Introduction
A series of advancements in pore translocation setups have brought this single-molecule technique to the forefront of numerous applications, far exceeding the originally envisioned purpose of sequencing nucleic acids (Kasianowicz et al., Reference Kasianowicz, Brandin, Branton and Deamer1996; Palyulin et al., Reference Palyulin, Ala-Nissila and Metzler2014; Deamer et al., Reference Deamer, Akeson and Branton2016). Recent applications include advanced molecular sensing (Rahman et al., Reference Rahman, Stott, Harrington, Li, Sampad, Lancaster, Yuzvinsky, Noller, Hawkins and Schmidt2019; Wang et al., Reference Wang, Sensale, Pan, Senapati and Chang2021; Leitao et al., Reference Leitao, Navikas, Miljkovic, Drake, Marion, Pistoletti Blanchet, Chen, Mayer, Keyser and Kuhn2023), out-of-equilibrium stochastic processes (Kantor and Kardar, Reference Kantor and Kardar2004; Grosberg et al., Reference Grosberg, Nechaev, Tamm and Vasilyev2006; Sarabadani and Ala-Nissila, Reference Sarabadani and Ala-Nissila2018; Suma et al., Reference Suma, Carnevale and Micheletti2023), RNA unfolding (Bandarkar et al., Reference Bandarkar, Yang, Henley, Wanunu and Whitford2020; Suma et al., Reference Suma, Coronel, Bussi and Micheletti2020), protein sequencing (Asandei et al., Reference Asandei, Di Muccio, Schiopu, Mereuta, Dragomir, Chinappi and Luchian2020; Yu et al., Reference Yu, Kang, Li, Mehrafrooz, Makhamreh, Fallahi, Foster, Aksimentiev, Chen and Wanunu2023), and probing of intra- and inter-molecular entanglement (Huang and Makarov, Reference Huang and Makarov2008; Rosa et al., Reference Rosa, Di Ventra and Micheletti2012; Suma et al., Reference Suma, Rosa and Micheletti2015; Narsimhan et al., Reference Narsimhan, Renner and Doyle2016; Plesa et al., Reference Plesa, Verschueren, Pud, van der Torre, Ruitenberg, Witteveen, Jonsson, Grosberg, Rabin and Dekker2016; Suma and Micheletti, Reference Suma and Micheletti2017; Marenda et al., Reference Marenda, Orlandini and Micheletti2017; Caraglio et al., Reference Caraglio, Orlandini and Whittington2017; Weiss et al., Reference Weiss, Marenda, Micheletti and Likos2019; Caraglio et al., Reference Caraglio, Orlandini and Whittington2020; Rheaume and Klotz, Reference Rheaume and Klotz2023).
One of the most exciting avenues for nanopore translocation is probing the structure and function of biological polymers. A notable example is offered by exonuclease-resistant RNAs (xrRNAs) (Pijlman et al., Reference Pijlman, Funk, Kondratieva, Leung, Torres, Van der Aa, Liu, Palmenberg, Shi, Hall and Khromykh2008; Chapman et al., Reference Chapman, Costantino, Rabe, Moon, Wilusz, Nix and Kieft2014; Akiyama et al., Reference Akiyama, Laurence, Massey, Costantino, Xie, Yang, Shi, Nix, Beckham and Kieft2016; MacFadden et al., Reference MacFadden, O’Donoghue, Silva, Chapman, Olsthoorn, Sterken, Pijlman, Bredenbeek and Kieft2018; Slonchak et al., Reference Slonchak, Hugo, Freney, Hall-Mendelin, Amarilla, Torres, Setoh, Peng, Sng, Hall, van den Hurk, Devine and Khromykh2020; Vicens and Kieft, Reference Vicens and Kieft2021). These modular elements, consisting of only a few dozen nucleotides, are located at the 5′ end of the RNA genome of flaviviruses and are responsible for infections such as Zika, dengue, and yellow fever (Slonchak et al., Reference Slonchak and Khromykh2018). xrRNAs are distinguished by their unique and diverse functional responses when pulled through the lumen of enzymes that process nucleic acids. Specifically, xrRNAs resist degradation by exonucleases that translocate nucleic acids from the 5′ end. However, they can be processed by replicases and reverse transcriptases, which translocate RNAs from the 3′ ends.
A mechanistic explanation for this behavior was provided by the theoretical and computational study of Suma et al. (Reference Suma, Coronel, Bussi and Micheletti2020), where a pore translocation setup, mimicking the action of processive enzymes, was used to unzip xrRNAs from both ends. The study, further supported by later work (Becchi et al., Reference Becchi, Chiarantoni, Suma and Micheletti2021; Niu et al., Reference Niu, Liu, Xu, Chen, Xu, Xu, Li and Fang2020), reported that the short and yet heavily entangled structure of xrRNAs, which includes several pseudoknots (Akiyama et al., Reference Akiyama, Laurence, Massey, Costantino, Xie, Yang, Shi, Nix, Beckham and Kieft2016), contributes to a strongly directional translocation response. Pulling xrRNAs from the 5′ end causes the molecule to close in on itself and resist further unzipping, explaining its resistance to exonucleases; conversely, when translocated from the 3′ end, the molecule unravels progressively, explaining its processability by replicases and helicases/reverse transcriptase (Suma et al., Reference Suma, Coronel, Bussi and Micheletti2020).
Differently from RNAs, double-stranded DNA (dsDNA) filaments are usually well described by general polymer models with torsional and bending rigidity (Chirico and Langowski, Reference Chirico and Langowski1994; Klenin et al., Reference Klenin, Merlitz and Langowski1998; Vologodskii and Cozzarelli, Reference Vologodskii and Cozzarelli1994). Although dsDNA does not form the complex architectures typical of RNAs, it can become knotted due to its spontaneous dynamics, both in bulk and under confinement (Rybenkov et al., Reference Rybenkov, Cozzarelli and Vologodskii1993; Arsuaga et al., Reference Arsuaga, V/azquez, Trigueros, Sumners and Roca2002; Marenduzzo et al., Reference Marenduzzo, Orlandini, Stasiak, Sumners, Tubiana and Micheletti2009). Additionally, dsDNA filaments can become knotted through the actions of type II topoisomerases, which perform strand crossings that can potentially alter the topological state of DNA, establishing a homeostatic level knotting that needs to be tightly regulated to avoid detrimental consequences for living cells (Portugal and Rodríguez-Campos, Reference Portugal and Rodríguez-Campos1996; Rybenkov et al., Reference Rybenkov, Ullsperger, Vologodskii and Cozzarelli1997; Olavarrieta et al., Reference Olavarrieta, Hernandez, Krimer and Schvartzman2002; Deibler et al., Reference Deibler, Mann, Sumners and Zechiedrich2007, p. 1; Valdés et al., Reference Valdés, Segura, Dyson, Martínez-García and Roca2018; Valdés et al., Reference Valdés, Coronel, Martínez-García, Segura, Dyson, Díaz-Ingelmo, Micheletti and Roca2019).
The emergence of DNA knots, be they formed spontaneously or introduced by topoisomerases, has been traditionally based on gel electrophoresis (Dröge and Cozzarelli, Reference Dröge and Cozzarelli1992; Trigueros et al., Reference Trigueros, Arsuaga, Vazquez, Sumners and Roca2001; Valdés et al., Reference Valdés, Martínez-García, Segura, Dyson, Díaz-Ingelmo and Roca2019). Such setups harness the different hindrances experienced by molecules with different knot types when moving through the gel mesh. Its main limitation regards the maximum length to which it can be practically applied, which is of the order of 10 kb.
Recent breakthroughs have opened the possibility of overcoming this practical limit by resorting to pore translocation setups (Plesa et al., Reference Plesa, Verschueren, Pud, van der Torre, Ruitenberg, Witteveen, Jonsson, Grosberg, Rabin and Dekker2016; Suma and Micheletti, Reference Suma and Micheletti2017; Sharma et al., Reference Sharma, Agrawal, Dai, Doyle and Garaj2019). Suitable choices of the pore diameter allow for translocating the DNA knots and reveal their passage from the drop of the ionic current, which depends on the obstruction of the pore caused by the passing knotted region and involves at least three dsDNA strands. While the technique may not be sensitive to the knot type and knot size (Suma and Micheletti, Reference Suma and Micheletti2017), it allows for probing the so-called topological friction (Rosa et al., Reference Rosa, Di Ventra and Micheletti2012; Suma et al., Reference Suma, Rosa and Micheletti2015). The latter can be revealed by using pores sufficiently narrow that only one dsDNA filament can pass through, causing the knot to remain localized at the pore entrance, hindering the translocation of the remainder of the filament that has to slide along the contour of the knotted region to pass through. In such a setup, the hindrance to translocation can depend on the knot type and the driving force (Rosa et al., Reference Rosa, Di Ventra and Micheletti2012; Suma et al., Reference Suma, Rosa and Micheletti2015; Narsimhan et al., Reference Narsimhan, Renner and Doyle2016). Increasing the driving force makes the knots tighter, enhancing the friction to the point that the translocation process can even be stalled indefinitely, as illustrated in Figure 1, which presents results from simulations specifically carried out for this study.
Figure 1. (a) Snapshot of a trefoil (
$ {3}_1 $
) knotted dsDNA translocating through a wide pore with a 4.25 nm diameter, allowing for the passage of a single double strand, thus blocking the knot. The total applied translocating force is 30 pN, sufficient to maintain the knot in a tight state near the pore entrance. (b) Time evolution of the number of base pairs,
$ n $
, which have translocated from the cis to the trans side of the slab where the nanopore is embedded. The trajectories are for a
$ {3}_1 $
-knotted dsDNA chain at three different driving forces. The translocation process speeds up when
$ f $
is increased from 10 to 20 pN and then slows down, and even stalls, at higher forces due to the topological friction in the tightened knotted region.
At the same time, dsDNA typically undergoes another type of in vivo transaction operated by, for example, helicases, namely unzipping. In the pore translocation setup, this effect can be mimicked by reducing the pore diameter so that only one strand of the DNA duplex can pass and is harnessed for fast and reliable genome sequencing (Manrao et al., Reference Manrao, Derrington, Laszlo, Langford, Hopper, Gillgren, Pavlenok, Niederweis and Gundlach2012; Jain et al., Reference Jain, Koren, Miga, Quick, Rand, Sasani, Tyson, Beggs, Dilthey, Fiddes, Malla, Marriott, Nieto, O’Grady, Olsen, Pedersen, Rhie, Richardson, Quinlan, Snutch, Tee, Paten, Phillippy, Simpson, Loman and Loose2018). This interesting out-of-equilibrium setup has been used before to explore fundamental aspects of the equilibrium thermodynamics (Dudko et al., Reference Dudko, Hummer and Szabo2008), from the sequence-dependent free energy profile (Huguet et al., Reference Huguet, Bizarro, Forns, Smith, Bustamante and Ritort2010) of unzipping to base pairing (Suma et al., Reference Suma, Carnevale and Micheletti2023) to the dynamical regimes appearing at different forces (Suma et al., Reference Suma, Carnevale and Micheletti2023), which differ considerably from those occurring without unzipping both in terms of typical translocation times and scaling behavior (Palyulin et al., Reference Palyulin, Ala-Nissila and Metzler2014; Chen et al., Reference Chen, Jou, Ermann, Muthukumar, Keyser and Bell2021; Suma et al., Reference Suma, Carnevale and Micheletti2023).
The examples above underscore three key points. First, the structural features of nucleic acids include physical entanglements, which can have complex and significant functional reverberations in vivo. Second, pore translocation setups are indispensable tools for mimicking the action of enzymes and probing the structural response of nucleic acid tangles at the single-molecule level. Third, the external control afforded by translocation setups, such as varying pore size and force application protocols (constant, time-ramped, oscillating), provides an ideal context for understanding the microscopic basis of the observed unzipping responses. This understanding offers crucial clues for decoding how nucleic acid architecture informs translocation.
One open problem that intersects all three aspects above is understanding how the statistically inevitable presence of knots can interfere with DNA unzipping by translocation. Studies have yet to be conducted on this process, which is qualitatively different from translocating knotted DNA without unzipping. For this reason, the insights gleaned from the pore translocation of knotted DNA cannot be directly applied to the unzipping scenario. This leaves fundamental questions about the unzipping of knotted dsDNA unanswered, such as: (i) how large must the driving force be to keep the knot tight at the pore entrance and prevent it from diffusing along the chain, (ii) what is the force-dependent topological friction, and (iii) how does this friction depend on the type of knot? These questions have implications also for in vivo DNA processing by enzymes, given that DNA knots not removed by defective topoisomerases can stall such processes, with negative consequences for the cell (Shishido et al., Reference Shishido, Komiyama and Ikawa1987; Postow et al., Reference Postow, Crisona, Peter, Hardy and Cozzarelli2001; Olavarrieta et al., Reference Olavarrieta, Hernandez, Krimer and Schvartzman2002; Deibler et al., Reference Deibler, Mann, Sumners and Zechiedrich2007, p. 44; Valdés et al., Reference Valdés, Segura, Dyson, Martínez-García and Roca2018; Valdés et al., Reference Valdés, Coronel, Martínez-García, Segura, Dyson, Díaz-Ingelmo, Micheletti and Roca2019). Although the interplay of DNA topology and unzipping is recognized as a key element of in vivo DNA transactions, the detailed characterization of the process has so far remained beyond the scope of single-molecule manipulation experiments.
Here, we address these questions with molecular dynamics simulations of a coarse-grained DNA model, oxDNA2 (Ouldridge et al., Reference Ouldridge, Louis and Doye2011; Snodin et al., Reference Snodin, Randisi, Mosayebi, Šulc, Schreck, Romano, Ouldridge, Tsukanov, Nir, Louis and Doye2015). We first consider the reference case of the nanopore unzipping of unknotted DNAs and study their translocation compliance at different forces. Next, we turn to knotted DNAs and discuss how the unzipping speed varies with knot type and applied force. Finally, we address the complementary aspect, namely how unzipping by translocation affects the knotted region, particularly its length and contour dynamics.
Notably, we do not observe significant effects related to knots at pulling forces of 50 pN, which is of the same order as the forces that can be generated by molecular motors (Smith et al., Reference Smith, Tans, Smith, Grimes, Anderson and Bustamante2001). The results are suggestive that topological entanglement may not significantly interfere with in vivo DNA unzipping operated by enzymes. However, the interplay of topology and unzipping is significantly different at 100 pN and larger forces, with major effects on the translocation process and knot sliding dynamics.
Results
To study the nanopore unzipping of knotted DNA filaments, we applied Langevin molecular dynamics simulations to 500-bp long DNA filaments described with the oxDNA2 model (Ouldridge et al., Reference Ouldridge, Louis and Doye2010, Reference Ouldridge, Louis and Doye2011; Snodin et al., Reference Snodin, Randisi, Mosayebi, Šulc, Schreck, Romano, Ouldridge, Tsukanov, Nir, Louis and Doye2015), a coarse-grained DNA representation with interactions parameters tuned to reproduce phenomenological data for DNA properties and interactions, including base pairing, stacking, and twist-bend couplings. The model’s predictive capabilities were validated in a variety of contexts, including the application of external mechanical forces (Romano et al., Reference Romano, Chakraborty, Doye, Ouldridge and Louis2013; Matek et al., Reference Matek, Ouldridge, Doye and Louis2015; Mosayebi et al., Reference Mosayebi, Louis, Doye and Ouldridge2015; Engel et al., Reference Engel, Smith, Jobst, Sajfutdinow, Liedl, Romano, Rovigatti, Louis and Doye2018).
The initial states were prepared from five different equilibrated (Monte Carlo generated) conformations of the 500 bp filaments. The five conformations were all unknotted because the 500 bp contour length, corresponding to about 10 DNA persistence lengths, is too short for significant spontaneous knotting in equilibrium (Rybenkov et al., Reference Rybenkov, Cozzarelli and Vologodskii1993; Tubiana et al., Reference Tubiana, Rosa, Fragiacomo and Micheletti2013; Uehara et al., Reference Uehara, Coronel, Micheletti and Deguchi2019). The 500-bp long filaments were next attached to leads that consisted of a double-stranded knotted region with
$ {3}_1 $
,
$ {4}_1 $
, and
$ {3}_1\#{3}_1 $
topology – the knotted region was omitted for unknotted (
$ {0}_1 $
) case – plus a 40-base long single-stranded stretch, pre-inserted into a pore (see Figure 2). The translocation process was driven by pulling the nucleotides inside the pore with a total longitudinal force,
$ f $
of 50, 100, and 150 pN. The pore diameter, 1.87 nm, was chosen small enough that only a single DNA strand can pass through it, causing translocating DNAs to unzip.
Figure 2. (a) Schematic illustration of the initial setup: an unknotted, equilibrated filament is attached to a lead consisting of a tightly-knotted double-stranded segment plus a single-stranded one pre-inserted into a cylindrical pore embedded in a slab. The narrow pore has a diameter of 1.87 nm, allowing only a single DNA strand to pass at a time. (b) Configurations of 500 bp-long DNA filaments during the simulated translocation-driven unzipping. The four snapshots are close-ups of the system near the pore and illustrate the different considered topologies: unknot (
$ {0}_1 $
), trefoil (
$ {3}_1 $
), figure-of-eight (
$ {4}_1 $
), and the composite granny knot (
$ {3}_1\#{3}_1 $
).
Nanopore unzipping of unknotted DNA
Figure 3a illustrates, for reference, the translocation response of unknotted DNA filaments. The traces show the number of translocated nucleotides as a function of time,
$ n(t) $
, for five independent trajectories at each indicated force. Note that traces start at about 40, corresponding to the length of the single-stranded DNA (ssDNA) segment of the lead that is already threaded inside the pore at
$ t=0 $
.
Figure 3. Number of translocated nucleotides,
$ n $
, as a function of time,
$ t $
for dsDNA filaments that (a) are unknotted and (b) have a
$ {3}_1 $
knot; see Figure 2 and methods. The traces are for pulling forces of 50, 100, and 150 pN, with five independent trajectories for each case. The dashed lines highlight two distinct velocity regimes in the 100 pN trajectories, a feature also present in some of the 150 pN traces. Configurations in panels (c) and (d) are snapshots at 100 pN for the
$ {3}_1 $
knot taken before and after the change in regime.
The traces at
$ f=50 $
pN have an overall linear appearance, indicative of an approximately constant unzipping velocity. However, the traces at the two largest forces, 100 and 150 pN, deviate noticeably from linearity. The convexity, or upward curvature of the late part of traces (
$ n(t)>300 $
), indicates that the average translocation speed increases in the second half of the translocation.
The translocation/unzipping speeds vary significantly across the forces. For comparison, average translocation times were computed at the 400 translocated bases mark, a convenient reference given the graphs’ range in Figure 3. The average times are equal to
$ 3.0\cdot {10}^6,\mathrm{6.9}\cdot {10}^5 $
and
$ 3.2\cdot {10}^5{\tau}_{MD} $
for
$ f=50 $
, 100, and 150 pN, respectively. In particular, we note that the above translocation/unzipping times do not follow the inverse force relationship expected for simple dissipative processes. Specifically, a twofold force variation from 50 to 100 pN produces an order-of-magnitude change in unzipping time.
The results parallel and expand those reported in Suma et al. (Reference Suma, Carnevale and Micheletti2023), where data for the out-of-equilibrium unzipping process of dsDNA were used within a theoretical framework that enabled reconstructing the free-energy profile of single base-pair formation. In that context, it was found that the unzipping process proceeded at relatively constant velocity for forces below
$ \sim 60 $
pN and could be modeled as a drift-diffusive process. At the same time, progressive speed-ups during translocation were observed at larger forces associated with an anomalous dynamics regime. By modeling the unzipping as a stochastic process in a one-dimensional tilted washboard (periodic) potential, it was shown that 60 pN force corresponded to lowering the barrier to unzip a base-pair to a value where advective transport becomes relevant over diffusion (Suma et al., Reference Suma, Carnevale and Micheletti2023). Additionally, we recall that DNA undergoes significant structural deformations, that is, overstretching, at about this same force when mechanically stretched (Smith et al., Reference Smith, Cui and Bustamante1996), and that the oxDNA2 model inherently accounts for these effects (Romano et al., Reference Romano, Chakraborty, Doye, Ouldridge and Louis2013). Thus, the crossover from linear to non-linear translocation/unzipping observed upon increasing
$ f $
from 50 to 100 pN is consistent with other qualitative changes of DNA properties in the same force range.
Nanopore unzipping of 3 1 -knotted DNA
The force-dependent translocation response is dramatically changed when the unknotted lead is replaced by a knotted one, even when the topology is the simplest non-trivial one. This emerges by inspecting Figure 3b, which shows the unzipping traces for DNA strands starting with a moderately tight trefoil-knotted (
$ {3}_1 $
) lead.
The comparison of the two panels in Figure 3 clarifies that at
$ f=50 $
pN, the unzipping of knotted and unknotted chains proceed almost undistinguishably. The average unzipping velocities of the two sets of traces, measured as nucleotides translocating per unit time, are compatible with statistical uncertainty,
$ 1.309\pm 0.028\cdot {10}^{-4}{\tau}_{MD}^{-1} $
for the
$ {0}_1 $
topology and
$ 1.288\pm 0.036\cdot {10}^{-4}{\tau}_{MD}^{-1} $
for the
$ {3}_1 $
case. The main perceived difference is the spread of the five traces, which is larger for the knotted cases.
However, increasing the force to 100 pN or more causes the unzipping of knotted chains to proceed more slowly and heterogeneously than unknotted DNAs. For
$ f=100 $
pN, the relative slowing down of the average velocity is approximately twofold, and the same holds for the largest considered force,
$ f=150 $
pN.
In addition, two different regimes are discernible, highlighted by the dashed lines for the
$ f=100 $
case, with snapshots before and after the change in regime presented in Figure 3c and d. Initially, the trefoil-knotted filament unzips at the same rate as the unknotted ones. Beyond this regime, which applies to the first 200 bp, the process slows down noticeably while also becoming more heterogeneous. An analogous effect is found for the
$ f=150 $
pN case, but with the important difference that the transient where the velocity is the same as in the unknotted case has a shorter duration and covers fewer base pairs (150). As we discuss later, the change in velocity is a consequence of the force-induced tightening of the knot near the pore entrance, which adds a significant hindrance – also termed topological friction – to the translocation process.
Effect of knot topology on DNA unzipping
We additionally considered leads with figure-of-eight (
$ {4}_1 $
) and granny (
$ {3}_1\#{3}_1 $
) knots to extend the range of topological complexity beyond the trivial (
$ {0}_1 $
) and trefoil (
$ {3}_1 $
) knot types. As a conventional measure of knot complexity, we consider the crossing number, corresponding to the minimum number of crossings in the simplest possible non-degenerate projection. This complexity measure equals 0, 3, 4, and 6 for the
$ {0}_1 $
,
$ {3}_1 $
,
$ {4}_1 $
, and
$ {3}_1\#{3}_1 $
knots, respectively.
The unzipping traces for all topologies are shown in Figure 4. We stress that we purposely attached the same set of equilibrated 500-bp long dsDNA conformations to the battery of differently knotted leads. With this choice, emerging systematic differences across the different topologies can be directly ascribed to the different knotted states of the lead and not to other effects, such as the initial DNA conformation on the cis side.
Figure 4. Number of translocated nucleotides,
$ n $
, as a function of time,
$ t $
, for DNA filaments with different knot types and at different driving forces, as indicated. The traces of five independent trajectories are shown for each case.
The data in panel (a) show that all traces are well-superposed and consistent with an approximate linear (constant velocity) behavior at the lowest considered force,
$ f=50 $
pN. This result confirms the earlier observation that the unzipping response is mainly independent of the knotted state at sufficiently small
$ f $
(Figure 3).
The data in panels (b) and (c), which refer to
$ f=100 $
and
$ 150 $
pN, respectively, are consistent with those of the trefoil knot case (Figure 3), too, in that the unzipping proceeds practically identically for all topologies of an initial tract, which spans 200 bp at
$ f=100 $
pN and 100 bp at
$ f=150 $
pN. Beyond this point, the unzipping slows down for all non-trivial knot types. At
$ f=100 $
pN, we observe that the highest unzipping hindrance is offered by the
$ {4}_1 $
knot, followed by the composite
$ {3}_1\#{3}_1 $
knot, and the
$ {3}_1 $
and
$ {0}_1 $
topologies. We recall that
$ {3}_1\#{3}_1 $
knot has the highest nominal complexity in the considered set, and yet it is not associated with the slowest unzipping at
$ f=100 $
pN, which is noteworthy. However, at 150 pN, the
$ {3}_1\#{3}_1 $
and
$ {4}_1 $
knots offer comparable hindrance, while the unzipping of the
$ {3}_1 $
case is faster and that of the unknot
$ {0}_1 $
remains the fastest.
The findings can be interpreted in terms of previously published results on the translocation – without unzipping – of knotted chains of beads (Suma et al., Reference Suma, Rosa and Micheletti2015). For such a system, it was shown that each prime knotted component behaves as a dissipative structural element that interferes with the mechanical tension propagating to the chain remainder by significantly reducing it. Without unzipping, the translocation velocity for the case of concatenated trefoil knots (
$ {3}_1\#{3}_1 $
) was mainly defined by the force dissipation within the first
$ {3}_1 $
-knotted component, which is less complex than the
$ {4}_1 $
knot. This observation helps rationalize that in specific force regimes, the hindrance of the
$ {3}_1\#{3}_1 $
case can be intermediate to the
$ {3}_1 $
and
$ {4}_1 $
ones.
The results of Figures 3 and 4 establish two points. First, the effects of DNA knots on the unzipping process are negligible, up to forces of at least 50 pN. This is a relatively large force for practical and biological purposes in that it is comparable to the force generated by the most powerful molecular motors (Smith et al., Reference Smith, Tans, Smith, Grimes, Anderson and Bustamante2001), and corresponds to the onset of the DNA overstretching transition observed in force spectroscopy (Smith et al., Reference Smith, Cui and Bustamante1996). Second, at forces of
$ 100 $
pN and beyond, the presence of knots is associated with significant slowing downs of the unzipping process depending on the interplay of knot topology and driving force.
Effect of the unzipped strand interfering with the knot
A noteworthy aspect of Figure 4 is the noticeable heterogeneity of the unzipping traces at
$ f=100 $
and
$ 150 $
pN. For instance, over the five
$ {4}_1 $
traces collected at
$ f=100 $
pN, the time required to reach the
$ n(t)=400 $
mark can range from
$ 1.2\cdot {10}^6{\tau}_{MD} $
to
$ 3.4\cdot {10}^6{\tau}_{MD} $
, a threefold ratio. For comparison, at
$ f=50 $
pN, the same ratio is only 1.02.
Visual inspection of the unzipping trajectories revealed that the heterogeneity is not only due to the presence of the knot but also to the hindrance arising from the unzipped ssDNA strand on the cis side becoming entangled with the knotted region. The effect is illustrated in Figure 5, which presents typical DNA conformations on the cis side of the pore.
Figure 5. Typical conformations of a
$ {4}_1 $
-knotted dsDNA filament at intermediate stages of translocation and increasing driving force, 50, 100, and 150 pN. At the two largest forces, one observes knot tightening and the wrapping of the cis unzipped strand around the dsDNA region proximal to the pore.
As illustrated, the knotted region typically leans against the pore entrance at the smallest considered force,
$ f=50 $
pN. However, at
$ f=100 $
pN and 150 pN, the knot is often not in direct contact with the pore but is kept at a finite distance from it by the cis unknotted strand that wraps around the dsDNA stem immediately below the knot. These wrappings arise from the torsional stress generated by the unzipping of double-helical DNA (Fosado et al., Reference Fosado, Michieletto, Brackley and Marenduzzo2021). When the stress is generated faster than it can be dissipated (Zheng et al., Reference Zheng, Suma, Maffeo, Chen, Alawami, Sha, Aksimentiev, Micheletti and Keyser2024), it can cause the relative rotation of the newly-unzipped and yet-to-unzip DNA strand, and hence their wrapping.
Like those of Figure 5, the wrapped conformations inevitably offer a multi-tier hindrance to nanopore unzipping. The translocating dsDNA experiences the combined friction from the knot and the wrapped unzipped filament to a degree that depends on the tightness and number of turns of the latter, thus increasing the heterogeneity of the unzipping process.
Knot dynamics
We next considered the sliding dynamics of the knots along the cis portion of the DNA chain, which we addressed by tracking in time the nucleotide indices corresponding to the two ends of each knot. We employed the method of Tubiana et al. (Reference Tubiana, Orlandini and Micheletti2011), which uses a bottom-up search scheme to identify the shortest segment of a chain that, once closed with a suitable arc, yields a ring with the sought knot topology (Tubiana et al., Reference Tubiana, Polles, Orlandini and Micheletti2018).
Figure 6a illustrates the typical evolution of the contour positions of
$ {3}_1 $
,
$ {4}_1 $
$ {3}_1\#{3}_1 $
knots for different forces. As indicated in the accompanying sketches, the
$ {n}_1 $
and
$ {n}_2 $
traces indicate the nucleotide indices of two ends of
$ {3}_1 $
and
$ {4}_1 $
knots and of the first (pore proximal) component of the
$ {3}_1\#{3}_1 $
composite knot. The indices for the second component of the composite knot are instead indicated as
$ {n}_3 $
and
$ {n}_4 $
. Additionally, the plots in Figure 6a show the traces of the index of the nucleotide at the pore entrance,
$ n $
.
Figure 6. (a) From top to bottom, three rows show the typical evolution of the contour positions of
$ {3}_1 $
,
$ {4}_1 $
, and
$ {3}_1\#{3}_1 $
knots in different setups. Sketches on the left provide the legend for the plotted nucleotide indices corresponding to the knot ends,
$ {n}_1,{n}_2 $
for
$ {3}_1 $
and
$ {4}_1 $
knots, and
$ {n}_1,{n}_2,{n}_3,{n}_4 $
for the
$ {3}_1\#{3}_1 $
knot. The
$ n(t) $
trace marks the index of the nucleotide at the pore entrance (or, equivalently, the number of translocated nucleotides, as in previous figures). The first column is for a setup where a base inside the pore is kept pinned. The second and third columns represent translocation cases at 50 and 100 pN, respectively. The traces in panel (b) illustrate the time evolution of the knot length,
$ {l}_k={n}_2-{n}_1 $
, for
$ {3}_1 $
,
$ {4}_1 $
topologies, and for each of the two prime components for the
$ {3}_1\#{3}_1 $
topology,
$ {l}_k={n}_2-{n}_1 $
and
$ {l}_k^{\prime }={n}_4-{n}_3 $
. The knot ends for prime and composite knots were detected using the software KymoKnot (Tubiana et al., Reference Tubiana, Polles, Orlandini and Micheletti2018, see Methods). Each plot shows the pinned case, as well as 50, 100, and 150 pN pulling forces. The traces of five independent trajectories are shown for each case.
The data in Figure 6a allows for tracking various quantities of interest as a function of time,
$ t $
. For instance,
$ n(t) $
is directly informative of the progress of the translocation/unzipping process. In contrast, the contour distance
$ {n}_1(t)-n(t) $
conveys how much the knotted region stays close to the pore during unzipping. In addition, the contour lengths of the prime knotted components are given by
$ {l}_k={n}_2(t)-{n}_1(t) $
and
$ {l}_k^{\prime }={n}_4(t)-{n}_3(t) $
and are shown in Figure 6b for the five independent trajectories of the considered cases.
Knot evolution in pinned DNA chains
The first column in Figure 6a is for the case where the ssDNA end inserted in the pore is not subject to a translocating force but is held in place by pinning a nucleotide inside the pore. The evolution of the pinned knotted configurations covers a time span of 3
$ \cdot {10}^6\hskip0.1em {\tau}_{MD} $
, comparable to the typical duration of unzipping processes at 100 pN. This case serves as a term of reference. Specifically, it establishes how the knotted DNA region evolves from its initial moderately tight state in the presence of the pore and slab but without any interference from a concurrent translocation/unzipping process and without mechanical tension propagating from the pore. The traces of the pinned case show a systematic increase in knot lengths across all three considered topologies; see also Figure 6b for
$ {l}_k $
and
$ {l}_k^{\prime } $
. The progressive loosening of knots reduces the system’s bending energy compared with the initial state, where knotted components are moderately tight (~150 bp) and significant curvature is thus packed into relatively short dsDNA stretches. The expansion of the knot is visibly asymmetric at the two ends because the knot cannot penetrate inside the pore and can only expand on the cis side.
The evolution of the
$ {3}_1\#{3}_1 $
case in the first column of Figure 6a is particularly interesting. The expansion is slowest for the first component (the one proximal to the pore), which is doubly constrained, being flanked by the slab and the pore on one side and the second
$ {3}_1 $
knot on the other. The second knotted component, pushed by the first one, eventually reaches the free DNA end and thus becomes untied. From this point, the dynamics proceeds with the remaining
$ {3}_1 $
knot, which reaches about the same size at the end of the simulated trajectory as the isolated
$ {3}_1 $
knot, about 200 bp (see also Figure 6b).
We conclude that knots in pinned DNA chains can evolve substantially, expanding and becoming untied over timespans comparable to the entire unzipping process at
$ f=100 $
pN.
Knot evolution during unzipping
The above dynamics is qualitatively modified when the pinning constraint is removed, and the DNA is forced to unzip by the driven translocation through the narrow pore.
The middle column of Figure 6a is for
$ f=50 $
pN. In the
$ {3}_1 $
case, the
$ n(t) $
,
$ {n}_1(t) $
, and
$ {n}_2(t) $
traces are overall parallel, with
$ {n}_1 $
staying close to
$ n $
at all times. These facts indicate that the
$ {3}_1 $
knot remains close to the pore entrance throughout unzipping and maintains its initial moderately tightened state (
$ {l}_k\sim 150 $
bp) as (from the relative ‘perspective of the cis chain’) it slides along the dsDNA contour at approximately constant velocity.
For the
$ {4}_1 $
and
$ {3}_1\#{3}_1 $
cases, the knots remain close to the pore entrance, and their lengths slightly increase over time, albeit to a lesser extent than for the pinned case, with the
$ {4}_1 $
reaching
$ {l}_k\sim 200 $
bp before escaping and
$ {3}_1\#{3}_1 $
reaching
$ {l}_k\sim 120 $
and
$ {l}_k^{\prime}\sim 150 $
bp for its prime components (Figure 6b).
Increasing the force to
$ f=100 $
pN introduces radical changes to knot evolution and sliding dynamics, as seen in the rightmost plots of Figure 6a. The
$ {3}_1 $
knot exhibits a substantial tightening at the pore entrance, and so does the first
$ {3}_1 $
component of the composite knot. Both values reach a stationary value of
$ {l}_k\sim 25 $
bp (Figure 6b). Instead, the length of the second component of the composite knot appears to be only modestly affected, with
$ {l}_k^{\prime } $
fluctuating over values of
$ \sim 120 $
bp. Interestingly, the length of the
$ {4}_1 $
knot also decreases with time, going from 200 nucleotides at
$ t=0 $
to 70 at
$ t=3\cdot {10}^6{\tau}_{MD} $
(Figure 6b), but never reaching the tightness observed at the late translocation stages of the
$ {3}_1 $
knot.
Finally, at 150 pN, the lengths of the
$ {3}_1 $
knot and the first
$ {3}_1 $
component of the composite knot both reach a similar asymptotic
$ {l}_k\sim 25 $
bp value as the ones of 100 pN, but at a much faster pace (Figure 6b). At this force, the
$ {4}_1 $
knot can become tighter than at 100 pN, reaching an asymptotic value of
$ {l}_k\sim 30 $
bp (Figure 6b).
The results clarify that the two dynamical regimes discussed for Figure 3b are directly connected to the degree of tightness of the knot. In fact, the
$ n(t) $
traces for
$ f=100 $
pN of Figure 6a indicate that unzipping of the chains does not proceed at a constant pace but progressively slows down. The latter occurs in correspondence with the knot length reduction, conveyed by the close approach of the
$ {n}_1(t) $
and
$ {n}_2(t) $
curves.
The slow down, as well as its dependence on the applied force and knot type, is analogous to the topological friction found in translocating knotted chains without unzipping, as observed in general polymer models in Rosa et al. (Reference Rosa, Di Ventra and Micheletti2012) and Suma et al. (Reference Suma, Rosa and Micheletti2015), and here confirmed for dsDNA, see Figure 1. Similarly to these cases, the knot slows down the process but does not necessarily halt it entirely, as the chain can still slide on its knotted contour unless the dynamics is jammed by extreme knot tightening. The degree of tightening and, in turn, the associated hindrance depends on the applied force and the knot characteristics, which can change how the tension force propagates along the chain on the cis side.
Conclusions
We used molecular dynamics simulations to study the nanopore unzipping of knotted DNA. In our study, we considered dsDNA filaments of about 500 bp prepared with different types of prearranged moderately tightened knots, namely the unknot (the trivial knot),
$ {3}_1 $
,
$ {4}_1 $
, and
$ {3}_1\#{3}_1 $
knots. The filaments were unzipped by pulling one single-stranded terminus through a narrow pore at three different forces,
$ f=50 $
, 100, and 150 pN. The progress of the unzipping process was characterized by analyzing the temporal traces of the number of translocated (hence unzipped) nucleotides and by tracking the position and length of the knotted region along the DNA contour.
The comparative analysis of the unzipping process across the considered knot types and forces enabled us to establish three main results. First, the DNA unzipping process at sufficiently low forces is virtually unaffected by the presence of knots. In fact, at
$ f=50 $
pN, the translocation traces of all three knot types were practically superposable to those of unknotted DNAs. Second, increasing the force to
$ f=100 $
and 150 pN caused knotted DNAs to unzip significantly more slowly and heterogeneously than unknotted ones. The highest hindrance was observed for
$ {4}_1 $
-knotted filaments, whose average unzipping at
$ f=150 $
pN was four times slower than the unknot. The corresponding dispersion of unzipping times was also substantial, accounting for a three-fold time difference between the slowest and fastest trajectories out of a set of five. Finally, analyzing the knotted DNA structure close to the pore revealed that the observed hindrance to unzipping involves at least two concurrent mechanisms: (i) the topological friction arising from the DNA chain sliding along its tightly knotted contour and (ii) the friction caused by the newly-unzipped cis DNA strand wrapping around the double-stranded DNA region between the knot and the pore.
The above results have implications in various physical and biological contexts. Because knots are statistically inevitable in sufficiently long DNA filaments, clarifying the impact of such forms of entanglement on how DNA unzips is relevant for polymer physics, particularly for developing predictive models for the complex force-dependent response of such processes. From the applicative point of view, the system and results discussed here could be used in prospective nanopore-based single-molecule unzipping experiments on long (hence knot-prone) DNAs, from interpreting the ionic current traces to designing such setups. Finally, DNA nanopore unzipping can be regarded as a gateway to elucidating the physical processes occurring in vivo, where genomic DNA is unzipped and translocated by various enzymes. It would thus be interesting to extend future considerations to DNA lengths and force regimes that match those relevant for in vivo DNA transactions as closely as possible, where molecular crowding may also play a role.
Model and numerical methods
We used a coarse-grained model of DNA, oxDNA2 (Ouldridge et al., Reference Ouldridge, Louis and Doye2010, Reference Ouldridge, Louis and Doye2011; Snodin et al., Reference Snodin, Randisi, Mosayebi, Šulc, Schreck, Romano, Ouldridge, Tsukanov, Nir, Louis and Doye2015), to simulate double-stranded DNA filaments of about 500 bp. Each nucleotide is treated as a rigid body with three interaction centers. The potential energy describing the interactions between nucleotides accounts for the chain connectivity, stacking effects, excluded volume interactions, twist-bend coupling, base pairing (with sequence-averaged binding interactions), and screened electrostatic interactions. The system was evolved with Langevin dynamics simulations using the LAMMPS simulation package (Henrich et al., Reference Henrich, Fosado, Curk and Ouldridge2018; Thompson et al., Reference Thompson, Aktulga, Berger, Bolintineanu, Brown, Crozier, in ’t Veld, Kohlmeyer, Moore, Nguyen, Shan, Stevens, Tranchida, Trott and Plimpton2022). The temperature was set to
$ T=300 $
K, and the monovalent salt concentration defining the Debye–Hueckel potential was set to 1 M NaCl, within the range adopted in vitro nanopore experiments. Other model parameters were set to the default values of the LAMMPS oxDNA2 implementation, except for the damp parameter, which was increased to
$ 5 $
as in Suma et al. (Reference Suma, Carnevale and Micheletti2023) to reduce inertial effects at the largest used forces. We used a timestep of
$ 0.01 $
$ {\tau}_{MD} $
, with the longest simulation lasting
$ 3.5\times {10}^6{\tau}_{MD} $
., where
$ {\tau}_{MD} $
is the characteristic simulation time.
The DNA strands have excluded volume interactions with a slab with an embedded cylindrical pore; see SI of Suma et al. (Reference Suma, Carnevale and Micheletti2023) for the potential. The pore length (slab thickness) is 8.52 nm. The nominal pore diameter was set to 1.87 nm (narrow pore) and 4.25 nm (wide pore) for translocations with and without unzipping of the double helix. The initial setup used in both situations, described hereafter, is the same. Note that 1.87 nm is a diameter sufficient to allow only a single ssDNA strand to pass at a time inside the pore. Given that the thickness (steric repulsion range) of the nanopore is 0.95 nm, the net diameter of the pore is about 1 nm, which is comparable, for instance, to the width of the lumen of biological nanopores used for unzipping, see for example, the MspA protein with a constriction of the order of
$ \sim 1 $
nm (Bhatti et al., Reference Bhatti, Jawed, Ali, Iqbal, Han, Lu and Liu2021). Instead 4.25 nm is sufficient to allow a dsDNA strand to pass, but not a knot, which is necessarily composed of
$ \ge 3 $
strands and hence bound to remain in the cis side of the pore.
To produce the initial conformation, we used an analogous scheme to Suma et al. (Reference Suma, Rosa and Micheletti2015): we first employed a Monte Carlo scheme to sample equilibrated configurations of coarse-grained semi-flexible chains with thickness, contour length, and persistence length corresponding to double-stranded DNA filaments of 500 bp. At the front of the chain, a tightened knot was attached of three different types,
$ {3}_1 $
,
$ {4}_1 $
,
$ {3}_1\#{3}_1 $
, taken from simulations of Suma et al. (Reference Suma, Rosa and Micheletti2015), and long about 50 bp. For the
$ {0}_1 $
unknotted case, we did not add anything.
The knotted terminus was then attached to a 40-base lead already threaded through the pore. The configuration was subsequently relaxed using an intermediate fine-grained model (see Suma and Micheletti (Reference Suma and Micheletti2017)) for the specifics, by pinning one nucleotide inside the pore. During this relaxation, the initially tightly knotted components expand to about 150 bp to lower the bending energy. The conformation was then mapped to the oxDNA2 representation of double-helical DNA with the tacoxDNA package (Suma et al., Reference Suma, Poppleton, Matthies, Šulc, Romano, Louis, Doye, Micheletti and Rovigatti2019), with the lead inside the pore mapped into a single-stranded DNA. The whole chain was again briefly relaxed by pinning one nucleotide inside the pore and letting the system evolve for a time span of
$ 200{\tau}_{MD} $
. Translocation was driven by a longitudinal force, f = 50, 100, 150 pN, acting exclusively on the DNA segment inside the pore and equally distributed among the nucleotides in the pore. This technical expedient is adopted to keep the driving force constant.
The relaxed filaments were translocated and unzipped by pulling the ssDNA stretch inside the pore with a constant total force,
$ f $
.
A resulting initial conformation is shown in Figure 2a. At variance with Suma et al. (Reference Suma, Carnevale and Micheletti2023), here we show the translocation process for this configuration instead of unzipping the first 200 bp bases, as our main interest is to study the knot positioning and effects. Five different Monte Carlo-generated configurations were used for each topology, and their sequence composition was also randomly picked at the oxDNA fine-graining step. The resulting conformations for the unknot and the three knot types are displayed in Figure 2b during translocation.
Detection of knots was carried out using the software KymoKnot (Tubiana et al., Reference Tubiana, Polles, Orlandini and Micheletti2018). From a mathematical point of view, knots are rigorously defined only for circular chains. Accordingly, to establish the knotted state of an open chain, it is necessary to close it into a ring by bridging its terminals with a suitable auxiliary arc (Tubiana et al., Reference Tubiana, Orlandini and Micheletti2011). This step was carried out with the so-called minimally interfering closing procedure, which selects the auxiliary arc that adds the least possible entanglement to the open chain. After closure, the knotted state of the chain is established using the standard Alexander determinants. This way, we assign a definite topological state to each configuration sampled in the MD trajectory and select the DNA nucleotide indexes that delimit the knotted region (further reducing the polymer region would result in not being able to detect the knot). For prime knots, these correspond to indexes
$ {n}_1 $
and
$ {n}_2 $
, while for composite knots, they correspond to indices
$ {n}_1 $
and
$ {n}_4 $
(Figure 6a). The prime components within a composite knot were identified by using a bottom-up search.
Open peer review
To view the open peer review materials for this article, please visit http://doi.org/10.1017/qrd.2024.26.
Acknowledgements
This study was funded in part by the European Union – NextGenerationEU, in the framework of the PRIN Project ‘The Physics of Chromosome Folding’ (code: 2022R8YXMR, CUP: G53D23000820006) and by PNRR Mission 4, Component 2, Investment 1.4_CN_00000013_CN-HPC: National Centre for HPC, Big Data and Quantum Computing – spoke 7 (CUP: G93C22000600001). The views and opinions expressed are solely those of the authors and do not necessarily reflect those of the European Union, or can the European Union be held responsible for them.
Competing interest
The authors declare no competing interest. All authors have contributed to this submission.
Open access
Comments
Dear Editors, <br>
We are pleased to submit our manuscript titled “Unzipping of knotted DNA via nanopore translocation”, for consideration for publication in “Quarterly Review of Biophysics Discovery”.
We are submitting this manuscript in response to the invitation by Prof. Felix Rirort and Prof. Fredrik Westerlund to contribute a research study to the article collection on the theme “Single Molecule Challenges in the 21st Century”.
<br>
In our theoretical and computational work, we consider nanopore translocation, one of the most powerful single-molecule probing techniques, and discuss its prospective use to study the unzipping of DNA filaments that are knotted.<br>
The combination of DNA knots and unzipping is a novel problem that has yet to be addressed before despite being relevant in biological contexts. Previous studies have addressed the pore translocation of knotted DNA without unzipping, as well as the unzipping of translocating DNA without knots. Both these contexts revealed unexpected complexities. For instance, when knotted DNA translocates through narrow pores without becoming unzipped, significant friction can be generated in the tightened knotted region, which may even stall the translocation process at sufficiently high driving forces. On the other hand, when unknotted DNA is unzipped by pulling only one of the DNA strands inside a pore, a remarkable transition from normal to anomalous dynamical regimes is observed upon increasing the driving force.<br>
Here, by addressing the translocation-driven unzipping of knotted DNA, we take a first step towards clarifying the unzipping action of enzymes such as helicases on DNA in vivo, which is inevitably knotted and entangled.
To this end, we report on a numerical study where we used molecular dynamics simulations on a coarse-grained model of DNA to study the nanopore-driven unzipping in the presence of various knot types.
We observe that, while at low forces knots do not interfere with unzipping, at large ones knotted DNA unzips more slowly and heterogeneously than unknotted DNA. As we discuss, the microscopic origin of the hindrance
typically involves two concurrent causes: the topological friction of the DNA chain sliding along its knotted contour and the additional friction originating from the entanglement with the newly unzipped DNA strand.<br>
These results reveal a previously unsuspected complexity of the interplay of DNA topology and unzipping, which we expect to appeal to the broad community of biological physicists and experimentalists, including those interested in advanced applications of single-molecule probing techniques. For these considerations, we believe that “Quarterly Review of Biophysics Discovery” would be the ideal avenue for our study.<br>
Yours sincerely,<br>
Antonio Suma and Cristian Micheletti