Topological Disentanglement Dynamics of Torus Knots on Open

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Letter Cite This: ACS Macro Lett. 2019, 8, 576−581

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Topological Disentanglement Dynamics of Torus Knots on Open Linear Polymers Michele Caraglio,†,‡ Fulvio Baldovin,§,∥ Boris Marcone,⊥ Enzo Orlandini,*,§,∥ and Attilio L. Stella*,§,∥ †

KU Leuven, Soft Matter and Biophysics Section, Celestijnenlaan 200D, 3001 Leuven, Belgium Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21A, A-6020 Innsbruck, Austria § Department of Physics and Astronomy, University of Padova, Via Marzolo 8, I-35131 Padova, Italy ∥ INFN, Sezione di Padova, Via Marzolo 8, I-35131 Padova, Italy ⊥ Center of Excellence for Stability Police Units, via Medici 87, 36100 Vicenza, Italy

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‡

S Supporting Information *

ABSTRACT: We simulate and study the topological disentanglement occurring when torus knots reach the ends of a semiflexible open polymer (decay into simpler knots or unknotting). Through a rescaling procedure and the application of appropriate boundary conditions, we show that the full unknotting process can be understood in terms of point-like particles representing essential crossings, diffusing on the support [0, 1]. We address the bending and configurational free energy drives on the diffusion process, together with the scaling properties of the effective diffusion and friction coefficients. Agreement with simulations suggests universal features for these two model parameters.

T

Here we adopt two complementary methods to shed light on processes involving the progressive modification of the internal structure of loose knots. First, we perform large-scale molecular dynamics (MD) simulations of semiflexible, tensionfree open chains with a knot initially tied in. These simulations allow us to pinpoint the main features of the knot dynamics on relaxed chains, in contrast to most studied situations with applied tension. Second, we propose a set of stochastic equations for knot dynamics, simplification, and full disentanglement, involving free energy drives. They are based only on two topological invariants of the knot k: the number of essential crossings nk and the ratio between the minimal knot length Sk0 and the diameter of the chain σ, Sk0/σ . To achieve this goal we restrict our considerations to the family of (p, 2) torus knots, which are univocally defined once an odd nk is given and for which all essential crossings share similar properties.34,35 Despite the drastic simplification implied by the projection of a collective three-dimensional (3D) process onto a local 1D Langevin dynamics for the essential crossings, model results are in good agreement with MD simulations. Remarkably, both global and local aspects are reflected by the peculiar scaling properties of effective friction and diffusion coefficients.

he elusive trait of topological entanglement is especially evident in processes such as the unknotting dynamics of long fluctuating open filaments,1−3 where entanglement properties may change in time and eventually disappear. A physical understanding about how polymers undergo disentanglement is required to determine the associated time scales4,5 and elastic properties6,7 in the perspective of getting insight into phenomena like the unfolding of knotted proteins8,9 and the effect of topological entanglement on the relaxation dynamics of DNA chains whose physical knots are formed by electric or mechanical compression.5,10−12 Ejection of viral DNAs from their capsids,13−16 the dynamics of DNAs under confinement,3,17,18 or their translocation through nanopores19,20 are further relevant instances of processes where unknotting may play a role. Due to inherent difficulties in detecting knotted portions in random chains,21,22 models for knot motion and disentanglement have focused so far on situations in which knots can be easily monitored. Examples include knotted polymers subject to extensile forces,7,23 under strong confinement17,18 or flattened on surfaces.24−26 In particular, when the length of the knot remains about constant (e.g., for strong pulling or for confinement in narrow channels), simple diffusion provides a reliable picture of the knot dynamics as a whole.17,23,27,27−31 However, if the knot length sensibly varies in time, one has to include in the description free energies that depend on the knot length. This is, for instance, the case in the recent debate about metastable knots in long open chains.32,33 © XXXX American Chemical Society

Received: January 19, 2019 Accepted: April 29, 2019

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DOI: 10.1021/acsmacrolett.9b00055 ACS Macro Lett. 2019, 8, 576−581

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this configurational entropy ΔFc(Sk) is expected to be 32,33,40 but it is negligible for small proportional to S1/3 k , 40,41 Since through our MD we could not achieve chains. chain lengths comparable to those studied by Monte Carlo methods in refs 33 and 41, we pragmatically consider the first S significant term in a Taylor expansion, ΔFc(Sk) = AckBT σk , regarding Ac as a free parameter, which may depend on L. Consistent with the above discussion, in our simulation we have found Ac ≃ 0 with N = 50 and 100, and Ac ≃ 1 with N = 20042 (see SI). Our theoretical model is a simplified setting that allows the reconstruction at any time of the typical location and average length of the knot, and the description of the various stages of its disentanglement. The basic entities representing the knot are the nk essential crossings, assumed to behave as Langevin particles. There are many reasons for this choice. First, nk is a knot invariant for closed loops and it univocally identifies the topology of simple torus knots.34 Moreover, nk has been shown to play a key role in the equilibrium thermodynamics of compact ring polymers, when constrained either by slip links or to move across pores.43,44 Indeed, while knots are weakly localized in good solvent,21,24 in globular polymers they are believed to be delocalized45,46 and their statistical properties can be interpreted in terms of the essential crossings behaving as point-like particles that compete for a share of the polymer backbone along which they are diffusing.43,44 Finally, the role of nk has been shown to be crucial also in the nonequilibrium dynamics of granular knotted chains flattened into 2D by gravity.25 Because of the genuine 3D character of our polymer dynamics, the nk particles have to encode subtle features. First, since Sk cannot be smaller than the minimal knot length Sk0 , we assign the size Sk0/nk to each particle.47 Second, once the fluctuating polymer is projected onto an arbitrary plane, the dynamics may give rise to Reidemeister moves34 in which crossings can be created or annihilated. Because of this, trajectories of essential crossings become discontinuous, and crossings may traverse each other, changing their order along the backbone (see Figure 2 for a schematic illustration). As a matter of fact, it has been recently pointed out that in the dynamics of chains hosting composite knots, entire prime components can pass through each other.48 This latter property conflicts with the single-file character of the 1D diffusion of ref 25 and with the need of assigning a minimal size to each essential crossings. A proper rescaling, mapping

MD simulations are based on a model for semiflexible chains of N beads of diameter σ and persistence length lp = 5σ.36 The knot is initially strongly tied at the center of the chain37 and the detection of the knotted portion relies on standard algorithms.21,38,39 See Supporting Information (SI) for details. Figure 1 shows a typical trajectory obtained for the trefoil (31) knot. At variance with the high-tension cases, Figure 1 and

Figure 1. Unknotting dynamics of a 31 knot within a free open polymer (x is the chemical coordinate along the backbone). Red color highlights the knotted portion; blue indicates dangling ends. A, B, and C display different snapshots. Simplification occurring at t ≃ 300τLJ marks the knot’s death.

Movie M1 clarify that a full description of the time evolution of the entangled portion of a loose knot in terms of either spatial or chemical coordinates is impracticable since topology escapes definite metric specifications. However, we claim that a complete dynamical characterization, including for example a precise localization of essential crossings, is not required to describe the statistics of knot decay and disentanglement. Let us start by identifying the two major contributions of the knot to the polymer’s free energy. The first is the bending energy stored in the knotted region, ΔFb(Sk). The latter is related to the knot configurational entropy, ΔFc(Sk) (see also, refs 32 and 33). Note that both contributions depend on the knot length Sk . Direct comparisons with calculations of ΔFb(Sk) from the MD simulations show consistency with (see SI) ÅÄÅ ÑÉÑ ÅÅ S jij Sk − Sk0 zyzÑÑÑ ÅÅ k0 zzzÑÑÑ ΔFb(Sk) = kBT ÅÅϵb0 expjjjj− ∼ ÅÅ σ j S − S zzÑÑÑ ÅÅ k0 {Ñ (1) k k Ç Ö In eq 1, ϵb0kBT represents the extra bending energy per monomer in the tightest knot configuration (Sk = Sk0) with respect to the equipartition value kBT characteristic of the ∼

∼

relaxed state Sk ≫ Sk ( Sk is the size of the knot beyond which the bending energy contained in the knotted portion of the chain relaxes to that of an unknotted chain of the same length). According to eq 1, a knot that is tightly placed in the middle of the chain swells from the length Sk0 by releasing the bending energy to the reservoir. From our MD, ϵb0 ≃ 0.74 and

∼

Sk /Sk0 ≃ 1.85 about independently of the knot type (see Figures S1−S3 and Table S1 in SI). If L = Nσ is large enough with respect to Sk , the dangling chains departing from the knotted region exert an entropic force on the knot which competes with the bending relaxation to possibly favor an optimal, metastable knot size.32,33 For sufficiently long chains,

Figure 2. Crossings traversing each other. (a−d) A sequence of Reidemeister moves in the knot diagram keeps red segments fixed. The red crossing within a trefoil knot is first (left to right) in (a) and second in (d). 577

DOI: 10.1021/acsmacrolett.9b00055 ACS Macro Lett. 2019, 8, 576−581

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Operatively, this is enforced by first computing the rescaled size S′k ≡ x′nk − x′1 and then inverting eq 2. While Sk0 is the length of the knot when it is maximally tight and the pair (ϵb0,

the 3D process onto the interval [0, 1] solves this puzzle (see Figure 3). Third, in the absence of applied tension, we must

∼

Sk ) is obtained by calculating the bending energy in MD simulations, best fitting the MD knots’ survival probability with the model parameters provides the estimate Ac ≃ 0 for N = 50, 100, and 0.5 < Ac < 1.0 for N = 200 (see SI). In eq 3, the coefficient αi is such to satisfy the physical requirements α1 f = −∂(ΔF)/∂x1, αnk f = −∂(ΔF)/∂xnk for the external particles, and to homogeneously share the free energy expansion or contraction among the internal ones according to the ordering of their coordinates x′i: αi = −1 + 2(i − 1)/(nk − 1). This drive has no effect on the central crossing and linearly increases as one moves toward the knot ends. Finally, D′ and ζ′ are the diffusion and the friction effective coefficients of the point-like particles, respectively. They are obtained as those that best fit the knots’ survival probability from the MD simulations. Estimates suggest the following scaling properties for D′ and ζ′ with the chain size: D′ =

Figure 3. Mathematical model for the unknotting dynamics. Explanations and details within the main text.

nk

distance between the beginning of the leftmost segment and the end of the rightmost one corresponds to the knot contour length Sk (Figure 3a). Moreover, in assigning the initial conditions, we assume that the internal particles are uniformly distributed within this length. Segments are allowed to cross each other but they cannot overlap: all this is achieved by rescaling the coordinates of their center xi onto the interval [0, 1], so that they become point-like particles: x′i ≡

2i − 1 Sk0 2 nk

L − Sk0

(i = 1, 2, ..., nk )

(2)

(refer again to Figure 3a). Simplifying notations, the labels “i” are assumed to be ordered according to increasing x′i ∈ [0, 1]; the point-like particles approximate the essential crossings’ behavior through a continuous dynamics described by the overdamped Langevin equations50 dx′i f′ = αi + dt ζ′

2D′ ηi(t )

, (L − Sk0)2

i L − Sk0 yz zz ζ′ = ζ0jjj k σ {

2

(4)

where D0 and ζ0 are independent of the chain length L and of the knot type k. Explicitly, we found D0 = 0.32 and ζ0 = 0.0032 (see also Table S1 in SI). Equation 4 for D′ is an obvious consequence of the rescaling in eq 2, and as such, it implies a size-independent diffusion coefficient D0 in the support [0, L] for the rigid segments, conferring them a local character. On the contrary, eq 4 for ζ′ is not related to the rescaling procedure, since ζ0 contains no length dimension. It represents instead the scaling with the system size that one would expect for the friction coefficient of a global quantity such as a normal mode.51 It further implies that, for large L, free energy terms of order ΔF ∼ (L − Sk0)3 or higher are necessary to oppose to an otherwise freely diffusing process. Within our scheme, an Ldependence for the free energy can enter through Ac or higher terms in the Taylor expansion of ΔFc. Complementing eq 3 with absorbing boundary conditions at x′ = 0 and x′ = 1, at any given time we can compute the survival probability Sk and the average size Sk for a torus knot k initially tightly placed at the center of the chain, x′i(0) = 1/2∀i (see SI). The main panel of Figure 4 compares such theoretical predictions with the MD results for a trefoil knot. Note that the time axis is rescaled by D′: in this scale, the survival probability of a free diffusion process (f = 0) for three particles amounts to a universal (D- and N-independent) curve, which is reported in Figure 4 with a black dashed-dot line. Note that knotted chains with N = 100 have a behavior very similar to this universal curve, meaning that the free energy drift effects are negligible in this case. On the contrary, for N = 50, the expansion term −∂(ΔFb)/∂Sk dominates the evolution leading to a faster knot decay, while the compression term −∂(ΔFc)/∂Sk becomes important for N = 200. In the latter case, a discrepancy between model predictions and numerical results is observed at short times: it is due to the fact that we rely on a first order approximation for the unknown functional form ΔFc(Sk). Figure 5 and Figure S6 in the SI confirm the agreement between MD results and theoretical estimates for the 51 and 71 knots, respectively.

take into account that in our problem there are neither privileged projection planes nor directions. As explained in the SI, the only objective measures one can rely on are the positions along the backbone of the crossings corresponding to the knot’s ends (which in turn also define the knot location and length Sk ), and one must abandon the idea of assigning a physical meaning to the location of internal crossings. To define our model, we map a given torus knot k, with nk essential crossings embedded into a chain of length L onto nk segments,49 each of length Sk0 , within the interval [0, L]. The

xi −

D0

(3)

where ηi are Gaussian white noises and f ′ = f /(L − Sk0) with f = −∂(ΔF )/∂Sk and ΔF = ΔFb + ΔFc. Given Ac, Sk0 , ϵb0, and

∼

Sk , the force f at each time is fully determined by the instantaneous knot size Sk , which is a measurable quantity. 578

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by rescaling the length before the decay by a factor δ, Δk̅ = ∼

δΔk. As for the parameters Sk0 , ϵb0, and Sk , δ is estimated from MD simulations by comparing the knot size just before and after the simplification. We measured δ = 0.75 in the simplification 71 → 51 and δ = 0.6 in the case 51 → 31. The inset of Figure 5 shows the probability of having a 31 knot when the initial configuration is given by a 51 knot tied in the middle of the chain. According to our protocol, initially the probability of having a 51 is one and that of having a 31 is zero. Then, 51s begin to decay into 31s, and eventually also 31s disappear. Again, model predictions fit very well simulations, and this is also true for the more complex disentanglement of the 71 knot (Figure 6 in the SI). Notice that in thinner, more flexible, chains than those considered here, the theory must be generalized to take into account the possible creation of new particles accounting for spontaneous knot formation in the chain. In summary, the paradigm of simple diffusion, underpinned by a proper account of the topological invariants nk and Sk0/σ and by the nontrivial scaling laws (eq 4), leads to a sound description of torus knot simplification and full disentanglement for unconstrained open chains. Our MD simulations depend on the parameters (σ, lp, T,N) (see SI), while the diffusion model is characterized by three free parameters (D0,

Figure 4. Survival probability (main figure) and length (inset) vs rescaled time of a 31 knot initially tied at the center, for three chain’s contour lengths L = Nσ. Symbols refer to the MD simulations; dashed curves to numerical solutions of eq 3 with parameters that are fitted to the data. The dashed-dotted curve is the pure diffusive case (f = 0), D- and N-independent in the rescaled axes.

∼

ζ0, Ac); indeed, parameters (ϵb0 , Sk0, Sk , δ) are fixed by direct comparison with the MD simulations. Although tests for other values of (σ, lp, T,N) may require model extensions, we introduced the basic ingredients for a simplified account of knot disentanglement, with universal features highlighted in the knot-type and chain-length independence of D0 and ζ0. Our approach is rather general and our model could describe the relaxation of knotted chain with different initial configurations without the need of changing the parameters’ set (see SI). In principle, the model can be generalized to encompass other knot families, like twist knots, more frequently found in proteins,53−55 and to deal with slipknots,56,57 for which the equivalence of all crossings with respect to knot decay does not hold. Another extension of the model can be realized by enriching the drift term with a free energy ΔF; (Sk , ;) ∝ Sk;θ(Sk − Sk0), with ; being the applied tension and θ is the Heaviside function (to guarantee Sk ≥ Sk0). This extension will allow to address the knot simplification dynamics and the underlying time scales of knotted, elongated DNA chains, systems that are nowadays experimentally studied by applying elongational flows,2,11,58,59 electric fields,30,31 and stretching microdevices.7,23 Note that, while here we discussed the case ; = 0, such a potential that in the high-tension limit constrains the nk points to rigidly move together, should allow the study of the crossover between loose and tight knots behaviors.

Figure 5. Disentanglement of a 51 knot. Survival probability for the 51 configurations (main figure) and probability for the 31 ones (inset) vs rescaled time. Symbols and lines as in Figure 4.

Boundary conditions, further specifying what happens when an external crossing reaches one of the backbone ends, endow the theoretical model with the possibility of describing the full knot simplification besides the first knot decay. Consider a torus knot initially tied in the middle of the chain. In this case, the decay sequence amounts to ... → 71 → 51 → 31 → unknotted. As one may have expected, in our MD simulations, we verified that each time a simplification occurs, the chain is left with a simpler knot k̅, constituted by nk̅ = nk − 2 essential crossings. An exception is made for the trefoil knot (nk = 3), which is characterized by unknotting number 1:52 in this case when a crossing touches a boundary unknotting occurs immediately. In terms of our model, we can thus specify the following scheme. With nk > 3, when either x′1 = 0 or x′nk = 1, we consistently assume that both x′1 and x′2 or both x′nk and x′nk−1 are simultaneously absorbed, respectively. Equation 3 can then be applied once more to describe the evolution of the resulting simpler knot k̅. However, as depicted in Figure 3b, the initial length Δk̅ of the survived knot cannot be simply thought in terms of the position of the survived crossings. Due to a topological effect explained in the SI, it is instead obtained

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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.9b00055. Details of MD simulations, knot detection, bending energy of the knotted chain, torus knot simplification, calculation of knot probability and knot size, decay of torus knot 71, functional form of the survival probability, probability distribution of knot decay times, summary of 579

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model parameters, comparison with time relaxation on unknotted chains, and different initial conditions (PDF)

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Movie of trefoil knot disentanglement (MPG)

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Enzo Orlandini: 0000-0003-3680-9488 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge support from the Research Project “Dynamical behavior of complex systems: from scaling symmetries to economic growth”, University of Padova.

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REFERENCES

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DOI: 10.1021/acsmacrolett.9b00055 ACS Macro Lett. 2019, 8, 576−581

Letter

ACS Macro Letters instance, notice that the derivative of S1/3 k changes rather fast for small Sk . (43) Baiesi, M.; Orlandini, E.; Stella, A. L.; Zonta, F. Topological signature of globular polymers. Phys. Rev. Lett. 2011, 106, 258301. (44) Baiesi, M.; Orlandini, E.; Stella, A. L. Knotted Globular Ring Polymers: How Topology Affects Statistics and Thermodynamics. Macromolecules 2014, 47, 8466−8476. (45) Virnau, P.; Kantor, Y.; Kardar, M. Knots in globule and coil phases of a model polyethylene. J. Am. Chem. Soc. 2005, 127, 15102− 15106. (46) Marcone, B.; Orlandini, E.; Stella, A. L.; Zonta, F. Size of knots in ring polymers. Phys. Rev. E 2007, 75, 041105−11. (47) In apportioning an equal size to the particles, we are implicitly take advantage of the fact that minimal crossings are topologically equivalent for torus knots. (48) Trefz, B.; Siebert, J.; Virnau, P. How molecular knots can pass through each other. Proc. Natl. Acad. Sci. U. S. A. 2014, 111, 7948− 7951. (49) An attempt we made to model the knot dynamics in terms of the diffusion of the two ends only revealed it to be unsuccessful in terms of obtaining simple stable parameters like those summarized in Table S1 of the SI. (50) In principle, one may consider a soft-core repulsion among the point-like particles by introducing a free energy penalty related to crossings that traverse each other.48 The dependency of crossings locations on the projection-plane makes, however, this activation barrier extremely hard to quantify by simulations. In our modeling, we found it sufficient to allow point-like particles to traverse each other without any energy cost. (51) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press, 1986. (52) The unknotting number is the smallest number of crossings to be reversed in order to obtain the unknot. (53) Sułkowska, J. I.; Rawdon, E. J.; Millett, K. C.; Onuchic, J. N.; Stasiak, A. Conservation of complex knotting and slipknotting patterns in proteins. Proc. Natl. Acad. Sci. U. S. A. 2012, 109, E1715−E1723. (54) Jamroz, M.; Niemyska, W.; Rawdon, E. J.; Stasiak, A.; Millett, K. C.; Sułkowski, P.; Sułkowska, J. I. KnotProt: a database of proteins with knots and slipknots. Nucleic Acids Res. 2015, 43, D306−D314. (55) Note, however, that most of the knotted proteins are trefoils, the knot type for which our theory fully holds. (56) King, N. P.; Yeates, E. O.; Yeates, T. O. Identification of rare slipknots in proteins and their implications for stability and folding. J. Mol. Biol. 2007, 373, 153−166. (57) Millett, K. C. Knots, slipknots, and ephemeral knots in random walks and equilateral polygons. J. Knot Theory Ramifications 2010, 19, 601−615. (58) Klotz, A. R.; Narsimhan, V.; Soh, B. W.; Doyle, P. S. Dynamics of DNA Knots during Chain Relaxation. Macromolecules 2017, 50, 4074. (59) Soh, B. W.; Klotz, A. R.; Doyle, P. S. Untying of Complex Knots on Stretched Polymers in Elongational Fields. Macromolecules 2018, 51, 9562.

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DOI: 10.1021/acsmacrolett.9b00055 ACS Macro Lett. 2019, 8, 576−581