Effects of knots on protein folding properties

Abstract

This work explores the impact of knots, knot depth and motif of the threading terminus in protein folding properties (kinetics, thermodynamics and mechanism) via extensive Monte Carlo simulations of lattice models. A knotted backbone has no effect on protein thermodynamic stability but it may affect key aspects of folding kinetics. In this regard, we found clear evidence for a functional advantage of knots: knots enhance kinetic stability because a knotted protein unfolds at a distinctively slower rate than its unknotted counterpart. However, an increase in knot deepness does not necessarily lead to more effective changes in folding properties. In this regard, a terminus with a non-trivial conformation (e.g. hairpin) can have a more dramatic effect in enhancing kinetic stability than knot depth. Nevertheless, our results suggest that the probability of the denatured ensemble to keep knotted is higher for proteins with deeper knots, indicating that knot depth plays a role in determining the topology of the denatured state. Refolding simulations starting from denatured knotted conformations show that not every knot is able to nucleate folding and further indicate that the formation of the knotting loop is a key event in the folding of knotted trefoils. They also show that there are specific native contacts within the knotted core that are crucial to keep a native knotting loop in denatured conformations which otherwise have no detectable structure. The study of the knotting mechanism reveals that the threading of the knotting loop generally occurs towards late folding in conformations that exhibit a significant degree of structural consolidation.

Figure 1. Lattice proteins used in this study that have a trefoil knot.

Lattice protein k0 is the reference system that was used to construct the other lattice proteins. The first bead is colored in black and the last bead in grey. The knotted core is located between bead three and bead 22 in k0, which classifies as a shallow knot. In systems k1 and k2 (a) we added one and two beads, respectively, to the first bead to extend the size of the first terminus thus increasing knot depth. The addition of three beads was done in two manners (b). In k3l they form a linear segment and in k3h they are arranged into a hairpin like conformation. In protein kd (c) the size of the first terminus was extended with seven beads and that of the last terminus with eight beads. We have also considered three knotted trefoils where the extended terminus does not establish any native contacts (d). These are used as control systems. Throughout this work we use the color code adopted in these figures to identify each considered lattice system.

Figure 2. The relation between knot depth and thermodynamic stability.

Thermodynamic stability is quantified by the value of the melting temperature T_m (which is the temperature at which the heat capacity peaks). Panels (a) to (e) show a comparison between the peaks of the heat capacity curves of the knotted systems and their unknotted counterparts. Panel (f) reports a direct comparison between the heat capacity curve of the control systems and the reference system k0. Throughout this work squares are used to identify the knotted proteins and circles are used to identify the unknotted ones. The control systems are identified with triangles.

Figure 3. Relation between knot depth, folding rate and unfolding rate.

The folding (unfolding) rate corresponds to the slope of each represented curve. The folding rate is determined at and the unfolding rate slightly above . Panel (a) reports the folding rate of the knotted systems and panel (b) reports the folding rate of their unknotted counterparts. Panel (c) reports the unfolding rate of knotted systems, while panel (c) reports the unfolding rate of their unknotted counterparts. The ratio between the folding rate of the deep knot kd and its unfolded counterpart ud is 0.13, and that between the knot k3h and u3h is only 0.05.

Figure 4. Impact of knot nativeness on folding and unfolding efficiency.

Panel (a) shows two knotted conformations with 8 (k8NC) and 12 (k12NC) native contacts that were sampled from unfolding simulations. Beads colored in yellow have at least two of its native contacts established. Three native contacts in k8NC pertain to the knotted core; k8NC also has an incorrect crossing and this malformed conformation classifies as a topological trap. On the other hand, the k12NC conformation has 7 native contacts established that belong to the knotted core and part of the knotting loop is already in its native position. This conformation classifies as a native knot. The two unknotted conformations shown in the figure, u8NC and u12NC, were also sampled from unfolding simulations and have 8 and 12 native contacts formed. Panel (b) reports the measurement of the folding () and unfolding () rates starting from these conformations. The unfolding rate of k12nc is six orders of magnitude smaller than those of the unknotted ones and four orders of magnitude smaller than that of k8nc. In the protein backbones is knotted as in 8knc, the unfolding rate becomes two orders of magnitude smaller with respect to the unknotted conformations.

Figure 5. Probability of knot formation as a function of the reaction coordinate, Q, the fraction of established native contacts for systems k3h and kd.

Also shown is the curve for the reference system k0.

Figure 6. Insights into the knotting mechanism of k3h from structural clustering.

Each conformation with fraction of native contacts Q is the closest to the cluster's centroid, and is taken as the cluster's representative. The residues colored in yellow have at least two of its native contacts formed. In parenthesis we show the ratio between the size of the cluster (i.e. its number of conformations) and the size of the initial ensemble of conformations with fraction of native contacts Q. In this mechanism knotting occurs in a conformation with fraction of native contacts . The chain terminus closest to the knotted core threads a knotting loop that is already in its native conformation. The hairpin-like terminus only acquires its native conformation after threading.

Figure 7. Insights into the knotting mechanism of kd from structural clustering.

There is an important conformational state with fraction of native contacts that corresponds to a malformed conformation with an incorrect crossing of the threading terminus, which nevertheless has a rather native-like knotting loop. Productive threading requires an enlarged and loosen knotting loop and occurs in conformations with fraction of native contacts .