A Topological Selection of Folding Pathways from Native States of Knotted Proteins

Abstract

Understanding how knotted proteins fold is a challenging problem in biology. Researchers have proposed several models for their folding pathways, based on theory, simulations and experiments. The geometry of proteins with the same knot type can vary substantially and recent simulations reveal different folding behaviour for deeply and shallow knotted proteins. We analyse proteins forming open-ended trefoil knots by introducing a topologically inspired statistical metric that measures their entanglement. By looking directly at the geometry and topology of their native states, we are able to probe different folding pathways for such proteins. In particular, the folding pathway of shallow knotted carbonic anhydrases involves the creation of a double-looped structure, contrary to what has been observed for other knotted trefoil proteins. We validate this with Molecular Dynamics simulations. By leveraging the geometry and local symmetries of knotted proteins’ native states, we provide the first numerical evidence of a double-loop folding mechanism in trefoil proteins.

1. Introduction

A small percentage of proteins deposited in the Protein Data Bank [1] (PDB) are known to fold into conformations that are non-trivially entangled. All the currently known knotted protein structures are catalogued in the online database KnotProt [2]. The vast majority of knotted proteins form simple knot-like structures. However, there are a few examples of proteins having more complex entanglement types [2]. Interestingly, even though they are not ubiquitous, knots in proteins have been conserved within species that are separated by millions of years of evolution [3]. This might imply that knottiness provides some advantages to proteins [4], but this theory is still being contested [5]. Besides the role of knots in proteins, there are many open questions about the pathway that the backbone of a knotted protein follows to reach its native folded state. To date, researchers have suggested several mechanisms for protein self-tying that are based on wet lab experiments, numerical simulations, mathematical theory or a combination of all the above [3,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21].

Recently, Flapan et al. [21] proposed a novel topological model that contains as special cases two of the most popular folding theories [13,22]. This theory is agnostic in terms of protein families, and it is based on the assumption that there exist two loops of different sizes, each containing up to two twists. It is shown that there are several possible pathways emerging from the different choices of parameters of the model and it is suggested that these pathways are enough to recover any knot-type found empirically thus far in proteins. As this model is theoretical, physical interactions and dynamics that may affect the folding pathway are not taken into consideration.

In this work, we systematically analyse the native states of all deposited positive trefoil knotted proteins, aiming to determine potential folding pathways. We focus on trefoil proteins as these make up the majority of knotted proteins. Considering that the global underlying topology of all proteins in our data set of proteins is a trefoil, the geometry of the conformation plays a key role in detecting the most probable folding motif.

Our analysis is based on KnotoEMD, a statistical distance that we define on distributions of protein backbone projections considered as knotoids. Knotoids are knot-like open-ended topological structures that extend knot theory to the case of open curves [23]. The knotoid approach is a probabilistic method that does not require an artificial closure of the protein backbone. Multiple projections of the backbone are considered to determine the predominant knotoid type [24]. It has been shown that knotoids provide a more refined overview of a protein’s topology [2]. KnotoEMD is a specialisation of the earth mover’s distance, i.e., the 1st Wasserstein metric [25], and it quantifies the distance between different knotoid distributions in the space of trefoil knotted proteins.

We observe the existence of a double-loop in the conformation of shallow knotted carbonic anhydrases, which suggest that their folding follows a pathway similar to the theoretical model of Flapan et al. We subsequently perform computer simulations for the carbonic anhydrase with PDB entry 4QEF. The results confirm that this protein follows a complex folding pathway involving a double loop formation at the beginning of the process, although the specific mechanisms proposed by Flapan et al. are not observed. In contrast, a much simpler geometry is revealed for other families of shallow knotted trefoil proteins and for all the deeply knotted ones, providing strong evidence against a double-loop hypothesis in these cases. This is in agreement with the current consensus that different folding pathways should be accessible by shallow knotted proteins [26,27] and with previous simulations of deeply knotted trefoil proteins [20]. We find that KnotoEMD extracts subtle geometric information from the knotoid distribution and groups together trefoil proteins by sequence similarity. This highlights the fact that KnotoEMD is a versatile and robust mathematical tool that captures subtle topological and geometric information of open-ended conformations and has potential beyond the scope of this research, as discussed in what follows.

2. Materials and Methods

2.1. The Knotoid Distribution Describes a Protein’s Entanglement

Mathematically, a knot is formally defined as a closed curve in 3-dimensional (3D) space, and two knots are deemed equivalent if one can be deformed into the other without cutting and pasting the curve [28]. In order to identify the topological type of a linear knotted protein, a classical approach consists in considering its backbone as a polygonal curve and then joining the two termini with a new segment. The resulting closed curve is then analysed as a knot, and its underlying knot type can be formally determined [29]. The obtained knot type is clearly dependent on the choice of segment connecting the end-points of the curve. There are several ways to create a closed curve out of an open one in an unbiased way, in order to approximate the topology of an open curve [30]. Perhaps the most popular approaches are those where one considers several different closures at once and then takes the probability distribution of knot types as a topological descriptor of the open curve [31,32]. In such a probabilistic approach, the most likely outcome in the knot distribution defines the protein’s dominant topology and characterises its global entanglement type, i.e., the topology of the entire curve. The position and the relative size of the knotted portion of the protein can be identified by the so-called knot fingerprints [33].

Recently, the introduction of new mathematical objects called knotoids inspired a more refined approach [34] aimed at detecting geometric and topological features of entangled open arcs subtler than their global knot type [2]. With this approach, the topology of a knotted protein is described by the distribution of its planar projections’ topological types [24,35], as shown in Figure 1.

A knotoid diagram is a possibly self-intersecting open curve in 2-dimensional (2D) space together with crossing information, i.e., the data on which portion of the curve is on top at each self-intersection. Two knotoid diagrams are considered equivalent if they differ by local transformations called Reidemeister moves, performed away from the endpoints [23]. A knotoid is then defined as an equivalence class of knotoid diagrams under this relation. Transformations that displace the endpoints of the diagram over or under other arcs can change the knotoid type, and any knotoid diagram can be unknotted by performing finitely many such moves. A knotoid can be interpreted as an open curve in 3D space with the additional data given by a fixed direction of projection [23,36].

Each planar projection of a curve in space yields a knotoid diagram. The shape of an open curve in 3D can thus be summarised by assigning the knotoid type of the 2D projection in each direction, as described in Figure 1A,B. This procedure assigns a knotoid distribution to an open curve in 3D, with the distribution given by the proportion of directions for which each knotoid is observed [24,34,35]. Crucially, the knotoid approach provides a richer representation of a protein’s entanglement than any knot closure technique [2]. Note that the knotoid distribution exhibits a natural antipodal symmetry.

The existence of open ends in a knotoid diagram enables another class of moves, whereby the endpoints pass under or over portions of the curve and potentially change the knotoid type, known as forbidden moves [37] (see Figure 1D). Forbidden moves allow every self-intersection of a knotoid diagram to be resolved and thus to untangle the diagram, in the same way that a tangled open curve in 3D can always be untangled without being cut. Counting the minimal number of forbidden moves required to pass between knotoid types quantifies the difference in their entanglement and defines a distance, $d_f$, known as the f-distance [37]. Similarly, the minimal number of forbidden moves needed to untangle a knotoid measures how complex the knotoid is. This notion of complexity can be encoded in the knotoid distributions to effectively differentiate between open curves sharing the same underlying global topology, i.e., with the same dominant knot type [37].

2.2. KnotoEMD: A Topological Distance to Distinguish Geometric Features of Knotted Proteins

Based on the above, it is reasonable to assume that knotoid distributions and pairwise f-distances of intra-distribution knotoids may reveal key information about the shape of knotted proteins. Motivated by this, we introduce KnotoEMD, a distance between the knotoid distributions of open curves in 3D. This distance incorporates both the topology (i.e., the global knot type) and geometry (i.e., local features and shape) of open curves. Given two such curves, KnotoEMD is defined as the earth mover’s distance [25] between their knotoid distributions, with cost matrix given by the f-distance. Our distance quantifies the cost of converting one knotoid distribution into another, where the cost per unit of converting one knotoid type to another is the f-distance between them. Intuitively, this captures the amount of untangling and tangling required to pass between knotoid distributions, and thus offers a way to meaningfully compare the entanglement of proteins with the same global topology, which was not possible with previous methods [2,24,29,31,32,33,38].

The KnotoEMD approach for knotted proteins incorporates the global topology of an open-ended conformation as well as its local geometric features and overall shape. For example, the presence of nugatory crossings—crossings that can be removed just by rotating a part of the curve—in a conformation will not dramatically change its global topology, as this is averaged out over a large number of projections. However, the knotoid distribution will likely differ from the same conformation but with the nugatory crossings removed. For this reason, we can consider that such geometric features of the conformation act as a confounding variable of our pipeline. Two curves are considered close by KnotoEMD if the lists of knotoids in their respective knotoid distributions are almost overlapping and with similar individual weights. For example, two curves differing only by local changes away from the endpoints, such as the ones in Figure 2C, are very close to each other according to KnotoEMD. At the same time, two sufficiently different embeddings of the same curve might be far from each other: a minimal realisation of an open trefoil, for example, would have a large KnotoEMD from any open trefoil with higher average crossing number.

2.3. Folding Hypotheses for Knotted Proteins

Results over the past decade arising from theory, simulations and experiments [3,6,7,8,9,10,11,12] reveal at least four potential folding mechanisms for trefoil proteins [20]. All of these pathways rely on the formation of a single twisted loop at one stage of the folding process, as shown in Figure 2A. They differ by the occurrence of N/C-terminus threading [13,14,15], loop-flipping (also known as mouse-trapping) [11,16,17] and by the presence of ribosomes and chaperones [12,18,19]. A detailed description of various folding mechanisms is beyond the scope of this paper. We point the interested reader to a review by Sułkowska and the references contained therein [20]. The single loop formation described by these folding mechanisms predicts a folded geometry resembling a somehow minimal open trefoil having only one pierced loop, as shown in Figure 2, referred to henceforth as single loop (1-L) trefoil configurations. In reality, the geometry of all proteins is far more complex than these idealised representations; however, the smoothed, coarse geometric structure of a trefoil protein should resemble the minimal, single-looped geometry proposed by the hypotheses, if the protein folds according to one of these pathways.

Flapan et al. [21] introduced a new theoretical model for the folding of knotted proteins based on loop flipping and numerical simulations of folding of complex knots [22]. A characteristic feature of this model is the formation of two initial loops, as opposed to one (see Figure 2B). Similarly to above, the formation of two loops predicted by this model should result in creating more complex, double-looped trefoil geometries. Knotting pathways in this model depend on the choice of several parameters, which regulate the number of twists in the loops and their relative positions. Twelve possible choices of parameters were identified, each yielding a different double-loop open positive trefoil (2-L) configuration. The twelve 2-L configurations are labelled by a two-letter word in $\{L,R\}$ followed by a ± and two (signed) integers, describing the relative positions of the loops and the number of twists [21]—a list of these names can be seen in Appendix B. Out of these twelve 2-L configurations, the least topologically complex are proposed as the most likely to occur as native states of trefoil proteins, while the more complex ones are discarded [21] based on an approximation to the knot fingerprint. We observe that the twelve proposed 2-L configurations can be divided into three groups, where configurations within each group are related by local transformations, as illustrated in Figure 2C. For more details, the reader is referred to Appendix B.

2.4. Methods

The folding pathways described in the previous section predict two different overarching trefoil geometries as an end result. Thus, we seek to differentiate trefoil proteins via their geometric entanglement in order to infer information about their respective folding pathways. As we are working with proteins in their native state deposited in the PDB, we work under the premise that different folding pathways leave different geometric artefacts in the knot cores, such as a single or double loop in the proteins’ native conformations. With this aim, we generate suitable curves of different lengths for the 1-L and twelve 2-L configurations predicted by the two folding models, and then we apply numerous rounds of numerical perturbations to produce several hundred piecewise-linear open curves resembling the end-states of each of the pathways described by the models to compare with positive trefoil proteins. Our final data set consists of all the positive trefoil (K$+3_1$) proteins taken from KnotProt [2] and a collection of generated open curves for the 1-L and 2-L configurations (see Figure 2D,E). More details on our data set can be found in Appendix A and Appendix B. We then use the knotoid approach for each curve in the data set and compute its knotoid distribution. The knotoid distribution of each curve analysed in this work was computed using Knoto-ID [35]. In practice, 5000 projections are computed for each curve, and this is known to be sufficient to properly approximate the continuous distribution [37]. Subsequently, we compute all pairwise KnotoEMD distances between distributions in the data set. All of these computed pairwise distances are contained in distance matrices that can be found in Appendix C; for data visualisation purposes, we display our results using the dimensionality reduction technique, UMAP [40], projecting the space of proteins and/or curves with distance given by KnotoEMD to a plane, while preserving the KnotoEMD distances as much as possible.

Our results suggest that the carbonic anhydrase 4QEF is entangled in a way similar to a 2-L configuration, so we selected it as a candidate protein for simulations to further probe its folding pathway. The coarse-grained simulations were performed in Gromacs 5 [41] using a $C\alpha$ structure-based model [42] with parameters proposed by the SMOG server [43] and shadow algorithm [44]. The native structure taken from PDB was first unfolded at a high temperature to obtain the starting frames for folding. Next, 200 folding trajectories with 300–500 millions of frames each were performed close to the folding temperature, where there is an equilibrium between folded and unfolded structures.

3. Results

We find that KnotoEMD is capable of detecting subtle geometric information from the knotoid distribution. This is reflected by the fact that it is able to group together trefoil proteins by sequence similarity. We differentiate between double-loop and single-loop open trefoils via this geometric and topological information, and detect fine structure in the space of trefoil proteins. The local geometric features detected suggest different folding pathways for trefoil proteins. In particular, we find that the proposed double-loop mechanism is the most likely for carbonic anhydrases forming a shallow trefoil.

3.1. Sequence Similarity from the Geometry of Proteins’ Native States

KnotoEMD is sensitive to the geometry and topology of the entire open curve, and is therefore influenced by the depth of the knot core, locations of the termini, and the structure and entanglement of its core. These features are often shared by proteins with similar sequences, and these proteins are therefore seen as similar by KnotoEMD, as shown in Figure 3B. To show this, we cluster all trefoil proteins by first computing the pairwise sequence similarity scores, using PDB, and then performing single-linkage hierarchical clustering and pruning clusters at a threshold of 30% similarity. More details on our subdivision can be found in Appendix A, and a dendrogram showing the clustering by sequence similarity is shown in Figure 3A.

The structural features of the dendrogram are reflected almost perfectly by KnotoEMD, for example, the proteins represented by 4QEF appear to form two separate groups in the UMAP [40] projection, coherently with the fact that two light blue clusters are merged quite late in the dendrogram to form the final one. Similarly, tight clusters in the dendrogram, such as the one represented by 6QQW, are compact in the UMAP projection. Moreover, we are able to detect a range of geometries and to efficiently separate trefoil proteins by their geometric features. Indeed, by design, KnotoEMD takes into account both the dominant topological type of a protein and its overall structure; thus, it is able to detect features of open curves not captured by previously defined methods [4,32,33,45] used to compare them. In particular, it is able to differentiate between proteins with the same dominant topology and at the same time distinguish pairs of homologous proteins with almost superimposable structures differing only by a strand passage, as in the case of the pair knotted AOTCases/unknotted OTCases [4] (see Appendix C). Within this study, we investigate the geometry and topology of protein structures and compare their entanglement to knotting geometries predicted by the hypothesised knotting pathways. As there is no restriction on the length of the two ends in the knotting hypotheses we consider, the pathways really only describe the entanglement of the core. In order to make the conformations proposed by the hypotheses comparable with the entangled proteins, we clip the ends of protein backbones up to their entangled core using a top-down approach [35]. When reducing the chain to the knotted core, the sequence similarity clustering is maintained, although it is slightly noisier in this case. The complete results on pairwise distances between protein cores can be found in Appendix C.

3.2. KnotoEMD Captures Subtle Geometric Differences between Double-Loop and Single-Loop Open Trefoils

In a recent study, Piejko et al. highlight different folding behaviours for deeply knotted and shallow proteins [26]. It is then natural to ask whether there are geometric features of trefoil proteins, other than depth, that could be used to infer further information about their knotting mechanisms.

The knotoid distribution can vary substantially for different geometric realisations of an open-ended trefoil. We quantify such differences in the knotoid distributions by computing the pairwise KnotoEMD between the cores of our 1-L and 2-L trajectories, and of trefoil proteins. As shown in Figure 4, the three groups of 2-L trajectories and the group of 1-L trajectories appear clearly separated, while 2-L trajectories within each of the three groups are close to each other. This demonstrates that KnotoEMD successfully captures subtle geometric structure, as configurations within the same group only differ by simple local moves (see Figure 2C), which only slightly impact the knotoid distribution and do not substantially change the overall geometry of the core.

3.3. Local Geometric Features Suggest Different Folding Pathways for Trefoil Proteins

When analysing pairwise distances between protein and configuration cores, each trefoil protein core appears to have a knotoid distribution highly similar to either those of 1-L trajectories or of simple 2-L trajectories. This is particularly evident when looking at the distances between our generated trajectories and the list of non-redundant trefoil proteins taken from KnotProt [2], as visualised using a UMAP [40] projection in Figure 4. Our results are coherent with the fact that complex 2-L configurations exhibit knot fingerprints that are not compatible with those of trefoil proteins [21]. Indeed, trefoil proteins are distributed unevenly between the two simple groups (1-L and simple 2-L trajectories), with the vast majority having a knotted core with topological complexity indistinguishable from that of simple open-ended trefoil cores, i.e., from the 1-L trajectories. In particular, all of the non-redundant deeply knotted proteins we considered share this feature. Again, this is coherent with previously simulated folding pathways for deep trefoil proteins [6,20,26], all involving the creation of a single twisted loop. This single loop can form either in a native state, as for the threading mechanisms, or unfolded, as in the loop flipping hypothesis. We emphasise that our computations demonstrate that the presence of two pierced loops in an open trefoil inevitably results in increased complexity of its knotoid distribution, making it substantially different from that of a simple trefoil; indeed, all three double-loop clusters appear clearly isolated in Figure 4. Therefore, our results provide strong evidence that the double loop folding mechanisms [21] can be excluded for deeply knotted trefoil proteins.

The same is true for the shallow protein 1FUG (see Figure 4), whose knot core appears very simple despite being considerably longer than the ones of deep proteins (~256 amino acids), and even longer than many shallow ones. For these proteins, the cores’ low topological complexity is shown in Figure 4, where the single-loop structure is clearly visible.

Note that there are two separate clusters of 1-L trajectories, each containing some proteins. For the 1-L trajectories, this stems from the fact that perturbing might have resulted in an over-simplification of the curve. Similarly, small errors in the knot core reduction of proteins with particularly simple core might give rise to an almost trivial knotoid distribution. In both cases, this happens to curves whose actual core is very simple, so this does not affect the interpretation of the results.

There is a consensus that multiple folding pathways are possible for shallow proteins [26,27,47], and this is reflected by our results. We find that, contrary to the aforementioned 1FUG which clusters with the 1-L trajectories, all carbonic anhydrase representatives, a group of proteins with very shallow trefoil knots, are clustered with simple 2-L trajectories. Therefore, their knotoid distributions have a complexity coherent with the double-looped trefoil geometries. This suggests that a single-loop formation followed by either threading a terminus or flipping a loop over a terminus are unlikely folding mechanisms for carbonic anhydrases. A pathway requiring double loop formation, such as those described by Flapan et al. [21] is the most likely among the proposed ones, as KnotoEMD shows that the knot core is tangled in a way most similar to a 2-L configuration.

To test this hypothesis, we performed coarse-grained simulations of folding for the carbonic anhydrase with PDB code 4QEF. This protein features a very interesting ‘spine’ of $\beta$-sheets, which joins together different parts of the protein. In particular, one can identify a small fragment of the chain extruding from the $\beta$-sheet core, which can be interpreted as one loop, shown in green in Figure 5, as well as a long fragment surrounding the entire $\beta$-core, which could be interpreted as the second loop, shown in red in Figure 5. The C-terminal tail protrudes from the inner loop and pierces the outer loop, eventually forming the knot. The folding of this protein starts from the formation of the $\beta$-sheet core and hence, the two loops are formed at the initial stage of the folding process. The $\beta$-sheet can be formed starting either from the formation of the outer red loop, or the internal green loop, as shown in Figure 5. Once both loops are formed, the tails are then placed and the simulations suggest the existence of two different pathways, depending on which tail is placed first. Similar behaviour was observed in the case of the $5_2$-knotted UCH-L [48], which was used as a example of a protein following a double-loop folding mechanism by Flapan et al. [21]. This mechanism, although not evidently confirming the mixed threading/mouse-trap movement event, is more complex than a simple threading mechanism observed for deeply knotted proteins.

The difficulties in unequivocal confirmation of the double-loop folding mechanism may be due to the fact that the mechanism detected is the result of some evolutionary optimisation. From a thermodynamic perspective, the folding speed is limited by the free energy barrier stemming from the knot formation. In particular, threading a free, dangling tail through an already formed loop is associated with a significant decrease of entropy, which is overcome by the concomitant formation of native contacts. This entropy–enthalpy interplay slows down the folding process, and would occur twice in a ‘pure’ 2-L folding mechanism (i.e., exactly like the one in Figure 2B), as there are two loops which have to be threaded. Therefore, from a kinetic point of view, such a mechanism could be optimised to reduce the threading burden. One obvious optimisation would be to make the folding similar to a single-threading mechanism, as noted also in [21]. In such a case, the observed double loop mechanism would be less evident, leaving only traces in the folding process or in the native structure. Our results indicate that these traces can be detected by KnotoEMD, giving material for follow-up studies, e.g., with mutation-based analysis.

4. Conclusions

Understanding how and why there are knots in proteins has been a challenging question in biology for the last few decades. One of the necessary steps to solving this question is to efficiently distinguish and characterise different types of entanglement. The knotoid approach makes significant progress towards characterising the entanglement of knotted proteins as the knotoid distribution gives a topological and geometric description of an open-ended curve, richer than any closure method can give [2]. However, until now, there has been no systematic way to extract information from this distribution about specific geometric features of a curve, with the exception of very recent work by a subgroup of the authors [37], in which the topological complexity given by the f-distance is used to separate deeply knotted trefoil proteins from shallow ones.

KnotoEMD provides a sophisticated tool able to precisely and efficiently compare the entanglement of open knotted curves. Remarkably, with KnotoEMD we are able to easily detect local geometric information within curves sharing the same global topology, which other methods have struggled to do. Importantly, we show that this geometric information can be used to reveal different folding pathways of trefoil proteins directly from their native states. Our methods are, of course, not restricted to studying folding hypotheses of proteins, as the topological and geometric information can be applied to other problems in protein science. In fact, they can be applied more generally to the study of entangled open polymers, such as DNA or RNA.