Open Access Article

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Beatrice W.
Soh
^{a},
Vivek
Narsimhan
^{b},
Alexander R.
Klotz
^{a} and
Patrick S.
Doyle
*^{a}
^{a}Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. E-mail: pdoyle@mit.edu
^{b}Department of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907, USA

Received
7th November 2017
, Accepted 4th February 2018

First published on 9th February 2018

We perform single-molecule DNA experiments to investigate the relaxation dynamics of knotted polymers and examine the steady-state behavior of knotted polymers in elongational fields. The occurrence of a knot reduces the relaxation time of a molecule and leads to a shift in the molecule's coil–stretch transition to larger strain rates. We measure chain extension and extension fluctuations as a function of strain rate for unknotted and knotted molecules. The curves for knotted molecules can be collapsed onto the unknotted curves by defining an effective Weissenberg number based on the measured knotted relaxation time in the low extension regime, or a relaxation time based on Rouse/Zimm scaling theories in the high extension regime. Because a knot reduces a molecule's relaxation time, we observe that knot untying near the coil–stretch transition can result in dramatic changes in the molecule's conformation. For example, a knotted molecule at a given strain rate can experience a stretch–coil transition, followed by a coil–stretch transition, after the knot partially or fully unties.

Theoretically, it has been proven that the knotting probability of a chain approaches unity as the chain length tends to infinity, hence knots are inevitably present in long polymer chains.^{21} The probability of knot formation in DNA has been studied both computationally^{22–24} and experimentally^{25,26} on length scales of relevance. Indeed, knots are known to occur in DNA,^{27,28} proteins^{29,30} and synthetic polymers.^{31} The presence of knots is known to have important ramifications for biology and biotechnology applications, from adding complexities to the folding energy landscapes of proteins^{32} to halting translocation of polymers through nanopores.^{33,34} From the perspective of polymer physics, knots have also been shown to affect overall polymer properties.^{19,35–39} For example, simulations have indicated that a knot reduces the mechanical strength of a polymer chain, and that a knotted chain under tension almost always breaks at the entrance to the knot.^{35}

Most studies on knotted polymers to date have been theoretical and computational in nature.^{35–40} Over the past decade, researchers have demonstrated experimental methods of introducing knots in biopolymers via chemical synthesis,^{41,42} optical tweezers^{43,44} and electric fields.^{45,46} The development of such techniques has ignited interest in the experimental probing of knotted polymer dynamics. Knots tied onto DNA chains held under tension were found to be mobile, and the diffusion of different knot types was observed and quantified.^{44} Previous work from our group investigated the swelling of knots in DNA molecules during chain relaxation and showed that the knot growth time scale is governed by the global chain relaxation time scale.^{20} Knotted DNA molecules subjected to elongational fields were shown to exhibit nucleation-type behavior of the coil–stretch transition and to stretch at a much slower rate compared to unknotted molecules.^{19} Experimentally, the dynamics of knotted polymers is a rich area for exploration. In particular, the effect of knots on the relaxation dynamics of polymers has yet to be investigated experimentally in detail.

Recently, our group performed a computational study of the steady-state and transient dynamics of knotted polymers in elongational flow and examined how knots modify the relaxation time of a chain.^{40} In the present work, we use single-molecule experiments to study the relaxation dynamics of knotted polymers and the effects of a knot on steady-state behavior of polymer chains in elongational fields. We use strong electric fields to induce knots in DNA molecules via an electrohydrodynamic instability^{45} and quantify the relaxation times of knotted molecules in relation to knot size. We then subject molecules with knots to planar elongational fields and observe the steady-state dynamics of the chain. Specifically, we focus on the steady-state fractional extension and extension fluctuations of the chain, and compare the dynamics of molecules with and without a knot. We find that the occurrence of a knot markedly changes the steady-state chain dynamics in elongational fields, which can be resolved by taking into account the faster relaxation time and reduction in free chain contour of the knotted molecule.

v_{x} = μE_{x} = x | (1) |

v_{y} = μE_{y} = −y | (2) |

The experimental buffer contained 4 vol% β-mercaptoethanol (BME, Calbiochem) and 0.1% 10 kDa polyvinylpyrrolidone (PVP, Polysciences) in 0.5× TBE solution. T4 DNA (165.6 kbp, Nippon Gene) was stained with fluorescent dye YOYO-1 (Invitrogen) at a base pair to dye ratio of 4:1, resulting in a 38% increase in DNA contour length^{51} and a stained contour length of 77 μm. The stained DNA solution was allowed to sit at least 12 hours prior to use and diluted in experimental buffer to an optimal viewing concentration of 0.03 μg mL^{−1} immediately before experiments. Single DNA molecules were observed using an inverted Zeiss Axiovert microscope with a 63× 1.4 NA oil-immersed objective and images were recorded using a Photometrics Prime 95B sCMOS camera. The extension of the molecule X_{ex} was determined by the projected distance along the primary axis of extension of the field.

(3) |

We consider the relaxations of an initially unknotted molecule that was induced to contain two knots, one of which eventually untied to leave the molecule with only one knot. Fig. 3 shows the scaled squared extension of the molecule with different number of knots as a function of time. The fractional knot sizes of the single and double knot at ∼30% chain extension are 0.07 ± 0.02 (95% confidence interval) and 0.16 ± 0.02 respectively. Each curve represents the average of three to four relaxations and the dotted lines are the fits to the last 30% extension. Based on the relaxation trajectories, it is evident that the relaxation process depends on the number of knots in the molecule. Specifically, the more knots the molecule has, the quicker the molecule relaxes back to equilibrium. The relaxation times of a molecule containing no knot, one knot and two knots are 1.46 ± 0.03 s, 1.28 ± 0.02 s and 1.07 ± 0.02 s respectively. We observe from the inset of Fig. 3 that the molecule containing two knots has a markedly smaller chain extension at equilibrium compared to the molecule with one knot, which in turn has a smaller chain extension compared to the molecule with no knot, consistent with there being more stored contour in two knots versus one knot, and one knot compared to no knot.

Qualitatively, we can rationalize the faster relaxation of knotted molecules by considering the free contour length of the molecule. Previous studies involving knots in T4 DNA molecules induced by an electrohydrodynamic collapse have estimated the knot contour in tight knots to be between 1 μm and 5 μm,^{19,20} which corresponds to between 1% and 6% of the contour length. Stored contour in the knot results in the molecule having less free chain contour for relaxation, and smaller molecules relax at a faster rate.^{54}

To further investigate the relaxation dynamics of knotted molecules, we quantify the relaxation process for an ensemble of molecules containing knots with a range of sizes. Fig. 4 shows the relaxation times of 24 knotted molecules relative to the relaxation time of unknotted molecules as a function of knot size at ∼30% chain extension. Each data point is obtained by averaging over at least four relaxations for the same knotted molecule. Although partial untying of the knot between stretch–relax cycles was possible, we do not observe any discernible trend in knot size over >10 relaxations (see ESI†). The relaxation time of unknotted molecules τ_{unknot} is extracted from the relaxation trajectory averaged over 50 relaxations from 24 molecules and determined to be 1.55 ± 0.02 s. The knot size for each knotted molecule is measured when the molecule is at ∼30% extension, corresponding to the onset of the linear force regime over which the relaxation is fit. It is important to highlight that the knot size continually increases during the relaxation process, thus the reported knot size is not an invariant. As seen from Fig. 4, there is a distinct relationship between relaxation time of a knotted molecule and knot size. In general, the larger the knot size at the onset of the linear force regime, the smaller the relaxation time of the molecule, with a Pearson correlation coefficient of −0.77.

Fig. 4 Ratio of relaxation times between a knotted and unknotted chain ( = τ_{knot}/τ_{unknot}) as a function of fractional knot size at ∼30% extension ( = L_{knot}/L_{chain}) for 24 T4 DNA molecules. Each data point corresponds to a different knotted T4 molecule, averaged over at least four relaxations. The filled circles represent molecules with one knot and the open circles represent molecules with two knots. The fractional knot size is measured as the fraction of chain occupied by the knot at the onset of the linear force regime. The curves correspond to Rouse theory = (1 − )^{1+2v} and Zimm theory = (1 − )^{3v} where ν = 0.588 is the excluded volume Flory exponent.^{55} Error bars represent 95% confidence interval (see ESI† for calculation). |

We can understand the faster relaxation times of knotted molecules by considering how the occurrence of knots reduces the free contour length of the chain. According to the Rouse model,^{56} the relaxation time of a freely-draining chain scales with chain length as τ_{rouse} ∝ L^{1+2v}, where L is the contour length and ν = 0.588 is the excluded volume Flory exponent.^{55} For a knotted molecule, the free chain contour that is allowed to relax is L_{free} = L_{chain} − L_{knot}. Hence, we have

τ_{rouse} ∝ (L_{chain} − L_{knot})^{1+2v} |

τ_{zimm} ∝ (L_{chain} − L_{knot})^{3v} |

Having observed a decrease in relaxation time with increasing knot size, we proceed to consider the effect of knot position on the relaxation process. We look at two molecules with similar knot sizes at ∼30% chain extension. Each molecule experienced 10 stretch–relax cycles, and the fractional knot sizes at ∼30% extension were between 0.12 and 0.14 across all relaxations. While the knot sizes remained largely constant, the knot positions varied through the stretch–relax cycles, with the knots being located at positions spanning about 20% of the chain for each molecule. Due to the difficulty in measuring an accurate knot position on a weakly extended chain, we quantify the knot position in the highly stretched chain prior to switching off the electric field. The knot position is measured as the fractional distance between the center of the knot and closer end of the chain, such that a knot position of 0.35 indicates a knot located 35% into the chain from one end and 65% into the chain from the other end. Although the initial knot position is not necessarily the position of the knot as the chain relaxes, we notice that the knot position shifts gradually (<0.05) between consecutive relaxations, hence the initial knot position is a reasonable estimate of the knot position during chain relaxation. Fig. 5 shows the averaged trajectories for relaxations of the molecules with different initial knot positions between 0.25 and 0.45. Each curve represents the average of between three and five relaxations. It is evident that the knotted relaxation trajectories are distinct from the unknotted relaxation trajectory. Between the relaxation trajectories of knotted molecules with different knot positions, however, there is no apparent trend. The measured relaxation times are 1.07 ± 0.03 s, 1.11 ± 0.02 s and 1.05 ± 0.03 s for molecules with an initial knot position of 0.25, 0.35 and 0.45 respectively.

We believe that the absent effect of knot position on relaxation time is largely due to the small knot sizes involved. As a knot can effectively be viewed as a bead along the chain with a larger drag coefficient, one would expect that the position of a large enough knot would play a role in determining how the free ends of the chain retract during relaxation. For example, consider the two extreme cases of a knot located in the center of a chain and a knot located near a chain end. The knot near the end of the chain can be seen as an effective tether, which would lead to a different relaxation process compared to the chain with a knot in the center. The sizes of knots generated in this study are likely not large enough for any such effect to be detected.

Wi ≡ τ_{unknot}. | (4) |

Fig. 6 displays images of a molecule without and with a knot stretched in planar elongational fields at a range of strain rates. At Wi > 2.0, there are small differences in the extensions of the unknotted and knotted molecules attributable to contour stored within the knot. At moderate Wi, apparent differences in the extensions of the unknotted and knotted molecules emerge, with the knotted molecules exhibiting noticeably smaller extensions at the same Wi. For Wi < 1.0, we observe stark differences in the configurations exhibited by the molecules. Specifically, the knotted molecule has retracted to a coiled state at Wi = 0.7, while the unknotted molecule remains in a stretched state. Evidently, the occurrence of a knot on a molecule reduces the chain relaxation time and consequently shifts the coil–stretch transition. This was also observed computationally in a recent study by Narsimhan et al.^{40} for a range of knot topologies.

The steady-state extension of unknotted and five different knotted molecules as a function of Wi is plotted in Fig. 7a. The steady-state extension is averaged over an ensemble of 10 molecules for the unknotted molecule and averaged over 30 s to 1 minute (∼30–60τ_{knot}) for each knotted molecule. Note that hysteresis in the extension-Wi curves was not observed (see ESI†). We observe an abrupt increase in steady state extension of unknotted molecules around Wi = 0.5, in good agreement with previous experimental studies.^{8,9,48} The knotted molecules are numbered in order of decreasing relaxation time (increasing knot size). The relaxation times and knot sizes of the knotted molecules are reported in Table S1 (see ESI†). As seen from Fig. 7a, at a given Wi, the knotted molecules always have smaller extensions compared to the unknotted molecules. Furthermore, the larger the knot on a molecule, the more shifted the extension-Wi curve becomes relative to the unknotted molecule. This signifies the importance of considering knotted molecules individually – a capability of single-molecule experiments – as the heterogeneity of knots can result in varying shifts of the coil–stretch transition.

Fig. 7 (a) Steady-state fractional extension as a function of Wi = τ for unknotted and five different knotted molecules with varying knot sizes. (b–d) Scaled steady-state fractional extension as a function of Wi_{eff} = τ_{knot}, Wi_{rouse} = Wi(1 − L_{knot}/L_{chain})^{1+2ν} and Wi_{zimm} = Wi(1 − L_{knot}/L_{chain})^{3ν} for unknotted and the five knotted molecules presented in (a). (e) Standard deviation of steady-state fractional extension as a function of Wi for unknotted and the five knotted molecules presented in (a). (f–h) Standard deviation of steady-state fractional extension as a function of Wi_{eff}, Wi_{rouse} and Wi_{zimm} for unknotted and the five knotted molecules presented in (a). The black lines are drawn to guide the eye. Error bars represent 95% confidence interval (see ESI† for calculation). |

To understand the shift in extension-Wi curves for knotted molecules, the change in relaxation time as a result of the occurrence of a knot needs to be considered. We can define an effective Weissenberg number

Wi_{eff} ≡ τ_{knot}. | (5) |

Close inspection of Fig. 7b reveals poor overlap of the data at high extensions, with the data points for the knotted molecules vertically shifted to noticeably larger extensions compared to the unknotted molecules. This is perhaps to be expected, as the longest relaxation mode is dominant only at small chain extensions. Rescaling Wi by the longest relaxation time of each knotted molecule can approximately collapse the extension-Wi curves, but there is generally better overlap in the low extension compared to the high extension regime. At higher chain extensions, the contribution of higher order relaxation modes cannot be neglected and, as shown previously, the relaxation of a knotted chain is dependent on the knot size. Inspired by Narsimhan et al.,^{40} we take into account the change in free chain contour at each strain rate when rescaling Wi. As discussed in Section 3.1, the knotted relaxation time based on Rouse scaling can be described as τ_{rouse} ∝ (L_{chain} − L_{knot})^{1+2v}. Hence, we have

(6) |

(7) |

A unique feature of the coil–stretch transition is the critical slowdown in polymer dynamics toward steady state that is accompanied by enhanced molecule extension fluctuations.^{11,62} The cause of this phenomenon is the large heterogeneity of polymer configurations corresponding to largely different chain extensions that are available close to the coil–stretch transition. As was done in previous studies,^{11,48} we can examine the steady-state extension fluctuations by measuring the standard deviation of the steady-state fractional extensions. Fig. 7e shows the standard deviation of steady-state fractional extensions as a function of Wi for the same unknotted and knotted molecules as in Fig. 7a–d. A peak in the standard deviation plot for unknotted molecules occurs at Wi ≈ 0.6, in alignment with the location of significant increase in chain extension observed in Fig. 7a and in good agreement with results reported in similar studies for T4 DNA.^{11,48} The magnitude of the peak is also consistent with that measured by Gerashchenko and Steinberg^{11} for T4 DNA in bulk elongational flow. Similar to the shift in extension-Wi curves seen in Fig. 7a, the peaks in standard deviation plots for the knotted molecules are shifted to higher Wi, with the larger knots leading to larger shifts relative to the unknot.

We can rescale Wi and standard deviation of steady-state extension in the standard deviation plots, as was performed for the extension-Wi curves in Fig. 7b–d. The standard deviation of scaled steady-state extension as a function of Wi_{eff}, Wi_{rouse} and Wi_{zimm} are plotted in Fig. 7f–h. As seen from Fig. 7f, rescaling Wi by the knotted relaxation time shifts the peaks in standard deviation for all knotted molecules to overlap with the peak corresponding to the unknotted molecules. The rescaling of Wi taking into account a changing free chain contour (Fig. 7g and h) leads to a collapse of the standard deviation curves onto the unknotted curve at high strain rates. However, the data points surrounding the peaks at low strain rates are shifted such that peaks in the rescaled standard deviation plots no longer overlap. The difficulty in measuring a representative knot size at low chain extensions manifests itself in the incorrect scaling of Wi near the coil–stretch transition. In the low extension regime, it is more suitable to rescale Wi by the knotted relaxation time that characterizes the dominant relaxation mode at small chain extensions.

A similar phenomenon is observed when a larger knot unties into a smaller knot. In Fig. 8b, a molecule containing a knot is initially stretched in an elongational field at Wi = 1.3 ( = 0.12 at Wi = 1.3) and the large extension fluctuations suggest proximity to the molecule's coil–stretch transition. The molecule retracts into a coiled conformation after Wi is lowered to 1.0. It remains arrested in a compact state for about 150 s, or ∼100τ_{unknot}, before it is stretched by the field and a significantly less bright spot along the chain corresponding to a smaller knot ( = 0.04 at Wi = 1.0) is observed. We note that the molecule undergoes extension fluctuations during the untying process. The long time required for the larger knot to partially untie in Fig. 8b relative to the time for a smaller knot to completely untie in Fig. 8a aligns with expectations that larger knots tend to be more complex topologically and hence cannot be untied as easily. This is in agreement with previous work demonstrating that molecules subjected to electric fields for longer durations undergo expansion processes characterized by longer time scales.^{45} The vast changes in chain extension as knots untie in the vicinity of the coil–stretch transition was also observed computationally by Narsimhan et al.^{40} for various knot topologies. The presence and subsequent untying of a knot can lead to a molecule experiencing both a stretch–coil and coil–stretch transition at a fixed strain rate in elongational fields, highlighting the important effect that knots can have on the dynamics of polymers in steady-state fields.

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## Footnote |

† Electronic supplementary information (ESI) available: Additional details regarding knot size measurements and analysis, estimate of maximum knotted molecule extension, additional extension-Wi and standard deviation plots, calculation of confidence intervals, and movies. See DOI: 10.1039/c7sm02195j |

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