Action at Hooked or Twisted–Hooked DNA Juxtapositions Rationalizes Unlinking Preference of Type-2 Topoisomerases

Linking number distribution depends on juxtaposition geometry

We began by investigating the distribution of Lk among conformations with a preformed juxtaposition (Fig. 2). The distribution was computed in two steps. We first determined the normalized distribution of writhe subject to the constraint of a preformed juxtaposition j for conformations with zero torsional energy, using the lattice method in Refs. 59 and 60 to calculate writhe. We denote this distribution as P_00,j(Wr), wherein the two zeros symbolize that torsional energy = 0 and Lk = 0. Examples are given in Fig. 2a. To simplify notation, here we use Lk to stand for ΔLk, where Δ represents the actual value of a given variable minus the value of the variable for a fully relaxed DNA circle. In other words, an Lk=0 lattice conformation in the present study corresponds to a fully relaxed DNA circle. Similarly, Tw (twist) in the present study stands for ΔTw, and Wr stands for ΔWr. To obtain a distribution of Lk, we applied the Călugăreanu–White–Fuller theorem^61–63

that relates Lk to Wr and Tw and introduced a torsional (twisting) energy

for Tw. Here, K_t=0.48 is equivalent to the variable C/l₀ in Liu and Chan⁴⁴ with l₀ being the Kuhn statistical length ≈1000 Å. The torsional energy in the present formulation is in units of k_BT, where k_B is the Boltzmann constant and T is the absolute temperature, and is identical with that in Ref. 44. By incorporating Boltzmann weighting into P_00,j(Wr), that is, P_00,j(Wr)→P_00,j(Wr) exp[−E_t(Wr; Lk)], then summing over Wr, and finally introducing an overall normalization factor, we obtained the normalized distribution P_j(Lk), which is subject to the constraint of a preformed j (see examples in Fig. 2b).

Using lattice circles of chain length n=100 as an example, we show in Fig. 2 that preformed juxtapositions shift the mean values of the Wr and Lk distributions. The preformed juxtaposition itself is the main contributor to the shift of the mean Wr or Lk relative to the zero value for circles without the constraint of a preformed juxtaposition (black curves in Fig. 2). Therefore, not surprisingly, the sign of the shift (negative or positive) is the same as the crossing sign of the preformed juxtaposition itself (color curves in Fig. 2). However, a preformed juxtaposition has very little effect on the width of the distribution. For the examples in Fig. 2, the mean value 〈Lk〉_j of the P_j(Lk) distribution is substantially different for a preformed hooked, half-hooked, and free juxtaposition, with 〈Lk〉_j=±1.10, ±0.93, and ±0.54, respectively (〈…〉 symbolizes conformational averaging, and the subscript j indicates the constraint of a preformed juxtaposition j). In contrast, the corresponding values for the standard deviation ${\sigma }_{Lk}^{(j)}={\sqrt{\langle {\left(Lk-{\langle Lk\rangle }_j\right)}^2\rangle }}_j$ of the distribution are essentially the same, with ${\sigma }_{Lk}^{(j)}$=2.67, 2.68, and 2.70, respectively. All of these values are also practically identical with the standard deviation σ_Lk=2.71 for the P₀₀(Lk) distribution (black curve) in Fig. 2b for circles without the constraint of a preformed juxtaposition.

Our next step is to use the P_j(Lk) distributions to determine the steady-state distribution of Lk maintained by selective segment passages. Type-2 topoisomerases apparently act on juxtapositions of both crossing signs.⁶⁴ Accordingly, here we performed model segment passages at a given juxtaposition geometry with a positive crossing as well as a juxtaposition with a negative crossing that reverses the direction of one of the given juxtaposition’s segments but is otherwise identical with it (Fig. 3a). In this way, the two juxtapositions acted upon by the topoisomerase in the present model are of the same shape and are merely “one-segment reverse” of each other.

In general, the steady-state Lk distribution P_st(Lk) for selective segment passages at any juxtaposition j(+) with a positive crossing and any other juxtaposition j(−) with a negative crossing may be computed from the equilibrium Lk distributions P_j(+)(Lk) and P_j(−)(Lk) with j(+) and j(−) preformed, respectively. Because Lk can only change to Lk±2 by one segment passage, the flux exiting an Lk population via segment passage is proportional to b_j(+)ρ_j(+)P_j(+)(Lk)/P_eq(Lk) + b_j(−)ρ_j(−)P_j(−)(Lk)/P_eq(Lk), where b_j(+) and b_j(−) are the segment-passage rates and ρ_j(+) and ρ_j(−) are the densities, respectively, of juxtapositions j(+) and j(−), following the general definitions for b_j and ρ_j in Materials and Methods, and P_eq(Lk) is the equilibrium unconstrained distribution of Lk. In the steady state, the flux exiting an Lk population must equal to the flux entering it. Thus, using the notation

the equality of incoming and outgoing fluxes requires

The only physically viable condition for satisfying this equation is that of detailed balance (i.e., there is no net flux between any pair of states), which entails a recursive relation

that is simpler than that in Eq. (4). It follows that S(Lk) for Lk≠0 can be solved:

where the top or bottom signs in ∓ and ± applies when Lk is, respectively, a positive or negative even number. Hence, by substituting simulated P_j(±)(Lk) values with preformed juxtapositions j(+) and j(−) into Eq. (6), S(Lk) for all Lk may be expressed in terms of S(0). The entire steady-state distribution P_st(Lk) can then be computed using the simulated equilibrium (unconstrained) distribution P_eq(Lk) via Eq. (3), with S(0) being the only remaining unknown, which can now be absorbed into the normalization for P_st(Lk).

Applying this formulation to the virtual segment passages in Fig. 1a for the j(+)′s and j(−)′s of the hooked, half-hooked, and free juxtapositions, we show in Fig. 3b that the steady-state P_st(Lk) distribution for the hooked juxtaposition is narrower [(σ_Lk)_st=2.58] than the equilibrium P_eq(Lk) distribution (σ_Lk=2.71). This result is qualitatively in line with experimentally observed narrowing of Lk distribution by type-2 topoisomerases,^25,30,58 although the ratio of the variances [σ_Lk/(σ_Lk)_st]²≈(2.71/2.58)² =1.10 from this model is less than the experimental value^25,30,58 of ~1.35–1.8. In contrast, the half-hooked and free juxtaposition geometries widen the steady-state Lk distribution relative to P_eq(Lk) in Fig. 3b.

Chain length dependence of supercoil distribution narrowing

Figure 4 shows the chain length dependence of the average writhe 〈Wr〉_j (among relaxed Lk=0 circles) and average linking number 〈Lk〉_j (over circles with all Lk values) in the presence of a preformed hooked, half-hooked, or free juxtaposition. The two averages 〈Wr〉_j and 〈Lk〉_j level off to an essentially constant value for n>200, suggesting that the effect of the preformed juxtaposition on Wr and Lk is local provided the circles are large. However, for small circles with n≲50, both 〈Wr〉_j and 〈Lk〉_j vary sharply with n. Furthermore, the roles of the mutually conjugate hooked and free juxtapositions for large n are apparently reversed when n becomes small, namely, the 〈Lk〉_j values for the hooked juxtaposition for small n are seen to approach 〈Lk〉_j values for the free juxtaposition for large n, and vice versa. As will be elucidated below after the necessary analytical formulation has been developed, this trend is critical in the accounting for how supercoil distribution narrowing by type-2 topoisomerases depends on DNA circle size.

Figure 5a provides the chain length dependence of the standard deviation of steady-state distribution P_st(Lk) maintained by selective segment passages. Naturally, the width of both the equilibrium and steady-state Lk distributions increases with circle size. Irrespective of this overall increase, Fig. 5a shows that for circle sizes larger than n≈50, P_st(Lk) is always significantly wider than the equilibrium Lk distribution P_eq(Lk) for the free juxtaposition, but P_st(Lk) is only slightly wider and slightly narrower than P_eq(Lk), respectively, for the half-hooked and hooked juxtapositions. Following Rybenkov et al.,²⁵ we define the supercoil distribution narrowing factor (Fig. 5b) as the variance of P_eq(Lk) divided by the variance of P_st(Lk):

This definition is identical with the R parameter in Stuchinskaya et al.⁵⁸ and is the reciprocal of the 〈ΔLk²〉^S/〈ΔLk²〉⁰ factor plotted in Fig. 2D of Trigueros et al.³⁰ Figure 5b shows the chain length dependence of R_Lk in our model. For the hooked juxtaposition, the narrowing of the Lk distribution (R_Lk>1 in the model) for circle sizes n≳50 is consistent with experiment. Moreover, the n dependence exhibited by the hooked juxtaposition (I) curve in Fig. 5b is remarkable in that it captures an intriguing, yet unexplained, experimental trend of decreasing R_Lk for small DNA circles as displayed first in Fig. 2D of Trigueros et al.³⁰ and more recently in Fig. 4a of Stuchinskaya et al.⁵⁸ This match between theory and experiment should provide important clues to the physics of type-2 topoisomerase action. We return to this point below.

Analytical treatment relates supercoil distribution narrowing with average Lk in the presence of the selected juxtaposition

Inspection of Figs. 4b and 5b indicates that, for the three juxtapositions considered, the supercoil distribution narrowing factor R_Lk by selective segment passage at a given juxtaposition j is essentially equal to the mean 〈Lk〉_j of the constrained Lk distribution with preformed j. Is this a fundamental relation generalizable to all juxtaposition geometries? We found through the following analytical derivation that R_Lk≈〈Lk〉_j is indeed generally valid and that this relation affords an important mathematical link between the constraint imposed on DNA conformations by the type-2 topoisomerase-preferred juxtaposition geometry on the one hand and type-2 topoisomerase-mediated supercoil distribution narrowing on the other.

Data such as those in Fig. 2a suggest that Wr distributions, independently of the presence or absence of a preformed juxtaposition, are generally well approximated by a Gaussian:

where 〈Wr〉 and σ_Wr are, respectively, the mean and standard deviation of the distribution that depend on chain length and the preformed juxtaposition. We use 〈Wr〉₀₀ and (σ_Wr)₀₀ to denote, respectively, the average writhe and the corresponding standard deviation for the “00” distribution with no torsional energy (K_t=0, black curve in Fig. 2a):

For the case when torsional energy E_t is operative (K_t>0), let 〈Wr〉₀ and (σ_Wr)₀ be, respectively, the mean and standard deviation of the writhe distribution among relaxed, Lk=0, chains, the Gaussian approximation of which is expressed here as

Because Eq. (2) is equal to 2K_tπ²(Wr)²/n for Lk=0, up to an overall normalization,

where

Therefore, it is clear from comparing Eq. (10) with Eqs. (11) and (12) that

In general, for Lk≠0, using Eq. (12) and completing squares for the variables Wr and Lk, and noting the relationship [(σ_Wr)₀₀]² = B[(σ_Wr)₀]² from Eq. (13),

After averaging the above expression for P(Wr; Lk) over Wr by performing ∫d(Wr), we arrive at the following expression for P(Lk) up to the standard normalization factor for a Gaussian distribution (note that both the first and second terms of the last line of the above equation yielded a constant independent of Lk after integration):

It follows that the average value of Lk over conformations with torsional energy is

where the second equality follows from Eq. (13), and the variance of the P (Lk) distribution

As noted in the starting point of our derivation in Eq. (8), the above relations apply regardless of the presence or absence of a preformed juxtaposition. Hence, more specifically, for the case with a preformed juxtaposition j, Eq. (16) becomes

where 〈Wr〉_00,j stands for the average writhe in the presence of preformed juxtaposition j over conformations with no torsional energy. The full Lk distribution with a preformed juxtaposition j may be expressed as

which can be further simplified because our simulation results in Fig. 2a indicate that, to a very good approximation, the standard deviation in Lk is independent of the existence of a preformed juxtaposition; that is, ${\sigma }_L^{(j)}={\sigma }_{Lk}$. Making use of this observation and combining Eqs. (6) and (19), we obtain

From the definition in Eq. (3) and the fact that average Lk is zero at topological equilibrium, viz., P_eq(Lk) ∝ exp[−(Lk)²/2(σ_Lk)²], we arrive at

Therefore,

and

Thus, by Eq. (23),

It is noteworthy that the segment-passage rates b_j(−), b_j(+) and the juxtaposition densities ρ_j(+), ρ_j(−) affect the mean [Eq. (22)] but not the standard deviation [Eq. (23)] of the steady-state Lk distribution. For special cases in which the juxtaposition satisfies 〈Lk〉_j(+)=−〈Lk〉_j(−), Eq. (24) reduces to R_Lk=〈Lk〉_j(+), which we denote simply as 〈Lk〉_j. This relation is verified in Fig. 6 for the hooked, free, and half-hooked juxtapositions, all of which satisfy 〈Lk〉_j(+)=−〈Lk〉_j(−). The result in Fig. 6 shows extremely good agreement between R_Lk computed via the recurrence relation in Eq. (6) and the analytical expression 〈Lk〉_j from Eq. (24).

Lk distribution narrowing by selective j*-symmetrized segment passages

The results so far (Figs. 3–6) were computed using our original virtual segment-passage operation (Fig. 1a). We now include the symmetrized operations (Fig. 1b) in the improved procedure to obtain j*-symmetrized P_st(Lk) and R_Lk values, as for the analysis for knots and catenanes in Materials and Methods. Let j(±)* be the juxtapositions conjugating to j(±). Because a juxtaposition and its conjugate have opposite signs, the process conjugating to that of an Lk flux passing through j(+) to become Lk−2 is that of an Lk−2 flux passing through j(+)* to become Lk. Similarly, the process conjugating to that of an Lk flux passing through j(−) to become Lk+2 is that of an Lk+2 flux passing through j(−)* to become Lk. Therefore, by stipulating that the statistical weights for the j(±) and j(±)* processes are equal, the j*-symmetrized P_st(Lk) may be computed by replacing every P_j(∓) or P_j(±) factor in Eq. (6) by a symmetrized factor, viz.,

where we have used Lk′ to represent any given variable for P_j(∓) or P_j(±) in Eq. (6) to distinguish it from the Lk variable defined by S(Lk) in that equation. Note that the expression in Eq. (25) also provides the replacement for P_j(∓), as may be made explicit by interchanging the “+” and “−” signs in Eq. (25). With all the P_j(±) values modified by Eq. (25), the j*-symmetrized P_st(Lk), and thus the j*-symmetrized R_Lk, can be determined using Eq. (3) and the recurrence relation Eq. (6) as described above.

Figure 7a compares the supercoil distribution narrowing factor R_Lk computed with and without j*-symmetrization. The scatter plot shows that the j*-symmetrized R_Lk values are higher, with the highest value reaching ≈1.4 instead of the highest R_Lk value of ≈1.1 in the absence of j*-symmetrization. j*-symmetrization also improves the correlation between R_Lk and the hookedness parameter H, raising the Pearson correlation coefficient to r=0.88 from r=0.70 for the R_Lk values obtained without j*-symmetrization. Figure 7b shows the scatter plot of j*-symmetrized R_Lk versus the difference in absolute values of the writhes of the juxtaposition and its conjugate. The quantity Wr_j in Fig. 7b is defined by Wr_j = ∑_i,i′∊{j} Wr(i, i′), where Wr(i, i′) is the writhe of two bonds i and i′ and the summation is restricted only to the bonds constituting the juxtaposition j. The degree of correlation between j*-symmetrized R_Lk with |Wr_j|−|Wr_j*| is similar to that between j*-symmetrized R_Lk and H.

Can we devise an equation with similar accuracy to that of Eq. (24) that relates j*-symmetrized R_Lk values with mean Lk values of constrained distributions? As illustrated in Fig. 1, j*-symmetrization entails including reverse virtual segment passages through conjugate juxtapositions. Because the initial state of the Lk→Lk+2 reverse virtual segment passages via juxtaposition j(+)* is equivalent to the final state of the Lk+2→Lk virtual segment passages via j(+)*, the mean of the Lk distribution for the initial state of the Lk→Lk+2 reverse virtual segment passages via juxtaposition j(+)* is equal to 〈Lk〉_j(+)*+2. Similarly, the mean of the Lk distribution for the initial state of Lk→Lk−2 reverse virtual segment passage via j(−)* is equal to 〈Lk〉_j(−)*−2. Heuristically, the selective segment passage under j*-symmetrization involving two original and their conjugate juxtapositions may be viewed as via a pair of effective positive and negative juxtapositions with the means of their respective Lk distributions equal to [〈Lk〉_j(+)+〈Lk〉_j(+)*+2]/2 and [〈Lk〉_j(−)+〈Lk〉_j(−)*−2]/2. Treating these expressions as the effective 〈Lk〉 quantities, respectively, for the positive and negative juxtapositions in Eq. (24), we arrive at the following ansatz for the j*-symmetrized supercoil distribution narrowing factor

We test this ansatz in Fig. 7c. The scatter plot in Fig. 7c shows a nearly perfect match between j*-symmetrized R_Lk values obtained by direct simulations using Eqs. (3), (6), and (25) on the one hand and the right side of Eq. (26) on the other. Consistent with Eq. (26), the horizontal variable in Fig. 7c is in good agreement with 2(R_Lk−1). Strictly speaking, the derivation of Eqs. (20)–(24) does not apply when segment passage is allowed to proceed via more than one juxtaposition of the same sign (as in the j*-symmetrized operation) because the sum of two Gaussian distributions is not a Gaussian distribution. Nonetheless, the sum of two Gaussian distributions whose difference in mean is small compared to their individual widths may be approximated not too badly by a single Gaussian distribution with a mean that equals to the average of the two original means. This is a likely reason for the success of Eq. (26). For example, for circles with chain length n=100, 〈Lk〉_j(+)=1.10 when j(+) is the hooked juxtaposition with a positive crossing, 2+〈Lk〉_j(+)*=2−0.54=1.46 when j(+)* is the free juxtaposition with a negative crossing; one can see that the difference 〈Lk〉_j(+)−(2+〈Lk〉_j(+)*)=1.46−1.10=0.36 is small compared to σ_Lk≈2.7.

Twisted–hooked and hooked juxtapositions have comparable disentangling and supercoil distribution narrowing potentials

The scatter plot for j*-symmetrized R_Lk in Fig. 7a (filled circles) shows that three R_Lk values at H=1/2 and H=1 are higher than that of the hooked juxtaposition at H=2. In other words, the exhaustive scanning of juxtaposition geometries in Fig. 7a has uncovered juxtapositions that can be even more effective in supercoil distribution narrowing than the hooked juxtaposition. Figure 8 shows the five juxtapositions besides the hooked juxtaposition that have the highest j*-symmetrized R_Lk values. Unexpectedly, not all juxtapositions with high R_Lk values were effective in unknotting and decatenating. Table 1 shows that juxtapositions (a) and (b) were clearly not effective in decatenating. Comparing the log₁₀ R_L and log₁₀ R_K values of the juxtapositions in Table 1 with that of log₁₀ R_L=1.21 and log₁₀ R_K=2.39 for the hooked juxtaposition (for n=100) indicates that juxtaposition (c) has both a higher unknotting potential and a slightly higher supercoil distribution narrowing potential (R_Lk=1.31) than that of the hooked juxtaposition (R_Lk=1.28), though the decatenating potential of juxtaposition (c) is somewhat lower than that of the hooked juxtaposition. Juxtapositions (d) and (e) in Fig. 8 should be of interest in future studies because although their R_Lk values are smaller than those of juxtapositions (a)–(c) and the hooked juxtaposition, their R_L and R_K values are quite high. We call (c) the twisted–hooked juxtaposition and it may be viewed as a repeating unit in a lattice supercoil (Fig. 9). For circle sizes up to n=500 we tested, the supercoil distribution narrowing of the twisted–hooked juxtaposition was consistently higher than that of the hooked juxtaposition; the difference increases with n for n≳200 (Fig. 10).

Sterics imposed by a hook-like juxtaposition reduce supercoil distribution narrowing in small circles

Figure 10 shows that the supercoil distribution narrowing factors R_Lk of the twisted–hooked juxtaposition (V) as well as the juxtapositions with one or two hooked segments [hooked (I) and half-hooked (IV)] exhibit a decreasing trend for small circle size n. As in Fig. 5b above, this trend is remarkably similar to that observed in experiments. Measured R_Lk values from two independent studies^30,58 show little dependence of DNA circle size for size ≳4 kb but a clear decreasing trend for DNA circle size ≲4 kb. At first glance, this behavior of R_Lk is peculiar because it is opposite to the observed experimental trend that type-2 topoisomerases are more effective in unknotting smaller DNA circles.²⁵ Why should the unknotting and supercoil distribution narrowing actions of type-2 topoisomerase show opposite dependence on DNA circle size? This question constitutes a challenge for various proposed models of type-2 topoisomerase action, as Stuchinskaya et al. recognized, commenting that “one might expect the models involving a bent G segment and detection of hooked juxtapositions to predict enhanced effects with small circles and reduced effects with larger ones”.⁵⁸

Contrary to the assumption of Stuchinskaya et al., our data demonstrate that the experimental R_Lk dependence on DNA circle size can be rationalized by selective segment passage through the hooked or twisted–hooked juxtaposition geometries (Fig. 10). R_Lk values for the half-hooked juxtaposition are smaller. Nonetheless, they show a similar n dependence, attesting to the robustness of the trend. However, the model unknotting potential of the half-hooked juxtaposition does not increase with decreasing n (Fig. 6 of Ref. 34). To our knowledge, the hooked juxtaposition hypothesis³¹ is the only theory that has been demonstrated quantitatively to reproduce both of the opposite experimental trends for an increasing R_K (unknotting)^24,34 and for a decreasing R_Lk with decreasing DNA circle size. What physics captured by our model is responsible for this success?

Figure 11 analyzes this question by comparing the constraining effects imposed by a hooked juxtaposition on the writhe of a small circle versus those imposed by a free juxtaposition. For the issue at hand, it suffices to focus on writhe because, by Eq. (24), the supercoil distribution narrowing factor of a juxtaposition is given by the average Lk in the presence of the juxtaposition (note that R_Lk=|〈Lk〉_j| for the hooked and free juxtapositions). In turn, by Eq. (18), 〈Lk〉_j is given by the average writhe 〈Wr〉_00,j in the presence of the same juxtaposition. Therefore, we may understand why the supercoil distribution narrowing effect of a juxtaposition has a certain dependence on DNA circle size by deciphering how the effect of the given juxtaposition on writhe depends on n. It will be clear from the analysis below that the geometric origin of the small-n behavior of the twisted–hooked juxtaposition should be very similar to that of the hooked juxtaposition, which we now consider.

As shown in Fig. 4, the averaged values of Wr and Lk with preformed hooked and free juxtapositions are nearly constant for large n, but show sharp variations for small n. For n≳200, 〈Lk〉_j=〈Wr〉_00,j≈1.11 and 0.49, respectively, for the hooked and free juxtapositions (Fig. 4b)^†. For smaller n, the 〈Lk〉_j values, and thus the 〈Wr〉_00,j values, of the two juxtapositions show an obvious converging trend. As n decreases, 〈Lk〉_j for the hooked juxtaposition remains larger than that for the free juxtaposition until the order reverses at n=22 (for which 〈Lk〉_j=0.88 for the hooked juxtaposition and 1.02 for the free juxtaposition). Ultimately, when n decreases further, for the unique smallest circles possible with either a preformed hooked juxtaposition or a preformed free juxtaposition (Fig. 11a and b), the 〈Wr〉_00,j=Wr=0.5 value for the hooked juxtaposition becomes significantly smaller than the Wr=1.5 value for the free juxtaposition^‡.

Intuitively, it is quite clear from an inspection of Fig. 11a why a hooked juxtaposition would constrain Wr in a small circle. The conformation in Fig. 11a is a limiting case but it serves to illustrate an important principle: A tightly connected small circle places such a severe restriction on conformational freedom that the first bond (pink, see below for color code) and second bond (blue) connected to one end of a segment of a hooked juxtaposition tend to run antiparallel with a nearby bond (red) belonging to the hooked juxtaposition itself. In the case of Fig. 11a, the bonds in such a pair run exactly antiparallel and are separated by the shortest nonzero distance on the simple-cubic lattice. This can be seen by following the contour of the circle in either directions. Mathematically, the contributions to Wr from a pair of antiparallel bonds in close spatial proximity always cancel to a large extent. There are eight such antiparallel bond pairs in Fig. 11a; hence, a small |Wr| ensues. In contrast, for a small circle containing a free juxtaposition, the limiting case in Fig. 11b shows that antiparallel bonds are, on average, farther apart; thus, their contributions to Wr do not cancel as much as antiparallel bond pairs in Fig. 11a. Moreover, in Fig. 11b, contributions to Wr from the bonds outside the free juxtaposition tend to be constructive to (has the same sign as) those from the juxtaposition itself (see below), thus adding to a larger |Wr|.

Considering these findings quantitatively, we label the beads and bonds of our circles by 0, 1, 2, 3, …, n−1, where bond i<n−1 connects beads i and i+1, and bond n−1 connects bead i−1 and bead 0. As in Ref. 44, we express Wr of a circle as

where the summations are over terms involving cross products of pairs of bonds. Note that j here is a bond summation index, not to be confused with the label for juxtapositions. We referred to the set of Wr(i, j) as the two-dimensional writhe map.⁴⁴ For our purpose of understanding supercoil distribution narrowing, we find it useful to first examine the one-dimensional writhe profile,

which is the contribution from the ith bond to the writhe of a circle.

The Wr(i) profiles for the smallest circles in the lattice model that contain either a hooked or a free juxtaposition (Fig. 11a and b) are plotted in Fig. 11c and d. For the n = 14 circle in Fig. 11a, the contributions to Wr from the hooked juxtaposition (plotted in blue in Fig. 11c) are largely compensated by contributions outside the juxtaposition (plotted in red in Fig. 11c). In contrast, for the n=18 circle in Fig. 11b, Fig. 11d shows that the Wr contributions from inside and outside the free juxtaposition partly reinforce one another. This notable difference, highlighted by the color of the bonds in Fig. 11a and b, elucidates why Wr of a very small circle with a hooked juxtaposition is significantly smaller than a similarly small circle with a free juxtaposition.

To better understand the influence of juxtaposition geometry on writhe, we further consider two circles constructed by adding two bonds to either a hooked or a free juxtaposition (Fig. 11e and f). In these constructs, each of the added bonds connects the end of one segment to another segment of the juxtaposition directly. The added bonds are longer than the unit lattice spacing, and they do not run along an edge of the simple-cubic lattice. The Wr(i) profiles of these constructs in Fig. 11e and f are similar, respectively, to those in Fig. 11c and d. Taken together, Fig. 11a–f demonstrate consistently that the difference in Wr between small circles containing a hooked juxtaposition and those containing a free juxtaposition is a consequence of writhe compensation and reinforcement (Fig. 12).

For a hooked juxtaposition in a larger circle (Fig. 11g), the contributions to Wr from the bonds outside the hooked juxtaposition (plotted in red) are small and cannot subtract much from the large contributions to Wr from the bonds inside the hooked juxtaposition itself (plotted in blue); thus, |Wr| is larger for large circles with a hooked juxtaposition. For a free juxtaposition in a large circle (Fig. 11h), the contributions to Wr from the cross products between the bonds within the juxtaposition and those outside the juxtapositions (plotted in red for i=1, 2) are much lower compared to the corresponding contributions in a small circle (Fig. 11d). Without the tight connectivity of a small circle, directions of the bonds immediately outside the free juxtaposition are more random with little bias to reinforce the contributions to Wr from the juxtaposition itself. Consequently, |Wr| is smaller for large circles with a free juxtaposition. Therefore, the differing effects of the juxtapositions on the writhe of a small circle versus that of a large circle are, in essence, consequences of differing steric effects of adopting these juxtapositions in small and large circles. Analysis of the corresponding Wr(i, j) maps is shown in Fig. S4.

Scaling relations among R_Lk, R_K, and R_L reflect experimental observations

Finally, we examine in Fig. 13 the relationship among R_Lk, R_K, and R_L. For the juxtapositions we studied, R_Lk correlates with both log R_K and log R_L. The data also show that j*-symmetrization led to better correlations (Fig. 13c and d), thus improving the qualitative agreement between our simulation and the experiment of Rybenkov et al.²⁵ In Ref. 25, reductions in knot and link populations were quantified, respectively, by R_kn=(P_K)_eq/(P_K)_st and R_cat = (P_L)_eq/(P_L)_st. These quantities are slightly different from our R_K and R_L. The definitions of R_K and R_L followed naturally from our formulation³⁴ because they do not require separate calculations of (P_K)_eq and (P_L)_eq [see Eqs. (38)–(40)], but (P_K)_eq and (P_L)_eq were needed to determine R_kn and R_cat. Nonetheless, as we have pointed out,³⁴ because (P_K)_eq and (P_L)_eq are small in experiment as well as in simulation, R_K≈R_kn and R_L≈R_cat. For instance, for the effect of topo IV on the 10 kb P4 DNA in Ref. 25, the experimental (P_K)_eq = 0.03 and (P_L)_eq = 0.064 imply that R_K=1.03R_kn and R_L=1.06R_cat.

The experimental data of Rybenkov et al.²⁵ on the effect of six type-2 topoisomerases from different organisms on the 7 kb pAB4 DNA are consistent with the scaling³⁴

Remarkably, both our lattice³⁴ and wormlike DNA chain⁴⁶ models predicted a similar scaling of R_K≈(R_L)^2.0. The different degrees of reduction in Lk variance determined experimentally for the same set of type-2 topoisomerases (Fig. 3B of Ref. 25) are consistent with

which, by virtue of Eq. (29), suggests that

The trend in Fig. 13c and d from our model, viz., R_Lk≈0.26(log₁₀ R_L)+1 and R_Lk≈0.10(log₁₀ R_K)+1, agrees qualitatively with these experimental observations, arguing that our juxtaposition-centric view is capturing an important part of the mechanism underlying real type-2 topoisomerase action. Indeed, the only physical rationalization to date of the experimental scaling behavior has been provided by our models. Nonetheless, the R_Lk versus log R_L and R_Lk versus log R_K slopes estimated in Fig. 13 are less steep than those observed in experiment. The differences in slope are proportionally larger than the ≈0.4 difference between the theoretical and experimental exponents for the R_K versus R_L scaling. In this comparison, it is also noteworthy that whereas there was little scatter in the R_K versus R_L plots of our models (r≈0.96 for the lattice model³⁴), there is considerable scatter in the R_Lk versus log R_K or log R_L plots in Fig. 13, suggesting that the corresponding slopes can be higher if the correlation were restricted to a subset of the juxtapositions. Elucidation of these issues awaits more extensive experimental data and further theoretical analyses.

Selective segment passage at multiple juxtaposition geometries

We begin with the master equation for knot and unknot populations undergoing selective segment passages through one specific juxtaposition geometry.³⁴ Although this formulation was introduced for lattice modeling, it is generally applicable to any chain representation.⁴⁶ We now generalize this formulation to allow for selective segment passages at multiple juxtaposition geometries. There are at least two reasons for this exercise: (i) In off-lattice continuum modeling, the geometrical characteristics selected by a type-2 topoisomerase would likely entail an ensemble of juxtaposition geometries. (ii) The extension is needed for expanding our lattice modeling effort. For example, segment passages can occur at two different juxtapositions when the shape but not the sign of the juxtaposition is selected by the topoisomerase (see Results and Discussion).

We use P_U and P_K to denote the unknot and knot populations, respectively, whereas $P_{\text{U}}^{(j)}$ and $P_{\text{K}}^{(j)}$ are the corresponding subpopulations in which each conformation has at least one instance of juxtaposition j. Generalizing our previous master equation [Eq. (2) in Ref. 34] to allow for segment passages at multiple juxtaposition geometries, we obtain the equation

where the summation over juxtaposition j is restricted to those through which segment passage can occur, t is time, and b_j is the rate of segment passage at j. The weight coefficients b_j can be different because segment passage may, a priori, proceed at different rates for different juxtaposition geometries. This feature is useful for treating the possibility that a topoisomerase may recognize and bind different juxtaposition geometries with different affinities or act to pass segments through different juxtaposition geometries at different kinetic rates. The quantities $T_{\text{U}\to \text{K}}^{(j)}$ and $T_{\text{K}\to \text{U}}^{(j)}$ are, respectively, unknot-to-knot and knot-to-unknot transition probabilities. Following Ref. 34, we use $c_{\text{U}}^{(j)}$ and $c_{\text{K}}^{(j)}$ to symbolize, respectively, the fraction of unknot and knot population that contain juxtaposition j; that is, $P_{\text{U}}^{(j)}=c_{\text{U}}^{(j)}{P}_{\text{U}}$ and $P_{\text{K}}^{(j)}=c_{\text{K}}^{(j)}{P}_{\text{K}}$. In this notation, Eq. (32) becomes

Now, let (P_U)_st and (P_K)_st be, respectively, the steady-state unknot and knot populations. Because they are steady-state populations, d(P_U)_st/dt = d(P_K)_st/dt = 0. Therefore, from Eq. (33),

Using the c^(j) notation, the transition probabilities

where N_j is the total number of instances of juxtaposition j in all conformations (j can occur more than once in a conformation). Each ${\mathcal{J}}^{(j)}$ is the fraction of N_j that entails a specific transition; thus, the ${\mathcal{J}}^{(j)}$ quantities satisfy the normalization condition ${\mathcal{J}}_{\text{K}\to \text{U}}^{(\text{j})}+{\mathcal{J}}_{\text{K}\to \text{K}}^{(\text{j})}+{\mathcal{J}}_{\text{U}\to \text{U}}^{(j)}+{\mathcal{J}}_{\text{U}\to \text{K}}^{(j)}=1$. Our knot reduction factor R_K is given by³⁴

Hence, combining Eqs. (34)–(36), and defining ρ_j≡N_j/P_eq as the density of juxtaposition j where P_eq=(P_U)_eq+(P_K)_eq is the total chain population, we obtain

as the generalized knot reduction factor. Our original formulation in Ref. 34 corresponds to having only one term in each of the summations in Eq. (38). In that case, the b_jρ_j factors cancel and Eq. (38) reduces to our previous equation for R_K for a single juxtaposition. Analogously, the generalized expression for the link (catenane) reduction factor is

where “L” symbolizes linked (catenated), “U” symbolizes unlinked, and ${\rho }_j^{[2]}$ is the density of juxtaposition j among two-circle configurations. When only one juxtaposition is selected, the $b_j{\rho }_j^{[2]}$ factors in Eq. (39) cancel and R_L is independent of DNA concentration.

The lattice model

The present effort is based on our lattice DNA model.^33,34 As a general approach, lattice modeling has proven since the 1930s^69,70 to be a powerful technique in statistical mechanics.⁷¹ It has contributed tremendously to polymer physics,^72,73 including protein biophysics.^74–81 Lattice techniques have been instrumental in developing important mathematical concepts used to elucidate the biophysics of DNA knots and links.^{4,33,34,82–86} Lattice techniques complement continuum coarse-grained approaches such as the “pearl necklace”,^87,88 freely jointed (random flight),^{14,41,42,89,90} and wormlike^6,44,53,91 chain models (reviewed in Ref. 92).

Each conformation in our study consists of n bonds joining n beads to form a closed circuit on the simple-cubic lattice. We refer to these conformations as circles, and n (circle size) as chain length (equivalent to “loop size” in Ref. 34). To ascertain how selective segment passages affect the steady-state distribution of the lattice DNA conformations, we first considered the hooked (I), free (III, referred to as “free nonplanar” in Ref. 34), and half-hooked (IV) juxtapositions (see Fig. 1a). We then broadened our study to all 680 5mer-on-5mer lattice juxtapositions that permit virtual segment passages.³⁴

Our lattice model is for exploring general biophysical principles, not for accurate structural modeling of DNA. One lattice bond is customarily identified with the Kuhn statistical length (l₀, ≈300 base pairs) or the persistence length (≈l₀/2 for DNA). Thus, a circle of chain length n in our lattice model may correspond approximately to a DNA circle with 150n–300n base pairs. However, because of the coarse-grained nature of the model, this relationship should only be regarded as a rough, order-of-magnitude estimate. For this reason, we focus on general trends of dependence of model properties on n but we do not attempt to match n directly with experimental DNA circle size.

Conformational statistics were computed using exact enumeration³⁴ for small circles (n ≤ 30) and Monte Carlo sampling for larger circles (up to n=500). As before,³⁴ conformational sampling was conducted using a combination of the Madras–Orlitsky–Shepp⁹³ (MOS) and the Berg–Foerster–Aragão de Carvalho–Caracciolo–Fröhlich^94,95 (BFACF) algorithms. For the hooked, half-hooked, and free juxtapositions, each n>30 data point was obtained from 8×10⁸ to 3.2×10⁹ simulation time steps. When scanning the comprehensive set of all possible juxtapositions (n=100), we used 3.6×10⁷ simulation time steps for each juxtaposition. Aside from the preformed juxtapositions, conformation freedom was not restricted in our simulations. This modeling situation was chosen to correspond to the conditions of the in vitro experiments we aim to rationalize here. In future studies, it would be interesting to extend the present methodology to investigate how the disentangling actions of type-2 topoisomerases might be affected by the compactness of the DNA conformations in confined spaces.^13–17

Symmetrized segment-passage operations in the lattice model likely provide a geometrically more accurate account of type-2 topoisomerase action

We have also developed an improved procedure for the virtual segment-passage operation.^33,34 As explained below and in Supplementary Data, the original virtual segment-passage operation likely gave rise to a larger fraction of conformations being entangled than is physically realistic (Figs. S1 and S2). Thus, the original procedure was biased toward underestimating the disentangling power of selective segment passages. Accordingly, although the link and knot reduction factors R_L and R_K for the hooked juxtaposition we computed previously were already significiant,^33,34 the corresponding reduction factors were even larger after the bias was removed (Fig. S3).

The rationale for the new procedure is as follows. Because each bond in a lattice chain corresponds roughly to l₀, the separation, d, between two segments of a juxtaposition in our self-avoiding lattice model is d≈l₀≈1000 Å. Considering that the width of a type-2 topoisomerase is typically ≈100 Å,⁴⁵ it is almost certain that d<l₀ in reality. In view of a DNA excluded-volume diameter ≈25 Å (0. 025l₀), it is instructive to consider an idealized situation in which the DNA chain is infinitely thin so that two segments of a juxtaposition have zero contact distance (d=0) and segment passage changes the crossing sign but leaves the conformation otherwise unchanged. One consequence of this hypothetical d=0 situation is that the passage at a juxtaposition with two straight segments would have no effect on knot population because, by symmetry, the conformational distributions before and after segment passage are exactly identical. Experiments to date do not provide a precise value for d but it is safe to assume that the value lies somewhere between the bounds of d=0–1000 Å. The considerations above suggest that it is closer to d=0 than to d=1000 Å.

In a separate study, we explored d values ranging from 25 to 200 Å (≈0.025–0.2 l₀) using a continuum wormlike DNA model and found that the knot reduction factor R_K increases with decreasing d for hook-like juxtapositions.⁴⁶ In an independent study using a random-flight model without excluded volume, passing of segments separated by as far as one Kuhn length was also considered.⁴¹ The physical basis of the variation of R_K with d may be understood by considering the original virtual segment passage in our lattice model (Fig. 1a), which transforms an on-lattice conformation to one not configured entirely on the lattice (dashed bonds in Fig. 1). As a result of their extra conformational freedom, chains created by virtual segment passages have a higher level of entanglement relative to those restricted strictly to the lattice (see Supplementary Data). Because the deformation caused by segment passage and thus the resulting bias toward relatively more entanglement increases with d, R_K decreases with increasing d.

Up to a degree, the bias observed in our lattice models can be a reflection of reality. Because DNA has a finite diameter (d>0) and is elastic, the transient conformational geometries created by local segment passages would be energetically slightly strained around the site of the passage. As a result, these conformational geometries have a slightly higher knotting probability than the conformations created by the corresponding ideal (d=0) segment passages that do not entail such an energetic strain. Indeed, wormlike DNA model studies^27,46 indicated that segment passages at a juxtaposition with two straight segments could result in an R_K≈0.7–0.8, which is slightly biased toward knotting, rather than the neutral R_K=1.0 expected for the d=0 ideal case. However, the corresponding bias in the lattice model was substantially higher, yielding for n=100 an R_K=0.18 for a juxtaposition with two straight segments.³⁴

Our new lattice procedure aims to compensate for this large bias so as to achieve results more akin to those for the d = 0 ideal situation. To do so, we augmented the contribution from the original virtual segment passage at a juxtaposition j (as in Fig. 1a) by the contribution from the reverse segment passage at the juxtaposition′s conjugate j* (as in Fig. 1b). We call this procedure j*-symmetrization. Intuitively, the conjugate j* of juxtaposition j is the “relaxed form” of the juxtaposition resulting from segment passage at j. For instance, the hooked and free juxtapositions are conjugate to each other. In general, let the position of any bead i of the two segments (1 and 2) of a juxtaposition j* be ${\overrightarrow{r}}_i(1)$ and ${\overrightarrow{r}}_i(2)$. Then the corresponding positions of its conjugate juxtaposition j* are ${\overrightarrow{r}}_i(1)$ (unchanged) and ${\overrightarrow{r}}_i(2)+2{\overrightarrow{r}}_{21}$, where ${\overrightarrow{r}}_{21}$ is the vector from the center bead of segment 2 to the center bead of segment 1; j and j* thus have opposite crossing signs. Because this construction can result in excluded volume violations, some juxtapositions do not have conjugates. Among the 680 5mer-on-5mer juxtapositions that can undergo virtual segment passages,³⁴ 425 have conjugates.

We define a j*-symmetrized segment passage as the forward transition via j (solid arrows in Fig. 1) plus the reverse transition via j* (dashed arrows in Fig. 1). This procedure is equivalent to extending the segment-passage operations to include all virtual conformations (those with dashed bonds in Fig. 1) created by forward transitions as well as the original, on-lattice conformations. Using Eq. (38) for a pair of selected juxtapositions and stipulating equal statistical weights for j and j*, we arrive at

as the j*-symmetrized knot reduction factor. The j*-symmetrized link reduction factor R_L is obtained by substituting “K” in the above expression with “L” (changing ${\mathcal{J}}_{\text{K}-\text{U}}^{(j)}$ to ${\mathcal{J}}_{\text{L}-\text{U}}^{(j)}$, etc). As noted above, j*-symmetrization increases R_K and R_L for all juxtapositions (Fig. S3)^§. Thus, our conclusion^24,33,34 in support of the hooked juxtaposition hypothesis³¹ is strengthened by the introduction of the symmetrized segment-passage operation. Further discussion of the procedure is provided in Supplementary Data.