Motion of Knots in DNA Stretched by Elongational Fields

Knots in DNA occur in biological systems, serve as a model system for polymer entanglement, and affect the efficacy of modern genomics technologies. We study the motion of complex knots in DNA by stretching molecules with a divergent electric field that provides an elongational force. We demonstrate that the motion of knots is nonisotropic and driven towards the closest end of the molecule. We show for the first time experimentally that knots can go from a mobile to a jammed state by varying an applied strain rate, and that this jamming is reversible. We measure the mobility of knots as a function of strain rate, demonstrating the conditions under which knots can be driven towards the ends of the molecule and untied.

Figure 1

(a) Drawing of the experimental device. A microfluidic $\mathsf{T}$ junction with a diverging electric field stretches knotted DNA at its stagnation point. Field lines show the trajectories of negatively charged test particles. (b) Representative images of a molecule at four time points as two knots translate towards one end of the molecule at $\mathrm{Wi}=0.85$. The scale bar is $5\mu \mathrm{m}$. Approximately 25% of stretched molecules have two knots.

Figure 2

(a) A kymograph of a knotted molecule stretched at $\mathrm{Wi}=1.6$. The white streaks are from other molecules passing through the field of view. (b) Averaged trajectory of translating knots along an ensemble of molecules as a function of accumulated strain, with individual trajectories underlaid. Green trajectories occurred at $\mathrm{Wi}=1.1–1.2$, blue trajectories at $\mathrm{Wi}=1.2–1.4$, and red trajectories at $\mathrm{Wi}=1.4–1.7$. The error bars represent standard error over a population average. The nearly vertical dotted line represents affine motion [Eq. (3)], and the curved line represents the hypothetical ensemble trajectory assuming a one-dimensional (1D) random walk with a diffusivity of $0.5\mu {\mathrm{m}}^2/\mathrm{s}$ [16] [Eq. (4)]. The thick blue and green curves represent simulations of $5_2$ and $6_3$ knots [15].

Figure 3

Time traces of knots moving towards the ends of the chain at $\mathrm{Wi}=1.1$ (gray) and halting as the strain rate is increased to $\mathrm{Wi}=1.9$ (red). (Inset) An example of a molecule with two knots, with the larger one moving towards the end and stopping as the molecule is stretched (corresponding to the bold curves), and the smaller one moving about the center of the molecule until halting when the local tension is increased.

Figure 4

(a) Kymograph of a molecule held at varying levels of the applied field ($\mathrm{Wi}=0.9–2.5$, 5–30 V) for a minute each, showing greater mobility at lower Wi. Note that time runs right to left in this image. The scale bar is $10\mu \mathrm{m}$. (b) Relationship between the measured effective knot mobility and the Weissenberg number. Each gray-scale trace represents a different molecule, and the red represents the population average. The vertical lines correspond to the high and low Wi in Fig. 3, and the gray bar corresponds to the range observed in Fig. 2. (Inset) The same data with a logarithmic $y$ axis.