(GL=6\))^{22}.
Figure 2 shows the result of this preliminary validation phase, presenting a comparison of the output produced by HyperBeta and rendered by HyperBeta ’s visualization tool by setting the angular threshold \(\alpha = 0.89\) and the distance threshold \(\varepsilon = 0.7\) nm (see Supplementary File #1 for further information and for comparison with Morphoscanner^{13}), against the structures identified by STRIDE^{23}, a wellestablished tool for secondary structure assignment, and rendered with VMD^{24}. The putative \(\beta\)sheets identified through HyperBeta in CG models correspond to the \(\beta\)sheets identified in unitedatom (UA) models through STRIDE and rendered with VMD.
Successively, five different SAPs MD trajectories were analyzed using HyperBeta with the same settings for \(\alpha\) and \(\varepsilon\). These trajectories were previously analyzed by Saracino et al.^{13}. These systems comprised a total of identical 100 peptides for BMHP1derived SAP sequences (B26: BtnGGGPFASTKT , GL = 10; B24: BtnGGGAFASTKT, GL = 10; 30: WGGGAFASTKT, GL = 10) and (LDLK)\(_3\) SAP (GL = 12), and 50 plus 50 opposite charged peptides for the complementary assembling peptides (CAPs), whose sequences are (LDLD)\(_3\) and (LKLK)\(_3\) (GL = 12).
Figure 3 shows that, after 500 ns (from top to bottom, left to right), the (LDLK)\(_3\) SAPs organize themselves into multiple independent cross\(\beta\) fibril seeds, which subsequently assemble into a “patchwork”like aggregate^{13}. Despite the huge variations of the number of triplets and components over time, the ratio between the different backbone grain types involved in the formation of the triplets remains stable. After the first 100 ns, the ratio between the numbers of Aspartic acid and Lysine backbone grains involved in the formation of the triplets is approximately equal to 1, as can be derived from the statistics at the left bottom corner of each panel. Indeed, the grain components of the triplets arise from the alternating alignment of opposite charge groups of Lysine and Aspartic Acid residues, resulting in \(\beta\)sheet rich aggregates. As a matter of fact, as shown by Saracino et al.^{13}, SAPs organization trend can be predicted by analyzing the first 500 ns of CGMD simulation trajectories.
As depicted in Fig. 4, SAP B24, the most promising sequence of the BMHP1derived SAPs for neural tissue engineering applications, shows the highest number of components and triplets. Large part of these triplets consists of hydrophobic grains such as Alanine, Glycine, Phenilalanine. The Biotin grains form a highly dynamic and unstable network of putative \(\beta\)sheets^{13}. On the contrary, B26 shows the lowest number of components and triplets. This tendency is ascribable to the \(\beta\)breaker effect of Pro residues and their favourable interactions with Biotin grains, which hampers the formation of stable \(\beta\)sheets. These are in turn related to the amphipathic features of each moieties of Biotin, such as hydrophilicity of the ureido ring and hydrophobicity of thiophene ring and valeryl chain. Despite the high similarity with SAP B24, SAPs 30 assemble into a less structured aggregate. Such difference is ascribable to the substitution of Biotin with the aromatic aminoacid Tryptophan, at the N terminus position. As depicted in Fig. 4, CAPs assemble into stable aggregates, such as (LDLK)\(_3\) SAPs. The ratio between the number of Lysine and Aspartic acid backbone grains, which are involved in the formation of triplets, is approximately equal to 2. Such feature is ascribable to the alternating arrangement of CAPs within bilayered aggregates. Indeed, each (LDLD)\(_3\) peptide is paired with two neighboring (LKLK)\(_3\) peptides.
Discussion
HyperBeta is a novel software for the analysis and rendering of MARTINI CGMD structures developed to monitor secondary structure patterns, obtained as a consequence of the establishment of noncovalent interactions between grains. The rationale is that in MARTINI CGMD simulations, the assignment of secondary structure patterns relies on the recognition of the threedimensional topological pattern of backbone atom groups. More in details, in MARTINI models, hydrogen bonds are implicitly described through the definition of particular bead types, such as N\(_{da}\) type^{10,11}, and protein secondary structures are constrained through the introduction of harmonic potentials among backbone grains. The harmonic potentials force the peptide chains to adopt extended conformations. Thus, in MARTINI CGMD simulations, peptide selfassembly may result into \(\beta\)sheet rich aggregates, whose geometries resemble those of a distorted lattice.
HyperBeta was also designed to provide a pleasant and productive user experience, by exploiting frustum culling, backface culling, and levelofdetail balancing to improve the reactivity of the realtime rendering even in the case of massive structures. Differently from any existing visualization tool, HyperBeta performs advanced visual cues like simulated depthoffield to improve the interpretation of distance and depth relationships^{17} between grains and peptides. Although HyperBeta was developed to investigate the relationships between grains and \(\beta\)structures, it also provides the possibility of rendering \(\beta\)sheet motifs, along with the other representations purely based on network connectivity, in order to simplify the interpretation of results. Examples of such representation are shown in Supplementary File #1.
HyperBeta was designed to render and analyze MARTINI CG biomolecular structures, reducing the number of steps required to track crucial structuring phenomena, as usually implemented with VMD and NAMD. Differently from VMD, which mandates the editing of tailored *tcl scripts to analyze and visualize MARTINI CG structures, HyperBeta does not require any additional scripts to analyze these structures. On the contrary, HyperBeta provides an intuitive, interactive and userfriendly interface. In particular, HyperBeta displays a variety of statistics about the identified \(\beta\)structures and the dynamic behavior of the system as graphical overlays; this information includes the number, and the type, of CG grains involved in the detected \(\beta\)structures. Specifically, HyperBeta detects the identified structures by considering the reciprocal distances and angles formed by CG grains belonging to different peptides, which are used to define the hypergraph of contacts. Such features will allow the investigation of even larger MARTINI CGMD outcomes, exploring size and time scales similar to those of laboratory experiments, such as NMR and cryoTEM. More in details, HyperBeta will allow to elucidate the interplay of intermolecular interactions completing the experimental theoretical workflow usually adopted for the investigation of selfassembling nanomaterials^{25}. We expect that HyperBeta will find immediate applications in the analysis of finer MD trajectories, mapped according to allatom (AA) and unitedatom (UA) model. This will be possible thanks to dedicated executable that allows the mapping and subsequent analysis of protein backbone according to the MARTINI model.
HyperBeta is available for download on GITHUB at the following address: https://github.com/aresio/hyperbeta.
Methods
In MARTINI CGMD simulations, peptide selfassembly may lead to \(\beta\)sheet rich aggregates, characterized by geometries similar to those of distorted lattices. The method employed by HyperBeta to discover the presence of \(\beta\)sheets, deploys the equivalent threedimensional topological pattern of CG grains. Given a collection of grains represented as a set of points in a three dimensional space, Hyperbeta’s functioning can be summarized into two main steps:

1.
all triples of grains belonging to different peptides (hampering the recognition of \(\alpha\)helix and randomcoil segments), which are “near enough” and “almost aligned”, are found;

2.
for each triple, we check which other triples overlap with it, thus suggesting that the grains in these triples belong to the same structure.
In what follows, we formalize these steps and provide an algorithm to discover possible \(\beta\)sheets in a collection of grains.
Basic notions
The L2norm, or Euclidean norm, of a point \(\vec {x} \in \mathbb {R}^3\)—in symbols, \(\vec {x}_2\)—is defined as \(\vec {x}_2 = \sqrt{\sum _{i=1}^3 x_i^2}\) and represents the Euclidean length of the point. Given two points \(\vec {x}\) and \(\vec {y}\), their distance is the Euclidean norm of their difference, i.e., \(\vec {x}  \vec {y}_2 = \sqrt{\sum _{i=1}^3 (x_i  y_i)^2}\).
Given three points \(\vec {x}, \vec {y}, \vec {z} \in \mathbb {R}^3\), the angle formed by \(\vec {x}\) and \(\vec {z}\) with respect to \(\vec {y}\) is defined as:
The value of the resulting angle is always between 0 and \(\pi\). Notice that the point \(\vec {x}\) and \(\vec {z}\) always form two angles with respect to \(\vec {y}\): either their are both \(\pi\), or one of them is acute and one is obtuse. The one that is considered hereby is always the acute one.
Aligned triples
The concepts of a triple of points (or grains, in our case) pertaining to different peptides that are “near enough” and “almost aligned” is formalized as follows.
Definition 1
Given \(\alpha \in [0,1]\), \(\varepsilon > 0\), and three points \(\vec {x}, \vec {y}, \vec {z} \in \mathbb {R}^3\), the ordered triple \((\vec {x}, \vec {y}, \vec {z})\) is defined to be \((\alpha , \varepsilon )\)aligned when the following two conditions hold:

\(\vec {x}  \vec {y}_2 \le \varepsilon\) and \(\vec {y}  \vec {z}_2 \le \varepsilon\). That is, the distances between the first and the second point, and between the second and the third point, are both smaller than or equal to \(\varepsilon\);

the angle formed by \(\ {x}\) and \(\vec {z}\) with respect to \(\vec {y}\) is larger than or equal to \(\alpha \pi\). Since the angle cannot be greater than \(\pi\), it means that the angle must be in \([\alpha \pi , \pi ]\).
An example of what are (and what are not) \((\alpha , \varepsilon )\)aligned triples is presented in Fig. 5.
Algorithm 1 shows how finding all \((\alpha ,\varepsilon )\)aligned triples in a set V of grains (represented as points in \(\mathbb {R}^3\)) can be performed. The time complexity of the algorithm is \(O(n^3)\), where n is the number of grains. The resulting set T of \((\alpha ,\varepsilon )\)aligned triples has cardinality bounded above by \(n^3\) but, in practical cases, we expect to obtain a set whose cardinality is way lower than this.
Connected components
Once all \((\alpha ,\varepsilon )\)aligned triples have been identified, we need to find a way of “gluing” them together if they overlap “enough”, that is, if they share at least two of the three grains.
Definition 2
Let \(a = (\vec {x}_a,\vec {y}_a,\vec {z}_a)\) and \(b = (\vec {x}_b,\vec {y}_b,\vec {z}_b)\) be two \((\alpha ,\varepsilon )\)aligned triples, for some \(\alpha \in [0,1]\) and \(\varepsilon > 0\). We say that a and b are overlapping when \(\{\vec {x}_a,\vec {y}_a,\vec {z}_a\} \cap \{\vec {x}_b,\vec {y}_b,\vec {z}_b\} \ge 2\).
We can now build a graph G where the vertices are the \((\alpha ,\varepsilon )\)aligned triples, and there exists an edge between two vertices if the corresponding triples are overlapping. Notice that the relation is symmetric (i.e., if a overlaps with b then also b overlaps with a), thus the resulting graph is undirected. As shown in Fig. 6, overlapping triples can be considered as “pieces” of the same structure once “glued together”. As shown in Fig. 7, the identification of aligned triples allows to easily discriminate the regular alternate \(\beta\)sheets domains from \(\alpha\)helix domains. When more triples are “glued together” regular \(\beta\)sheets are identified. The process of finding which triples can be “glued together” can then be expressed as finding the set of connected components in the graph G. The entire process is described by Algorithm 2, and the structure of the resulting algorithm is illustrated with an example in Fig. 8.
The resulting time complexity is given by iterating across all pairs of \((\alpha ,\varepsilon )\)aligned triples, which has quadratic complexity with respect to the number of triples, thus resulting in a time complexity of \(O(n^6)\). Depending on the representation employed for the graph, the time needed to compute the connected components is either linear in the number of vertices and edges, or quadratic in the number of vertices, resulting in both cases in a worstcase time complexity of \(O(n^6)\). While this time complexity seems high with respect to the number of grains, we remark that this is a worstcase scenario and, in practice, we expect the number of \((\alpha ,\varepsilon )\)triples to be much lower than cubic with respect to the number of grains.
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Acknowledgements
F.F., F.G. and G.A.A.S. were funded by the “Ministero della Salute Ricerca Corrente 20182020” funding granted by the Italian Ministry of Health and by the “5x1000” voluntary contributions. Financial support for computational analysis, performed by F.F.. G.A.A.S. and F.G., also came from Revert Onlus.
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M.S.N., F.F., L.M. and G.A.A.S. conceived and developed HyperBeta, M.S.N. implemented the HyperBeta Visualization Tool, M.S.N., F.F., and L.M. conducted the experiments, M.S.N., F.F. and L.M. created the figures, all authors provided critical feedback and approved the final version of the manuscript.
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Nobile, M.S., Fontana, F., Manzoni, L. et al. HyperBeta: characterizing the structural dynamics of proteins and selfassembling peptides. Sci Rep 11, 7783 (2021). https://doi.org/10.1038/s41598021870870
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DOI: https://doi.org/10.1038/s41598021870870
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