## Abstract

Collagen fibrils are the major constituents of the extracellular matrix, which provides structural support to vertebrate connective tissues. It is widely assumed that the superstructure of collagen fibrils is encoded in the primary sequences of the molecular building blocks. However, the interplay between large-scale architecture and small-scale molecular interactions makes the ab initio prediction of collagen structure challenging. Here, we propose a model that allows us to predict the periodic structure of collagen fibers and the axial offset between the molecules, purely on the basis of simple predictive rules for the interaction between amino acid residues. With our model, we identify the sequence-dependent collagen fiber geometries with the lowest free energy and validate the predicted geometries against the available experimental data. We propose a procedure for searching for optimal staggering distances. Finally, we build a classification algorithm and use it to scan 11 data sets of vertebrate fibrillar collagens, and predict the periodicity of the resulting assemblies. We analyzed the experimentally observed variance of the optimal stagger distances across species, and find that these distances, and the resulting fibrillar phenotypes, are evolutionary well preserved. Moreover, we observed that the energy minimum at the optimal stagger distance is broad in all cases, suggesting a further evolutionary adaptation designed to improve the assembly kinetics. Our periodicity predictions are not only in good agreement with the experimental data on collagen molecular staggering for all collagen types analyzed, but also for synthetic peptides. We argue that, with our model, it becomes possible to design tailor-made, periodic collagen structures, thereby enabling the design of novel biomimetic materials based on collagen-mimetic trimers.

## Significance

The pathway for protein self-assembly is determined by the free energy landscape coded in the noncovalent interactions between the building blocks. We use this basic principle to develop a model that describes the mechanisms involved in the staggering of collagen molecules in fibrillar assemblies. In this work we present a simple, parameter-free model for collagen fibril design that allows us to predict the structure of self-assembling collagen fibers on the basis of the amino acid sequence of the constituent

*α*-chain subunits. We develop a classification algorithm and use it to scan through large data sets of collagen molecules to predict the periodicity of the resulting assemblies. We argue that the interaction model presented in this work provides a foundation for engineering of novel collagen molecules with specific material properties for targeted applications.## Introduction

The material properties of connective tissues, such as tendon, skin, bone, and cartilage, are largely controlled by fibrillar assemblies of collagen proteins. Collagen molecules are long ($\approx $ 300 nm), rope-like structures, formed from three monomeric

*α*-chains twisted together into a triple helix (1

). In vertebrates, there are at least 10 distinct collagen molecules, each comprising 3 monomers, drawn from 12 different *α*-chains, encoded by 11 genes. The primary structure of the individual*α*-chains determines the geometrical and biophysical parameters of the collagen helix, which in turn govern the organization of molecules within the fibril, thereby establishing interactions necessary for quaternary structures to form.Collagen fibrils are composed of hundreds of aligned helices. The major collagens, types I, II, and III, form wide, long, unbranched fibrils, which are the dominant components of structural tissue, typically in conjunction with smaller quantities of the minor collagens, types V and XI, which are thought to act as fibril nucleators (

1

). TEM studies of these fibrils show periodic dark-light bands along their length with periodicity $D\approx 67\phantom{\rule{0.5em}{0ex}}\text{nm}$, attributed to the constituent molecules being longitudinally staggered relative to their neighbors by integer multiples of *D*(2

, 3

, 4

, 5

). Such fibrils are found in tendons, cornea, skin, and cartilage (6

, 7

, 8

). However, not all collagen molecular species assemble into these classical periodic fibrils. Regulatory or developmental collagen proteins do not form wide, striated fibrils under physiological conditions. These polymers are incorporated into the structurally defined suprastructure as a result of heterotypic interactions (collagen type V and XI) (9

). In addition, some collagens form thin, nonbanded assemblies (type XXIV and XXVII) (10

, 11

, 12

, 13

).To unravel the design principles of collagen assembly, we must find a mapping between the primary sequence of the collagen trimer and the phenotypic, structural features of the collagen fibril. Given the primary sequence of the

*α*-chain subunits, is it possible to predict the value of the axial offset between assembled polymers? Previous work has provided evidence for a link between sequence and the supramolecular structure of collagen assemblies (14

, 15

, 16

, 17

). In fact, interaction-based scoring systems for linear sequences have been proposed in (14

,15

,17

). In what follows, we use a more physically detailed model to arrive at a simple theoretical tool to predict the observed molecular geometry. Given the size of each collagen monomer of around 3000 amino acid residues, and the lack of detailed structural data, a fully atomistic (free) energy optimization procedure to model collagenous assemblies would be prohibitively expensive. Consequently, we take a coarse-grained approach to estimate the free energy of assembly. We make use of well-established empirical estimates of the strength of residue-residue interactions, based on so-called statistical contact potentials (CPs). We integrate these CPs in a simplified representation of collagen molecular structure. The resulting model allows us to estimate the relative stability of various collagen arrangements. We analyzed the primary structures of collagen proteins that can be classified into various functional types, across several vertebrate organisms. We used primary sequence data for collagen types for which experimental data regarding the phenotype of higher-order structure are available (Table 1), to establish a procedure for periodicity prediction. Here, we show that the axial staggering is indeed fully encoded in the collagen helix and can be predicted solely based on the primary structure of the trimer *α*-chain subunits. Moreover, we provide evidence that the stagger distance between collagen molecules in their fibrils and as a result, the phenotypic features of those fibrils, are well preserved across evolutionary time.Table 1The 11 types of fibrillar proteins analyzed in this work and their corresponding experimentally determined molecular aggregations

ID | Type | Trimer composition | Length $\left[s\right]$ | Fibril periodicity D [nm] | D [s] | Species |
---|---|---|---|---|---|---|

1 | I | ${\left[\alpha 1\left(I\right)\right]}_{3}$ | 1012 | 67 | 234 | calf ( 18 ) (bovine) |

2 | I | ${\left[\alpha 1\left(I\right)\right]}_{2}\alpha 2\left(I\right)$ | 1012 | 67 | 234 | human, rat, bovine ( 19 ,20 ) |

3 | II | ${\left[\alpha 1\left(II\right)\right]}_{3}$ | 1012 | 67 | 234 | human, lamprey, bovine ( 5 ) |

4 | III | ${\left[\alpha 1\left(III\right)\right]}_{3}$ | 1027 | 66.7 ± 0.2 | 234 | calf ( 21 ) (bovine) |

5 | V | $\alpha 1\left(V\right)\alpha 2\left(V\right)\alpha 3\left(V\right)$ | 1012 | unknown | – | – |

6 | V | ${\left[\alpha 1\left(V\right)\right]}_{2}\alpha 2\left(V\right)$ | 1012 | periodic, 67 | 234 | rat ( 22 ) |

7 | V | ${\left[\alpha 1\left(V\right)\right]}_{3}$ | 1012 | nonperiodic | – | calf ( 23 ) |

8 | XI | $\alpha 1\left(XI\right)\alpha 2\left(XI\right)\alpha 3\left(XI\right)$ | 1012 | periodic, 67 | 234 | chick ( 24 ) |

9 | XXIV | ${\left[\alpha 1\left(XXIV\right)\right]}_{3}$ | 979 | unknown | – | – |

10 | XXVII | ${\left[\alpha 1\left(XXVII\right)\right]}_{3}$ | 988 | nonperiodic | – | mouse ( 11 ) |

11 | I | ${\left[\alpha 2\left(I\right)\right]}_{3}$ | 1012 | – | – | – |

IDs 1–10 are natural molecular types of collagen. Some collagen types are known to build periodic assemblies, but not others (see

*Fibril periodicity*column). The length of a trimer is given in helical segments*s*(see text).a Variant of collagen type I found in development and disease.

b Not found in vivo, but present in vitro when collagen type I heterotrimers are denatured and then renatured (

25

).c Test set sequences used for the estimation of the

*α*-parameter (see in text).## Methods

### Model

Our model aims to strike a balance between simplicity at the level of the description of the structure of the collagen triple helices, and realism in the description of the intercollagen interactions. We achieve this compromise by using a

*knowledge-based*representation of the interaction between individual amino acid residues between the pairs of triple helices arranged as in the three-dimensional fibril structure.#### Representation of collagen molecules

On the basis of the available experimental data regarding the structure of collagen molecules, and inspired by earlier models (

26

,27

), we use the following representation of trimeric collagen molecules:- 1We consider the triple helical domains of the collagen proteins only, based on the widely held understanding that the triple helix domain is the main driver in determining the fibrillar structure. Each triple helix is modeled as a rigid rod denoted by
*T*. - 2The triple helical rigid rod is considered to be composed of elementary subunits, hereafter referred to as
*segments*$\left\{{s}_{n}\right\}$, where*n*describes the position of a segment along the rod axis, $\text{n}\in \left\{1,\dots ,N\right\}$ and*N*is the total number of segments. In what follows, we denote distances along the collagen triple helix in terms of segments, such that the length of the triple helix, $L=N$. The segments are thus separated by a distance $l\approx 0.29\phantom{\rule{0.5em}{0ex}}\text{nm}$ along the rod axis, the expected rise per residue of the collagen triple helix. - 3Each segment comprises a group of three coplanar amino acids, i.e., a cross section of the respective triple helix.

#### Interaction between collagen helices

Consider two parallel triple helices, ${T}_{a}$ and ${T}_{b}$, each composed of N $\approx $ 1000 segments (Fig. 1). We will consider all possible relative displacements of the two helices. We number these displacements using the index p; we let $\Delta {x}_{p}$ denotes the distance by which one helix is shifted with respect to the other, such that $\Delta {x}_{p}$ will change as p is changed. We measure $\Delta {x}_{p}$ in terms of integer numbers of helical segments, so that $\Delta {x}_{p}\in \left\{1,\dots ,N\right\}$. For p = 0, the two helices are completely aligned (no relative shift). In this case, $\Delta {x}_{0}$ = 0. Below we explain how we compute the interaction energy as $\Delta {x}_{p}$ is increased.

#### Calculation of pairwise interactions between the segments

Firstly, we need to decide how to score interactions between the helical segments. To a first approximation, we assume that the interaction energy between two segments is given by the average over all possible inter-residue contacts (see Section S1.1 for details). If two segments belonging to neighboring molecules: ${s}_{i}$ = $\left(G,{X}_{1},{Y}_{1}\right)$ and ${s}_{j}$ = $\left(G,{X}_{2},{Y}_{2}\right)$ are in proximity, the total interaction energy between these two segments is approximated as:

where the ${\epsilon}_{{A}_{x},{A}_{y}}$ denotes the energy of contact between the amino acid ${A}_{x}\in \left\{{X}_{1},{Y}_{1}\right\}$ and ${A}_{y}\in \left\{{X}_{2},{Y}_{2}\right\}$. For each of four possible ${A}_{x}{A}_{y}$ interactions, ${\epsilon}_{{A}_{x},{A}_{y}}$ is selected from a CP matrix discussed below.

${e}_{\left({s}_{i},{s}_{j}\right)}=\frac{1}{4}\sum _{x,y}{\mathrm{\epsilon}}_{\left({A}_{x},{A}_{y}\right)},$

(1)

where the ${\epsilon}_{{A}_{x},{A}_{y}}$ denotes the energy of contact between the amino acid ${A}_{x}\in \left\{{X}_{1},{Y}_{1}\right\}$ and ${A}_{y}\in \left\{{X}_{2},{Y}_{2}\right\}$. For each of four possible ${A}_{x}{A}_{y}$ interactions, ${\epsilon}_{{A}_{x},{A}_{y}}$ is selected from a CP matrix discussed below.

#### Energy of intermolecular interactions

Given two molecular trimers and a specific value of the offset $\Delta {x}_{p}$, we need an energy function $H\left(\Delta {x}_{p}\right)$ that provides a reasonable estimate of the intermolecular interaction energy. To calculate this total interaction energy, we defined the matrix ${E}^{\Delta {x}_{p}}\in {\mathbb{R}}^{L\times L}$, such that its elements ${E}_{ij}^{\Delta {x}_{p}}$ = ${e}_{\left({s}_{i},{s}_{j}\right)}$ describe the interaction energies that result when

*i*-th segment of trimer ${T}_{a}$ makes contact with the*j*-th segment of trimer ${T}_{b}$ (Fig. 1). The array ${E}^{\Delta {x}_{p}}$ contains all pairwise interactions between the segments, calculated for the molecular alignment $\Delta {x}_{p}$. Note that for $\Delta {x}_{0}$= 0, E is symmetric.Having defined ${E}^{\Delta {x}_{p}}$, we must now specify which segments are in fact interacting for a given value $\Delta {x}_{p}$ of the stagger. When mapping real-space protein structures on to lattice models, it is commonly assumed that a pair of residues form a contact if the distance between their C

_{α}atoms is less than 0.75 nm. The lateral intermolecular distances in collagen fibrils vary between 1.1 and 1.6 nm (28

) and the internal radius of a triple helix falls in the range 0.1–0.2 nm (29

,30

). Thus, we ignore interactions between pairs of amino acids that are separated by more than one segment (see Fig. S1, *A*and*B*). This information is encoded in the binary contact matrix $Q\in {\mathbb{R}}^{L\times L}$, where:${Q}_{ij}=\{\begin{array}{ll}1\hfill & \text{if}|i-j|\u2a7d1\hfill \\ 0\hfill & \text{elsewhere}\hfill \end{array}.$

(2)

Finally, we compute the total energy of two interacting trimers by simply adding contributions from the relevant elements of ${E}^{\Delta {x}_{p}}$, selected for range defined above by applying the matrix

*Q*. Following the notation described above, this can be simply written as:$H\left(\Delta {x}_{p}\right)=\frac{1}{3}\sum _{\left(i,j\right):|i-j|\le 1}\text{energy}\left({s}_{i},{s}_{j}\right)=\frac{1}{3}\sum _{i,j}{E}_{ij}{Q}_{ji}=\frac{1}{3}\text{Tr}\left({E}^{\Delta {x}_{p}}Q\right).$

(3)

To summarize, Eq. 3 describes the following: for each given configuration ($\Delta {x}_{p}$), we check current contacts between neighboring helical segments of interacting trimers and add together their energies. At every iteration, matrix ${E}^{\Delta {x}_{p}}$, which stores pairwise energy quantities is updated. The $H\left(\Delta {x}_{p}\right)$, given by a sum of pair interactions, can be interpreted as the free energy of the molecular complex, computed for the given conformation directly from the amino acid sequences of interacting trimers. The factor 1/3 in (

3

) accounts for the intersegmental interaction range, as given by the matrix *Q*.#### Constraints imposed on the staggering distance

In this work we made a few assumptions that constrain the possible values of the stagger $\Delta {x}_{p}$. These constraints, described in the two subsequent paragraphs, result from known features of collagen fibrillar structures, and specific choices that we made to facilitate comparison of our findings with experimental data.

#### The longitudinal repeating structural unit

The energy calculation described above for two interacting triple helices can be easily extended to describe the amino acid-residue interactions that exist in the collagen fibrillar environment. The total interaction energy in the fibril results from the pairwise interactions within one longitudinal repeating structural unit (LRU) of the fibril. The LRU is the minimum set of molecules that have mutual longitudinal overlap. Each molecule in the LRU is defined as a cyclic permutation of the collagen triple helix segments such that the N-terminal end of the triple helix is offset, respectively, by $m\Delta {x}_{p}$, where $m=\mathrm{1,2},\cdots ,\left(M-1\right)$. Here,

*M*denotes the number of the members of the LRU, which depends on the molecular length and the stagger $\Delta {x}_{p}$ (see Fig. 1). The possible intermolecular staggers between the*M*molecules of the LRU are thus $\Delta {x}_{p},2\Delta {x}_{p},\cdots ,\left(M-1\right)\Delta {x}_{p}$. Here, we compute the individual interaction energies for all*M*possible intermolecular interactions in the LRU, making no assumptions about how the collagen molecules are laterally arranged within the fibril.In this work we model interactions between

*identical*collagen proteins. This constraint causes a reduction in the number of stagger possibilities since staggers can become mirror reflections of each other due to the imposed periodic boundary conditions. Therefore, the interaction energies for staggers of $m\Delta {x}_{p}$ and $\left(M-m\right)\Delta {x}_{p}$ are energetically identical (see supporting material for details).#### Gap insertion

The stagger in the fibril enforces a longitudinal break

*g*between trimers along a common axis when the stagger $\Delta {x}_{p}$ is not a divisor of the molecular length*L*. Concretely, if $\left(q,r\right)$ denotes, respectively, the quotient and remainder of $\frac{L}{\Delta x}$ and if $r\ne $ 0 and $\Delta x\ne $ 0, then the gap*g*(measured in helical segments) is computed as $\left(q+1\right)\times \Delta x-L$. It then follows that*g*depends linearly on $\Delta x$ and is equal to 0 when*L*can be written as an integer number times $\Delta x$.#### Amino acid contact forms

We select the pairwise energies ${\epsilon}_{{A}_{x},{A}_{y}}$ with the aid of a statistical protein CP. In general, CPs are free energy functions derived from protein structural data by various principles, and are typically used in protein folding (e.g., to distinguish between the native fold and decoys) or docking (

31

, 32

, 33

, 34

). CPs can differ according to the assumptions made in deriving them, and according to the details of the estimation (e.g., the number of parameters). Empirical CPs are called *knowledge-based*potentials because they are derived under the assumption that the frequency of observed contacts between a given pair of amino acid residues across protein structures reflects the strength of interaction between those residues. Miyazawa and Jernigan (MJ) used a quasi-chemical approximation to relate the effective contact energy between amino acids*x*and*y*to the frequency with which nonbonded contacts between*x*and*y*are observed in known protein structures (31

,35

, 36

, 37

). MJ introduced two types of CPs: one containing strong hydrophobic components is intended to describe the energy of amino acid transfer from solvent to the protein internal environment (MJh), the second describes the interactions between amino acid residues within the average protein environment (MJb).In this work, we assumed that the MJb matrices are better suited to study interactions between collagen trimers. The environment within a collagen fibril mimics the internal environment of a globular protein because it is dominated by protein-protein interactions while solvent-protein contacts are limited. The MJ statistical CP matrices used in this work were obtained from the AAindex database (https://www.genome.jp/aaindex/) (

38

) using the indexes: MIYS850103, MIYS960102, MIYS990107.### Detection of periodicity signals

The goal of our analysis is to discriminate among collagen trimer types that can self-assemble into periodic fibrillar structures. To detect periodicity signals across polypeptide sequences, we developed a simple method to search for the local minima of the interaction energy, which is computed as a function of a staggering position $H=f\left(\Delta {x}_{p}\right)$, defined by Eq. 3. Firstly, we consider a point

where ${\overline{H}}_{lu}=\frac{{\sum}_{i=l}^{u}H\left(\Delta {x}_{i}\right)}{u-l+1}$ is the average value of the interaction energy computed over the indicated range of stagger positions $\Delta {x}_{p}$, where $\left\{l,u\right\}$ are, respectively, the lower and an upper bound of that range. In this work we chose the range of $\Delta {x}_{p}$ values such that the number of molecules in the LRU,

*p*corresponding to the interhelical distance $\Delta {x}_{p}$ (see Fig. 1*A*) as a potential periodicity signal if the interaction energy computed at this point—$H\left(\Delta {x}_{p}\right)$—is significantly lower than the average interaction energy computed over the sampled range of staggering positions. Specifically we ask whether a stagger $\Delta {x}_{p}$ exists for which $H\left(\Delta {x}_{p}\right)$ satisfies the following inequality:$H\left(\Delta {x}_{p}\right)\le {\overline{H}}_{lu}-\alpha {\sigma}_{lu},$

(4)

where ${\overline{H}}_{lu}=\frac{{\sum}_{i=l}^{u}H\left(\Delta {x}_{i}\right)}{u-l+1}$ is the average value of the interaction energy computed over the indicated range of stagger positions $\Delta {x}_{p}$, where $\left\{l,u\right\}$ are, respectively, the lower and an upper bound of that range. In this work we chose the range of $\Delta {x}_{p}$ values such that the number of molecules in the LRU,

*M*, is between 3 and 7. The parameter*σ*is the standard deviation of the energy $H\left(\Delta {x}_{p}\right)$ from mean ${\overline{H}}_{lu}$, and*α*is a parameter we specify that selects the dispersion range, and is chosen empirically (see Fig. 3*B*). We tested the performance of different values of*α*using the validation sets and selected $\alpha \approx 2.15$. For validation, we extracted the subset of trimers from the species for which the periodicity is known experimentally (according to Table 1). The selected value of the*α*parameter used in the calculations is found as the maximal value for which we make the correct prediction for all known labeled examples: low*α*finds even small minima (makes false-positive predictions), whereas a high*α*value requires a very deep minimum (makes false-negative prediction, see Fig. 3*B*).Given that we have identified a set of points $p\in \left\{{p}_{1},\dots ,{p}_{n}\right\}$ where the function $H\left(\Delta {x}_{p}\right)$ reaches a local minimum for each trimer type, we next check if any of these candidate points satisfy the requirement that the above relation is satisfied by each integral

*m*multiple of the selected offset distance $\Delta {x}_{p}$. In summary, we assign periodicity*p*to a given collagen trimer under the following condition:$H\left(m\Delta {x}_{p}\right)\le {\overline{H}}_{ml,mu}-\alpha {\sigma}_{ml,mu},\forall m=1,\dots ,M.$

(5)

The above requirement states that if we identify an energy minimum at position

*p*(distance $\Delta {x}_{p}$), all of its integral multiples indexed here by the index*m*, which runs over the possible intermolecular displacement choices, $m=1,\dots ,\left(M-1\right)$, are also local minima of the interaction energy. This asserts that the pairwise interaction energy between the trimers shifted by the distance $m\Delta {x}_{p}$ reaches a minimum, whichever the*m*value is. For practical reasons, we restricted our analysis to consideration of $M\le 7$.### Data sets

To carry out this analysis we used the data sets of homologous fibrillar collagen

*α*-chain sequences as described below. A multiple sequence alignment (MSA) is built for each fibrillar*α*-chain allowing the helical region of each pro-collagen*α*-chain sequence to be identified and extracted. The MSA can be viewed as an array where each sequence occupies a row and the columns correspond to the sequence sites. We then used the extracted regions of each sequence record to build various types of collagen trimer found in each species.#### Data acquisition

For each collagen

*α*-chain, the set of sequence orthologs was obtained using the National Center for Biotechnology Information Protein Reference Sequences Database resource (RefSeq) (https://www.ncbi.nlm.nih.gov/protein/). We identified homologous sequences using BLAST. This approach makes the assumption that orthologous sequences originate from a shared ancestor via an evolutionary process that includes mutations, insertions, and deletions. The algorithm implemented in the software program MUSCLE (http://www.drive5.com/muscle/) aims to reproduce the pattern of these events by maximizing similarity between aligned sequences (39

,40

).#### Data processing

Each data set of sequences was filtered to remove duplicates, leaving a single representative sequence for each collagen

*α*-chain for each species. For each*α*-chain type we built an MSA with MUSCLE using its default parameters.#### Hetero-trimer construction

There are various ways in which each hetero-trimer can be constructed by alternating the relative positions of the three component

*α*-chains among leading, middle, and trailing. In cases where a hetero-trimer is composed of three unique*α*-chains, there are six possible variants, whereas if the hetero-trimer contains two distinct*α*-chains, then there are just three possible variants. In this study we analyze four known collagen hetero-trimers (Table 1). Each hetero-trimer is constructed assuming the following order of*α*-chains: type I: $\alpha 1\left(I\right)\alpha 1\left(I\right)\alpha 2\left(I\right)$; type V (a): $\alpha 1\left(V\right)\alpha 2\left(V\right)\alpha 1\left(V\right)$; type V (b): $\alpha 2\left(V\right)\alpha 3\left(V\right)\alpha 1\left(V\right)$; type XI: $\alpha 1\left(II\right)\alpha 1\left(XI\right)\alpha 2\left(XI\right)$.## Results

The self-assembly of supramolecular collagen structures is mediated by a large number of interhelix interactions. Our hypothesis is that the preferred arrangements of two (or more) collagen triple helices should have a more favorable interaction energy than alignments that are not observed in experiments.

To examine how the relative stagger of collagen triple helices affects the energy of fibrillar ensembles, we constructed the energy function (Eq. 3) that provides a first-order estimate of the strength of the sequence-dependent pairwise interaction between two trimers. Figs. 2 and S3–S6 show estimates of pairwise interaction energy between identical collagen trimers for different

*M*values across several trimer types listed in Table 1. Here, we investigate the interaction energies obtained using Eq. 3 for each of the*m*differently aligned trimers and take the average across the data sets of orthologs from diverse species, such that the error bands show the standard deviation at each $\Delta {x}_{p}$ distance. By comparing these curves, we have identified that a well-defined energy minimum exists universally among both the major and minor collagen types for $M=5$. The arrangement with M = 7 ($\frac{L}{7}\le \Delta {x}_{p}\le \frac{L}{6}$, Fig. S3) does not seem to be probable—we do not detect an energy minimum at any offset from this range. Similarly, for M = 6 (Fig. S4) and M = 4 (Fig. S5) staggers we do not observe significant signals suggesting periodic fibrils structures in the interaction curves. For the arrangement with three staggers, which spans a broad range of offset values, $\frac{L}{3}\le \Delta {x}_{p}\le \frac{L}{2}$, we observe a drop at $\Delta {x}_{352}$ for mutual interactions between type III collagen trimers, but not for any other collagen types (Fig. S6). Moreover, we noticed a lack of correlation between the energy curves reported for each of the $\Delta {x}_{p}$ multiples for the arrangements constructed by*M*= 4, 6, and 7 staggers across all the collagen types analyzed. Conversely, the suprastructure with M = 5 gives correlated patterns of interaction, with a clear energetic minimum reported for all $\Delta {x}_{m\times p}$ neighbors, where $m\in \left\{1,\dots ,4\right\}$, for 9 out of 11 trimer types (Fig. 2).The second factor determining the arrangement topology are the entropic effects: the entropy of the system (lattice) increases with the number of degrees of freedom to distribute trimers in the fibril lattice. Intuitively, the higher the number of molecules per repeating unit,

*M*, the greater the number of possible molecular configurations, and so the greater the entropy. However, what is the exact relation between the entropy and*M*? To address this question, we examined the entropy gain per triple helix assuming different*M*stagger values. If all staggers are equiprobable, the entropy of a macrostate (system with*M*staggers) can be evaluated with the Boltzmann formula. Our approach is illustrated in Fig. S2 (see figure caption for details). The slow (logarithmic) growth of Sn from S3 = 0 to S7 = 3.22 R/mol is shown in Fig. S2*B*. Clearly, this analysis shows that the entropy gain per triple helix is too small to justify the selection of any particular arrangement (*M*) over others. Furthermore, it is sub-extensive: if we increase the length of the trimers and build a fiber twice the original length, we would not change the stagger entropy.### Periodicity prediction

#### Fibrillar collagens

For each sample trimer of vertebrate species, we use Eq. 5 to examine the interaction curves presented in Fig. 2 across various collagen types. The results are summarized in Table 2 and visualized in Fig. 3

*C*. In general, we detect periodicity signals for 10 out of 11 of the data sets of trimers analyzed, among which 7 trimer types exhibit no or marginal signal variance across the different species. We predict that the analyzed molecular trimers of collagen type I (Fig. 3*D*a, b), type II (Fig. 3*D*c), type XI (Fig. 3*D*h), and heterotrimers of collagen type V (Fig. 3*D*f, g) upon self-interactions are capable of assembly into periodic fibrils, regardless of their phylogenetic origin. Conversely, we anticipate that all studied collagen type XXIV trimers do not aggregate into periodic structures. As for the second developmental collagen, type XXVII, we do not find periodicity signals for the high percentage of analyzed trimers (91%), although some exceptions are provided by some species of birds (Fig. 3*D*j).Table 2Summary of periodicity prediction for orthologs of collagen triple helices of various types

Trimer type/molecular composition | Species with predicted periodicity (%) | $\Delta {x}_{p}$ values [s] | No. of records | Percentage of records for the $\Delta {x}_{p}$ given in ( ) | Predicted periodicity p [s] |
---|---|---|---|---|---|

$\left[\alpha 1{\left(I\right)}_{3}\right]$ | 100 | 235, 234 | 125 | 99.2 (235) 96.80 (234) | 235, 234 |

$\left[\alpha 1{\left(I\right)}_{2}\alpha 2\left(I\right)\right]$ | 100 | 235, 234 | 118 | 100 (235) 97.46 (234) | 235, 234 |

$\left[\alpha 1{\left(II\right)}_{3}\right]$ | 100 | 235, 234, 236 | 116 | 100 (235) 28.45 (234) 24.14 (236) | 235 |

$\left[\alpha 1{\left(III\right)}_{3}\right]$ | 84.72 | 235, 234 | 144 | 82.64 (235) 66.67 (234) | 235, 234 |

$\left[\alpha 1{\left(V\right)}_{3}\right]$ | 65.04 | 235, 234 | 123 | 63.41 (235) 2.44 (234) | 235 |

$\left[\alpha 1{\left(V\right)}_{2}\alpha 2\left(V\right)\right]$ | 96.4 | 235, 234 | 111 | 94.59 (235) 7.21 (234) | 235 |

$\left[\alpha 1\left(V\right)\alpha 2\left(V\right)\alpha 3\left(V\right)\right]$ | 92.19 | 235, 234 | 64 | 92.19 (235) 32.81 (234) 3.12 (236) | 235 |

$\left[\alpha 1\left(XI\right)\alpha 2\left(XI\right)\alpha 1\left(II\right)\right]$ | 100 | 235, 234 | 75 | 65.33 (235) 100 (234) | 234 |

$\left[\alpha 1{\left(XXIV\right)}_{3}\right]$ | 0 | – | 148 | – | nonperiodic |

$\left[\alpha 1{\left(XXVII\right)}_{3}\right]$ | 8.33 | 240, 241 | 156 | 7.05 (240) 1.28 (241) | nonperiodic |

$\left[\alpha 2{\left(I\right)}_{3}\right]$ | 83.08 | 236, 235, 234 | 201 | 8.96 (236) 82.09 (235) 7.46 (234) | 235 |

Bold values are intended to highlight the collagen types for which fibril periodicity is completely conserved across species.

a A collagen type I homotrimer not known to occur in vivo but detected in vitro (

25

).For the remaining collagen types, we noticed periodicity signals for a fraction of trimers (Table 2). This raises the possibility that the ability to form periodic structure could be, at least to some extent, species dependent for these trimer types. On the other hand, this could arise as a result of sequencing errors or could be an artifact of the simple binary classification scheme used (our model only returns the decision, we do not have any information about the distance form the decision boundaries to the actual value measured). To explore further, within each collagen type, we classify trimers into nine taxonomic groups, taking into account the class of the corresponding organisms. The results of this analysis are shown in Fig. 3

*C*and*D*. Here, different phylogenetic groups are marked by colors according to the legend. We detect periodicity signals among the majority of collagen type III trimers (86%), except for some species of birds and reptiles (Fig. 3*D*d). The analysis of interactions between type V homotrimers suggests that most trimers of fish and birds do not encode any periodicity signals, whereas more evolutionarily advanced mammalian species are likely to confer signals of periodic assembly (Fig. 3*D*e). At present we do not know the meaning of this observation. We extended our analysis to include a second type of collagen type I homotrimer: $\alpha 2{\left(I\right)}_{3}$. The structure of the C-terminal propeptide of pro-$\alpha 2\left(I\right)$-chain prevents this trimer from occurring in vivo (41

,42

). However, this homotrimer has been detected in vitro in collagen renaturation experiments (25

). We include here an analysis of its putative propensity to assemble into periodic fibrils as the $\alpha 2$-chain is known to influence the biophysical and molecular properties of the collagen type I heterotrimers, such as denaturation temperature or binding to matrix proteins (43

), and thus future research may be aimed at engineering this molecule. For this homotrimer, we detect putative indicators of periodic assembly for 83% of the tested trimers, where the remaining 17% of nonperiodic results comes mostly from different species of fish (Fig. 3 *D*k).The predicted values of the stagger distance

*p*($\Delta {x}_{p}$) which corresponds to the periodic energy minimum of the interhelical interaction energy (i.e., periodicity) defined with Eq. 5 are listed for each type of trimer in Table 2 and shown across phylogenetic groups in Fig. 3*A*. Among major and regulatory collagen types (Fig. 3*A*a–h, k) we detected three possible values of an interhelical stagger: $p=\left\{\mathrm{234,235,236}\right\}\left[s\right]$. For a marginal percentage of collagen type XXVII trimers of birds species (8%) we predicted a single value $p=240\left[s\right]$.#### Collagen model peptides

The prediction of fiber periodicity is one of the most prominent challenges in the design of fibrous proteins for biomedical applications. Recent studies show advancements in collagen mimetic peptide design, which has succeeded in obtaining collagen-mimetic trimers that are capable of self-assembly into periodic mini-fibers (

44

, 45

, 46

, 47

). In these cases, sequence design is based on the selection of a fragment of a collagen helix, which is subsequently repeated in tandem (44

, 45

, 46

, 47

). Chen et al. (45

) have produced three collagen-mimetic helical peptides, two of which (designated by the authors as COL108 and COL877) have been shown experimentally to self-assemble into periodic structures. For these two constructs, the authors designed primary sequences that repeat a 108 amino-acid-long triple-helical motif extracted from the human $\alpha 1\left(I\right)$-chain three times. These 378-residue-long peptides form a stable helix which self-assembles into periodic mini-fibrils with a periodicity of 35 nm (44

). For the reference (negative control), the authors constructed a third peptide (named COL108rr), which in contrast contains a randomized fragment of $\alpha 1\left(I\right)$-helical sequence in the middle (45

). These helices do not form periodic fibrils but instead build nonspecific aggregates (45

).We tested our model predictions on these three collagen-mimetic helices. First, we retrieved the peptide sequences provided in (

45

,44

) and constructed corresponding homotrimer models. We then examined the pairwise interaction energies between the trimers for different offset values using the same approach as for the natural fibrillar proteins. The resulting interaction energy curves as a function of $\Delta {x}_{p}$ are shown in Figs. 4, S7 and S8. Finally, given the computed energy patterns, we used Eq. 5 to predict periodicity.We found that our predictions are in agreement with experimentally measured periodicity values for collagen mimetic peptides. The homotrimers built by assembling COL108 peptides are predicted to form fibrils with a periodicity of $\approx $35 nm (0.3 gap, 0.7 overlap), as observed by TEM (

44

). Our model predicts a periodicity value of 122 helical segments $\approx $35 nm (Fig. S7), reproducing the estimate made by authors using a linear scoring model (44

). For the peptide COL877, the measured periodicity value equals $32\pm 1.4$ nm (45

), whereas our model predicts $p=123$ helical segments, which corresponds to $\approx $35 nm (Fig. 4). Given the predicted offset value $p=\left\{\mathrm{122,123}\right\}$, we anticipate that these collagen-mimetic trimers self-assemble into an arrangement with M = 4 in the LRU, instead of M = 5 as found in the case of natural collagens.#### Evaluation with randomized sequences

To evaluate the likelihood of detecting periodicity signals by chance we carried out three experiments. First, we generated a set of 1000 pseudorandom sequences of length

*L*= 1014 [s], where the amino acid at each position was drawn from the uniform distribution. We did not detect periodicity signals for any of these samples. To explore further, we generated additional two sets of sequences: 1) 1000 samples obtained by sampling the amino acid at X and Y position from the distribution of amino acid occurrences at X and Y position estimated for the human $\alpha 1\left(I\right)$-chain, and 2) 1000 samples obtained by permuting the order of $GXY$ tripeptides of human $\alpha 1\left(I\right)$-chain. This*α*variant is known to contain a periodicity signal. Therefore if the sequence composition has a predominant impact on the reported signal, we would expect to be identified in cases 1) and 2). For the first group, we found that just 1.3% of samples are predicted to encode some level of periodicity signal. Finally, if we only permute the triplets $GXY$ that occur in the $\alpha 1\left(I\right)$ sequence, we find that only 1.4% of samples contain some level of periodicity signal across a statistical sample of 1000 records.## Discussion

The energy of noncovalent interactions drives collagen proteins to form contacts and stabilizes their initial register, while the formation of intermolecular covalent bonds finalizes quaternary structure formation. Thermodynamically, the formation of collagen fibrils in normal conditions (physiological salt, 293–310 K) is an endothermic process, which occurs due to the great positive value of assembly entropy, resulting from solvent rearrangement (

48

,49

).- Kadler K.E.
- Hojima Y.
- Prockop D.J.

Assembly of collagen fibrils de novo by cleavage of the type I pC-collagen with procollagen C-proteinase. Assay of critical concentration demonstrates that collagen self-assembly is a classical example of an entropy-driven process.

*J. Biol. Chem.*1987; 262: 15696-15701

After the initial spontaneous association of trimers, the exact details of the quaternary structure are established by the gain in stabilization energy. This further energy gain can be attributed to the optimal alignment of collagen trimers in the fibril interior. To infer the most optimal alignment, we first asked about the constraints imposed on the arrangement of trimers within the fibril. Here, we have provided evidence that the value of molecular stagger is encoded in the sequence of collagen trimers and that the entropic effects that result from the selection of specific

*M*are negligible. We examined the free energy change as a function of the mutual orientation between the trimers using a pairwise approximation to the possible intersegmental interactions that result if two trimers are in contact. Our analysis suggests that the emergence of intermolecular stagger and, as a consequence, detectable structural features of collagen fibrils, results from the drop of the interaction free energy observed when the trimers are shifted by the optimal distance.In genomes of modern vertebrate species, 11 fibrillar collagen genes encode for distinct

*α*-polypeptides which combine to form collagen molecules of seven types (see Table 1). These genes have been grouped into three clades—A, B, and C—by similarity (50

,51

). Major fibrillar collagens (I, II, and III, built exclusively from chains of subclass A) and minor fibrillar collagens (V and XI, combining chains from clade A and B) constitute the main component of fibrils, forming the core of the extracellular matrix (52

). By comparing the interaction energy curves averaged over the data sets of protein orthologs, we identified that a well-defined energy minimum exists universally among the major and minor collagen types for $M=5$ (Fig. 2 *A*–*H*). Moreover, for $M=5$ the patterns are correlated, such that the energy minimum exists for all possible trimer interactions in the LRU, i.e., for all integer multiples of the $\Delta {x}_{p}$ stagger. Interestingly, the value of the energy minima are similar for all integer multiples of the $\Delta {x}_{p}$ stagger across all collagen types, except for the collagen type II $\alpha 1{\left(II\right)}_{3}$ trimer, implying that, with this exception, there is little energy penalty for altering the lateral arrangement of molecules within collagen fibrils. This suggests that lateral compression of collagen fibrils, for instance, is likely to be relatively facile. We find that the computed energy minima are broad and funnel shaped in all cases, suggesting the adaptation of collagen assemblies to recover from longitudinal strains without compromising their molecular registry and that a component of the fibril elasticity is encoded in the protein sequences. Moreover, it is plausible that the funnel-shaped interaction curves increase the robustness of the assembly kinetics.We then employed Eq. 5 to predict periodicity for analyzed collagen trimers across the species. The results are summarized in Table 2. Since the specific stagger distance can clearly be selected over competing ones, we predict that collagen trimer types I, II, III, V, and XI self-assemble into periodic supra structures. This has been confirmed experimentally for some species (see Table 1 and Fig. 2). Conversely, for the developmental collagen types (XXIV and XXVII, made up of chains from subclass C) we predict the formation of nonspecific aggregates since no energetically favorable alignment can be identified (see Fig. 2 for comparison). It has been hypothesized that the partial processing and retention of the N-terminal globular extension by the mature form of the protein gives rise to the lack of experimentally observed banding pattern (

12

,53

). Our analysis reveals that the lack of structural features observed for these trimer types is encoded, to a large extent, in the helical region of underlying *α*-polypeptides.## Conclusions

The unique structural properties of collagen triple helices endow them with the capability to encode information about self-assembly mechanisms (

54

). In this work we presented a simple, parameter-free model for collagen fibril design that predicts the structure of self-assembling collagen fibers on the basis of the amino acid sequence of the constituent *α*-chain subunits. The increasing availability of genomic data allows us to test the idea that optimal molecular alignment is dictated by the free energy of intermolecular interactions. Using our simple model we can estimate the free energy of interactions between each pair of trimers from the 11 data sets of vertebrate fibrillar collagens. We have analyzed the variance in the reported optimal stagger value across the species for each trimer type, and conclude that the stagger distance between collagen molecules in their fibrils and as a result, the phenotypic features of those fibrils, are well preserved across evolutionary time. Our predicted periodicities are in good agreement with experimental findings concerning the structural features of collagenous fibrils. We believe that the interaction model presented in this work provides a foundation for the future studies which aim to design the new*α*-peptide sequences for targeted applications. Collagen-mimetic trimers capable of assembly into the fibrillar suprastructures with desirable structural features are currently in high demand for medicine and material engineering and the understanding given by our model should allow simple prediction of the ability of a sequence both to form periodic fibrils itself and to design optimal interaction with other collagenous proteins, e.g., in vivo prosthetic applications.## Author contributions

M.J.D., L.J.C., D.F., and A.M.P. designed the research. A.M.P. collected and processed the data and carried out all calculations. A.M.P., M.J.D., L.J.C., and D.F. wrote the article.

## Acknowledgments

We thank Ieva Goldberga for helpful discussions related to the collagen TEM studies. A.M.P. was funded by a Raymond and Beverly Sackler Fund for Physics of Medicine (University of Cambridge), the European Research Council , and the Simons Foundation .

### Declaration of interests

The authors declare no competing interests.

## Supporting material

- Document S1. Figures S1–S8

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## Article Info

### Publication History

Editor: Markus Buehler.

Accepted:
July 12,
2022

Received:
March 9,
2022

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