Topologically driven swelling of a polymer loop

Although knots in polymers have been studied for several decades, they remain perhaps the least understood subject in polymer physics. Most of the work in this area has been directed at classification of knots, finding efficient topological invariants, and the probabilistic questions, e.g., “What is the probability to obtain a certain knot type under given conditions (e.g., on loop closure)?” Much less is known about the more physical aspects, which are how knots influence the properties of polymers. The simplest question to ask about physical properties is what the average spatial size is of a polymer loop whose knot type is quenched. To this end, des Cloizeaux (1) conjectured as early as 1981 that the size of a trivially knotted loop (i.e., an unknot) scales with the number of segments, N, in the same way as in the case of a self-avoiding walk, which is N ^ν, where ν = ν_SAW ≈ 0.589 ≈ 3/5. We should emphasize that the polymer in question is not phantom in the sense that segments cannot cross each other, but it is assumed to have a negligible excluded volume (or thickness). Thus, according to des Cloizeaux's conjecture, exclusion of all knots acts effectively as volume exclusion. More systematic arguments, albeit still only at a scaling level, to support this conjecture were presented more recently (2), yielding the following prediction for the (mean square) average gyration radius of a trivially knotted loop. Here, is the segment length, and the parameter N ₀ is sometimes called the characteristic length of random knotting; it appears in the probability of observing a trivially knotted conformation (an unknot) in a fluctuating phantom (i.e., freely crossing itself) loop: When N is smaller than N ₀, a phantom loop has few conformations with nontrivial knots. Therefore, the set of allowed conformations for an unknotted nonphantom loop is not significantly different from that of a phantom loop. This condition is why at N < N ₀ the Gaussian scaling of gyration radius is expected. For this case, the prefactor results from the facts that (i) the mean square gyration radius for the linear chain is 1/6 of its mean square end-to-end distance, , and (ii) for the loop is half that for the linear chain (3). Prefactor A for the N > N ₀ regime in Eq. 1 must provide for smooth crossover between regimes at N ∼ N ₀, which means Given the fundamental character of the problem, and given that theoretical arguments remain far short of mathematically rigorous, it is vital to look at the simulation data. At present, the situation on this front is contradictory. The difficulty is that the N^ν scaling is only expected at N » N ₀, whereas N ₀, according to all simulations (4, 5), is as large as N ₀ ∼ 300, although somewhat model-dependent (e.g., segments of fixed length vs. segments of Gaussian distributed length). Deutsch (6) claimed a few data points consistent with the prediction (Eq. 1). In ref. 8, the authors came to the contradictory conclusion that the N ^ν scaling is observed upon fitting the R_g dependence on N over the entire interval of N from well below to well above N ₀, whereas in the N > N ₀ range the Gaussian behavior N ^1/2 is recovered. This conclusion appears to suggest that the loops with N < N ₀, which experience virtually no topological constraints, swell most strongly because of these constraints, which does not seem possible. Also, this result is not supported by the other earlier work from the same group (9), where the authors mostly looked at the role of excluded volume but also formulated the conclusion that “when N is large enough... the value of the exponent ” (for the given knot 𝒦) “should be consistent with that of self-avoiding walks.” Finally, in a recent work (7), the authors examined polymers of up to N = 500 segments and claimed to observe the N ^ν scaling.

In all the works (7–9), to extract the value of scaling exponent from the data, which (particularly in ref. 7) is almost entirely restricted to the crossover range, authors fitted the data using the formula: with adjustable parameters A, B, and ν, usually assuming for simplicity Δ= 0.5 (although the fit parameters A and B were not given in ref. 7). This approach, suggested in ref. 10, is motivated by the analogy with the renormalization group treatment of the excluded volume problem. Unfortunately, this analogy itself hinges on the idea that the power ν in Eq. 1 is the same as that for self-avoiding walks, which is precisely the idea to be tested. Furthermore, Eq. 4, even if valid, is not the interpolation working across the crossover range from trivial to nontrivial scaling. Indeed, this formula does not approach the N ^1/2 in any range of N, which, for trivial knots with N « N ₀, is not the conclusion of a complicated unreliable theory, but is just the matter of common sense (when there are few knots, prohibition of knots cannot affect average loop size).

In this article, we systematically test the prediction (Eq. 1) for the length up to N = 3,000; this length is determined by our current computational capabilities, but it is also about the threshold, above which excluded volume effects become significant for DNA. (Indeed, for the polymer with segments of the length and diameter d, the excluded volume effect does not lead to appreciable swelling as long as N « (3); for double-stranded DNA at a reasonable ionic strength, taking into account the electrostatic contribution to the effective diameter, this argument implies chain length up to ≈1,000 segments or 300,000 bp. Consistent with the predicted value, we find ν ≈ 0.58 ± 0.02. Furthermore, we were able to examine the probability distribution of the gyration radius and to find, for instance, that trivial knots are noticeably less compressible than the average of all loops.

The plan of our simulation is as follows. First, we generated loops of the length N divisible by 3. To produce one loop, we generated N/3 randomly oriented equilateral triangles of perimeter . We considered each triangle a triplet of head-to-tail connected vectors. Collecting all N vectors from N/3 triangles, we reshuffled them and connected them all together, again in the head-to-tail manner, thus obtaining the desirable closed loop. A similar simpler method applicable for even N and reshuffling vectors obtained from zero-sum pairs yields the loops with overlapping nodes. Overlapping nodes happen when the reshuffling results in the succession of some 2m < N vectors belonging to exactly m pairs and thus forming the zero sum (i.e., closed) subloop. The probability of such an event is of the order unity, because the probability for the two vectors from the same pair to be next to each other after the reshuffling scales as 1/N, and there are ≈N such pairs; more accurate calculation (11) shows that this probability approaches 1/e as N → ∞. For the triangles, no such problem occurs, because the probability for the three vectors of the triplet being next to each other scales as 1/N ², whereas the number of triangles is still ≈N, so the overlapping loops are rare as 1/N (and the probability to have two, or, in general, m triplets to occupy completely the 3m stretch of the reshuffled sequence does not change the 1/N estimate). This problem of overlapping is why we chose to generate loops from triplets of vectors. For each loop, we compute the gyration radius: Second, once a loop is generated, we determined its topology by computing the topological invariants. For the loops with N < 300, we used Alexander invariant Δ(–1) and Vassiliev degree 2 and degree 3 invariants v ₂ and v ₃ (12). The loop was identified as a trivial knot when it yielded |Δ(–1)|\ = 1, v ₂ = 0, and v ₃ = 0. For longer loops with N > 300, for reasons of computational impotence, we only used Δ(–1) and v ₂ invariants, assigning trivial knot status to the loops with |Δ(–1)| = 1 and v ₂ = 0. The details of our computational implementation of these invariants are described elsewhere (13). Of course, because of the incomplete nature of topological invariants, our trivial knot assignment was only an approximation and surely was sometimes in error.

At every N, we continued generating loops until collecting the desirable number of presumably trivial knots as specified in Table 1. Collecting this amount of statistics required >10⁵ CPU hours, roughly 11 CPU years. This extraordinarily long execution is the painful result of the exponentially rare nature of trivially knotted loops (2).

The first result of our simulations, presented in Fig. 1, is the fraction of trivial knots among all loops, w _triv, as it depends on N. Overall, our data agree well with the exponential formula (Eq. 2) and the data of earlier simulations (4, 5). However, deviation from the exponential is apparent in the region N > 1,500. We believe, although we have no rigorous proof, that it results from the use of insufficiently powerful invariants in assigning trivial status to a loop and reflects the contamination of the supposedly trivial pool with some nontrivial knots. Accordingly, to extract the parameters N ₀ and w ₀ (see Eq. 2), we used only data in the range 50 ≤ N ≤ 300, where the occurrence of mistakenly identified knots is lower and where we could rely on the third Vassiliev invariant in addition to the other two. This process yields the best-fit parameters N ₀ = 241 ± 0.6 and w ₀ = 1.07 ± 0.01. Our result for the characteristic length of random knotting is somewhat smaller than reported in the previous works (4, 5), which we interpret as due to the subtle difference in the models examined. Indeed, in ref. 4, the authors look at the chain with some excluded volume, or thickness d; their results for N ₀ dependence on d approach ours when extrapolated to d = 0. Ref. 5 examined Gaussian random polygons, i.e., objects with Gaussian distributed segment length, whereas in our model all segments are of identical length.

We now approach the central issue of this article, which is our data on the gyration radius of loops, as plotted in Fig. 2. As a consistency check, at each N we look at the averaged over all generated loops, irrespective of their topology. As Fig. 2 Upper Inset indicates, the swelling parameter, defined as , is practically independent of N. Since , as we pointed out before, is the mean square gyration radius for Gaussian loops, this result confirms the statistically representative character of our loop ensemble.

The similar swelling parameter for trivial knots, , is also shown in Fig. 2 Upper Inset. A few points shown as stars were independently collected by A. Vologodskii (personal communication) by using only the Alexander invariant, and they agree with our data. The data demonstrate clearly that loops with trivial knot topology are on average much more extended than other loops.

To move beyond this qualitative conclusion to the quantitative characterization of topology-driven swelling, we found it necessary to look more closely at the errors caused by the contamination of the trivial knots pool due to mistaken assignment of some nontrivial knots as trivial because of insufficiently powerful topological invariants. We used the following procedure to correct for this problem of trivial ensemble contamination.

Let w = w _triv = w ₀ exp(–N/N ₀), the true probability of finding a trivial knot. Then, the averaged gyration radius for all loops (which is equal to ) reads Now let δ be the probability of a nontrivial knot mistakenly assigned as trivial. In Fig. 1, δ is visible as the vertical distance the data points rise above the fit line. The fraction of loops to which we assign, correctly or mistakenly, the trivial status is w + δ. The conditional probabilities of the loop to be a trivial or nontrivial knot provided it is assigned trivial status by our imperfect topological invariants are w/(w + δ) and δ/(w + δ), respectively. Accordingly, the gyration radius averaged over this contaminated trivial pool, , can be described as the weighted average of loops that either possess or lack trivial knot topology; thus, Implicit here is an assumption that mistakenly identified knots have the same average gyration radius as all other nontrivial knots. Accepting it, we observe that in Eqs. 6 and 7 we know everything except and . We solve these coupled equations to find where the later simplification makes use of the observation that δ « w everywhere, and that w « 1 when the correction in question is noticeable (say, at N >1,000 or so). Thus, we obtain that is somewhat larger than directly measured quantity (because ) by the amount proportional to δ/w.

Corrected data for are presented in the main part of Fig. 2 as diamonds (⋄). The data fit well to the simple power law at N > 500, with best-fit parameters A = 0.44 ± 0.03 and ν = 0.58 ± 0.02. This result is fully consistent with theoretical prediction (Eq. 2) in several respects. First and foremost is the very fact of power-law dependence of on N. Second, the value of exponent ν matches well with that of the self-avoiding walks, thus confirming the des Cloizeaux conjecture (1). Third, the range of N where the power law is observed supports the idea that it should start at N > N ₀, as in Eq. 1. Fourth, the value of prefactor A agrees with prediction (Eq. 3), which is A ≈ 0.42, thus providing for smooth crossover at N close to N ₀, as expected.

The fit quality is addressed in Fig. 2 Lower Inset, where data/fit is plotted against N. Overall, data remain within ±5% of the fit. Importantly, the difference between corrected (see Eq. 8) and uncorrected data are within the 5% error corridor, suggesting that the fit result is reliable and is not affected dramatically by the inevitable errors of knot identification.

At the same time, we should point out that within the 5% corridor, our data exhibit a small but systematic bend upward. Formally, this curvature leads to the observation that power-law fitting of only part of our data, starting from a larger N, say N > 1,000 or N > 1,500, yields increasing ν, up to the physically absurd values of 0.9 or so at very large N. Of course, these unphysical results come from the narrowing windows of data where the statistics are increasingly poor. Nevertheless, currently we do not know if the upward bend of the data curve in Fig. 2 is entirely due to the measurement errors, or if it hints to something more serious. In particular, this bend prevents us from meaningfully fitting the data with Eq. 4. Further work is needed to understand whether data improvement, formula modification, or both is required.

Within our current capabilities, we can use our data to address quite a few more interesting issues. One possibility is to look at the mean square gyration radius of nontrivial knots. Such data are presented in Fig. 3. Apart from being pulled to much larger values of N, our data in this respect are quite similar to those presented earlier (7). For every nontrivial knot, the mean square gyration radius is initially smaller than the topology blind average over all loops, , and becomes larger at certain values of N characteristic for every knot. On theoretical grounds, it was hypothesized (10) that the leading term in N → ∞ asymptotics should be valid for every particular knot type, with both scaling power ν and “amplitude” A independent of the knot type. Indeed, this conclusion seems inevitable considering the fact that any given knot at sufficiently large N is dominated by the stretches that effectively look like parts of a trivial knot (see also refs. 2 and 14). Looking at the data, Fig. 3, with this theoretical concept in mind, we see that the sizes of all knots considered do approach each other with increasing N. However, this approach happens quite slowly even when N is as large as, say, n = 1,500 ≈ 6N ₀.

Our data allow us to make one more step and to look not only at the averaged value of for trivial and some nontrivial knots, but also at the probability distributions of R ² _g. We were able to generate and analyze histograms of quality (i.e., looking smooth when plotted) for loops of size N ≤ 600, where contamination of the trivial pool and the corresponding correction (Eq. 8) are totally insignificant. Predictably, the probability distributions are different for different topological classes, such as all loops versus loops of a certain knot type, 𝒦. Also predictably, the probability distributions of spread out as N increases. The latter observation suggests the idea of looking at the probability distributions of the rescaled variable , where the normalization factor, , is taken separately for each N and for each topological entity.

Our main findings are summarized in Fig. 4, where we present probability distributions P(ρ) for the trivial knots 0₁ (⋄), trefoils 3₁ (▵), and 4₁ knots (□). The data shown are for n = 90, where high-quality statistics were most easily attainable; each histogram is the result of >20 million loops.

In the same Fig. 4, we plot also for comparison the analytically computed probability distributions for linear chains and for all loops. For linear chains, the necessary distribution P _linear(ρ) was found by Fixman a long time ago (15). He showed that the corresponding characteristic function (Fourier transform of the probability density) is equal to K _linear(s) = (sin z/z)^–3/2, where z ² = 4_1s, and where s is conjugate to ρ (i.e., Fourier transform involves e ^1sρ). We were able to derive a similar expression for the probability distribution over all loops, irrespective of topology. In this case, the characteristic function reads K _{all loops}(s) = (2 sin(z/2)/z)^–3, where z ² = 8_1s, with the same definition of s. Numerical inversion of Fourier transforms yield the curves presented in Fig. 4. To avoid overloading the figure, we do not show the corresponding data points obtained for linear chains and for all loops, but they all sit essentially on top of the theoretical curves (which is comforting, because it confirms once again the ergodicity of our loop-generation routine).

Comparing the shapes of probability distributions for all loops and those with identified topology, we notice that the latter distributions are somewhat more narrow. Although the effect looks small for the eye, it is certainly there and it is well above the error bars of our measurements. This effect means simple knots are less likely to swell much above their average size than other knots, and they are also less likely to shrink far below their average than other knots.

The latter point is of particular interest given its relation to all problems involving collapsed polymers, such as proteins. A closer look at the small R_g region of the probability distribution is presented in Fig. 4 Upper Inset. There, probability distributions are plotted in the semilog scale against 1/ρ. This scale can be understood as the plot of “confinement” entropy, which corresponds to the squeezing the polymer to within certain (small) radius. The reason we plot the data against 1/ρ is because both P _linear(ρ) and P _{all loops}(ρ) at small ρ have asymptotics ≈exp(–const/ρ), which corresponds to confinement entropy ≈1/ρ, and which can be established by a simple scaling argument, as described in ref. 3 (p. 42). This 1/ρ behavior is seen clearly in Fig. 4 Upper Inset. Furthermore, we see indeed that compressing any specific knot, trivial or otherwise, is significantly more difficult than compressing a phantom loop. Analytical expression of entropy for knots is not known, only the scaling at small ρ was conjectured in ref. 16. Although our work is qualitatively consistent with this prediction in terms of the direction of the trend, more data are needed for quantitative conclusion.

To conclude, we want to speculate on some broader implications of our findings. To this end, one can conjecture that for any given knot, 𝒦, final asymptotics of at N → ∞ is governed by the same exponent ν as for the trivial knots; that is, according to our results above, . To explain this point, and also to understand at which N this asymptote takes over, it is useful to compare the polymers with quenched and annealed topology, the latter being simply phantom. At every N, the polymer with annealed topology samples a certain ensemble of knots; as N grows longer, the set of knots that are sampled becomes more diverse. In a loose sense, we can imagine a certain average for this set of knots, something like average number of knots or average knot complexity. For instance, we can determine the average diameter of a maximally inflated tube (17) or the minimal rope length (18). Let us denote as the average or typical knot for the given length N. Clearly, as N increases, the typical knot gets more complex, its rope length increases, and its inflated diameter decreases. Now let us go back and consider the real polymer of the given length N with real quenched knot topology, 𝒦. We should recognize the important difference between the cases when the given knot 𝒦 is more complex (with a longer rope length or smaller tube diameter) or simpler (with a shorter rope length or larger tube diameter) than . In the former case, our real polymer can be called overknotted, because it contains a larger amount of knots than it would contain spontaneously if allowed. In the latter case, the polymer can be called underknotted, because it has fewer knots than typical for its length. Overknotted polymers should be more compact than the annealed or phantom loop; in other words, for them, we expect . By contrast, underknotted polymers should be more swollen than their phantom counterparts, . In light of this consideration, we can now understand what happens if we have a given nontrivial knot 𝒦 and we increase N. At the beginning, N is small and we are in the overknotted regime. Eventually, with growing N we expect to cross over into the underknotted regime, and it is in this regime that we expect the size of the knot to scale as N ^ν, because every underknotted loop consists mainly of very long pieces that are not entangled with each other and are not knotted themselves. The number of such pieces depends on the knot 𝒦, but it does not depend on N, such that their length scales as N and their size, therefore, must scale as N ^ν. Much work is needed to make these considerations more quantitative and less speculative; a start in this direction would be to define in a more rigorous fashion. Among other things relevant here, the issue of knot localization (19–21) must be quantified and incorporated. One relevant development in this direction is contained in ref. 22, in which the authors examine what they call the “equilibrium length” of a knot.

To summarize, we have presented simulation data on the sizes of loops with the topologies of trivial or nontrivial knots, for the lengths of up to 3,000 segments. We found that topological constraints have a marginal effect on the loop size as long as the loop is shorter than the characteristic length of random knotting, which is ≈250. At larger N, our results for trivial knots are consistent with crossing over into the scaling regime R_g ∼ N ^ν analogous to self-avoiding walks statistics, for which ν ≈ 0.59. Our findings are also consistent with the idea that the size of any particular nontrivial knot becomes asymptotically equal to that of the trivial knot at very large N, although our data suggest the slow decaying approach to this asymptotic regime. Finally, looking at the probability distributions of the (properly rescaled) loop sizes, we found that topologically complex loops are less likely to adopt either strongly collapsed or strongly expanded configurations.