System Setup.

We carried out various Langevin dynamics simulations of knotted DNA rings translocating through a nanopore embedded in a slab (Fig. 1). The rings, modeled mesoscopically with oxDNA (36⇓⇓–39), were 10 kbp long and were nicked, to allow relaxation of torsional stress, as in typical experiments (34, 40). The translocation is driven by a longitudinal electric field exerting a force of 0.2 pN on each nucleotide inside the pore. For simplicity, we neglect the action of the field outside the pore (41, 42) that, in actual realizations, can facilitate the capture and pore insertion of the knotted chains (43⇓–45). The pore is 10 nm wide and 10 nm long, so that each of the two dsDNA filaments inside it (∼30 bp long) is pulled with a total force of 12 pN.

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Fig. 1.

Typical configuration of a knotted double-stranded DNA ring translocating through a nanopore. Inset shows the knot approaching the pore entrance. The ring is 10 kbp long and is modeled with oxDNA. The pore, which is 10 nm wide, is embedded in an impenetrable 10-nm–thick slab. A translocating force of 0.2 pN is applied to each nucleotide inside the pore.

The translocation dynamics were studied for 50 different equilibrated knotted DNA configurations. These were generated with a Monte Carlo scheme applied to a coarse-grained DNA model and were subsequently refined and relaxed with the oxDNA model. All configurations featured a trefoil or 31 knot, which is by far the dominant topology at the considered DNA length (17, 32, 46). These initial configurations were primed at the pore entrance at a random point lying on their convex hull.

Translocation Dynamics Overview.

For a first, general characterization of the process we profiled the translocated fraction of the chain, x, as a function of the elapsed simulation time, t. This dependence is shown in Fig. 2A, where, as customary, it is presented as a t vs. x plot. The red curves cover the individual trajectories whereas the black points represent the average curve.

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Fig. 2.

(A) The time required to translocate a fraction x of the knotted DNA rings is shown for 50 independent simulations (red curves). The black points show the average ⟨t⟩ vs. x curve. (B) The waiting-time curve, w=d⟨t⟩dx, highlights two main regimes corresponding to the tension propagation along the chain (x≲0.5) followed by the translocation of the rectified chain tail (x≳0.75). The dashed and dotted lines are best fits based, respectively, on w∝xα (yielding α∼0.32) and w∝(1−x).

Fig. 2B shows, instead, the so-called waiting time (47), w=dt/dx, i.e., the inverse of the translocation velocity, whose average profile clearly outlines the two known main translocation regimes (45, 47, 48).

The first part of the curve, for x≲0.5, corresponds to the tension propagation along the chain, which itself presents an articulate phenomenology (49⇓–51). For chains that are asymptotically long and free of entanglement, theoretical scaling arguments predict a power-law behavior, w∝xν, where ν=0.588 is the metric exponent for self-avoiding walks (47, 49, 52⇓⇓–55), whereas smaller effective exponents are expected for chains of finite length (47, 56). Fig. 2B shows that data points for the 10-kbp chains are indeed well fitted by a power law, and the effective exponent, 0.32, is close to that previously reported at comparable DNA lengths (57).

At x∼0.6 the tension propagation regime crosses over to the tail retraction regime. In this stage the still untranslocated remainder of the chain, which is fully rectified, accelerates toward the pore. Because the pore is large enough to let the whole knot pass through, this second stage also follows the behavior expected theoretically, w∝(1−x) (47).

Statistics of Knot Translocation Events.

Inspired by experiments, we detect the passage of the knot through the pore by monitoring the degree of obstruction of the latter. During such an event, in fact, the pore lumen must accommodate up to four double-stranded filaments, instead of the usual two (sketches in Fig. 3A).

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Fig. 3.

(A) Time evolution of the chain fraction that is inside the pore, Δx (red), and that has already translocated through it, x (blue). An absolute scale in base pairs for x and Δx is also provided for the semilog plot. The knot passage event is highlighted in B and its time of occurrence, t∗, is defined as the midpoint of the time interval Δt∗ during which Δx exceeds by more than 30% its baseline value. C and D show the probability distributions, computed over the 50 independent runs, of the translocated chain fraction at the passage (pore obstruction) event, x∗, and of the event duration, Δt∗, respectively.

We accordingly monitored the time evolution of the chain fraction inside the pore, Δx, which is shown in Fig. 3A. The pore obstruction caused by the passing knot is indeed signaled by a bump that stands out against the Δx baseline (Fig. 3B). Note that this major pore obstruction event is preceded by a smaller signal burst caused by the knot partially entering the pore and then retracting from it. Such translocation attempts, illustrated in Movies S1 and S2, affect about 50% of the trajectories. Their occurrence arguably depends on frictional effects arising from the geometry of the knot and the direction with which it engages the pore.

Various observables of interest, related to those monitored in the experiments of ref. 34, can be derived from the analysis of the Δx profile: the fraction of the translocated chain at which the pore-obstruction event takes place, x∗; the elapsed time at which it occurs, t∗; and the temporal duration of the event, Δt∗. Because t∗ and x∗ are monotonically related, we focus on x∗ and Δt∗, whose probability distributions are shown in Fig. 3 C and D.

The key features are two. First, the distribution of x∗ is skewed toward large values of x∗; see Fig. S1 for the same effect in the companion distribution of t∗. In fact, passage events are virtually absent for x∗<0.3 and the distribution is prominently peaked at x∗∼1. Second, the distribution of the obstruction duration has an overall decreasing trend, with the shortest obstruction events (which have a minimum duration of 300 τLJ) being the most probable also. Both these features match the ones reported by Plesa et al. (34).

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Fig. S1.

Probability distribution of the normalized time of knot passage, t∗/ttot. t∗ is the time of occurrence of the pore-obstruction event and ttot is the total translocation time. The distribution is normalized and the data are obtained over 50 independent runs. Given the monotonic relationship between t∗ and x∗, this distribution complements the one for x∗ shown in Fig. 3C of the main text.

This consistency of the experimental and theoretical distributions for x∗ and Δt∗ is noteworthy given the different contour lengths considered here (10 kbp) and in the experiment (20 kbp or longer). This underscores the robustness of the effects addressed with either of the two approaches. The agreement also gives confidence for using the model to gain insight into aspects that cannot be directly accessed with current experiments. These primarily include various properties of the knotted region, which we discuss in the following.

Knot Translocation Modes.

As we discuss, both the position of the knot along the chain contour and its size affect the Δt∗ and x∗ distributions in ways that are much richer than previously suspected.

A particularly intriguing relationship is found between the pore obstruction duration, Δt∗, and the size of the knotted region, lk, when it just enters the pore. We recall that, as customary, the knotted portion is identified as the shortest portion of the chain that, upon closure, has the same topology as the entire ring. A scatter plot of the two quantities is presented in Fig. 4A, where two relevant features are noted. First, the datapoints occupy an L-shaped region. Second, for either arm of this region the correlation between Δt∗ and lk is rather weak. Both aspects are not intuitive and, in fact, had not been previously predicted or envisaged.

The analysis of the trajectories showed that the distinct arms in the diagram of Fig. 4A originate from two different modes of knot translocation, as described below.

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Fig. 4.

(A) Scatter plot of the knot length when the pore obstruction event starts, lk(x∗), vs. the duration of the event itself, Δt∗. (B) Knot length at the beginning of the translocation process, lk(0), and at the pore obstruction event, lk(x∗). Data points are divided into two classes based on whether the knot at the beginning of the translocation straddled (green) or did not straddle (blue) the site antipodal (on the ring contour) to the translocation initiation site (main text). (C) For the first group, the pore obstruction event practically involves only the essential crossings of the knotted region, which spans both translocating filaments. (D) For the second group, the pore obstruction is caused by a single-filament knot. (E) Knot length lk during pore obstruction at x∗. Data points for double-filament knots follow closely the curve lk=lchain(1−x∗) (solid green line), whereas for the other points lk is about constant and equal to 160 bp. (F) Scatter plot of Δt∗ against the waiting time w∗ at the time of passage.

In the first mode the knot is tight and localized on one of the two translocating filaments (Fig. 4D). This is the most intuitive type of knot passage and, in fact, it was the mode of choice used in ref. 34 to interpret the experimental data on Δt∗ and thus obtain a mapping between pore-obstruction time and knot length. By using a linear mapping, Plesa et al. (34) were able to conclude that knots could be rather tight upon translocation, spanning an arc length of tens of nanometers and hence comparable to the DNA persistence length. This result was further put in the context of the elegant theory of metastable knots, which predicts knot localization based on the fact that, in the otherwise broad distribution of knot lengths, the most probable one is about constant—rather than growing—with chain length, lchain (35).

Our results vividly confirm the significant occurrence of tight knots. Indeed, we observe that the average knot length at the passage event is about 54 nm, which is in full accord with the estimate of Plesa et al. (34). This knot length is reached independently of the initial one due to tightening of the knot caused by the propagating chain tension (Fig. 4B). We also note that the lk vs. Δt∗ profile in Fig. 4A is rather flat for this translocation mode and hence is different from the linear relationship expected intuitively. An explanation of this effect is discussed later in this article.

The second, and new mode is associated to the green points in Fig. 4. It involves knots that span a significant portion of the ring, consistent with the theoretical results of ref. 58 on DNA chains of comparable size, which indicated that the most probable knot length is about 2,200 bp. In fact, these knots experience significantly less tightening during translocation than those discussed above (Fig. 4B). Intriguingly, these knots are large and yet their pore-obstruction times are not at all dissimilar from the tight knot case discussed before.

This conundrum is solved by considering the actual conformation of such rings when the knot is presented at the pore entrance. A typical configuration is shown in Fig. 4C. The accompanying sketch clarifies that the knotted portion now spans the entire cis part of the ring. This is quantitatively shown in the semilog plot of Fig. 4E, where we observe that for this class of knots, the relative chain fraction occupied by the knot is lk/lchain∼(1−x∗).

However, a significant obstruction of the pore occurs only when the region accommodating the essential crossings passes through it. As seen in Fig. 4, this region is typically small, involving 123 bp (42 nm) on average upon entering the pore (and is slightly reduced when fully inside it).

It is therefore this short, essentially entangled portion of “double-filament” knots, and not their entire contour lengths, lk, that is captured by Δt∗.

To our knowledge, the possible occurrence of a second mode of translocation, although rather natural a posteriori, has not been considered or foreseen in previous translocation studies, either for dsDNA ring experiments or for simulations of linear, open chains where it can also occur if translocation starts from inside the knot loop region.

Note that, because the essentially entangled region is comparable in size to the tight, single-filament knots, the two modes of translocation cannot be distinguished from the sole analysis of Δt∗. This has direct bearings on the interpretation of experimental data. In fact, it poses the necessity to devise suitable means of discriminating or controlling the incidence of the two modes. In this way one could relate more reliably the measured observables to the spontaneous knotting properties of DNA. Our results suggest that this could be achieved, for instance, by suitably choosing the DNA length. The latter, in fact, affects the balance of the two modes, as we discuss later in connection with Fig. 5A.

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Fig. 5.

(A) Model estimate of relative percentage of single- and double-filament knots in DNA rings of different length. The estimate considers the length and positioning of the knotted region (highlighted in red in the sketches) with respect to the point (marked with a cross) antipodal to the root (marking the point where translocation initiates). (B) The same model is used to predict the translocated chain fraction at the time of the pore obstruction event, x∗. (C) Accounting for the sliding of the knot along the chain brings the model distribution into good quantitative agreement with the actual simulation data.

We conclude the analysis of the data in Fig. 4A by discussing the second notable feature, namely the lack of a noticeable correlation between the pore obstruction duration, Δt∗, and knot length, lk. For the second mode of translocation, it is now clear that no obvious relationship between lk and Δt∗ can be expected, because the lk is not directly informative for the pore obstruction caused by the essentially entangled region. The case is different, however, for the first mode, i.e., tight single-filament knots, where a proportionality relationship between knot size and passage time appears plausible and was previously surmised (34).

This point is clarified by the plot of Fig. 4F, which shows the relationship between Δt∗ and w∗, the inverse translocation velocity at the passage event. The two quantities are visibly correlated for both knot translocation modes. Together with the plot in Fig. 4A the data clarify that of these two properties relatable to the passage time, knot length and knot translocation velocity, the dominant one is the latter. Note that upon entering the pore the contour lengths of single-filament knots and of double-filament essential crossings span a limited range, typically from 120 bp to 160 bp. As a result, Δt∗ and w∗ have an approximate linear proportionality.

This observation might be harnessed to extract further knot-related properties from Δt∗. Because the average translocation velocity depends on the translocated chain fraction, the observed Δt∗ vs. w∗ correlation should effectively subsume a dependence of Δt∗ on the knot position along the chain contour, x∗, which could be recovered with sufficient statistics.

Interpretative Model.

We now consider the origin of the two different translocation modes and of the skewed distribution of the knot position at passage time, x∗.

Both aspects are best illustrated with the following schematic model of the translocation process. In this purposely simplified scheme we assume that the rooting point where the translocation process initiates is equally likely to lie anywhere on the ring contour. We also assume that tension propagates in the same way along the two ring arms departing from the root, so that they meet at the antipodal midpoint, that is, at the point at half ring contour length from the root.

The main discriminator for the two translocation modes is whether the knot is entirely located on only one of the two arms or whether it straddles the antipodal midpoint and hence spans both arms (sketches in Fig. 5).

In the former case the sliding of the progressively tensioned ring arm causes the knotted region to tighten toward the distal knot end (i.e., the end that is farthest from the rooting point) while it is dragged to the pore (Movie S3). The tightened single-filament knot will then pass through the pore.

Instead, when the knotted region straddles the antipodal midpoint, the knot will be pulled from both sides and will be dragged toward the pore by both tensioned arms. Such double pulling typically causes the essential crossings to become interlocked, trapping the knot in a moderate degree of tightening (Movie S4). The pore obstruction event is then associated to the passage of the essential crossings.

These two cases are directly associated to the two different translocation modes highlighted in Fig. 4, so much so that the two sets in Fig. 4 were not assigned from an a posteriori supervised inspection of the trajectories, but rather a priori based on the aforementioned distinction. In fact, the two sets precisely correspond to knots that span a single ring arm or both ring arms at the beginning of translocation. The neat separation of the two sets of points in Fig. 4 A, B, E, and F supports the viability and usefulness of such a discriminatory criterion.

The same criterion can be also used to estimate how the relative incidence of single- vs. double-filament entanglement varies with ring length. We considered an ensemble of Monte Carlo-generated rings of length 10 kbp, 20 kbp, and 50 kbp; picked a rooting point randomly along their contour; and then located the knot on the ring, which we considered open in correspondence to the rooting point. The rings were next assigned to one of the two classes based on whether the knotted region straddled the antipodal midpoint or not. The results, given in Fig. 5A, show that the incidence of single-filament entanglement increases steadily with chain length and goes from 35% for 10 kbp to 70% for 50 kbp. Based on this result, which reflects the interplay of knot and chain lengths (35, 58⇓⇓–61), we speculate that most of the knot passage events detected in the experiment of Plesa et al. (34) pertained to single-stranded entanglements, as implicitly assumed by the authors.

The same schematic framework can account for the qualitative features of the distribution of x∗, the knot positioning at time of translocation. We considered, again, the Monte Carlo-generated ensemble of rings for which we stochastically picked the rooting points. Next, for double-filament knots (straddling the antipodal midpoint) we picked x∗ uniformly between the two knot ends. For the other, single-filament knots, we picked x∗ as the distal end of the knot, the one farthest from the pore. These criteria embody in the simplest possible way the phenomenology described in the previous paragraphs.

The resulting probability distributions for x∗, shown in Fig. 5B, are in qualitative agreement with simulation and experimental data. It is seen that, at all ring lengths, the distributions are skewed toward x∗=1. As for the balance of single- and double-filament knots, the skewness also depends on the interplay of knot and ring length. Indeed, the x∗ distribution becomes flatter for longer rings, where the knotted region occupies a smaller fraction of the chain contour.

The above modeling scheme neglects the possibility that tight knots may slide on the filament contour. As was clarified in the theoretical study of refs. 27, 30, and 31, such sliding can occur for fully flexible chains and, in fact, make it possible for individual knotted filaments to fully translocate through very narrow pores, as long as the driving force is not high enough to cause jamming (30, 62, 63). As a matter of fact, we observed the same knot sliding phenomenology for the present dsDNA system too (Movies S1–S3).

To account for such a sliding effect for single-filament knots, we accordingly adjusted the model. Specifically, we assumed that x∗ could fall with equal probability between the distal knot end and the antipodal midpoint. The x∗ probability distribution predicted by such a model is shown in Fig. 5C. It presents a noticeably stronger shift toward x∗ values that follows closely the data from the simulated trajectories. This good level of agreement is somewhat surprising given the simplicity of the model, which does not account for frictional effects related to pore size and force magnitude. However, the good accord further corroborates the relevance of sliding effects for dsDNA. We believe this would be an important avenue to explore further, especially by seeking a quantitative comparison against experimental data. For this, it would probably become essential to take into account the finite resolution of time measurements that could account for the observed effective dependence of the distribution of t∗ (related monotonically, but nonlinearly, to x∗) on the driving force.