Figure 1

Motion and tightening of a knot on a homopolymer (first panel) and on protein 1j85 (second panel) during stretching with constant velocity. Squares and circles indicate positions of the ends of the knot along the chain. Knots typically slide off homopolymeric chains. Here, however, we have chosen a less frequent example in which a knot tightens close to one end of the chain and may remain immobile in the absence of an adequate thermal activation [17, 22, 23]. In contrast, knots in proteins always tighten in a specific position inside its initial configuration, after a series of jumps. Each jump corresponds to a definite force peak in the force-displacement curve shown in the third panel. Reducing the pulling speed by an order of magnitude makes only small shifts in the curves shown in all panels. A video pertaining to knot tightening is available in the supplementary material [24].Reuse & Permissions

Figure 2

The ends of a knot in the 1j85 protein in the native state are located at amino acids $n_1=75$ and $n_2=119$. In a tightened configuration, the ends of the knot are located between $n_1$ and $n_2$, with one end either in a sharp turn or at the end of a helix. The arrows indicate these characteristic places. The numbers show percentages of situations (based on 700 trajectories) in which a knot’s end is pinned at the feature after moving from the native state. The innermost features correspond to the tightest knot.Reuse & Permissions

Figure 3

The preferred final locations of a knot’s ends in the 2etl protein found in 700 trajectories. The darker peaks indicate the most likely outcome: $n_1\sim 111$ (the end of a helix) and $n_2\sim 126$ (a turn). The corresponding tightened knot conformation is shown on the bottom right. The relevant sequential segment is shown on the top right in the native conformation where the arrows indicate the values of $n_i$. The less probable outcomes are shown by the lighter peaks. Here $n_1\sim 101$ or 106, whereas $n_2\sim 119$.Reuse & Permissions

Figure 4

After terminating the pulling process at $d=500\AA$ (indicated by the arrow) 1j85 returns to its native state in most cases. Sometimes, it ends up in a metastable state (such as shown on the right).Reuse & Permissions

Figure 5

Distribution of waiting distances $\delta d$ of the left end of a knot in protein 1j85 at various temperatures $T$. Pathways 1 and 2 are indicated by symbols in circles. The panels are explained in the text.Reuse & Permissions