Table 1.
Protein structures investigated in this work.
Lseq denotes the number of residues in the structure, “cont.” the number of native contacts, Ntail and Ctail the length of N- and C-terminal 52 knot tails. The knotted core of the innermost 31 knot inside 52 knot is also shown. PDB code 4WLR denotes protein with the deepest 52 knot in the UCH family. More detailed comparison incl the sequence similarity is given in SI.
Fig 1.
Molecular structures and a topological fingerprint for Ubiquitin C-terminal Hydrolases, UCH.
A: The topological fingerprint—presented as a matrix—for UCH shows that this protein forms the Gordian knot (52) as a whole but some of its subchains form two trefoil slipknots (described in detail in the text). B: Ribbon representation of backbone superposition of UCH-L1 (PDB code 2LEN and 3IRT) and UCH-L5 (PDB code 4I6N), where red and blue color indicate the N- and C-terminus, respectively. C: Model of the chaperonin cage, a cylindrical box (closed from both sides, with height equal to the diameter), which encapsulates the UCH-L5 protein.
Fig 2.
Schematic representation of possible folding pathways of UCHs.
The folding starts with a collapse, next, depending on which terminus becomes structured first, two topologically distinct pathways are possible. Lower arrows: the pathway where N-terminus is structured after C-terminus (FN), protein self-ties in one step (direct 01 → 52 transition). The upper arrows: pathway where C-terminus is structured last (FC)—the protein follows 01 → 31 → 52 pathway, where in the 31 knotted intermediate the N-terminus is in its native position, while C-terminus is not. Different background denotes different topology.
Table 2.
Probabilities of (un)folding pathways in different conditions.
and
denote the folding probability of 3IRT via pathways FN and FC at different temperatures T. Pmisfold denotes the probability of misfolded structures during the folding of 3IRT.
and
denote the unfolding probabilities of 3IRT via pathways UN and UC.
Fig 3.
Plots of knotting probability as a function of Q during folding of UCH-L1 protein (PDB code 3IRT) in different conditions and temperatures.
A: The total knotting probability K(Q)T (all possible type of knots) in the confinement. B: The knotting probability of a 31 knot () in the confinement. C: The knotting probability of 52 knot (
) in the confinement. D: The comparison of K(Q),
and
in bulk and the confinement at T = 110.
Fig 4.
Misfolded structures obtained in UCH-L1 folding.
Left: Changing the crossings in 52 knot results either in 31 or 01 knot (the crossings changed are marked with a circle). Middle: Comparison of the native structure of UCH-L1 (rainbow) with its misfolded structure (sandy)—upper panel—31, bottom—01 knot. Encircled is the crucial change in topology. Right: Enlargement of the topology-changing fragment. Note the different arrangement of sandy part of the chain.
Fig 5.
Simulated chevron plot for UCH-L1.
A: Representative mean Q as the function of time (red) and smoothed curve (green). B: Chevron plot obtained for bulk. C: Chevron plot obtained for the confinement. B and C comes from fitting od sum of exponentials to plot in A. D: comparison of one slow and one fast phase for bulk (red) and confinement (green). The dashed lines present the expected chevron plot. Fitting the equation describing chevron plot results in an approximate Tf equal 114 (−ϵ/kbT = −1.05) for bulk and 120 (−ϵ/kbT = −0.99) for the confinement. In B, C, D the fitting error bars higher than 5% of value were shown.
Fig 6.
Random knots observed in the unfolded basin of UCH-L1.
A: Schematic drawings of the type of observed random knots. B: Autocorrelation time in the unfolded basin as a function of temperature T. C: Random knot probability (for details see main text). D: preferred locations of a 31 and E: 41 knot’s termini. In the plots, data for the confinement (green) and bulk (red).
Fig 7.
Short-lived knots during folding and unfolding.
A: Exemplary unfolding trajectory with the topology indicated by color stripes. B: The retying probability for structures differing in knot tail length. C: The probability of formation of short-lived knots on folding/unfolding pathway. For the confinement at T = 120 bottom dot denotes short-lived knots on folding, top—unfolding pathway.