Behavior of Complex Knots in Single DNA Molecules

We used optical tweezers to tie individual DNA molecules in knots. Although these knots become highly localized under tension, they remain surprisingly mobile and undergo thermal diffusion with classical random walk statistics. The diffusion constants of knots with different complexities correlate with theoretical calculations of knot sizes. We show that this correlation can be explained by a simple hydrodynamical model of “self-reptation” of the knot along a polymer.

Figure 1

Observing knot diffusion. (a) The stained, knotted DNA appears as a diffraction-limited contour between two beads, with an increase in fluorescence at the knot (arrow). The path of the DNA is found by software and its intensity profile computed; tiling these profiles from sequential frames (b) reveals a diffusive trace which indicates the knot’s trajectory. Scale bars, $5\mu \mathrm{m}$ (horizontal) and 5 s (vertical); scale of (a) is the same as the horizontal scale of (b), The trace is quantified (offset white trace) and its mean squared traveled distance as a function of time $t$ computed (c) which obeys a power law with exponent $\alpha \approx 1.06$ (see text). (d) We have tied and analyzed open knots of types (from left to right) $3_1$, $4_1$, $5_1$, $5_2$, and $7_1$ [18].

Figure 2

Knot diffusion constants $D$ vary dramatically with complexity (a); their respective friction coefficients $\zeta =k_{B}T/D$ correlate well to theoretical knot lengths (b). The slope of the best fit line is $0.29\pm 0.003\mathrm{p}\mathrm{N}/(\mathrm{m}\mathrm{m}/\mathrm{s})=(2.9\pm 0.3)\times {10}^{-10}\mathrm{N}/(\mathrm{m}/\mathrm{s})$. Only data points with relative extensions between 0.55 and 0.75 were used. The numbers of knots observed were 23, 9, 3, 4, and 3 for knot types $3_1$, $4_1$, $5_1$, $5_2$, and $7_1$, respectively.

Figure 3

Averaged extra knot lengths correlate with those determined from numerical simulations of ideal open knots [23]; the slope of the best fit line gives a DNA effective diameter of $26\pm 6\mathrm{n}\mathrm{m}$. The ordinate axis represents the difference between the lengths required for a knot and an unknotted segment.