Polymer uncrossing and knotting in protein folding, and their role in minimal folding pathways

Abstract

We introduce a method for calculating the extent to which chain non-crossing is important in the most efficient, optimal trajectories or pathways for a protein to fold. This involves recording all unphysical crossing events of a ghost chain, and calculating the minimal uncrossing cost that would have been required to avoid such events. A depth-first tree search algorithm is applied to find minimal transformations to fold [Formula: see text], [Formula: see text], [Formula: see text], and knotted proteins. In all cases, the extra uncrossing/non-crossing distance is a small fraction of the total distance travelled by a ghost chain. Different structural classes may be distinguished by the amount of extra uncrossing distance, and the effectiveness of such discrimination is compared with other order parameters. It was seen that non-crossing distance over chain length provided the best discrimination between structural and kinetic classes. The scaling of non-crossing distance with chain length implies an inevitable crossover to entanglement-dominated folding mechanisms for sufficiently long chains. We further quantify the minimal folding pathways by collecting the sequence of uncrossing moves, which generally involve leg, loop, and elbow-like uncrossing moves, and rendering the collection of these moves over the unfolded ensemble as a multiple-transformation "alignment". The consensus minimal pathway is constructed and shown schematically for representative cases of an [Formula: see text], [Formula: see text], and knotted protein. An overlap parameter is defined between pathways; we find that [Formula: see text] proteins have minimal overlap indicating diverse folding pathways, knotted proteins are highly constrained to follow a dominant pathway, and [Formula: see text] proteins are somewhere in between. Thus we have shown how topological chain constraints can induce dominant pathway mechanisms in protein folding.

Figure 1. Approximate minimal transformation for a simple conformation pair, and the degree to which link length changes.

(a) Several intermediate conformations for a transformation (A–G proceeding along the color sequence red, green, yellow, cyan, magenta, gray, and blue) are shown. The step-size delta is shown. Note the step in which the first bead of the chain () is “snapped” into the final conformation because its distance to the destination is less than (going from D to E). In the intermediate conformation F (Gray), beads 0 to 3 have reached their final locations and no longer move. Note also the link length violation of link 4 in conformation F, due to the approximation that ignores end point rotations, for this intermediate figure. A milder violation is observed when going from D (cyan) to E (magenta), since bead 1 through all assume a step size of while bead 0 moved a step size . (b) Panel b shows a surface plot showing link length as a function of link number and step number during transformation. For the whole process, mean link length is 0.98 units and standard deviation is 0.063.

Figure 2. Link length statistics for randomly generated transformation pairs.

Histogram of the average link length over the course of a transformation, for transformations between 200 randomly generated structures of 9 links and the (randomly generated) reference structure shown in the inset to the figure. The “native” or reference state is shown in the inset, along with several of the 200 initial states. For the ensemble of transformations shown, the ensemble average of the mean link length is 0.96.

Figure 3. Crossing detection using projections.

(a) A 3 link chain with its vertical projection. A crossing in the projection is shown with a green circle. The crossing in the projection occurs at points and , where the chain is parametrized uniformly from 0 to 3. Since link 1 is under link 3 at the point of projection crossing, 0.29 will appear with a negative sign in the corresponding (eqn 1). (b) The blue chain and the red chain have the exact same vertical projection, however their corresponding matrices are different in sign, as given in Eq. 2. This indicates that the over-under sense has changed for the links whose projections are crossing. This in turn indicates that a true crossing has occurred when going from the red conformation to the blue conformation, as opposed to a series of conformations where the chain has navigated to conformations having the opposite crossing sense without passing through itself.

Figure 4. Two possible minimal uncrossing transformations.

Two possible untangling transformations. The top transformation involves twisting of the loop. The lower transformation involves a snake like movement of the vertical leg. A third one would involve moving the horizontal leg, in a similar snake-like fashion. Note that the moves represented here are not necessarily the most efficient ones in their topological class, but rather the most intuitive ones. There are transformations that are topologically equivalent but generally involve less total motion of the chain (see for example Figures 11(a), 11(b)).

Figure 5. Accounting for history-dependence in minimal uncrossing transformations.

The minimal untangling movement in going from A to C (through B′) is less than the sum of the minimum untangling movements going from A to B and then from B to C.

Figure 6. Snapshots of a transformation with two crossings.

A few snapshots during a transformation involving 2 instances of chain crossing. The transformation occurs clockwise starting from initial configuration I and proceeding to final configuration F.

Figure 7. Identification of leg-uncrossing.

For the crossing points indicated by the green circles, two legs, colored blue and red, can be identified. Each leg starts at the crossing and terminates at an end.

Figure 8. Crossing substructures.

(a) A single leg structure, (b) A loop structure, (c) An elbow structure.

Figure 9. Schematic illustration of the canonical leg movement.

Schematic illustration of the canonical leg movement, either from left to right as in (a) or effectively its time reverse as in (b). Both transformations traverse the same distance. The transformation in (a) is equivalent to the “plug” transformation analyzed in the context of folding simulations for trefoil knotted proteins , while the transformation in (b) (see ref. for a detailed description of this transformation) is equivalent to the “slipknotting” transformation more often observed in the folding of knotted proteins .

Figure 10. A single leg movement can undo several crossings.

One can reverse the over-under nature of all the crossings that have occurred on a leg, through a single leg movement.

Figure 11. Relation of minimal loop uncrossing to Reidemeister type I moves.

(a)Reversing the over-under nature of a crossing through a topological loop twist: Reidemeister move type I. (b) By “pinching” the loop before the twist, the cost in distance for changing the crossing nature is reduced.

Figure 12. Schematic of the canonical elbow move.

Schematic of the canonical elbow move. From left to right.

Figure 13. A simple example depicting various crossing substructures.

A chain with several self-crossing points before and after untangling. Various topological substructures that are discussed in the text are color coded. For the case of the legs (red and cyan) note that various other legs can be identified, for example a leg that starts at crossing 2 and ends at the red terminus. Here we color only the shortest legs from crossing 1 to the terminus as red, and crossing 2 to the opposite terminus as cyan.

Figure 14. Illustration of the depth-first tree search algorithm for the given crossing structure shown.

An example (subset) tree of possible transformations for a given crossing structure. Accumulated distances are given inside the circles representing nodes of the tree; the non-crossing transformations and their corresponding distances are shown next to the branches of the tree. The algorithm starts from the bottom node and proceeds to the top nodes, starting in this case along the right-most branch. The possible transformations to be considered as candidate minimal transformations are : , , which then terminates because the accumulated distance exceeds the minimum so far of 25, and .

Figure 15. Clustering of protein classes depending on order parameter.

(A) Scatter plot of all proteins as a function of and . Knotted proteins are indicated as green circles and are clustered; unknotted proteins are clustered using with the black closed curve, and contain -helical proteins clustered in red, and mixed - proteins clustered in magenta. Beta proteins are indicated in blue. Two and three state proteins are indicated as triangles and squares respectively. LRO provides a strong discriminant agains and mixed proteins, but not knotted and unknotted proteins, while discriminates knotted from unknotted proteins, and moderately discriminates proteins from mixed proteins. (B) Scatter plot of all proteins as a function of and . The rendering scheme for protein classes is the same as in panel (A). Kinetic 2-state folders are indicated by the black dashed curve. Both and distinguish knotted from unknotted proteins, and 2-state from 3-state proteins. By projecting proteins and either mixed / or all- proteins onto each order parameter, one can see how can discriminate proteins from both mixed or proteins, while cannot. This is despite the significant correlation between and .

Figure 16. Statistical significance for all order parameters in distinguishing between different classes of proteins.

The -log of the statistical significance is plotted as a function of pairs of protein classes, so that a higher number indicates better ability to distinguish between different classes. The blue horizontal line indicates a threshold of for statistical significance.

Figure 17. Approximate scaling laws for MRSD and non-crossing distance per residue

, across proteins and for domains within a single protein. (A) MRSD (blue circles) as a function of chain length for our protein dataset. The slope of the best fit line on the log-log plot gives the power law scaling: . Non-crossing distance per residue (red circles) vs. shows much larger scatter across native topologies, but follows an approximate scaling law which is superextensive, indicating an increasing importance of chain non-crossing per residue as system size is increased. At system sizes larger than , even minimal motion is dominated by entanglement. (B) Same quantities as in panel (A), but for the domains in proteins 2A5E and 2HA8. The scaling laws are different than in panel (A), and show stronger chain-length dependence. For 2HA8, domains 1, 2 and 1-2 together (the full protein) are considered; for 2A5E domains 1,2,3,4, 1-2, 2-3, 3-4, 1-2-3, 2-3-4, and 1-2-3-4 (the full protein) are considered. Based on these scaling laws found by building up proteins from subdomains, at system sizes larger than , minimal pathways become entanglement-dominated. (C) Schematic renderings of the domains, color-coded in 2HA8 (left) and 2A5E (right).

Figure 18. Schematic renderings of the three proteins whose minimal transformations we investigate in detail.

(A) acyl-coenzyme A binding protein, PDB id 2ABD , an all- protein; (B) Src homology 3 (SH3) domain of phosphatidylinositol 3-kinase, PDB id 1PKS , a largely protein; (C) The designed knotted protein 2ouf-knot, PDB id 3MLG .

Figure 19. Bar plots for the uncrossing operations involved in minimal transformations from an unfolded ensemble, for the protein 2ABD.

The sequence of noncrossing operations the transformation corresponding to a given pair of conformations is represented as a color-coded series of bars, with the sequence of moves going from right to left, and the length of the bar indicating the non-crossing distance undertaken by a particular move. Red bars indicate N-terminal leg () uncrossing, green bars indicate C-terminal leg () uncrossing, blue bars indicate Reidemeister “pinch and twist” loop uncrossing moves, and cyan bars indicate elbow uncrossing moves. The same set of 172 transformations is shown in panels A and B. Panel A sorts uncrossing transformations by rank ordering the following move types, largest to smallest: , , loop uncrossing, elbow move. Panel B sorts moves by , , loop uncrossing, elbow move. The scale bar underneath each panel indicates a distance of 100 in units of the link length. The arrow in each panel denotes the “most representative” transformation, as defined in the text.

Figure 20. Bar plots of the uncrossing operations involved in minimal transformations for the -sheet protein 1PKS.

See Figure 19 and the text for more details. Red bars: uncrossing moves; green bars: uncrossing moves; Blue bars: loop uncrossing moves; Cyan bars: elbow uncrossing moves. The same set of 195 transformations is shown in panels A and B, sorted as in Figure 19. The scale bar underneath each panel indicates a distance of 100 in units of the link length.

Figure 21. Bar plots of the uncrossing operations involved in the minimal transformations for the knotted protein 3MLG.

See Figure 19 and the text for more details. Red bars: uncrossing moves; green bars: uncrossing moves; Blue bars: loop uncrossing moves; Cyan bars: elbow uncrossing moves. The same set of 90 transformations is shown in panels A and B, sorted as in Figure 19. The scale bar underneath each panel indicates a distance of 100 in units of the link length. The arrow in each panel denotes the “most representative” transformation, as defined in the text. The transformation located 8 bars up from the bottom of Panel A requires both and moves, however both leg motions are very small.

Figure 22. Consensus histograms of the transformations described in Figures 19, 20, 21.

See text for a description of the construction. Each bar represents the distance of a corresponding move type, N or C leg ( or ), elbow , or loop . The order of the sequence of moves is taken from right to left along the x-axis. An all- protein (2ABD), an all- protein (1PKS), and a knotted protein (3MLG) are considered. (A) Transformations with leg as the largest move. These encompass 15% of the transformations those in the protein, 16% of the transformations in the protein, and 73% of the transformations for the knotted protein. (B) Transformations with leg as the largest move, which encompass 13% of the protein transformations, 54% of protein transformations, and 18% of knotted protein transformations. (C) Transformations with either an elbow E or loop R as the largest move, which encompass 71% of the protein transformations, 29% of protein transformations, and 9% of knotted protein transformations.

Figure 23. Schematic of the most representative transformation for the protein 2ABD.

Figure 24. Schematic of the most representative transformation for the protein 1PKS.

Figure 25. Schematic of the most representative transformation for the knotted protein 3MLG.

Figure 26. Overlap between minimal transformations.

Schematic diagram for the residues involved in uncrossing operations for two minimal transformations labelled by and , to illustrate the sequence overlap between transformations.

Figure 27. Distribution of pathway overlap between minimal transformations, for an , , and knotted protein.

Pathway overlap () distributions for the 3 proteins in Figure 18, as defined by Equation (8), operating on the transformations in Figure 19, 20, 21. (a) The pathway overlap distribution for the all- protein 2ABD indicates a large contribution for (the peak height in the distribution is ), indicating a diverse set of minimal transformations fold the protein. The average for these transformations is . (b) The pathway overlap distribution for the -protein shows the emergence of a peak around , indicating partial restriction of folding pathways. The peak near still carries more weight in the distribution. The average . (c) The peak around becomes dominant for the pathway overlap distribution of the knotted protein, indicating the emergence of a dominant restricted minimal folding pathway. The average .