## Abstract

In the present article, we investigate and review the influence of chain stiffness on self-entanglements and knots in a single polymer chain with Monte Carlo simulations spanning good solvent, theta and globular phases. The last-named are of particular importance as a model system for DNA in viral capsids. Intriguingly, the dependence of knot occurrence and complexity with increasing stiffness is non-trivial, but can be understood with a few simple concepts outlined in the present article.

- knot
- Monte Carlo simulation
- polymer
- stiffness

## Introduction

Day-to-day experience reveals the unfortunate tendency of cables and cords to embrace entanglements and knots when left unattended, particularly after exposure to confinement in various forms. As such, knots provide an excellent gauge for self-entanglement, and concepts from knot theory are used to analyse ‘molecular cords’ ranging from synthetic polymers [1–3] to proteins [4–19] and DNA [20–23].

Mathematically, knots are only well-defined in closed curves [24] and classified according to the minimum number of crossings in a projection on to a plane: a ring without crossings is referred to as an ‘unknot’, whereas the simplest, so-called ‘trefoil knot’ (3_{1}) has three crossings. There is one knot with four crossings (4_{1}), two with five (5_{1}, 5_{2}), three with six (6_{1}, 6_{2}, 6_{3}) and thereafter the number of distinct knots as a function of crossings increases exponentially [25]. In the present mini-review, we apply a more physical definition which is more in tune with our everyday experiences and allows us to consider knots in open linear chains [1,26]. The two termini of the chain are connected in a well-defined manner (thus closing the chain) before knot detection algorithms, e.g. based on the evaluation of the Alexander polynomial [24,27], are applied. Preferably, the closure applied should minimize the chances of introducing additional crossings that could change the knot type [28,29]. Alternatively, one could also apply many random closures [4,30] and select the knot type which occurs most frequently.

In the present article, we discuss knottedness in several single polymer phases. We take a somewhat unusual, but nevertheless instructive, approach and investigate the emergence of self-entanglements and knots as a function of stiffness. Even though such an investigation would be difficult in experiments, it is straightforward to implement in computer simulations. Typical bond angle potentials [23] between adjacent bonds have the form *V*/*k*_{B}*T*=*B*(1−cosθ) with θ being the angle between the tangent vectors of two adjacent chain segments. By modifying the amplitude *B*, stiffness and persistence length of the polymer can be adjusted.

## Ideal chains and good solvent conditions

The random walk or freely jointed chain (which describes a polymer under melt or theta conditions) is the simplest model of an ideal polymer and one of a few whose structural properties can actually be calculated analytically [31]. It was also the first model for which knots were determined in a computer simulation [32]. Random walks tend to be fairly knotted by construction (as neighbouring monomers have zero volume and connecting bonds are therefore completely uncorrelated). This behaviour changes completely once excluded volume is taken into account [33,34]: adjacent monomers can no longer be placed in the direct vicinity of each other, correlations emerge and knots become rare (compare with Figure 1, *B*=0).

Even though these two cases have been studied extensively, we are only aware of a single study which investigates the influence of stiffness on knots in a self-avoiding lattice model [35]. In the present article, we discuss these effects within a Monte Carlo framework where excluded volume between non-bonded monomers (implemented via a repulsive *r*^{−12} potential) is easily turned on or off. In our model, adjacent beads are connected by bonds of fixed length, the previously introduced bond angle potential (with amplitude *B*) allows us to continuously adjust the stiffness of the polymer chain.

In Figure 1, the probability of observing an unknotted configuration of a freely moving single polymer chain of length *N*=150 is shown as a function of chain stiffness for good solvent conditions (with excluded volume) as well as for ideal chains (without excluded volume): whereas the probability for an unknot increases monotonously as a function of chain stiffness for the latter, there is a non-trivial functional dependence for good solvent conditions: here, the unknot probability of completely flexible chains (*B*≪1) is relatively high, followed by a rather fast and monotone decrease when we pass over to the regime of semi-flexible chains where the unknot probability reaches a minimum at *B*≈4 before it increases again, and approaches *P*=1 in the limiting case of a rigid rod. Although the overall effect is rather small for chains of length *N*=150 (see Figure 1), the same qualitative behaviour can also be observed for significantly larger chains. These results also corroborate findings in [35] where the knotting probability of lattice polygons (of sizes up to *N*=3200) has been studied as a function of curvature fugacity γ.

The study of freely moving chains under good solvent conditions illustrates that the non-trivial behaviour of the knotting probability is due to the influence of excluded volume interactions on the topology of typical chain conformations: the prerequisite for the appearance of knotted chain conformations is the formation of loops through which the chain can thread itself. For small stiffnesses, these loops are strongly suppressed by excluded volume interactions. For large values of the parameter *B*, loops vanish again as the chain becomes more and more rod-like. Only in the semi-flexible case (*B*≈4) can neither excluded volume interactions nor the stiffening of the chain prohibit the formation of looped chain conformations.

## Spherical confinement

In contrast with the study of semi-flexible polymers in good and theta solvent, the case of spherical confinement has attracted considerable attention in recent years as was also investigated experimentally. In a series of groundbreaking experiments, it was established that DNA in phage capsids tends to be knotted [36–38]. This is particularly true for artificial mutations of the virus in which both sticky ends of the DNA may enter the capsid and cyclize in confinement [38,39]. (In the wild-type, one end remains in the loading channel which serves as an injection needle.) Theoretical approaches [21,22,40–42] mostly focused on describing the arrangement in a coarse-grained manner. The most sophisticated models [22] include not only the correct persistence length of DNA, but also a cholesteric potential which correctly reproduces the toroidal structure of DNA and the experimentally observed [39] preference for torus-type knots (e.g. 3_{1}, 5_{1}). In the present article, we follow a more generic approach detailed in [23], and discuss the change in structure and the emergence of entanglements at a fixed chain length as stiffness increases.

First, we inspect visually the influence of stiffness on structure of a freely moving single polymer chain confined to a spherical capsid [23,43] (Figure 2). Colours change gradually from red to blue as we progress from the first to the last monomer to illustrate local order.

Although a fully flexible chain (Figure 2, left, *B*=0) is only ordered locally, an emerging bending rigidity (Figure 2, middle, *B*=10) straightens the chain. For even larger stiffnesses (Figure 2, right), the chain is spooled up inside the capsid in configurations which resemble those observed in viral DNA (for which *B*≈20 [44]). Even though the toroidal structure on the right appears more orderly than the flexible chain, the topological analysis tells its own tale. Figure 3 shows the fraction of unknotted configurations and of configurations which contain complex knots with more than five fundamental crossings. For *N*=200, the locally ordered flexible chains are almost always unknotted. (A significant fraction of knotted chains appears for *N*≈500 and similar density in agreement with [1]). Intriguingly, a moderate increase in stiffness reduces the number of unknotted configurations significantly to 15% and, at the same time, the fraction of complex knots increases to 65%. At *B*≈15, the persistence length of the chain roughly corresponds to the diameter of the enclosing sphere [43,44]. Consequently, the chain is ‘reflected’ back and forth at the capsid wall and large loops appear through which the chain can thread itself. These loops are a necessary prerequisite and are ultimately responsible for the formation of knots. Interestingly, both the fraction of unknotted configurations and of configurations with complex knots saturate for large stiffnesses even though an increasingly ordered structure is imposed. Again, this phenomenon can be explained by a careful inspection of Figure 1 (right). Whereas the structure is globally ordered, the toroidal structure is not perfect and the chain crosses itself close to the capsid which again leads to the formation of entanglements and knots. Intriguingly, if the stiffness is high enough, torus knots are once again preferred, even though no cholesteric potential is applied [23], which indicates that the emergence of toroidal structure is always accompanied by a preference towards torus knots. As indicated above, in wild-type phage DNA, one end of the chain typically stays at the loading channel. To mimic this situation, we fixed one end at the surface. However, knotting probabilities are almost identical with the (confined) freely moving case.

## Conclusions

In the present mini-review, we discuss the effect of chain stiffness on knotting in various single-chain polymer phases. Whereas knots in flexible random walks are abundant, they become less and less likely with increasing stiffness until they vanish in the limit of a stiff rod. In contrast, the behaviour of a (swollen) polymer coil with excluded volume interactions (good solvent conditions) is quite non-trivial as the dependence of knotting probabilities on chain stiffness is non-monotonous. Even though knots are rare for flexible chains and in the stiff rod limit, there exists a stiffness regime for semi-flexible chains where knots become more abundant. In this regime, the chain is able to form loops through which the chain can thread itself, a prerequisite for the appearance of knots. A qualitatively similar effect can be observed for polymers in spherical confinement. Although knots are not unlikely for flexible chains, the probabilities of finding knots increase considerably as the stiffness increases. Once the persistence length of the chain is similar to the diameter of the containing sphere, the chain is ‘reflected’ at the wall end and loops occur. Intriguingly, the percentage of knotted configurations remains high in toroidal structures which are obtained for even larger stiffnesses. In these structures, torus knots are preferred, whereas twist knots are suppressed.

In conclusion, we observe that the topological state of a single polymer chain can exhibit a rather non-trivial dependence on chain stiffness if physical interactions, namely excluded volume interactions within the chain or constraints which are imposed by the environment, are taken into account: the appearance of knots is connected to the emergence of looped chain conformations, the formation of which are regulated by chain stiffness.

## Funding

This work received financial support from the Deutsche Forschungsgemeinschaft [grant number SFB 625-A17] and the Graduate School of Excellency Materials Science in Mainz.

## Acknowledgments

P.V. thanks Peter Cifra and Andrzej Stasiak for their contributions to the study of confined polymers. We are grateful to Hsiao-Ping Hsu and Benjamin Trefz for their involvement in the initial stages of the good solvent case. P.V. also thanks Friederike Schmid and Kurt Binder for support and stimulating discussions.

## Footnotes

Topological Aspects of DNA Function and Protein Folding: An Independent Meeting held at the Isaac Newton Institute for Mathematical Sciences, Cambridge, U.K., 3–7 September 2012, as part of the Isaac Newton Institute Programme Topological Dynamics in the Physical and Biological Sciences (16 July–21 December 2012). Organized and Edited by Andrew Bates (University of Liverpool, U.K.), Dorothy Buck (Imperial College London, U.K.), Sarah Harris (University of Leeds, U.K.), Andrzej Stasiak (University of Lausanne, Switzerland) and De Witt Sumners (Florida State University, U.S.A.).

- © The Authors Journal compilation © 2013 Biochemical Society