Possible applications

A lot of information can be deduced from the results of trajectory analysis. For example, the results of the “fast analysis” (existence of a knot at each time step) can be used to characterize probability of knot occurrence in various reaction coordinates, such as native contacts (Q, number of contact existing in protein in native state) or RMSD. As an example, the figure below shows a plot of probability of knot occurrence P(K)=0.5 (black line) superimposed on free energy F(Q,RMSD) (right panel) for a knotted protein 2ouf (constructed in [4]) shown in the middle (figures from the paper [1]).

Fig 1. Schematic representation (left) of a knot in 2ouf protein (middle). Panel on the right presents free energy landscape of this protein measured by the fraction of native contacts formed Q and RMSD for 2ouf; black curve represents a probability of finding a knot P(K)=0,5.

As another application, the results of detailed analysis can be used to analyze knots and slipknots from the viewpoint of thermodynamics processes, as discussed in [1]. Based on such analysis, figures like those below (from [1]) can be plot. These figures show knotting, folding, and unfolding events observed for 2ouf. Trajectory insets show (top left) random knotting by the C-terminal, (top middle) folding via knotting at low Q (unique to 2ouf), and (top right) unknotting and unfolding by unplugging the N-terminal. Corresponding protein configurations are shown next to the insets. In the bottom inset, the topological signature of the protein measured by the position of the knot along the sequence is shown, with knot termini denoted by blue and red dots.

Fig 2. Knotting, folding, and unfolding events observed for 2ouf.

Yet another application amounts to describing the evolution of a knot when a protein is stretched by AFM or optical tweezers, as discussed in [2], and shown in figure below. This figure (from [2]) shows unraveling of the protein (left panel) at constant speed (found based on structure based model). The upper part presents unfolding curves of force versus pulling spring F(d). The lower part of the figure presents sequential movement of knot’s ends during the knot-tightening process corresponding to the unfolding curve shown above.

Fig 3. Unfolding curves of force versus pulling spring F(d) and corresponding sequential movement of knot’s ends during the knot-tightening process.

Finally, the trajectory analysis can be used to find optimal direction of pulling the protein to untie it. Behavior of protein configuration pulled in different directions is shown in figure below (from [3]), where, first of all, attachments points (p2, p1) are shown that may lead to an unknotted configuration at various temperatures (denoted by colors, as indicated on the scale above). Below on the right, a distribution of the knot end locations, (k2,k1) is shown. The color scale indicates the probability of attaining particular values of k2 and k1. The framed part in the right panel indicates the region covered by the left panel.

Fig 4. Stretching protein in various directions by pulling different pairs of aminoacids.


[1] Sulkowska JI, Noel JK, Onuchic JN (2012) Energy landscape of knotted protein folding. Proc. Nat. Acad. Sci. USA 109(44):17783-8.
[2] Sulkowska JI, Sulkowski P, Szymczak P, Cieplak M (2008) Stabilizing effect of knots on proteins. Proc. Nat. Acad. Sci. USA 105(50):19714-9.
[3] Sulkowska JI, Sulkowski P, Szymczak P, Cieplak M (2010) Untying knots in proteins. J. Am. Chem. Soc. 132(40):13954-6.
[4] King NP, Jacobitz AW, Sawaya MR, Goldschmidt L, Yeates TO (2010) Structure and folding of a designed knotted protein. Proc Natl Acad Sci USA 107:20732–20737.


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