Letter pubs.acs.org/macroletters
Minimal Surfaces on Unconcatenated Polymer Rings in Melt Jan Smrek*,† and Alexander Y. Grosberg Center for Soft Matter Research and Department of Physics, New York University, New York, New York 10003, United States S Supporting Information *
ABSTRACT: In order to quantify the effect of mutual threading on conformations and dynamics of unconcatenated and unknotted rings in the melt we computationally examine their minimal surfaces. We found a linear scaling of the surface area with the ring length. Minimal surfaces allow for an unambiguous algorithmic definition of mutual threading between rings. Based on it, we found that, although ring threading is frequent, majority of cases correspond to short loops. These findings explain why approximate theories that neglect threading are so unexpectedly successful despite having no small parameter justification. We also examine threading dynamics and identify the threading order parameter that reflects the ring diffusivity.
call a “topological glass” (following the earlier suggestion by Obukhov26). The authors even illustrated their ideas with custom-made, ring-shaped pasta (named anelloni, after anello, which is Italian for a ring).27,28 Thus, there is a contradiction: the agreement of the simple annealed tree model with the simulations suggests that the role of ring−ring threading is quantitatively marginal, at least for the properties studied so far. Yet, the topological glass model says that threading is the key. To address this controversy, we need a method to detect the threading. Previous works on threading used indirect observation methods14 or had to modify the system (by piercing the whole melt by a palisade of straight immobile spikes or by freezing a large fraction of all rings, for example, all rings but one).8,9,15 Though these methods allow the observation and investigation of threading, they also dramatically affect the system itself. In this work, we found that a natural, noninvasive tool to observe threading is provided by the minimal surfaces. We consider a disk-like, two-sided (orientable) surface with a single connected fixed boundary: this boundary is the polymer ring, which is topologically equivalent (homeomorphic) to a circle, because our rings are unknots. We do not consider surfaces of other topologies described in refs 29−31. The piercing of one ring’s surface by another (unconcatenated) ring gives rise to the desired objective definition of ring−ring threading. The study of minimal surfaces of the rings was pioneered by M. Lang;32 here we use longer rings, crucial for accessing the regime where average ring is compact (size ∼ N1/3). The minimal surfaces for each ring length N = 100, 200, 400, 800, and 1600 are spanned on a sample of M = 200 rings taken
M
otivated by both theoretical challenges and applications in genome folding,1−3 concentrated systems of long, flexible, unknotted, and unconcatenated polymer rings have attracted much attention.4−18 The melt of rings is the simplest of polymer systems where the topological noncrossing constraints restrict the available domain of the conformation space. The main difficulty of the problem stems from the failure of the field-theoretic methods19 when topological constraints are concerned.20 No systematic analytical approach exists for topologically constrained polymers; as such, recourse to computationally expensive simulations is frequently necessary (although recent experiments are catching up10,16,17). The growing availability of data from these sources has prompted theorists to search for simplified empirical models. In this direction, three lines of thought can be identified, to some extent contradicting one another. First, the “annealed tree model”,21,22 based on earlier studies,23,24 neglects the threading of one ring by others, assuming each ring to be a double-folded annealed branched polymer (see also25). The neglect of threading is certainly a poorly controlled (i.e., not parameter-based) approximation; yet, it yields a good agreement with simulation data on all measured static and dynamic scaling exponents. Second, Obukhov et al.12 and Ge et al.18 developed certain ad hoc constructions of space-filling fractals, which do not prohibit threading, but do not study it explicitly either. These models offer some conceptual advantages and an important finite-sized analysis;12 but as far as scaling exponents are concerned, their results are close to those of the annealed tree model and agree with simulations equally well. Third, Michieletto et al.8,9,15 and Lee et al.14 pursued a sharply contrasting approach and argued that mutual ring threading must be the most important property of the system, leading eventually to the phenomenon that they imaginatively © XXXX American Chemical Society
Received: April 14, 2016 Accepted: May 25, 2016
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DOI: 10.1021/acsmacrolett.6b00289 ACS Macro Lett. 2016, 5, 750−754
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methods of the following refs: 37−42 (see SI for details). Because they are unknotted, these rings obey self-avoiding walk (SAW) statistics,40,43,44 where the gyration radius Rg ∼ NνSAW, with νSAW ≃ 0.588. We found the scaling of the minimal surface area A ∼ NxSAW, with exponent xSAW ≃ 1.25 ± 0.07. Our numerical finding is consistent, though does not agree perfectly, with the hypothesis that xSAW = 2νSAW ≈ 1.176, justified by the idea that Rg ∼ NνSAW is the only length scale and so A ∼ R2g. Ring threading: We say a ring A penetrates (threads) ring B if ring A crosses the minimal surface of ring B (see Figure in the Abstract or Figure S5 in SI). This relation is not symmetric in general, but frequently it is (Figure S6 in SI). The mean number of rings threaded by a single ring, nt(N), grows from about nt(100) = 2 to about nt(1600) = 19 (Figure 2a). The rate of growth of nt is decreasing with N, because we
from a well-equilibrated MD simulation of a melt of unconcatenated and unknotted rings of Halverson et al;5 the polymer model employed in these simulations has entanglement length Ne ≃ 28 ± 1. The surface-minimization is performed using the Surface Evolver program33 on each ring separately. An initial triangulated surface is spanned on the ring, as described in Supporting Information (SI), and evolves under surface tension by moving the free vertices. Note that the minimal surface can be (and frequently is) self-intersecting. The fixed topology, coupled with the simple evolution described, raises the issue of whether the global area minimum is reachable.31,33 As reported in the SI, we implemented simulated annealing with several different protocols to ensure that the surfaces are sufficiently close to the minimum, and that competing minima do not have an effect on the area distribution. An example of a final minimal surface is shown in the top inset of Figure 1.
Figure 1. Histogram of area of minimal surfaces, plotted for various lengths N normalized by Nx, with x = 1.03. Top inset: Example of a minimal surface spanned on a ring with N = 1600. For details, please zoom into the electronic version of this figure. Bottom inset: Mean area, in units of bond length normalized by N, as a function of N.
Figure 2. Ring threading statistics. (a) Number of rings penetrated by a single ring nt as a function of N in log−log scale. (b) Histogram of number of penetrations np of a single ring caused by a single other ring in linear-log scale. (c) Mean np as a function of N in log−log scale. (d) Area per one penetration as a function of N in log−log scale. All error bars correspond to the ensemble standard deviations.
In Figure 1 we plot the histogram of areas of minimal surfaces normalized by Nx, where x = 1.03 is the best fit of mean area versus N for N ≥ 400. Of course, we can not determine the exponent x very accurately: we fit only three values of N and, moreover, even the largest ring system is not yet completely in the asymptotic regime.5 However, the bottom inset of Figure 1, shows steadily decreasing slope of the mean area normalized by N, which suggests the decrease of the effective exponent x toward 1. Moreover, a significant fraction of the total area lies within a short distance of the boundary, as is shown in SI. This is consistent with a tree-like structure of the conformation. The same scaling of minimal surfaces (x ≈ 1.03) is found also for (a ring version of) the Hilbert space-filling curve (see SI). Hilbert curves and polymers in the ring melt are similar in terms of gyration radius exponent (Rg ∼ Nν, ν = 1/3), but sharply different in terms of the contact exponent γ, which defines the probability s−γ of spatial contact between two monomers chain length s apart (the contact exponent is typically measured in genomic “C experiments”). Our finding suggests that the minimal surface exponent x is related to ν, but not to γ, because γ = 4/3 for a Hilbert curve and γ ≈ 1.05 for ring melt.3,34−36 For a comparison, we also spanned the minimal surfaces on free unknotted rings obtained from our lattice simulation using
approach the asymptotic compact regime (i.e., ν = 1/3 as N → ∞). This is consistent with the trend of the average number of rings K1(N) sharing a common volume, computed in Halverson et al.5 For large rings, we observe nt ≥ K1, pointing to the ramified nature of the ring periphery reflected by γ ≃ 1 (see ref 3). Interestingly, short rings have nt < K1, despite the fact that their conformations are more open. A ring can penetrate surface of another ring multiple times, but this number must be even because the rings are unlinked. The distribution of number of penetrations π(np) of a ring caused by another single ring is found to be exponential (Figure 2b) for all N. The corresponding mean number of crossings seems to grow with N, although with increasing error bars (Figure 2c). For the longer rings, N ≥ 400, the mean total number of penetrations caused by a single ring, equal to npnt, scales linearly with area (and, hence, linearly with N as well), as expected in a dense system (Figure 2d). For shorter rings the mean area per single penetration is greater than for longer rings as the shorter rings are swollen not only due to penetrations. A more detailed view of threading: We define the threading length Lt as the number of monomers of a penetrating segment between subsequent penetrations of a minimal surface of a threaded ring (top inset A of Figure 3). Note that the threading lengths are examined for each pair of threaded rings independently of other rings. The distribution of Lt for any 751
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In SI we also analyzed the distribution of the number of threaded neighbors as a function of the separation length (Figure S10). Dynamics and threading: To investigate the effect of the threading on the ring dynamics, we analyzed 10 successive snapshots of the ring melt system of Halverson et al.5,6 with N = 800 separated by time δt = 106τ, where τ is the time unit of the Kremer-Grest model used in the simulations. The time interval δt corresponds to the mean square displacement of the rings about 0.85R2g and the whole 10δt captures the crossover from the subdiffusive to the diffusive regime reported in ref 6. At first, we looked at the ring pairs that were found to be threaded at all the 10 instances. In the inset of Figure 4, the
Figure 3. Distribution of threading lengths Lt. The distribution is plotted for different N in log−log scale for all numbers of penetrations np. Slope −1.5 is shown that fits the data quite well, even for N/2 > Lt > Ne. Top inset A: An example of threading ring (black) being divided by the threaded ring (red) into np = 4 segments of threading lengths Lt1, Lt2, Lt3, and Lt4. Bottom inset B: Example of separation length as the total length of two black threading length segments, corresponding to Lt1 and Lt3 in top inset. The remaining part (dashed gray) has a total length N − Lsep.
number of penetrations np is depicted in Figure 3. On short scales 10 < Lt < Ne ≃ 30, the probability p(Lt) scales . This can be understood as a distribution approximately as L−1.5 t of length of a 3D random walk between subsequent intersections of a planar surface, because on the short scales, the ring exhibits random walk statistics and the surface is locally flat. The probability p(Lt) reaches a broad minimum around Lt ≃ N/2. If there were only two penetrations for each ring, the distribution p(Lt) would be symmetric, corresponding to segments of Lt and N − Lt monomers long, which only partly explains the increase of p(Lt) at large Lt. More detailed analysis (see SI) suggests that it is a signature of correlations between subsequent surface crossings. It will require more work to understand their nature. In any case, we are able to collapse the distributions for different N, by normalizing Lt by N, and p by a length-dependent factor k(N) that scales roughly as k(N) ∼ N−1.5 (Figure S8 in SI). A measure of entanglement of two rings can be constructed from the threading lengths in the following way. Starting at an arbitrary segment, one labels sequentially all threading segments by integers and sums up separately all threading lengths with even and odd labels. This gives two numbers, the smaller of which we call separation length Lsep (the other is trivially N − Lsep, see Figure 3, bottom inset B). The separation length reflects the amount of material of the penetrating ring that is located on one side of the penetrated surface. In strict geometric terms, this is not true, as when the surface is highly curved, two subsequent surface penetrations could go both in the direction of the local normal surface vector. Indeed, there can be more severe entanglements, such as one ring wound multiple times around another. Our simple approach does not capture these cases. From observation we do not expect such complicated entanglements to be frequent; nevertheless, we should consider Lsep as a lower bound on the amount of entanglement. For values of Lsep close to N/2, we expect a significant rearrangement of the two rings to be necessary for them to separate. The separation length, is a global measure for any two overlapping rings and is distributed on the ring pairs (compare at a local “centrality”45).
Figure 4. Relative ring−ring mean square displacement as a function of time for different ring pair ensembles. Filled triangles represent ring pairs with separation length between one and two Ne and so on. Inset: Separation length distribution for all ring pairs (green circles) and for ring pairs that remained threaded at all ten time snapshots (black circles). Natural units of Kremer-Grest model are used.
distribution of the separation lengths for these pairs is compared with the (static) distribution of separation lengths for all threaded pairs (see also Figure S9 in SI). The distribution for the “persistently” threaded pairs is significantly broadened at longer lengths, while the threading length distribution was only mildly affected (not shown). This suggests that the separation length can serve as a sensitive measure of interpenetrating rings’ ability to separate. Although it is possible that the threading dissolves and reappears again between the snapshots, it is unlikely at this time scale. To explore this further, we measured the relative mean square displacement g of the rings as a function of time and separation length. As shown in Figure 4, the greater is the separation length for two rings, the slower is their mutual separation. This trend is clearly visible, although our ensemble size is not large enough to conclusively extract more detailed information such as the scaling exponents of g with time as a function of the separation length (see statistics details in SI). Conclusions and outlook: We have examined the minimal surfaces spanned on polymer rings. For a well-equilibrated unconcatenated melt,5 we found minimal surface area to scale linearly with the ring length N. This suggests the presence of an underlying double-folded, but not tightly double-folded, treelike structure and explains the accuracy of the predictions of the tree-based theories. An interesting open question here is to identify the branching statistics of these trees. At the same time, loose double-folded branches do allow for ring−ring threading, and we found many protrusions even as long as 5Ne (see SI). These long protrusions represent a natural candidate for explaining the slow crossover between subdiffusion and normal diffusion of individual rings in the melt.6,14 They should also be 752
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taken into account in constructing the computationally efficient “annealed trees” representations of rings.25 Since we examined rings of the equilibrium melt simulated in ref 6, it is not surprising that we did not observe the topological glass phase. However, using minimal surfaces, we were able to identify separation length Lsep as a promising order parameter characterizing the slowing down of the relative diffusion for pairs of deeply threaded rings. This encourages further work to pinpoint the threading-sensitive properties, as well as to explain the surprising lack of such sensitivity for many observables.46 One possible direction of such analysis would be to span surfaces on every ring and minimize them all simultaneously, with the constraint of mutual avoidance. We have also found an intriguing contrast between the linear scaling of minimal surface area for unknotted rings in an unconcatenated melt and the significantly sharper scaling for unknotted rings freely fluctuating in space (N1.25±0.07). Since unknots in free space are governed by self-avoiding size exponent νSAW ≈ 0.588, we hypothesized that our numerical result 1.25 is 2νSAW in disguise. This raises an interesting question concerning the minimal surface area for a freely fluctuating ring, not restricted to be an unknot. In this case, the ring size is Gaussian, leading to the hypothesis that the minimal area should be linear in N. In this context it is interesting to note the theorem, due to van Rensburg and Whittington,47 that the perimeter of a random 2D polygon scales linearly with its area. To conclude, our study of minimal surfaces for polymer rings uncovered a rich source of insight into the static and dynamic characteristics of topologically constrained polymers.
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REFERENCES
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsmacrolett.6b00289.
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Surface preparation procedures and statistics details (PDF).
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Present Address †
Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported partially by the MRSEC Program of the National Science Foundation under Award Number DMR1420073. We thank J. Halverson, K. Kremer, M. Lang, G. Oshanin, and S. Whittington for fruitful discussions and K. Brakke for his help and advice with the Surface Evolver. We thank C. Sandford for his help in manuscript writing and useful suggestions and L. Smrekova for her help and patience. J.S. acknowledges the hospitality of T. Blažek at the Department of Theoretical Physics at Comenius University where part of this work was done. 753
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