(A) Crystal structure of TrmD-Tm1570 monomeric protein (Protein Data Bank ID code: 8B1N). The double trefoil-knot, 31#31, is shown in blue for the TrmD domain and in yellow for the Tm1570 domain. The secondary structure representations for TrmD, residues 1-240, and Tm1570, residues 241-433, are represented in the bottom of panel A. The β-sheets and α-helices are represented by rectangles and arrows, respectively. (B) Dimeric crystal structures of TrmD-Tm1570 (PDB ID: 8B1N), TrmD (PDB ID: 8BYH), and Tm1570 (PDB ID: 8RI0).

Schematic representation of the unfolding mechanism of the double-knotted protein TrmD-Tm1570.

The figure shows possible unfolding pathways observed in the structure-based model simulations. Starting from a native double-knotted topology (A), the system undergoes a series of intermediate processes (B, C, D, E, F, and G) before reaching its unfolded state (I), which is represented by a random unknotted/trivial conformation. Orange and blue colors represent the reversible and non-reversible processes, respectively.

The unfolding trajectories and the probabilities of contact formation per residue.

A, B, C, D display the fraction of native contacts, Q, as function of the number of frames. E, F, G, and H show the probability of residue i being in contact with its native contacts as function of Q. Panels A/E, B/F, C/G, and D/H correspond to the unfolding pathways 1, 2, 3, and 4, respectively. Figure 3—figure supplement 1. Unfolding trajectories observed less frequently, rare events. Figure 3—figure supplement 2. Refolding dynamics of a fusion protein, visualized through snapshots at key time steps

Experimental unfolding and refolding of Tm1570, TrmD and the Tm1570-TrmD Fusion using single-molecule optical tweezers.

In each trace the unfolding is colored (orange for Tm1570, dark blue for TrmD and black for the Fusion), while the refolding trace is in grey. The traces were measured at a constant velocity of 20 nms−1 for both the retraction and approach segments (unfolding and refolding). Figure 4—figure supplement 1. Experimental unfolding under constant velocity of 20 and 500 nms−1.

Force versus extension of Tm1570 (A) and TrmD (B).

The original data collected from the simulation are shown in light grey and the smoothed data by moving average algorithm in red. Each peak is represented by an index. Panels C and D show cartoon representations of Tm1570 and TrmD native structures and snapshots from the simulation. The indices (1, 2, 3, and 4) and (1, 2, 3, 4, and 5) correspond to the same indices in A and B, respectively. The knot core is highlighted in yellow and in blue for the Tm1570 and TrmD domains, respectively. Figure 5—figure supplement 1. Clustering of the unfolding trajectories for Tm1570 obtained via SOM analysis. Figure 5—figure supplement 2. Clustering of unfolding trajectories for TrmD obtained via SOM analysis. Figure 5—figure supplement 3. Distribution of simulations for each protein across clusters Figure 5—figure supplement 4. End-to-end cysteine distances, all atoms MD - probability density and free energy landscapes for TrmD, Tm1570, and the TrmD–Tm1570.

(A) Force versus extension of TrmD-Tm1570 protein. The original data from the simulation is shown in light grey and the smoothed data by moving average algorithm in red. Each peak is represented by an index. (B) Cartoon representation of the TrmD-Tm1570 fusion protein’s native structure and snapshots from the simulation. The indices 1, 2, 3, 4, 5, 6, and 7 correspond to those in panel A. The knot core is highlighted in blue and yellow for TrmD and Tm1570, respectively. The knot core length for all conformations is always pictured as encompassing the same number of residues as that in the native structure for clarity. (C) Probability of native contacts - upper and lower triangle based on 1000 conformations before (B) and after (A) of each index, respectively, where red points indicate a high probability of there being a contact and blue points indicate a low probability of there being a contact (see colour map). Figure 6—figure supplement 1. Clustering of unfolding trajectories for Tm1570-TrmD obtained via SOM analysis.

Self-Organizing Map (SOM) representation.

The high-dimensional input space represents the input dataset, fraction of native contacts (Q). Each input vector, xi, is associate with weight vector, wi. The neurons are in a k × ℓ lattice, SOM grid, 2D output space. The SOM utilizes a geometric layout, typically rectangular or hexagonal, where each individual unit within the grid is a neuron.

Unfolding trajectories observed less frequently, rare events.

Panel A – .

Panel B – .

Panel C – .

Panel D – .

Panel E – .

Panel F – .

Panel G – .

Panel H – .

Panel I – .

Refolding dynamics of a fusion protein, visualized through snapshots at key time steps (frames 1, 19000, 38800, 46000, 54800, and 72800).

The knot cores, localized within residues 85-129 (TrmD domain, blue) and 352-397 (Tm1570 domain, yellow), remain invariant during the refolding process.

Experimental unfolding of Tm1570, TrmD and the Tm1570-TrmD Fusion using single-molecule optical tweezers.

In each trace the unfolding is colored (dark blue for TrmD, orange for Tm1570, and black for the TrmD-Tm1570). The traces were measured at a constant velocity of 20 and 50 nms−1.

Clustering of the unfolding trajectories for Tm1570 obtained via SOM analysis.

Each subplot (Clusters 1–16) displays the average unfolding profile (red line) of a cluster, plotted as the fraction of native contacts (Q) versus simulation frames. Grey traces represent individual trajectories within each cluster.

Clustering of unfolding trajectories for TrmD obtained via SOM analysis.

Each subplot (Clusters 1–16) displays the average unfolding profile (red line) of a cluster, plotted as the fraction of native contacts (Q) versus simulation frames. Grey traces represent individual trajectories within each cluster.

Distribution of simulations for each protein across clusters.

Panels A, B, and C show Tm1570, TrmD, and TrmD-Tm1570, respectively. The most populated cluster for each protein is highlighted in red: cluster 1 for Tm1570, cluster 11 for TrmD, and cluster 7 for TrmD-Tm1570.

Probability densities and free energy landscapes for TrmD, Tm1570, and the TrmD–Tm1570 proteins.

Left panels: probability density of end-to-end distances between cysteine residues. Right panels: free energy surfaces as a function of end-to-end distances and RMSDs, with color indicating free energy in units of kT. Probability density and free energy were computed using the package PyEMMA in python version 3.8.8. Details about the MD simulation in Methods and Materials.

Clustering of unfolding trajectories for Tm1570-TrmD obtained via SOM analysis.

Each subplot (Clusters 1–16) displays the average unfolding profile (red line) of a cluster, plotted as the fraction of native contacts (Q) versus simulation frames. Grey traces represent individual trajectories within each cluster.

Thermal Denaturation of TrmD, Tm1570, and TrmD-Tm1570.

A, C, and E - Fluorescence melting curves of TrmD, Tm1570, and TrmD-Tm1570, obtained by monitoring the decrease in fluorescence intensity as the temperature increases. The analysis were conducted under different protein concentrations, from 0 (blank) to 10 μM. The decrease in fluorescence intensity is due to the protein unfolding. B, D, and F - First derivative of the melting curves, highlighting the inflection point corresponding to the melting temperature (Tm). Results shown here are from one replicate.

Melting temperature (Tm) under different protein concentrations (1-10 μM).

TrmD, Tm1570, and TrmD-Tm1570 are shown in pentagon, triangle down, and hexagon markers in colors blue, orange, and black, respectively. The fluorescence melting curves measure by DSF in Figure S1. Appendix 1—figure 1—figure supplement 1. Thermal Denaturation of TrmD, Tm1570, and TrmD-Tm1570.