Linking of Ring Polymers in Slit-Like Confinement - Macromolecules

Article pubs.acs.org/Macromolecules

Linking of Ring Polymers in Slit-Like Confinement G. D’Adamo,*,† E. Orlandini,‡ and C. Micheletti*,† †

SISSA, International School for Advanced Studies, via Bonomea 265, I-34136 Trieste, Italy Dipartimento di Fisica e Astronomia, Università di Padova and Sezione INFN, Via Marzolo 8, I-35100 Padova, Italy

‡

S Supporting Information *

ABSTRACT: Stochastic simulations are used to study the linking properties of solutions of circular polymers in slit confinement. Specifically, we consider dispersions of semiflexible rings at various densities ϕ and slit height, H. The competing length scales in the system have significant effects on the interchain entanglement. We observe that the linking probability is largest for a specific slit height that is about independent of solution density. However, when ϕ is large, links with given number of components can significantly depart from the overall linking trend with H. In this case, binary links are found to be least probable when the overall incidence of links is maximum. We show that this intriguing dichotomy and other properties, including upper bounds on links abundance, can be quantitatively captured with an approximate model based on continuum percolation theory. Our results suggest that slit-like confinement could be used in applicative contexts to control independently both the average abundance and the topological complexity of annealed molecular links.

I. INTRODUCTION The current interest in molecular linking is motivated by its ubiquity in physical, chemical and biological contexts. Interlocked molecules are a main target of supramolecular synthetic chemistry too,1−4 and catenanes of DNA rings spontaneously form during the cell cycle of bacteria5 and other microorganisms, such as trypanosome where thousands of short (1− 2.5 kbp) DNA loops establish a chain mail network.6−9 The linking of rings in solutions is, likewise, one of the classic and open problems in polymer physics.10,11 Yet, this form of intermolecular entanglement has been much less investigated than the intramolecular counterpart, namely knotting.12,13 This is because the notion of linking, unlike that of knotting, involves at least two rings, making the problem less tractable analytically and more demanding computationally for the higher complexity of the multivariable polynomial invariants used for link detection.14 For these reasons, linking has been mostly addressed for the primary, but still challenging case of pairs of unconstrained rings in bulk,11,15,16 though solutions of linkable rings has been theoretically studied too in bulk or within boxes, observing a rich phenomenology.17−19 These considerations motivate the present work where we consider a still virtually unexplored problem, namely the effect of slit-like confinement on the linking properties of a solution of semiflexible chains. Such system is particularly suited to understand the general problem of how linking depends on various competing length scales: chain contour length, persistence length, average interchain separation and size of the confining region. Indeed, we show that the resulting phenomenology is not only very rich, but presents unexpected and counterintuitive features too, notably the fact that the © 2017 American Chemical Society

incidence of links is maximized at a specific level of slit confinement. The article is organized as follows. The model for the solution of confined ring polymers and the methods used to detect their linking properties are described in section II. The results are next presented in section III starting from the characterization of the metric properties of confined rings, and then moving on to profile their linking probability as a function of both the degree of slit confinement and solution density. We next discuss the articulated properties of the system and show that it can be transparently captured by using concept first introduced in the continuum percolation theory of overlapping spheres. Promising perspectives for future developments are finally outlined in the conclusions.

II. MODEL AND METHODS II.A. The Model. We considered a solution of N semiflexible equilateral polygons of L segments of unitary length placed inside a periodic square slit of height H and volume V, see Figure 1. The semiflexible character of the chains is accounted for by the following potential, N

L

βVintra = − ∑ ∑ κb biα ·biα+ 1 α=1 i=1

(1)

Received: October 22, 2016 Revised: January 22, 2017 Published: February 6, 2017 1713

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slit preserves the intra- and interchain topologies, the sampled ensemble of equilibrated rings is topologically unrestricted. II.B. Detection of Links. The detection of the homotopical linking between a pair of chains is performed by computing the two-variable Alexander polynomial Δ12(s, t) at s = −1, t = −1; see refs 14, 21, and 22. Specifically, the criterion Δ12(−1,−1) ≠ 0 allows us to identity unambiguously any topologically linked pair with less than nine minimal crossings. When dealing with a solution of rings, we first test any pair of rings for their homotopical linking and then process the binarylinking matrix with the clustering procedure of ref 23 to identify multicomponent links. Clearly, this scheme is inherently tied to pairwise links and hence it cannot detect Borromean and other types of multicomponent Brunnian links that become unlinked if any of their components is removed. This class of links of links could, in principle, be detected by mechanically pulling the rings away one from another while enforcing the uncrossability of the ring bonds. This pulling dynamics would, in fact, allow to separate all distinct topological components and thus detect any kind of links including the Brunnian ones. However, the bond uncrossability constraints, requires using a very small integration time step and hence is not very practical in situations, such as the present one, where one deals with hundreds of rings.

Figure 1. (a) Typical configuration of the considered Kratky−Porod rings confined inside a periodic slit of height H at 15% volume fraction (ϕ = 0.15). The dispersion consists of N = 256 rings, each made of L = 340 unitary bonds. Different colors are used for links that are either isolated or take part to links with 2, 3, or more components.

where κb is the bending rigidity, bαi is the ith bond vector (of unitary length) of the αth chain, and with the proviso that bαL+1 ≡ bα1 because the chains are closed. For this first systematic study no further intra- and interchain interactions, such as excluded volume effects, are considered and the system therefore consists of a dispersion of semiflexible phantom rings with persistence length given by lp b

=−

III. RESULTS AND DISCUSSION III.A. Metric Properties of a Single Confined Ring. As a first step we characterized the average (root-mean-square) radius of gyration for one chain confined in slits of different height, H. The results are shown in Figure 2 and are consistent with what

1

(

log coth(κb) −

1 κb

)

(2)

For definiteness, we set the parameters of the model chains equal to L = 340 and lp/b = 5. This choice corresponds to typical discretizations of P4 DNA rings20 (L = 3.4 μm, lp = 50 nm, b = 10 nm), but for which we presently neglect the finite thickness. For the dispersion, we consider N = 256 chains at various concentra tions. Specifically, the volume fraction 3 4π ϕ = 3 R̂ g N /V , where R̂ g is the gyration radius of a chain in bulk, was varied in the 0.05−0.65 range. The slit is modeled by two impenetrable nonadbsorbing walls placed along the z−direction at z = 0 and z = H. The parallelepiped simulation box is periodic in the other two Cartesian directions, see Figure 1. The canonical ensemble of the solution was next sampled with a two-tier stochastic approach. First a Metropolis Monte Carlo scheme, based on unrestricted crankshaft moves, was used to generate single-ring uncorrelated configurations inside the slit. The typical number of Monte Carlo sweeps between two samples have been chosen to be proportional to the estimated decorrelation time of Rg, which increases significantly as the slit height decreases. The samples were next subdivided in batches of N rings and the elements of each batch were stochastically placed inside the slit. Specifically the xy components of each ring center of mass (i.e., the slit plane) were sampled from a uniform distribution by keeping its z component fixed. Because the rings are phantom, the insertion move proposed in the Monte Carlo scheme does not introduce any bias in the statistics and is always accepted. For each batch, this procedure was typically repeated 50 times, and to get sufficiently accurate estimates of the linking probability, we considered a minimum number of batches of the order of 104. Note that, since neither the single-ring Monte Carlo moves nor the ring insertion in the

Figure 2. (a) Effect of slit confinement on the transverse and longitudinal components of the radius of gyration of a single chain. (b) H-dependence of the overall (root-mean-square) radius of gyration.

reported in previous metric studies of slit-confined linear and ring polymers.18,24−28 Specifically, upon reducing H, one has that the component of Rg that is transverse to the slit plane decreases monotonically, as intuitively expected. At the same time, the increasing confinement causes the ring to spread in the slit plane, and hence the parallel component of Rg increases upon lowering H, see Figure 2a. The competition between these two trends produces the nonmonotonic H-dependence of Rg that is visible in Figure 2b. Similarly to what previously found for chains with excluded volume interactions,27 the maximum overall compression is achieved for H ≈ 1.35 R̂ g, where R̂ g is the gyration radius for an isolated unconfined chain. 1714

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Macromolecules III.B. Linking Properties of Several Confined Rings. It is interesting to relate the results of Figure 2 to the effects of confinement on the linking probability of a solution of rings at fixed density. In particular, for increasing mild confinement (i.e., H/R̂ g decreasing from 10 to 4 in Figure 2b) one could expect that the reduction of Rg would make it more difficult for two chains to interpenetrate. Accordingly an initial decrease of the linking probability with confinement is anticipated. Figure 3 shows the actual results of the linking probability, Plink, and its dependence on the slit height, H, and the solution volume

Figure 4. Linking probability density, ρ(2) link, for a system of only two slitconfined rings as a function of their center-of-mass distances measured perpendicularly, d⊥, and longitudinally to the slit plane, d∥. For simplicity, the center of mass of one of the rings is rooted in the slit mid plane. The three panels, drawn to scale, correspond to three different degrees of confinement. The indicated total linking probability, Plink was obtained by integrating ρ(2) link over the parallel and orthogonal directions.

The crossover occurring at H ∼ 4R̂ g is understood by considering Figure 5, which shows the probability distributions,

Figure 3. Linking probability, Plink, as a function of (H/R̂ g)−1 for various values of the solution volume fraction, ϕ.

fraction, ϕ. The iso-density profiles for Plink show a systematic, qualitative departure from the trend expected from the argument given above. In fact, at each considered value of ϕ, the initial progressive reduction of H causes the linking probability to increase and not decrease. This trend holds up to (H*/R̂ g)−1 ≈ 0.25, i.e., H ≈ 4R̂ g, when Plink reaches is maximum before dropping for narrower slits. A key element of this nonintuitive trend is that the slit height that maximizes the linking probability, H*, is about three times larger than the one giving the maximum chain compression, H ≈ 1.35 R̂ g, see Figure 2b. As a matter of fact, at H* the slit is still too wide to appreciably alter the chain metric properties with respect to the unconstrained case. To understand the origin of the Plink versus H trend of Figure 3, we consider the linking probability of only two rings, P(2) link, and discuss how it depends on the position of their centers of mass within the slit. For simplicity, we restrict ourselves to the case where the center of mass of one ring is rooted in the slit mid plane. Figure 4 shows a contour plot of the corresponding linking probability density, ρ(2) link, as a function of distance of the two rings measured either perpendicularly to the slit plane, d⊥, or longitudinally to it, d∥. To better convey the interplay of metric and linking properties, the average projected gyration ellipsoid for the rooted ring is outlined with a dashed line. The density plots shows that, as the slit height is reduced, the region where ρ(2) link is appreciable decrease in transverse size too. More interestingly, Figure 4 shows that such link suppressing (2) mechanism, is counteracted by two other effects: the ρlink distribution becomes broader longitudinally, and its peak value increases too. For intermediate confinement, H ≈ 4R̂ g, the latter effects overcompensate the former one and, for this minimal system of two rings only, they produce a nonmonotonic dependence of the linking probability profile analogous to that shown in Figure 3 for the ring solution.

Figure 5. Probability density that the center of a segment is at a transverse height z in slits of various heights. For clarity, the probability profile, shown with a black curve, has been normalized to its maximum value for each slit height. The depletion layers near the walls have width Δ ∼ R̂ g. The region beyond the depletion layer(s) is indicated with the dashed red line. Note that the maximum of Plink occurs for H ∼ 4R̂ g, when the slit transverse span is half covered by the depletion layers.

ρ(z), to find a ring at various transverse heights, z in the slit. For ease of comparison, the ρ(z) profiles have been normalized to their maximum value, ρmax for each slight height, H. Because of conformational restrictions near the walls, one observes that the abundance of rings is suppressed near the slit boundaries. The width of this depletion region, Δ ≡ 1/2∫ H0 dz (1 − ρ(z)/ρmax), is found to be very close to Rg for all the cases shown in Figure 5. At H < 4Δ ∼ 4R̂ g the two depleted layers cover most of transverse span of the slit. Therefore, as H is lower below 4Rg, rings become unlikely to “stack” and link in the transverse direction and, at fixed overall density, must increase their longitudinal spreading in the slit plane. As a result, for H < 4 Rg, the linking probability has to decrease concomitantly with H, which explains why the maximum of Plink occurs for H ∼ 4R̂ g, see Figure 3. To our knowledge, this transverse and longitudinal competition of linking in a slit-confined solution of rings has not been 1715

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Macromolecules discussed or reported before. The closest previous studies are arguably for a lattice model of two spatially constrained polygons11 and for rigid self-assembled constructs within a slab.3 III.C. Analogies with the Continuum Percolation of Overlapping Spheres. Beyond the insight provided by Figure 4, the overall incidence of links as well as the abundance of various link types can be captured, and even predicted, with good quantitative control by generalizing concepts developed for the continuum percolation of overlapping spheres. To this end, we recall that, for a bulk solution of fully penetrable spheres of radius R at number density n, the probability that a sphere is part of an overlapping cluster is29,30 Poverlap = 1 − e−nI2

(3)

where I2 is the characteristic volume over which two spheres overlap. This quantity is twice the second virial coefficient of a system of hard-spheres of radius R, I2 ≡

∫ dr fov (r),

⎧1, if|r| < 2R fov = ⎨ ⎩ 0, otherwise

Figure 6. (a) Topological virial coefficient for two slit-confined rings, 02 as a function of the reduced slit height. (b) Scatter plot of the linking probability data of Figure 3 (all ϕ’s and H’s and links with any number of components) versus 02ϕ. The dashed curve is the theoretical approximation of eq 5 based on the continuum percolation analogy.

(4)

where r is the distance of the centers of the two spheres. It is physically appealing to draw an analogy between the case of linkable rings considered here and a collection of penetrable spheres (or ellipsoids). From this standpoint, a set of m rings that are linked, and hence necessarily interpenetrating, should correspond to a cluster of m overlapping spheres. Accordingly, by generalizing expression 3, we surmise that the probability that a ring is involved in a link is given by theory Plink = 1 − exp[−ϕ 02]

(5)

where 02 is proportional to the integral of the linking probability density ρ(2) link(r1, r2; H) taken over all the accessible center of mass positions of two rings, r1 and r2, 02(H ) =

3 3 4πR̂ g V

∫ d3r1d3r2 ρlink(2)(r1, r2; H) (6) Figure 7. Scatter plot of the binary linking probability, P(2) link, versus the overall linking proability, Plink for various slit heights and solution densities. The dashed curve represents the analytical estimate of Ptheory link obtained from the percolation theory of overlapping spheres, see eq 7.

see also Supporting Information for more details. Generalizing from the case of overlapping spheres, 02 can be interpreted as the typical volume within which two ring polymers can be actually linked and hence corresponds to twice the topological second virial coefficient of unlinked phantom rings.21,31 The resulting profile of 02(H ) is shown in Figure 6a and exhibits the same nonmonotonic dependence on the slit height previously discussed. Moreover, as it is shown in panel b of the same figure, the theoretical curve of eq 5 gives a very satisfactory account of the actual measurements of P(2) link. In fact, data points collected at all considered slit heights and solution densities fall on the theoretical curve, which is parameter free. III.D. Total versus Binary-Only Linking Probabilities. The theoretical expression of eq 5, which is based on an integral property of two rings solely, holds well even though most links involve more than two components, see Supporting Information, section 2. This is clarified by the scatter plot in Figure 7 of probabilities that a randomly chosen ring takes part to a proper link with any number of components, Plink, or with just two components, P(2) link. Again, the data in Figure 7 are for all considered densities and slit heights.

The plot presents an unexpectedly complex covariation of the two quantities with ϕ and H. First, by inspecting the span of the coordinate axis, one notes that no more than 25% of the rings are ever involved in binary links. Second, the data points have an approximately quadratic envelope. Third, the traces connecting data points at fixed density are curled so that there cannot be a general consensual covariation of the total and binary-only linking probabilities. More precisely, the data in Figure 7 show that a positive correlation of the two types of probabilities does exists, but only at low densities. For ϕ < 0.1, in fact, P(2) link and Plink increase and decrease concomitantly. However, as the density is increased beyond 0.1, the maximum of P(2) link is attained for slit heights that become rapidly smaller than 4R̂ g. Instead, the maximum of the overall linking probability (either binary or not) is consistently found at H ≈ 4R̂ g for all densities, see Figure 3. As a result, when ϕ increases, the mismatch of the maxima of P(2) link and Plink grows too. For ϕ > 0.35, it becomes large enough that a further intriguing effect ensues: the overall linking 1716

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0.3, one has that Plink is larger than 50% already in bulk and hence will stay so until H is reduced sufficiently below 4R̂ g. Only at that stage, the suppression of the more complex links allows P(2) link to approach the theoretical maximum of 25%. Further reductions of the slit height will then restore the correlated, concomitant reduction of both P(2) link and Plink, as observed in Figure 7.

probability, Plink, is maximum for about the same slit heights for which the binary incidence, P(2) link, is minimum. This rich phenomenology, which would have been difficult to anticipate from a priori considerations, can again be rationalized with the help of the continuum percolation analogy. Within such framework, the total and pairwise linking probabilities are approximately tied by32,33 (see also section 1 of Supporting Information), (2)theory theory theory Plink = Plink ·(1 − Plink )

IV. CONCLUSIONS Our results, and the interpretative framework of the continuum percolation theory demonstrate that slit confinement can be a valuable means for modulating the abundance and the number of components of links, and even for tuning both features independently to a certain extent. We envisage that our study could be extended in several directions to address problems of theoretical and possibly applicative interest. On the one hand, it would be relevant to extend the treatment to more detailed and accurate models for polymers or biopolymers that could be used in actual experiments within nanoslits. On the other hand, it would be interesting to characterize the precise topological spectrum by using more powerful link invariants, whose systematic applicability is still limited by the demanding computational resources. This point is particularly interesting because it is known that slit confinement can significantly affect the knotting properties of individual rings.27,34 This ought to generate a particularly rich interplay of intra- and interchain entanglement. This possibility, is indeed suggested by the results of Figure 9 which shows the abundance of binary links with one

(7)

This quadratic relationship is shown with a dashed line in the scatter plot of Figure 7. It provides an upper bound for the actual data points which, on average, deviate from it by 5%. According to the theoretical approximation of eq 7 the largest possible incidence of binary links is 25%, and it should occur when the total linking probability is 50%. Both conditions are in good agreement with the actual data points in Figure 7, as well as with the heatmap plots of Figure 8.

Figure 8. (a) Heatmap of P(2) link as a function of slit height and solution density. The solid white line connects the maxima of P(2) link as a function of ϕ at fixed H. The dashed line is the theoretical prediction based on eq 5. (b) Heatmap of Plink as a function of ϕ and H/R̂ g. Figure 9. Heatmap of the probability to observe binary links involving a knotted ring in one or both components as a function of slit height and solution density.

Panel a was obtained by combining the density-dependent ̂ profiles of P(2) link at different confinement, H/Rg. The locus of points corresponding to the theoretical and actual maxima of P(2) link are shown with dashed and solid curves, respectively. The closeness of the two curves further supports the viability of the theoretical approximation. More importantly, the same theoretical framework, and in particular eq 7, can be used to rationalize the qualitatively different regimes covariation of P(2) link and Plink at low and high densities seen in Figure 7. In fact, one notes that at sufficiently low densities, ϕ < 0.2, the total linking probability is smaller than 50% at all levels of confinement, see Figure 3. On the basis of eq 7 this implies that theory theory binary links are the dominant species (P(2) /Plink > 0.5), link (2) which explains the correlated covariations of Plink and Plink observed at low densities. The situation changes qualitatively when Plink is pushed beyond 50% by varying either H or ϕ. In this case, eq 7 clarifies that P(2) link does not increase anymore with Plink but rather decrease. This happens because links with more than two components become favored. In fact, the probability of an m-component link predicted from the continuum percolation analogy32,33 can be theory m−1 approximated as Ptheory . link · (1 − Plink ) These observations explain why at sufficiently high density, the maxima of Plink along the iso-ϕ curves in Figure 8 correspond to the minima, rather that the maxima of P(2) link. In particular, for ϕ >

or two knotted components as a function of H and ϕ. The comparison with the overall binary link population in Figure 7 reveals major variations in the balance of knotted and unknotted links populations and hence worthy of being studied further. Finally one could extend this analysis to ring polymers and biopolymers confined in nanochannels. Such setups are nowadays accessible to microfluidic experiments and have been recently studied theoretically for the special case of a lattice model a pair of rings spanning a tube.35

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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/acs.macromol.6b02293. Derivation of the linking probability for a bulk solution of noninteracting rings and characterization of the average number of link components (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*(G.D.) E-mail: [email protected]. 1717

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Macromolecules *(C.M.) E-mail: [email protected].

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ORCID

G. D’Adamo: 0000-0002-4423-9510 C. Micheletti: 0000-0002-1022-1638 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge support from the Italian Ministry of Education, Grant PRIN No. 2010HXAW77.

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REFERENCES

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