Long-Lived Self-Entanglements in Ring Polymers

The entanglement of ring polymers remains mysterious in many aspects. In this Letter, we use electric fields to induce self-entanglements in circular DNA molecules, which serve as a minimal system for studying chain entanglements. We show that self-threadings give rise to entanglements in ring polymers and can slow down polymer dynamics significantly. We find that strongly entangled circular molecules remain kinetically arrested in a compact state for very long times, thereby providing experimental evidence for the severe topological constraints imposed by threadings.

Figure 1

Ensemble average expansion of (a) linear and (b) circular DNA molecules compressed by a 10 Hz ac square-wave electric field of various field strengths and durations. The dashed line indicates the equilibrium average radius of gyration ${⟨R_g⟩}_{\mathrm{eq}}=0.95\mu \mathrm{m}$ for linear molecules and ${⟨R_g⟩}_{\mathrm{eq}}=0.75\mu \mathrm{m}$ for circular molecules.

Figure 2

Expansion of individual circular molecules compressed by ac square-wave electric fields. (a) Representative individual traces of $R_g$ vs time for different expansion pathways. The dashed lines indicate the equilibrium average radius of gyration ${⟨R_g⟩}_{\mathrm{eq}}=0.75\mu \mathrm{m}$. The type II traces are underlaid by the representative type I trace to facilitate comparison of timescales. (b) Snapshots of circular DNA conformation during expansion corresponding to the time points labeled in (a). Scale bars represent $5\mu \mathrm{m}$.

Figure 3

(a) Normalized mean square $R_g$ as a function of time for linear and circular molecules that are compressed with various field strengths for 1 s and exhibit type I expansion. The dashed lines are single exponential fits to the data. (b) Time constants extracted from single exponential fits to data in (a) as a function of field strength for circular (blue circle) and linear (red square) molecules. The dotted lines represent the relaxation times ${\tau }_c=1.3\mathrm{s}$ (blue) and ${\tau }_l=4.9\mathrm{s}$ (red) for circular and linear molecules respectively. Error bars represent 95% confidence interval. If not visible, the error bar is smaller than the symbol.

Figure 4

(a) Distributions of time required to expand back to equilibrium for the ensemble of linear and circular molecules subjected to an electric field $E_{\mathrm{rms}}=900\mathrm{V}/\mathrm{cm}$ for 10 s. (b) Snapshots of three linear (top) and five circular (bottom) molecules expanding back to equilibrium over time following compaction by an electric field $E_{\mathrm{rms}}=900\mathrm{V}/\mathrm{cm}$ for 10 s. The electric field is turned off at time $t=0$. Four of the five circular molecules expand back to equilibrium only after a $900\mathrm{V}/\mathrm{cm}$, 0.5 s pulse is applied (indicated by arrows).