Disentangling entanglements in biopolymer solutions

Reptation theory has been highly successful in explaining the unusual material properties of entangled polymer solutions. It reduces the complex many-body dynamics to a single-polymer description, where each polymer is envisaged to be confined to a tube through which it moves in a snake-like fashion. For flexible polymers, reptation theory has been amply confirmed by both experiments and simulations. In contrast, for semiflexible polymers, experimental and numerical tests are either limited to the onset of reptation, or were performed for tracer polymers in a fixed, static matrix. Here, we report Brownian dynamics simulations of entangled solutions of semiflexible polymers, which show that curvilinear motion along a tube (reptation) is no longer the dominant mode of dynamics. Instead, we find that polymers disentangle due to correlated constraint release, which leads to equilibration of internal bending modes before polymers diffuse the full tube length. The physical mechanism underlying terminal stress relaxation is rotational diffusion mediated by disentanglement rather than curvilinear motion along a tube.

) is shown together with the results from Brownian dynamics simulations. We show the results of the bead-spring algorithm used in this work (solid yellow circles) and compare it with the theoretical prediction (Eq. (2)) and simulations employing a bead-rod algorithm (green solid squares) [1,2]. In the bead-spring case, the numerical data were obtained for polymers with p/L = 5 in a rather dilute solution with ξ/L = 0.34. The bead-rod algorithm employs free polymers of identical characteristics. The deviations between simulations and between simulations and analytical results are all comparable to the statistical error which are about symbol size.        Figure 15. Scaling of the mean-square end-to-end distance with friction coefficients. A single polymer with a friction coefficient ζ is considered in an entangled solution of polymers whose friction coefficient ζn differs from ζ. Simulations were performed for polymers with lp/L = 1 in solutions with a mesh size ξ/L = 0.086, and a range of values for ζn and ζ as indicated in the graph. In the upper panel, the friction coefficient of the surrounding polymers is fixed, ζn = 1, and ζ is varied, and vice versa in the lower panel. Upon changing the friction of the tracer filament, rescaling the time with (ζn/ζ) 2/3 results in a good collapse of the internal relaxation time. Consistently using (ζn/ζ) 1/3 achieves a data collapse, if the friction coefficient of the surrounding filaments is altered.

SUPPLEMENTARY NOTE 1: TESTS OF BROWNIAN DYNAMICS SIMULATIONS
We performed extensive tests to ensure the reliability of our Brownian dynamics simulations. First, we tested some known static quantities, in particular the tangent-tangent correlation function of an individual polymer chain. Then we tested the dynamics of the system by analysing the results of our simulation in dilute solution. Finally, we analysed the effect of system size and filament thickness on the dynamics to find parameter ranges where these have only negligible effects on the dynamics of the entangled polymer solution.
Tangent-tangent correlation function. One of the most basic quantities characterising the statistical properties of the worm-like chain model is the tangent-tangent correlation function which is given by [3,4] where p denotes the polymer's persistence length. As shown in Supplementary Fig. 1, our simulations agree very well with the expected exponential decay, both for dilute and a dense entangled polymer solutions.
End-to-end distance fluctuations. A known possible issue of bead-spring algorithms like the one used in this work is the finite extensibility of the springs. It can be avoided by using a constrained algorithm simulating an inextensible chain [1,2,[5][6][7]. Alternatively, as done in this work, a proper choice of the computational parameters is necessary such that the finite extensibility only affects the dynamics at small times but leaves the dynamics at long times unchanged. To carefully check for effects from chain extensibility we simulated the dynamics of the end-to-end distance fluctuations δR 2 (t) in dilute solutions and tested it against known analytical results for free polymers [8][9][10][11][12]: Additionally, we tested our simulations against the results of a bead-rod algorithm of free polymers [1,7], which fully respects the inextensibility of the polymer chain and was implemented by our group as well [2]. As shown in Supplementary Fig. 2, there is excellent agreement between our Brownian dynamics simulations and both the analytical predictions and the bead-rod algorithm. Thus we conclude that on the timescales of interest our algorithm gives correct results within the error margin.
Test of chain crossing. In this work, we are interested in the dynamics of entangled solutions where topological constraints hinder the dynamics of the polymers. Therefore, it is of utmost importance to ensure that these constraints are faithfully implemented, and the Brownian dynamic simulations do not show any spurious chain crossings. In order to test that polymer chains do not cross in our simulations we studied the scenario illustrated in Supplementary Fig. 3: Two initially stretched chains are positioned in the simulation box to form a cross, one along the x-axis and the other parallel to the y-axis such that the center of both chains is on the z-axis. Initially, the shortest distance between the chains is set to 1.5 σ. We considered a situation where the ends (outermost beads) of these chains are pulled with a (spring) forces of magnitude 10 k B T /a 0 against the other parallel to the z-direction; see arrows in Supplementary   Fig. 3. These forces are larger than the thermal forces acting on a single bead, which are of order k B T /a 0 . In three runs of the Brownian dynamics simulations, which each lasted for a time period of 3τ d , we did not find any chain crossings. Thus we concluded that the implementations of both the excluded volume and the connectivity of the chain are chosen such that chain crossings are prevented. As an additional precaution, we included an abort condition in all of our simulations. If the distance between any two beads decreases below 0.75 σ or the bond length in a chain increases above 1.2 L/N the simulation was aborted with an error.
Simulation volume and finite size effects. The size of the simulation volume was determined such as to avoid finite size artefacts and still achieve acceptable run times (less than 14 days on an Intel Xeon E5-2670v2 CPU) for each individual simulation run. Supplementary Fig. 4 compares simulation results for the transverse dynamics of the center monomer for different values of X/L (indicated in the graph) for an exemplary system with p /L = 1 and density ξ/L = 0.086. We obtained similarly good agreement over many decades in simulation time for other observables, and concluded that X = 1.35 L is a good choice for the size of our simulation box. In particular, since X > L, this makes it impossible for one end of a polymer to come into contact with its other end.
Filament thickness (aspect ratio). Finally, we wanted to test for the effect of the choice of filament thickness on the dynamics of the entangled polymer solution. Since the mutual interaction between beads is given in terms of the WCA potential, we used the range σ of that potential as a measure for the filament diameter. For biopolymers like F-actin the ratio of filament length to filament thickness (aspect ratio) is approximately L/σ ≈ 10 3 . This value is significantly smaller compared to L/σ ≈ 10 2 which we used in our simulations; note that the choice of the aspect ratio is dictated by the degree of discretisation of the polymer chain (number of beads N ). Therefore, we tested how quantities like We performed simulations in two very different parameter ranges of the polymer stiffness for a range of aspect ratios . Supplementary Fig. 5 shows the dynamics of transverse contour fluctuations, g 1,⊥ (t), of a rather stiff tracer polymer with p /L = 25 in a solution with mesh size of ξ/L = 0.17, and Supplementary Fig. 6 shows the relaxation of a tracer filament's orientation, δe 2 R (t), with p /L = 1 in a solution with mesh size ξ/L = 0.12. These simulations show that the dynamics depend only weakly on the choice of the aspect ratio (chain discretisation) once the aspect ratio becomes larger than 50. Therefore, for the results presented in this study, we used chains with N = 45 (corresponding to L/σ ≈ 52) and for higher densities N = 60 (corresponding to L/σ ≈ 70).
As there is hardly any dependence on the aspect ratio for systems with 35, we concluded that we have reached the slender filament limit in our simulation, such that only the topology is relevant for the dynamics. Due to this observation, the mesh size ξ suffices to uniquely describe the system, whereas the aspect ratio or the occupied volume fraction is irrelevant for the dynamics of an entangled polymer solution, the aim of the current study. Especially, this strongly suggests that the results of our simulations remain valid for filaments of a much higher aspect ratio.

SUPPLEMENTARY NOTE 2: ODIJK-SEMENOV SCALING
For semiflexible polymers, Odijk and Semenov [13,14], developed scaling theories to explain the effect of topological constraints on the bending modes of semiflexible polymers in entangled solutions. They proposed that these topological constraints are effectively described by a tube of diameter d which limits the amplitude of a polymer's bending fluctuations. Specifically, they predicted that the tube diameter should scale as d ∼ ξ 6/5 −1/5 p [13,14], and that the ensuing 'entanglement length' (collision length with the tube) should scale as L 3 e ∼ d 2 p [13]. Since the tube is a phenomenological concept to characterize the transient effect of topological constraints, there is some degree of ambiguity in the definition of the tube diameter. If one wants to define it as the width of a polymer's bending fluctuations one has to define a proper time interval during which those fluctuations are sampled [15]. This time interval has to be of the order of the entanglement time τ e ∼ L 4 e / p [12], but is otherwise not well-defined. Therefore, we chose to define the tube diameter d as that value of the transverse fluctuations g 1,⊥ (t) where it starts to deviate from the t 3/4 -behaviour of free polymers. As illustrated in Supplementary Fig. 7, we fitted the dynamics of The resulting values of the tube diameter d for various values of the mesh size ξ are shown in Supplementary Fig. 8.
We find good agreement with the scaling law predicted by Semenov and Odijk [13,14], in accordance with results from earlier simulations [16] as well as experiments [15]. Given the expression for the tube diameter, the entanglement length may also be written as L e ∼ ξ 4/5 1/5 p . We extracted the value for the entanglement length L e from data as shown in Supplementary Fig. 7 using τ e ∼ L 4 e / p . As shown in Supplementary Fig. 9, our data are again consistent with Odijk-Semenov scaling.

Scaling in a monodisperse solution
In the main text we report the existence of a new time scale τ eq governing the relaxation of the internal bending modes of polymers in an entangled solution. It is best accessible by observing the time evolution of the mean-square end-to-end distance δR 2 (t). The equilibration time τ eq is defined as the time at which the internal modes saturate to the value of unconstrained fluctuations δR 2 eq = 1 45 L 4 / 2 p [12,17]. In order to analyse how τ eq depends on polymer length L, persistence length p and mesh size ξ, we used a procedure analogous to the determination of the entanglement time τ e : As illustrated in Supplementary Fig. 10, we fitted the intermediate regime of hindered relaxation of δR 2 (t) at times larger than τ e with an effective power law, and obtained τ eq by the intersection of that curve with the saturation value We have used this procedure to determine τ eq for a broad parameter regime spanning mesh sizes in the range 0.3 ξ/L 0.075, and persistence lengths in the range 0.2 p /L 100. By varying the polymer stiffness for fixed mesh size, we find that τ eq is independent of p ; Supplementary Fig. 11 shows a characteristic example for ξ/L = 0.086. Recall that we only consider systems with ξ p .
Supplementary Fig. 12 shows τ eq for a set of mesh sizes ξ, and a relative persistence length p /L = 1. We checked several possible scaling laws for the mesh size dependence of τ eq , and found that the best fit is achieved with an exponent close to 2. From the fit of with exponent 2, we also determined the numerical prefactor of τ eq to be As a additional test of the scaling law, Eq.
(3), we tested how the numerical data for the end-to-end distance fluctuations collapse upon rescaling the time axis as shown in Supplementary Fig. 13. We find excellent data collapse in the whole terminal relaxation regime, further substantiating our scaling law for the equilibration time τ eq .

Scaling in a polydisperse solution
To elucidate the significance of classical constraint release mechanisms, we also studied a system where a single filament of length L (tracer filament) is embedded in an entangled solution of filaments of size L n , and the same persistence length p as the tracer filament. The mesh size ξ of the solution is now defined as ξ = 3/(νL n ). Invoking a classical constraint release argument one would argue that constraint release is mediated by filaments (of length L n ) in the entangled solution moving past the tracer filament by curvilinear center of mass motion [18][19][20]. The corresponding characteristic time scale is given by the disengagement time τ cr × ∼ L 3 n . However, as shown in Supplementary Fig. 14, in the length range 0.5L n < L < 3L n , the best data collapse of the end-to-end distance fluctuations is achieved by rescaling time t not by a factor proportional to L 3 n but linear in L n . These results clearly rule out a classical constraint release mechanisms mediated by curvilinear motion of surrounding chains of length L n .

Scaling in a solution of polymers with different friction coefficients
The above findings raised the questions: What is or what are the basic time scales that determine the relaxation time of the internal bending modes in an entangled polymer solution? Are the topological constraints mainly released by the motion of the considered filaments or the dynamics of filaments in its surroundings? To answer these questions we considered a chain with friction coefficient ζ (tracer filament) embedded in a solution of filaments whose friction coefficient ζ n differs from ζ. This enabled us to differentiate between the contribution of the filament's dynamics and the surrounding filaments to the constraint release. In the upper panel of Supplementary Fig. 15 we kept the friction coefficients of the surrounding polymers fixed at ζ n = 1 and varied the friction coefficient ζ of the tracer polymer. A good data collapse of the tube relaxation time is achieved by rescaling time with (ζ n /ζ) 2/3 , indicating that τ eq ∼ ζ 2/3 . In contrast, varying ζ n for fixed ζ = 11 we find good data collapse upon rescaling with a factor (ζ n /ζ) 2/3 ; see lower panel of Supplementary Fig. 15. This suggests τ eq ∼ ζ 1/3 n . Taken together, these results are consistent with the following scaling form of the equilibration time with α + β = 1. From the data collapse in Supplementary Fig. 15 we estimate that α ≈ 2/3 and β ≈ 1/3. Both of these exponents for the dependence of the equilibration time on the friction coefficients of the tracer polymer and the surrounding polymers have a certain range of error, and should therefore not be taken too literally. In any case, however, Eq. (4) clearly shows that the equilibration of internal mode results from an interplay between the motion of the tracer filament and its surrounding filaments. Moreover, as the equilibration time is proportional to the square of the number of constraints, N 2 × ∼ (L/ξ) 2 , the correlated dynamics of the filament and its surroundings still amounts to a statistically independent release of the topological constraints. However, each of these constraint release events comes about by the correlated motion of the filament and its neighbouring filaments. These correlations are encoded in the exponents of the scaling form of τ eq , Eq. (4).

SUPPLEMENTARY NOTE 4: TERMINAL RELAXATION
We quantified the rotational relaxation by observing the mean-square deviation of the normalized end-to-end vector δe 2 R . Supplementary Fig. 16 shows results from our Brownian dynamics simulations for solutions with different polymer stiffnesses and different mesh sizes, as indicated in the graph. There are three clearly distinct regimes: (1) The initial relaxation of a tracer polymer in an entangled solution is identical to the relaxation of a freely diffusing polymer, and given by δe 2 R ∼ t 3/4 (which is due to internal bending modes [4]).
(2) This is followed by an intermediate regime which coincides with the intermediate regime for the internal modes (δR 2 ) discussed in the main text: While the rotational relaxation is significantly slowed down as compared to a free polymer it does not come to a halt, again indicating that the Odijk-Semenov tube is only transient. The speed of relaxation, slope of δe 2 R in the double logarithmic representation of Supplementary Fig. 16, decreases with decreasing mesh size or increasing polymer stiffness. (3) In the final regimes, we find that all curves are well fitted by an expression describing the rotational relaxation of a free rigid rod [4]: For a free polymer (red circles in Supplementary Fig. 16), this rigid rod regime is reached for times larger that the relaxation time of internal modes ∼ L 4 / p . Recently, it has been shown by extensive Brownian dynamics simulations of a two-dimensional system complemented by analytical theory that an effective Perrin theory describes the terminal relaxation dynamics of rigid rods in a dense array of obstacles [21] suggesting that at asymptotically long time scales the dynamics of a rigid rod diffusing in a fixed array is well described by free diffusion though with renormalized diffusion constant parallel and perpendicular to its long axis. Our simulations show that such an effective free diffusion even describes the asymptotic terminal dynamics of a semiflexible polymer in an entangled solution, as implicitly assumed in all previous scaling theories of terminal relaxation [13,14,22]. Thus we concluded to use the expression 2 − 2 exp(−t/τ r ) to determine τ r , and thereby determined the dependence of the terminal relaxation time on polymer stiffness, polymer length and mesh size; see data in the main text.
One of our main conclusions in the main text is that in the semiflexible regime the terminal relaxation time scales as This is distinct from both Doi's and Odijk's results. To validate it we have performed simulations over a broad parameter range, covering 0.2 ≤ p /L ≤ 10 and 0.075 ≤ ξ/L ≤ 0.16. A representative scaling analysis for a typical set of raw data (in the semiflexible regime) is shown in Supplementary Fig. 17. There is a nice data collapse upon rescaling time t by τ r ∼ p L 4 /ξ 2 .