Fig 1.
A typical DNA model chain with an embedded trefoil knot.
The chain consists of 1,000 beads and corresponds to a double-stranded DNA molecule of ≈13,000 base pairs. This is roughly the size which can still be sequenced in current nanopore devices. The trefoil (light green) consists of ≈4,000 base pairs (N≈300) and has a diameter of ≈160nm. It is thus a typical representative of a knot in a DNA chain even if longer chains are considered. For comparison, the DNA chain is pictured in relation to nanopores with diameters of 5nm, 10nm and 20nm.
Fig 2.
Probability of observing knots in dsDNA as a function of DNA length (in base pairs) for physiological salt concentrations (0.15M NaCl).
The total knotting probability as well as the probability of trefoil knots are shown. The probability of complex and composite knots, i.e. knots with more than three crossings, is displayed to illustrate the increase of knot complexity with chain size. Error bars are smaller than symbol size. All data points and errors are documented in S1 Table. One bead of the coarse-grained chain corresponds to ≈13 base pairs.
Fig 3.
Probabilities of dominant knot types in dsDNA as a function of DNA length (in base pairs) for physiological salt concentrations (0.15M NaCl).
The knot spectrum is dominated by composite knots which have trefoil knots as prime factors. The likelihood of observing a prime knot other than the trefoil 31 or figure-eight knot 41 is very small for DNA chains of up to ≈525,000bp (N = 40,000 beads), see S1 Fig. Data points are documented in S2 Table.
Fig 4.
Probability distributions of the trefoil contour length for various DNA sizes.
Displayed values are (conditional) probabilities of trefoil sizes of up to 13,000bp (N = 1000), given that a chain contains a trefoil knot in the first place. Therefore, the area below each individual curve equals one (if each curve is integrated over its entire domain). Irrespective of the actual DNA length, the most likely trefoil size is close to ≈3,000bp (N≈230). After reaching its maximum, each distribution decreases until the trefoil size is equal to the total DNA length. The original data has been smoothed by a (two-sided) running average. For comparison, the original data is shown for a total DNA length of ≈66,000bp (N = 5000). Inset: Distribution of trefoil diameters for a DNA strand of ≈13,000bp (N = 1000). The most likely diameter is ≈200nm.