The selected 31 trajectories were analyzed by monitoring the evolution of several geometrical and topological parameters during the folding process.

As a first step we identified the folding stage at which the backbone self-ties into knot. Accordingly, for each trajectory we calculated the percentage of native contacts (overlap) that are formed when the first knotting event occurs. The distribution of these overlaps for the considered trajectories is shown in Fig. 2. The distribution is peaked at about 90% overlap. This indicates that the knot is typically formed at a rather late stage of the folding process.

Next, to characterize the diversity of the folding pathways and the implications for the knotting mechanism, we computed the average “path similarity parameter”, . As explained in the Materials and Methods section, this quantity measures the consistency of the temporal succession in which the native contacts are formed in two given pathways. The parameter takes on values ranging from 0, for no similarity, to 1 when all native contacts form with exactly the same succession in the two trajectories. We emphasize that depends only on the time order of native contact formation events (and not their exact timing).

To have a robust indication of the degree of heterogeneity of the selected trajectories, we computed the distribution of over all possible pairs of trajectories, see Fig. 3. As a term of reference, the same Figure shows the distribution computed over previously-studied folding trajectories of the unknotted WW domain FIP35 [16]. It is seen that the distribution of MJ0366 is narrower and shifted towards significantly higher values of than for the unknotted protein. Indeed the former has a peak at while the latter has it at . This relatively low value of and the distribution broadness is typical of folding processes that proceed by multiple pathways [16], [24], as FIP35 is known to do. The different characteristics of the distribution for MJ0366 therefore strongly suggest the existence of one dominant folding pathway for MJ0366.

We accordingly sought to analyze in detail the folding process to verify that knotting occurs via one dominant mechanism and characterize it.

In this regard a valuable term of reference is given by the earlier study of Noel et al. [12] where the folding thermodynamics of MJ0366 was systematically characterised with both atomistic and coarse-grained native-centric models. When the atomistic model was employed, it was seen that knotting preferentially occurred via slipknotting. Specifically, in most of the productive trajectories obtained at the folding temperature of the structure-based model, the C-terminal attained a hairpin-bent conformation and established the knot by threading the open region involving residues 17–54. The slipknotting mechanism was found to occur more frequently than that of other knotting modes, such as the threading of the open region by a non-bent C-terminus, or knot formation at the N terminus. Interestingly, the coarse-grained native-centric model was more prone to unproductive kinetic traps and displayed significant heterogeneity for knotting mechanisms too. These aspects indicated that the realistic treatment of protein structural detailed significantly helped reduce the impact of unproductive routes in the folding process [12].

Here, by addressing the same protein folding process with a realistic, non native-centric force-field, it is possible to examine to what extent various aspects of the knotting process are sensitive to the treatment of inter-atomic interactions.

As a first step of the analysis, we profiled the folding trajectories along two relevant order parameters: the root mean square distance (RMSD) to the native structure and the RMSD to the native -sheet. The first collective variable monitors the overall progress towards the native geometry. The second one, instead, carries information about one of the expected entropic bottlenecks of the folding process, namely the formation of the native antiparallel -sheet which involves amino acid pairs with a sequence separation as large as 38.

Since in the native MJ0366 structure the C-terminal helix protrudes through the region intervening between the two paired -strands, monitoring the formation of the -sheet is relevant to understand whether the sheet is formed before or after the knot.

The results shown in the left panel of Fig. 4 indicate that the -sheet is fully formed rather early, when the total RMSD to native of the chain is about 15 Å. At this stage the fraction of formed native contacts is about 40–50. The self-tying of the molecule into a trefoil knot typically occurs after the formation of the -sheet. This is evident from the placement of the diamond symbols in Fig. 4 which mark the first occurrence of knots for each of the 31 trajectories. It is seen that all first-knotting events occur when the -sheet is fully formed, with only two exceptions that will be discussed later.

The detailed inspection of the trajectories indicates that the knotting process almost invariably occurs through the so-called “threading” mechanism, where the -terminal -helix (residues 74–87) directly enters, without bending, the open region between amino acids 17–54 involving helices and and the intervening loop, see the sketch in the left panel of Fig. 5. In this case, the threaded region and the -sheet (respectively shown in blue and red in Fig. 1) establish a tertiary contact before the terminal helix penetrates into the open region in between the helices and (see left panel in Fig. 5). This mechanism accounts for as many as 26 of the 31 rMD trajectories.

In three other cases, the folding was found to occur through the so-called “slipknot” mechanism [12] (i.e. where the open region is entered by the backward-bent C-terminus). In all three instances the terminus entered the loose – region producing a shallow slipknotted trefoil, as shown in the central panel of Fig. 5.

Finally, in two further cases we observed another knotting mechanism which involves a concerted backbone movement that had not been previously reported for MJ0366. Specifically, in two trajectories when the -sheet and the terminal -helix are already formed and juxtaposed in an unknotted configuration the loop performs a “mousetrap-like” movement establishing the native knotted topology. This movement, which bears some analogies with the suggested knotting mechanism for an unrelated protein with a non-trefoil topology [25], is schematically represented in the right panel Fig. 5. The mousetrap knotting events correspond to the two outlying diamonds reported in Fig. 4, with collective coordinates (6 Å, 8 Å) and (12 Å, 10 Å).

Videos obtained from the atomistic DRP trajectories which illustrate the three observed knotting mechanisms are included in the on-line SI.

It is important to notice that the trajectories associated to the various knotting modes do not present significant quantitative differences regarding the overall solvent accessibility of polar and non-polar residues during the folding process. This point is illustrated in Fig. 6 where the number of exposed hydrophobic and hydrophilic residues are profiled versus the RMSD to the native state. The consistency of the various profiles provides a quantitative basis for expecting that the relative weight of the knotting mechanisms should not depend critically on the specific model adopted to describe the solvent-induced interactions.

To understand how the interplay of amino acid interaction captured by the realistic force field favours knotting by threading, we have carried out a comparative analysis of the reaction mechanism in successful and unsuccessful folding trajectories. Specifically, the productive, successful set consisted of the 26 trajectories displaying the dominant (threading) knotting mechanism. The non-productive one included an equal number of trajectories that reached an unknotted configuration and nevertheless had a good native similarity (namely an RMSD to the crystal structure less than 5 Å).

The projection of the unsuccessful trajectories along the two collective coordinates considered before is shown in Fig. 4B. The qualitative difference with respect to the analogous plot for the successful ones (panel A) is striking. In particular, it is seen that in successful trajectories the formation of the sheet involving strands and occurs rather early on and prior to the establishment of the overall tertiary organization of MJ0366. In fact, the total RMSD to native decreases appreciably only after the -sheet is established. By converse, for unsuccessful trajectories, this hierarchy of contacts formation is not observed, and the -sheet formation proceeds in parallel with the acquiring of the overall native structure. One therefore concludes that the early formation of the -sheet provides the most appropriate conditions for knotting by leaving the region delimited by the sheet accessible to threading events.

This conclusion is supported by the detailed inspection of the unsuccessful trajectories, which are exemplified in the sequence of snapshots shown in Fig. 7. As it is visible in this figure, the C-terminal helix threads the correct region between strands and prior to the formation of the -sheet. When the latter is finally establishes, the -terminal segment remains trapped on the wrong side of the loop bridging and and, for steric reasons cannot go past it and attain the native knotted topology.

The relevance of this mechanism for misfolding is highlighted by the fact that all unsuccessful trajectories displayed a late formation of the -sheet. We emphasize again that, according to our simulations, the correct knotting of the chain is not promoted by the formation specific contacts which fail to form in misfolding events. Rather, for the chain to acquire the native topology, it is essential that the native contacts form in the correct order.

The fact that the observed dominant knotting mode differs from the one reported previously using pure native-centric force fields suggests that non-native interactions could be relevant for favouring the correct succession of contacts leading to self-knotting (or avoiding unproductive ones). This possibility is particularly interesting in connection with the ongoing discussion about the role that non-native interactions can have in aiding the knotting process even during the early folding [13], [14].

To investigate this aspect we generated several folding trajectories for MJ0366 using simplified models where the effect of non-native interactions could be easily turned on or off. Specifically, we considered two different coarse-grained models: one with only native-centric interactions and the other additionally incorporating non-native interactions. The latter included quasi-chemical and screened electrostatic pairwise interactions, as in the recent study of the early folding stages of a trefoil-knotted carbamoyltransferases [14].

The folding process presents major differences in the two models. First, they differ significantly in terms of knotting probability. Specifically, for each model we considered an extensive set of 10,000 uncorrelated configurations, equilibrated at the nominal temperature of 300 K. In the native-only case, 12% of the sampled configurations were knotted, while this number had a sixfold increased, to 75%, in presence of non-native interactions. This result aptly complements the atomistic DRP simulations, for highlighting the role of non-native interactions in aiding the formation of the native knotted topology of MJ0366.

Secondly, productive trajectories follow different dominant mechanisms in the two models. In fact, when the pure native-centric model is used, 8 out of the 10 trajectories involved the slipknotting mechanism, while the threading one was observed in all trajectories (10 out of 10) with the additional non-native interactions. The latter result, which is in full accord with the atomistic DRP simulations, reinforces the concept that non-native interaction can promote the correct order of contact formation required for self-knotting.

This point is further supported by the inspection of the density plots in Fig. 8. In fact, non-native interactions are more clearly associated to the early formation of the -sheet than for the native-only case. Furthermore, the path outlined in panel B bears more analogies than the one in panel a with the density plot of Fig. 4A, which captured the successful folding events obtained from atomistic DRP simulations. Indeed, in the simplified model, the early formation of the sheet is promoted by the fact that the non-native quasi-chemical interaction generates an overall attractive interaction between the residues in and those in . Consistently with the misfolding events discussed previously, one can therefore argue that the weaker drive of the native-centric model to form early on the -sheet, is also responsible for its lower knotting propensity.

Based on these results, we can argue that mutations in the sheet regions with residues characterized by a weaker effective attraction, would delay the formation of the -hairpin in the folding process and would make the chain more prone to reach the unknotted mis-folded state. This prediction may be verified experimentally.

In conclusion, the DRP simulations presented here provided the first systematic attempt to characterize the folding process of a natively-knotted protein, MJ0366, using a realistic atomistic force field. MJ0366 knotting is observed to occur via threading at the C-terminal. The comparison of productive and unproductive trajectories (which respectively end up in natively-knotted and unknotted states) further indicates that knotting is aided by the early formation of the native -sheet. By comparing the MJ0366 knotting propensity and mechanisms in simplified folding models it is argued that non-native interactions are important for aiding knotting by promoting the correct order of contact formation.

While there is no a priori reason to expect that non-native interactions are crucial for guiding the folding process of knotted proteins in general, it is interesting to notice that their important role has been previously suggested for another trefoil-knotted protein carbamoyltransferases [13], [14]. In our view, it would be most interesting to further examine this effect, in future studies on MJ0366 or other proteins either through experiments (e.g. involving mutagenesis) or with more extensive simulations, possibly involving explicit solvent treatment or unbiased dynamics.

In order to generate an ensemble of trial trajectories connecting a given initial configuration to the native state we used the following variant of the rMD algorithm. At each integration step, we evaluated a collective coordinate (CC) which measures the distance of between the instantaneous contact map and the native contact map:(1)with with a distance cutoff of Å. In this equation, is the 3N-dimensional vector in configuration space, and and are the instantaneous and native contact map, respectively. The entries of the contact map C are chosen to interpolate smoothly between 0 and 1, depending on the relative distance of the atoms and :(2)where r₀ = 7.5 Å is a fixed reference distance.

In the rMD algorithm, no bias is applied to the chain when it spontaneously diffuses towards the bottom of the folding funnel, i.e. any time the value of the CC at time is smaller than the minimum value so far. On the other hand, fluctuations which would drive the contact map further from the native one (hence increasing ) are hindered by introducing a biasing force, defined by the time-dependent potential(3)In these equations, kcal/mol is the so-called ratchet constant and is the minimum value assumed by the collective variable along the rMD trajectory, up to time .

In the original formulation of the rMD algorithm [19], the variable is updated only when the system visits a configuration with . With this choice, monotonically decreases during the course of the simulation. In this work, we choose to significantly weaken the effect of the bias by allowing the system to backtrack along the direction defined by the CC. This is done by occasionally updating also when it increases, according to a Metropolis accept/reject criterium. Namely, is updated to if , where is a random number sampled from a uniform distribution and is an artificial “inverse thermal energy”. This modification of the original rMD algorithm is required to escape from kinetic traps. Without it the folding efficiency to the correct topologically non-trivial native state is strongly suppressed. Each trial trajectory consisted of steps of rMD with a nominal integration time step of fs.

The DRP algorithm is used to identify the most probable path in each set of trial rMD trajectories sharing the same boundary conditions. This is done by evaluating the relative probability for each path to be realized in the unbiased over-damped Langevin dynamics:(4)In this equation, the index runs over the different time-step in the trajectory, the index runs over the atoms in the protein, is the Boltzmann's constant and is the diffusion coefficient of the -th atom.

In both rMD and standard high-temperature MD simulations we used the AMBER ff99SB force field [15] in implicit solvent, within the Generalized Born formalism implemented in GROMACS 4.5.2 [26]. In such an approach, the Born radii are calculated according to the Onufriev-Bashford-Case algorithm [27]. The hydrophobic tendency of non-polar residues is taken into account through an interaction term proportional to the solvent-accessible-surface-area (SASA). The solvent-exposed surface of the different atoms is calculated from the Born-radii, according to the approximation developed by Schaefer, Bartels and Karplus in [28].

The CG folding simulations were based on the model developed in Ref.s [29], [30]. In this approach, amino acids are represented by spherical beads centered at the positions. The non-bonded part of the potential energy contains both native and non-native interactions. The former are the same used in the G-type model of Ref. [31], while the latter consist of a quasi-chemical potential, which accounts for the statistical propensity of different amino-acids to form contact and of a Debye-screened electrostatic term. A detailed description of the force field of this model can be found in Ref. [14]. In our previous work, we have shown that such non-native interactions are able to strongly promote the knot formation in natively knotted polypeptides [14]. Folding simulations for protein MJ0366 in this CG model were performed using a MC algorithm described in detail in Ref. [14]. This type of crankshaft-based MC algorithm is commonly employed in polymer physics [32] to study dynamic properties, since it is was shown that they mimic the intrinsic dynamics of a polymer in solution [33] at a much lower computational cost than standard MD simulations [34]. The folding dynamics of CG model with native and non-native interactions was simulated by generating 200 MC trajectories, while the dynamics of the model with only native interactions was investigated by generating 500 MC trajectories. For both CG models, trajectories consist of 1.510⁸ attempted MC moves, corresponding to 1.510⁴ saved frames. MC moves that we have employed were the local crank-shaft and Cartesian moves, whose boldness was chosen such that the acceptance rate was nearly constant and approximately equal to 50. In both cases, we have collected a total of 10 folding transitions, leading to native configurations with the correct knotted topology. In order to compute the frequency of knotted configurations at thermal equilibrium we performed MC simulations which combine local moves and global pivot moves.

The conformations visited during the MC dynamics were analysed to establish their global and local knotted state. The global topological state was established and assigned by computing the Alexander determinants after suitable closure of the whole protein chain into a ring. For each configuration, this entailed 100 alternative closures where each terminus is prolonged far out of the protein along a stochastically chosen direction, and the end of the prolonged segments are closed by an arc “at infinity” (i.e. not intersecting the protein). As in ref. [14], to avoid considering back-turning closures, stochastic exit directions are picked uniformly among those which form an angle of more than 90° with the oriented segment going from each terminus to the C_α at a sequence distance of 10. If the majority of the 100 stochastic closures return non-trivial Alexander determinants, then the whole conformation can be considered as globally knotted. Because protein knotting can occur through slipknot formation [35], the global topology investigation was complemented by a local one. In fact, a slipknot can be detecting by identifying a non-trivially knotted portion of a chain that has a different global topology, in our case the unknotted one. To this purpose, we repeated the above-mentioned statistical closure scheme for all possible subportions of length 20, 30, 40, … of the protein chain so to identify the smallest knotted, or pseudo-knotted, chain portion [36], [37].

To quantitatively measure the folding pathways diversity we implemented the analysis described in Camilloni et al. [19], that will be shortly summarized in the following. A folding mechanism is here considered to be a specific sequence of native contacts formation. Hence, for each path we measured the time of formation of each native contact, as the frame of the trajectory where the contact is first formed. Given as the time of formation of the native contact in the trajectory, we computed for each path the matrix defined as:(5)containing all the information regarding the folding mechanism as defined above. For each pair of pathways it is possible to compute the similarity defined as(6) being the total number of native contacts. The similarity ranges from for completely different mechanisms, to for completely identical mechanisms. Finally, we consider the distribution(7)of the similarity parameter, evaluated from all pairs of the folding pathways.