Linking topology of tethered polymer rings with applications to chromosome segregation and estimation of the knotting length

The Gauss linking number (Ca) of two flexible polymer rings which are tethered to one another is investigated. For ideal random walks, mean linking-squared varies with the square root of polymer length while for self-avoiding walks, linking-squared increases logarithmically with polymer length. The free-energy cost of linking of polymer rings is therefore strongly dependent on degree of self-avoidance, i.e., on intersegment excluded volume. Scaling arguments and numerical data are used to determine the free-energy cost of fixed linking number in both the fluctuation and large-Ca regimes; for ideal random walks, for $\left|\text{Ca}\right|>N^{1/4}$, the free energy of catenation is found to grow $\propto {\left|\text{Ca}/N^{1/4}\right|}^{4/3}$. When excluded volume interactions between segments are present, the free energy rapidly approaches a linear dependence on Gauss linking $\left(dF/d\text{Ca}\approx 3.7k_{B}T\right)$, suggestive of a novel “catenation condensation” effect. These results are used to show that condensation of long entangled polymers along their length, so as to increase excluded volume while decreasing number of statistical segments, can drive disentanglement if a mechanism is present to permit topology change. For chromosomal DNA molecules, lengthwise condensation is therefore an effective means to bias topoisomerases to eliminate catenations between replicated chromatids. The results for mean-square catenation are also used to provide a simple approximate estimate for the “knotting length,” or number of segments required to have a knot along a single circular polymer, explaining why the knotting length ranges from $\approx 300$ for an ideal random walk to ${10}^6$ for a self-avoiding walk.

Figure 1

(a) Tethered polymer rings. Two flexible polymers each $N$ segments in length are attached by a connector segment. (b) Polymers undergo close encounters where two segments are separated by approximately one segment length. At these points the sign of the crossing may be changed by a small shift in position of either segment, without changing the conformation of the remainder of the chains.

Figure 2

(a) Monte Carlo (MC) simulation results for mean-square catenation $⟨{\text{Ca}}^2⟩$ of tethered rings compared with analytical and scaling formulas. Circles show results for ideal random walks (IRW, segment diameter $d=0$); for $N>500$ the results converge to $\propto N^{1/2}$ (straight line). The smooth curve through the data points is the result of an exact calculation for the Gaussian polymer model (see text), which passes through all the MC data. Squares indicate self-avoiding walk (SAW) results obtained by increasing the segment diameter to $d=0.2$; the magnitude of $⟨\text{Ca}⟩$ is strongly suppressed, and follows a logarithmic behavior (smooth curve through the data is $a\text{ln}N+b$, see text). Intermediate values of segment diameter ($d=0.01$, $+$; $d=0.05$, $\times$) give results where IRW-like $\left(\approx N^{1/2}\right)$ behavior is seen for small $N$, but which for larger $N$ gives way to a slowdown toward SAW $\left(\approx \text{ln}N\right)$ behavior. (b) Same data as in (a), plotted on linear-log scale, showing that for nonzero excluded volume ($d=0.01$, $+$; 0.05, $\times$; 0.2, ◻) mean-square catenation grows logarithmically with $N$.

Figure 3

Catenation free energy for tethered ideal random walks. Data are shown for $N=20$, 40, 80, 160, 320, 640, 1280, and 2560, in terms of the scaled catenation $x=\text{Ca}/{⟨{\text{Ca}}^2⟩}_0^{1/2}$. Data for all lengths collapse onto a single curve; for $x<1$ the free-energy increase is consistent with a $x^2$ law (line, lower left); for $x>1$ the free-energy increase slows down and is consistent with $\approx x^{4/3}$ (line, upper right).

Figure 4

(a) Catenation free energy versus Ca for tethered polymers with self-avoidance $\left(d=0.2\right)$ in the small-Ca regime, obtained from unbiased Ca fluctuations $\left(\tau =0\right)$. Data are shown for $N=20$ (◇), 40 (*), 80 $\left(\Delta \right)$, 160 $(\nabla )$, 320 (○), 640 $(\times )$, 1280 $(+)$, and 2560 (◻). As $N$ increases, the free energies for given Ca go down, but only slowly. (b) Free energies of (a), multiplied by ${\left({⟨{\text{Ca}}^2⟩}_0\right)}^{1/2}\approx 0.095\text{ln}N-0.24$, for $\left|\text{Ca}\right|=1$ (○), 2 $(\times )$, 3 (◻), and 4 (◇). For $N>100$, the scaled free energies are nearly constant, indicating that for $N$ up to $\approx 3000$, $\beta F\approx g\left(\text{Ca}\right)/{\left(⟨{\text{Ca}}^2⟩\right)}^{1/2}$, where $g\left(\text{Ca}\right)$ is an $N$-independent constant.

Figure 5

(Color online) Free energy of catenation over wide range of Ca obtained from biased-Ca computations $\left(\tau >0\right)$. (a) Free energy obtained from computations with $\tau =1$, 2, 3, 3.5, 3.6, and 3.75, for $N=640$. After an initial sublinear increase from zero, $F\left(\text{Ca}\right)$ grows nearly linearly with Ca. (b) Free-energy change per added catenation $\left(dF/d\text{Ca}\right)$ versus catenation density $\left(\text{Ca}/N\right)$. Results for $N=160$ (*) and $N=640$ (○) rapidly jump to near $3.7k_{B}T$ per added catenation. The lack of $N$ dependence indicates that the free energy is dominated by local effects, i.e., by formation of a tightly entangled state. (c) Sample configuration of $N=640$ chains with $\tau =3.5$, showing separated disentangled and entangled domains.

Figure 6

Time course of removal of catenation in MC simulations with only local vertex diffusion, for various polymer lengths (legend gives $N$ values). In all cases $\text{Ca}/N=0.16$ at $t=0$; at subsequent times Ca can change as a result of intermittently allowed segment overlaps. Each data set shows results averaged over four independent runs. (a) $⟨\text{Ca}⟩$ as a function of time, for $d=0.2$. Catenations are smoothly removed for $t<3\times {10}^4$; at later times fluctuations begin to dominate over free-energy-driven catenation reduction. (b) ${⟨{\text{Ca}}^2⟩}^{1/2}$ as a function of time for the runs of (a). In the smooth relaxation regime the results are nearly the same as in (a) indicating that there is very little variation from run to run. (c) Catenation density $⟨\text{Ca}⟩$ of (a) and (b) as a function of time. In the smooth removal regime $t<3\times {10}^4$, the data collapse onto a single curve terminating near $\text{Ca}/N=0.03$. The results indicate that the dynamics in the smooth removal regime are dominated by local relaxation processes. (d) Data for $⟨\text{Ca}⟩/N$ versus time for $d=0.05$ collapse for times $t<{10}^4$.

Figure 7

Lengthwise condensation mechanism. A segment of length $b$ of polymer of thickness $d$ is lengthwise-compacted into a new segment of length $b^{\prime }$ and thickness $d^{\prime }$. Successive segments are connected by short flexible linkers of the original polymer. As the segments become longer and thicker, entanglements are driven out of the polymer, both with itself and with other nearby polymers undergoing the same type of condensation.

Figure 8

(a) Close encounter of two segments spaced by roughly $N/2$ along a large circular polymer of $N$ segments allows it to be considered as two tethered rings. (b) A reconnection converting a trefoil knot on a single ring of $N$ segments to a $\text{Ca}=2$ link of two unknotted rings of $N/2$ segments. (c) Results for unknot probability $P_{\text{unknot}}$ as a function of $N$ for polymers with $d=0$ (IRW), $d=0.01$, $d=0.05$, and $d=0.2$. Exponential decays $\left(P_{\text{unknot}}\propto e^{-N/N_0}\right)$ determine knotting lengths $N_0$.