Research ArticleBIOPHYSICS

Grid diagrams as tools to investigate knot spaces and topoisomerase-mediated simplification of DNA topology

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Science Advances  26 Feb 2020:
Vol. 6, no. 9, eaay1458
DOI: 10.1126/sciadv.aay1458
  • Fig. 1 A standard knot diagram and a grid diagram of a knot.

    (A) Standard diagram of a left-handed trefoil knot. We use Alexander-Briggs’ notation of knots, where the first number indicates the minimal crossing number of a given knot type and the subscript number denotes its tabular position among all knots with that crossing number. Standard knot diagrams are scale free and therefore do not inform about the number of statistical segments of a knotted polymer that they represent. (B to D) Generation of a grid diagram of a trefoil knot. Grid diagram formalism requires that the square grid with n rows and n columns has exactly one segment in each row and column of the represented polygonal chain. A grid diagram of a knot configuration is generated in three steps. (B) Place 2n markings (dots) corresponding to ends of the modeled polygonal chain segments. Markings are placed following a “sudoku” rule, requiring that each column and each row of the grid contain exactly two markings in distinct squares. (C) Each pair of markings in the same row/column is connected by a segment. (D) For the segments that intersect, we follow the convention that the vertical segment passes over the horizontal segment.

  • Fig. 2 Interleaving commutations as a model for strand passages.

    (A) Intersegmental passages between grid diagrams representing the (i) unknot 01, the (ii) twist knot 52, and the (iii) torus knot 51. Intersegmental passages are achieved by interleaving commutation of two consecutive rows/columns such that the interiors of their corresponding intervals intersect nontrivially but neither is contained in the other. The exchanged segments are highlighted in red and purple. When the area delimited by two parts of the diagram between two consecutive crossings forms a rectangle, a hooked juxtaposition is formed. A juxtaposition can be “strongly hooked” if the rectangle is a square. The diagram of 52 contains one strongly hooked juxtaposition. Performing an interleaving commutation at that hooked juxtaposition transforms the 52 into the trivial knot. This exchange transforms the hooked juxtaposition into a “free” one. Juxtapositions that are neither hooked nor free are called mixed. Performing an interleaving commutation at the highlighted mixed juxtaposition on 52 transforms this knot into the 51 (shown on the right). (B) The local deformations that transform the three grid diagrams in (A) into their respective standard diagrams. Thick gray lines highlight the arcs involved in the various deformations.

  • Fig. 3 Visualization of strand passage–mediated knot interconversion fluxes using circos plots.

    The three layers of thin external arcs, progressing from inside to outside, represent outgoing, incoming, and total fluxes involving a given knot type, respectively. These external arcs are segmented to indicate how the respective fluxes were redistributed. The thickness of the (interior) chords connecting different knot types reflects the fraction of outgoing and incoming knot interconversion fluxes between given types of knots. The chords representing knot interconversion fluxes are colored as the knot type that these fluxes originate from, with the exception of chords starting and ending in the same knot type, which are in gray. These correspond to the fluxes resulting from strand passages not changing the knot type. The bases of the chords representing outgoing fluxes are colored according to the knot type that a given flux leads to. The bases of chords representing incoming fluxes are left white. The length of the various thick arcs around the circumference, colored as the corresponding knot diagrams, indicates the sum of fluxes outgoing from and incoming to a given knot type. The global observed flux in a given system is normalized to 1, which corresponds to the circumference of the circle.

  • Fig. 4 Knot interconversions occurring at hooked juxtapositions lead to preferential unknotting.

    (A) Color guide informing which colors correspond to which knots in circos plots shown in (B) and (C). A comparison of the topological consequences of unbiased strand passages (B) with the ones resulting from strand passages occurring only at strongly hooked juxtapositions (C). Comparison of circos plots for small grid diagrams (i) with those for larger grid diagrams (ii and iii) shows that as the system gets more complex, more types of knots are formed and they contribute stronger to the knot interconversion fluxes. In the circos plots representing fluxes resulting from unbiased strand passages (A), the incoming and outgoing fluxes connecting any pair of knots are of the same intensity. This indicates that the generated set of grid diagrams represents the topological equilibrium. When the same set of grid diagrams undergoes intersegmental passages involving only strongly hooked juxtaposition, the interconversion fluxes from the trefoil to the unknot are much more intense than the opposite fluxes. This effect is especially strong for smaller GNs.

  • Fig. 5 Occurrence probability of prime knots as a function of GN.

    The curves give the occurrence probabilities of knots with minimal crossing number up to 6, plotted as a function of the grid number. The occurrence probability of a given knot type is defined as the ratio between the configurations representing that knot type and the total number of configurations. We consider also the composite knots obtained as the connected sum of two trefoils.

  • Fig. 6 Topological simplification due to hooked juxtapositions.

    The plots show the transition probabilities of each knot type toward simpler knots, as a function of the grid number. In each plot, the dotted line refers to strand passages happening at strongly hooked juxtaposition, while for the other, we consider unbiased interleaving commutations. In the case of the unknot, the 31 and the 41, we consider only the unknotting probabilities. For the 51 and the 52, we consider passages toward the 31 and toward the 31 and the unknot, respectively. Last, for 6 crossings knots, we plot the transition probabilities toward knots with lower “length-over-diameter ratio” (thus, for the 61, we consider only passages toward knots with crossing number less than or equal to 5, while for the 62 and the 63, we also consider passages toward the 61 and toward the 61 and 62, respectively) (37).

Supplementary Materials

  • Supplementary material for this article is available at

    Supplementary Materials and Methods

    Computations and results

    Fig. S1. Local isotopy and crossing change.

    Fig. S2. The network of knot diagrams.

    Fig. S3. Grid diagrams.

    Fig. S4. Side-by-side comparison of circos plots in the 2D and 3D models.

    Fig. S5. The knot reduction factor increases with the tightness of hooked juxtapositions.

    References (38, 39)

  • Supplementary Materials

    This PDF file includes:

    • Supplementary Materials and Methods
    • Computations and results
    • Fig. S1. Local isotopy and crossing change.
    • Fig. S2. The network of knot diagrams.
    • Fig. S3. Grid diagrams.
    • Fig. S4. Side-by-side comparison of circos plots in the 2D and 3D models.
    • Fig. S5. The knot reduction factor increases with the tightness of hooked juxtapositions.
    • References (38, 39)

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