# How molecular knots can pass through each other

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Edited by Michael L. Klein, Temple University, Philadelphia, PA, and approved April 24, 2014 (received for review October 23, 2013)

## Significance

We suggest and explain a mechanism in which two molecular knots on a DNA strand can pass through each other and swap positions along the strand: One of the two knots expands in size and the other diffuses along the contour of the former. This peculiar mechanism, which only requires a few *k _{B}T*, is not only interesting from an aesthetic point of view but may also play a role in future technological applications such as nanopore sequencing once strand sizes exceed 100,000 base pairs.

## Abstract

We propose a mechanism in which two molecular knots pass through each other and swap positions along a polymer strand. Associated free energy barriers in our simulations only amount to a few *k _{B}T*, which may enable the interchange of knots on a single DNA strand.

Ever since Kelvin conjectured atoms to be composed of knots in the ether (1), knots have stimulated the imagination of natural scientists and mathematicians alike. In recent years, the field went through a renaissance and progressed considerably, spurred by the realization that topology may not only diversify structure but can also have a profound impact on the function of biological macromolecules. Knots in proteins have been reported (2⇓⇓⇓⇓⇓–8) and even created artificially (9). Topoisomerases can remove (10) or create (11) knots in DNA, which may otherwise inhibit transcription and replication, and viral DNA is known to be highly knotted in the capsid (12⇓⇓⇓–16). Artificial knots have also been tied in single DNA molecules with optical tweezers, and dynamics have been studied both experimentally and with computer simulations (17, 18). Knots are also known to weaken strands, which tend to rupture at the entrance to the knot (19, 20). Even though most of these examples are not knotted in a strict mathematical sense (21), which only defines knots in closed curves, they nevertheless raise fundamental questions and challenge our understanding of topics as diverse as DNA ejection (22) and protein folding (23). Knots may also play a role in future technological applications, particularly in the advent of DNA nanopore sequencing (24). While the probability of observing a knot in a DNA strand of 10,000 base pairs in good solvent conditions only amounts to a few percent (25, 26), knots and even multiple knots will become abundant once strand sizes exceed 100,000 base pairs in the near future. Part of this problem was recently addressed in a simulation study (27): A single knot will not necessarily jam the channel once it arrives at the pore but may slide along its entrance.

In the following, we would like to elucidate a fascinating and little-known property of composite knots: Two knots can diffuse through each other. In our simulations, we use a standard bead-spring polymer model (28), which does not allow for bond crossings if local dynamics are applied. Furthermore, we apply an additional angular potential and tune the stiffness of the chain such that in good solvent (high salt) conditions, the persistence length of DNA is reproduced (for *κ* = 20 *k*_{B}*T*). Our polymer consists of 250 monomers, which corresponds to roughly 1,875 base pairs. Details on our coarse-grained model, the mapping onto DNA, and determination of knot sizes are given in *Materials and Methods*. Note that no bias was applied and our simulations are solely driven by thermal fluctuations. A video of one “tunneling” event is provided (Movie S1).

In Fig. 1 we have prepared a starting configuration with a trefoil knot

## Results

In Fig. 2*A*, we follow the location of the knot centers with respect to each other, and record their distance (in units of monomers) as a function of simulation time. In this framework, the two knots are separated when the two centers are around 100 monomers apart. At *A*, knots may pass through each other over and over again via an entangled intermediate state. Can we understand this peculiar diffusion mechanism? In Fig. 2*C* (which shows the same section as in Fig. 2*A*), we record the size of each knot. When two knots are separated, the trefoil knot occupies around 60 monomers, whereas the figure-eight knot is slightly larger at around 80 monomers. In the entangled intermediate state, one of the knots suddenly expands to a bit less than the combined size of the two knots in the separated state, whereas the size of the other knot grows only marginally. Intriguingly, it is not always the larger figure-eight knot that expands, even though its expansion is a bit more likely, as can be seen in the accumulated histogram in Fig. 2*D*. As the two knot centers more or less coincide in the entangled state, we conclude that the smaller knot diffuses along the strand of the enlarged knot (as depicted in Fig. 1*B*) until the two are separated again. They may then either occupy the same positions as before or have interchanged positions along the strand. At large stiffness, the probability distribution of the intertwined state is split up into a triple peak (Fig. 2*B*), which emerges from two separate contributions. If the *B*).

We can also derive an estimate for the “topological” free energy barrier, which needs to be overcome in a “knot swapping” event. This barrier essentially accounts for the obstruction caused by entanglements. In Fig. 2*B* we have accumulated data from simulations as shown in Fig. 2*A* to obtain a histogram of the time series and a corresponding probability distribution. For *κ* = 20 *k*_{B}*T*, the most likely state is the combined state, whereas the separated states are metastable.

From Fig. 2*B*, the “topological” free energy is derived as *κ* = 0 *k*_{B}*T* in Fig. 3*A*), the system first needs to overcome a barrier, *κ* = 20 *k*_{B}*T* in Fig. 3*B*), the system needs to overcome *k*_{B}*T*, which would be accessible in experiments. Can we alter this barrier? Fig. 3*A* shows free energy profiles from simulations with different angular stiffness at the same wall distance. While, in the case of the lowest stiffness, the separated states are more likely, the intertwined state is more probable at larger stiffnesses as indicated above. Fig. 3*B* also shows free energy profiles from simulations in which the walls were placed closer together (to

## Discussion and Conclusion

In conclusion, we present a mechanism that allows for two molecular knots to diffuse through each other and swap positions along a strand. The corresponding free energy barrier in our simulations only amounts to a few *k*_{B}*T* and should be attainable in experiments similar to ref. 17 (with loose composite knots) and, potentially, in vivo. The barrier can be altered by changing the chain stiffness as well as the wall distance to make the “tunneling” event more or less probable. To what extent this peculiar diffusion mechanism might affect DNA behavior in nano-manipulation experiments will be investigated in future studies.

## Materials and Methods

### Model and Simulation Details.

The model we apply is essentially a discrete variant of the well-known worm-like chain model (with excluded volume interactions), which has been used extensively to characterize mechanical properties of DNA (25, 26, 32, 33). We start with a standard bead-spring polymer model from ref. 28, which does not allow for bond crossings. All beads interact via a cut and shifted Lennard−Jones potential (Eq. **1**). Adjacent monomers interact via the finitely extensible nonlinear elastic (FENE) potential (Eq. **2**). Chain stiffness is implemented via a bond angle potential (Eq. **3**), where angle *i* − 1, *i*, and **4**), where *i*. For simplicity, we define the normal vector of the walls to coincide with the *x* axis of our system.

with *y* and *z* coordinates. The simulations are run with the CPU version of HOOMD (29) and use the implemented Langevin dynamics thermostat at *κ* = 10 *k*_{B}*T*,

### Mapping onto DNA.

In the context of knots, a similar model was applied in ref. 25 where the parameters were obtained from mapping the probability for obtaining trefoil knots in the polymer model onto the experimental probability observed for a DNA strand of 11,600 base pairs as a function of NaCl concentration. Our model (which is based on this model) has essentially two parameters, which can be fitted to mimic real DNA: the chain stiffness κ and the diameter of the chain (σ in Lennard−Jones units). σ is taken from ref. 25. For high salt concentration (1 M NaCl), the effective diameter of the chain is slightly larger then the locus of DNA

The relevant energy scale of our model is defined by κ in Eq. **3**. For the discrete worm-like chain model,

As our model features excluded volume interactions, variable bond lengths, and angles, we have verified this relation by measuring the persistence length (from the decay of the bond angle autocorrelation function) as a function of κ in simulations of unbound chains. Hence, for high salt conditions

From Eq. **5**, we also obtain the persistence length in simulation units. For

To confirm the validity of our model, we have undertaken extensive Monte Carlo simulations (with fixed bond lengths). We have obtained the probability of observing trefoil knots in an 11.6-kilobase DNA strand in high salt concentration (1 M NaCl,

### Detection and Localization of Knots.

To be able to detect knots, the chain has to be closed first. This is done by drawing a line outwards and parallel to the walls from the fixed beads. Then we connect these lines with a large half-circle. After the closure, we calculate the products of the Alexander polynomials

For each configuration we confirm that there was no bond-crossing by computing *x* axis by using the arithmetic mean.

### Data Analysis.

The computational determination of knot sizes as described above typically results in strongly fluctuating data, even if underlying structures are similar. The method's immanent noise covers up relevant features of the transition such as the triple peak in Fig. 2*B* and the slightly increased size of the translocating knot in the intertwined state (Fig. 2*D*; compare with Fig. S1). It also (artificially) broadens the peaks of the probability distribution at the expense of the transition states. For this reason, it is not recommended to apply the data analysis directly to raw data. Instead, we have chosen to smoothen the data by applying a running average over 100 adjacent data points. Note that the length of this interval has a minor influence on the barrier height as shown in Fig. S2.

## Acknowledgments

P.V. would like to thank M. Kardar for pointing out that two knots on a rope may change their position and G. Dietler, C. Micheletti, and E. Rawdon for helpful discussions. We would also like to thank F. Rieger for performing the Monte Carlo simulations in *Materials and Methods* and D. Richard for his work on the analysis method in *Supporting Information* (Figs. S1 and S2). B.T. and P.V. would like to acknowledge the MAINZ Graduate School of Excellence for financial support.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: virnau{at}uni-mainz.de.

Author contributions: P.V. designed research; B.T. and J.S. performed research; B.T., J.S., and P.V. analyzed data; and B.T. and P.V. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1319376111/-/DCSupplemental.

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