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Electric-Field-Driven Trapping of Polyelectrolytes in Needle-like Backfolded States Antonio Suma,† Marco Di Stefano,‡ and Cristian Micheletti*,† †

International School for Advanced Studies (SISSA), via Bonomea 265, I-34136 Trieste, Italy Centre for Genomic Regulation (CRG), Barcelona Institute of Science and Technology (BIST), Baldiri i Reixac 4, 08028 Barcelona, Spain

‡

S Supporting Information *

ABSTRACT: Polyelectrolytes with strong counterion couplings are compact at low electric fields but switch to elongated forms at high ones. As yet, little is known from both nanomanipulation experiments and theory about the transition mechanisms between compact and extended states. Here, we systematically address this out-of-equilibrium conversion using molecular dynamics simulations of coarse-grained polyelectrolyte chains of 1000 and 1500 monomers in a salt buffer of about 20 000 and 80 000 charged particles, respectively. We find that compact-to-elongated transitions are smooth and fast in only half of the trajectories. In the other half, the elongation is jammed halfway by the formation of backfolded, needle-like states. These states, which have not been envisioned before, have lifetimes that are orders of magnitude larger than the chain’s relaxation time and ought to be detectable with current singlemolecule experimental setups.

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INTRODUCTION

pointed out the slow relaxation dynamics and poor reconfiguration compliance of PEs in AC fields. Here, to advance the understanding of these open issues, we present a systematic theoretical study of how a long PE chain in salt solution responds when an electric field is suddenly switched on. Specifically, we consider long flexible PE chains in a trivalent salt solution and study the out-of-equilibrium transition from globular to elongated elicited by the sudden action of an electric field. We show that long chains of 1000 and 1500 monomers are prone to be stalled and trapped in bridged backfolded states whose lifetimes can exceed the PE relaxation time by orders of magnitude. Possible experimental setups to detect the predicted longlived states are suggested.

Polyelectrolytes are the systems of choice in single-molecule manipulation contexts for their relevance in biological contexts1−17 and for their unexpected complexity when strongly interacting with counterions in solution.18−28 A key example is the response of polyelectrolytes (PEs) to external electric fields, which was first studied by Netz in ref 29. This theoretical study predicted that PE chains that strongly interact with surrounding counterions can transition from being compact to extended when the field is increased beyond a length-dependent threshold. Experimental evidence for this type of transition, by now addressed by several theoretical studies,29−35 has been recently gathered by Klotz et al.,36 who considered 165kb long DNA filaments in TBE buffer and elicited a reconfiguration from globular to extend using fields of about 30 V/cm. At the same time, it has also been experimentally shown that in monovalent salt and even stronger fields (hundreds of V/cm) DNA filaments can became compact again rather than more extended, a still largely unexplained phenomenon.37−41 Several studies have aimed at recapitulating such varied phenomenology within seamless theoretical frameworks.38,41−48 Despite these ongoing efforts, the out-of-equilibrium response of polyelectrolytes in electric fields has eluded a systematic characterization and has been addressed by only few studies. These include the experiment of Tang et al.40 demonstrating that DNA coils can stay trapped in knotted states even for tens of seconds and the theoretical work of refs 33 and 35 that © XXXX American Chemical Society

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MODEL AND METHODS Most of our study is focused on a model salt solution with a dispersed flexible PE chain comprising a total of 25 752 particles, at least 10 times larger than considered in previous computational studies of the collapse-to-globule transition. The polyion is made of N = 1000 monovalent beads, each of diameter σ and electric charge −q, and is placed inside a periodic simulation box together with an equal number of dissociated ions, each of size σ, but opposite charge q. The box is a 1100σ × 210σ × 210σ parallelepiped that additionally Received: January 4, 2018 Revised: May 23, 2018

A

DOI: 10.1021/acs.macromol.8b00019 Macromolecules XXXX, XXX, XXX−XXX

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Figure 1. (a) Snapshot of the polyelectrolyte and ionic buffer at steady state in a strong electric field, E = 0.6ϵ/qσ. The field is directed longitudinally, i.e., parallel to the long axis of the simulation box, see arrow, and the negatively charged polyelectrolyte moves in the opposite direction. The legend to the right gives the color code, number, and charge of chain monomers and ionic particles. (b) Steady-state longitudinal span, x,̅ as a function of the field strength, E. The dashed line marks the critical field Ec ∼ 0.14ϵ/qσ. (c) Typical steady-state configurations at different (indicated) values of E. Ions that are closer than 2σ to the chain monomers are also shown. Note that these bound ions are mostly trivalent counterions. For visual clarity a tube representation for the PE chain is used in the close-ups in panel (c) and the PE monomers’ size is enlarged by 4 times in the small snapshots of panel (c) and the one of panel (a).

time step was set equal to 6 × 10−3τLJ, where τLJ ≡ σ(m/ϵ)1/2 is the characteristic Lennard-Jones time. On average, a single production run for the reference system (N = 1000 and trivalent counterions) had duration 106τLJ and required 5 × 104 CPU hours on the Intel-based high-performance computing cluster (Ulysses) based in SISSA, Trieste. Further methodological details are provided in section I of the Supporting Information.

contains a buffer of trivalent salt dissociated in 5938 counterions of charge 3q and 17 814 co-ions of charge −q (see Figure 1a). All ions have diameter σ. In our electrically neutral solution the counterion density, ρc = 10−4σ−3, exceeds by several times the one of PE monomers, ρm = 2 × 10−5σ−3. We sought this condition, which is computationally more onerous than the customary matching of the absolute total charge of counterions and PE monomers, because it renders realistically both typical experimental conditions (a standard σ = 0.23 nm corresponds to about 20 mM salt solution; see section I in the Supporting Information) as well as the plug flow of counterions. In addition to the reference case of PE chains with N = 1000 monomers with trivalent counterions, we shall also present results for three other cases: N = 1500 with trivalent counterions and N = 1000 with both di- and monovalent counterions. The total number of particles in these systems is respectively 80 392, 28 721, and 37 628. The potential energy of the system comprises a truncated and shifted Lennard-Jones potential and a Coulomb term, respectively, for the steric and electrostatic interactions of all pairs of beads and a FENE term for the bonding of consecutive beads along the chain. All charged particles additionally interact with an external field, E, parallel to the long side of the simulation box. The time evolution of the system is described by a standard Langevin dynamics, which was integrated with the LAMMPS simulation package49 using default50 values for the friction coefficient and the mass of the particles. The system thermal energy κBT was set equal to the amplitude of the Lennard-Jones potential, ϵ, and the dielectric constant of the medium was set so that the unit charges, q, are associated with a typical value of the Bjerrum length, lB = 3σ.31,32,35,51,52 The long-range Coulomb interactions were computed with the particle−particle particle-mesh solver,53 and the integration

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RESULTS AND DISCUSSION Critical Elongating Field. Globule-to-extended transitions are observed in polyelectrolytes that strongly interact with surrounding counterions. Customarily, the parameter used to capture the strength of such electrostatic coupling is |q | |q | lB

Ξ = mσ / c2 , where |qm| and |qc| denote the absolute valence of the chain monomers and counterions, respectively.29 Clearcut globule-to-extended transitions are observed for Ξ about equal to 15 or larger. In our system, where |qm| = 1 and lB = 3σ this condition can be met by using trivalent counterions, | qc| = 3, yielding Ξ = 18, a typical value for strong coupling.22,31,32,35 To determine the critical field for our system, we collected tens of trajectories started from fully elongated states, a setting we discuss later, at various values of E in the range 0−0.8ϵ/qσ. As a primary observable we monitored the evolution of the chain longitudinal span, x. The field dependence of its steadystate value, which we denote with an overbar, x,̅ is shown in Figure 1b and has the typical sigmoidal shape of globule-toextended transitions. x̅ is negligible below the critical field Ec ∼ 0.14ϵ/qσ and increases systematically above it, reaching x̅ = 870σ at the largest considered field, 0.8ϵ/qσ (see also Figure S1). Note that even at this field, the span is much shorter than the longitudinal side of the box, so that periodic copies of the B

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Macromolecules chain are separated by about 200σ across the wrapped boundaries in all directions. This distance exceeds by more than an order of magnitude the nominal Debye screening length, lD ∼ 12σ, thus ruling out artifactual boundary effects. Figure 1c shows typical snapshots of the chain and bound counterions for fields smaller and larger than Ec. One notes that at the largest fields, E = 0.4−0.8ϵ/qσ, the span is not only substantial (>70% of the chain contour) but also uniform along the chain. This contrasts with the nonhomogeneous elongation found at fields that are only moderately larger than Ec. For instance, in the typical steady-state conformation at E = 0.2ϵ/qσ the chain is mostly globular except for a terminal stretch directed along the plug flow of the counterions. To our knowledge this effect has not been reported before for polyelectrolytes in electric fields. At the same time it is reminiscent of the tadpole states observed during the anomalous sedimentation of flexible polymers, where in the strong-stretching regime chains have a compact leading end followed by a stretched trailing part.54,55 Elongation of Collapsed States. The trajectories discussed above were purposely started from fully extended states because these expectedly offer the lowest kinetic hindrance toward reaching steady-state conformations, be they globular or extended. By contrast, the reverse process of reaching the elongated steady state from an equilibrated globule is expectedly riddled with kinetic traps arising from topological constraints, or selfentanglements, analogous to those affecting the reconfiguration compliance observed of long and densely packed biopolymers.36,40 We therefore sought to assess whether and how such effects are at play in our general model of long flexible PEs with strong counterion interactions. We accordingly studied how globular states that have been equilibrated in zero field respond to the sudden action of electric fields that are strong. By this we intend fields large enough to elicit uniformly elongated PE conformations at steady state. We hence obtained a set of equilibrated globular PE conformations by evolving in zero field a set of stochastically prepared initial states. The duration was set equal to 1.5 × 105τLJ, which is larger than the characteristic relaxation time (∼104τLJ) of metric observables (see Figures S2 and S3). These equilibrated states, which are compactified by the effective selfattraction mediated by the counterions, are significantly entangled. In fact, in an ensemble of hundreds of independent realizations, about 55% of the equilibrated PE chains contained physical knots (see section IV in the Supporting Information). Figure 2 shows the time evolution of the longitudinal span, x, for various equilibrated globules that were suddenly exposed at t = 0 to strong fields, E = 0.4 and E = 0.6ϵ/qσ. One sees that across the ensemble of independent trajectories the x(t) traces are practically superposed only in the initial elongation phase and specifically for x increasing up to 370σ for E = 0.4ϵ/qσ and up to 440σ for E = 0.6ϵ/qσ. Intriguingly, these values correspond to about half the asymptotic, steady-state span in both fields, a result that we shall discuss later. After this initial phase, the traces become highly heterogeneous. The variability is so high that across the trajectories in Figure 2 the duration of the elongation process is seen to vary by more than 2 orders of magnitude. In fact, at both fields only about half of the elongations proceed without noticeable hindrance and reach the steady

Figure 2. Time evolution of the longitudinal span, x, after a longitudinal electric field, E, is suddenly switched on at t = 0. The traces in panels (a) and (b) correspond to independent trajectories at E = 0.4ϵ/qσ and E = 0.6ϵ/qσ, respectively. A blue to red color code is used to distinguish trajectories that evolve rapidly (blue) or slowly (red) toward the steady-state elongation. The latter, x,̅ is indicated with a horizontal dashed line for reference. Analogous curves for E = 0.8 are shown in Figure S4.

state in about 4 × 103τLJ at E = 0.6ϵ/qσ and 1.5 × 104τLJ at E = 0.4ϵ/qσ. These time scales are indeed comparable to the aforementioned characteristic relaxation time. The other half of the trajectories instead are clearly stalled for timespans in the 104−105τLJ range for E = 0.6ϵ/qσ and even for longer times at the lower field E = 0.4ϵ/qσ. For the latter case, in fact, two of the trajectories are still stalled at 106τLJ, which is a timespan about 100 times larger than the intrinsic relaxation time of ∼104τLJ required to revert from extended to globular state in zero field. Stalling Mechanism. The hindrance to elongation is largely independent of the presence of physical knots in the initial state. In fact, knotted and unknotted conformations can both evolve toward stalled or nonstalled states, as shown in Figure S5. This observation alone clarifies that the frequent and long-lived jamming of Figure 2 have an altogether different origin from the kinetic trapping due to self-knotting observed experimentally for DNA.36,40 We thus looked for systematic differences in the conformational evolution of stalled and nonstalled trajectories. The results are best discussed in terms of the distinct kinetic stages visible in elongation traces. These stages are marked out in Figure 3a,b for representative unhindered and jammed processes, along with the corresponding PE conformations and the time evolution of the chain longitudinal velocity. Stage I is the transient during which the chain still retains the globular state induced by the condensed trivalent counterions. Because these effectively neutralize the charge of the polyion, the latter has no appreciable net velocity along the field direction.30−32,34,56 This first phase lasts only for about 102τLJ, a brief timespan compared to the chain relaxation time, and is followed by the actual elongation process, II. During this stage, stalled and C

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Figure 3. (a) Time evolution of the longitudinal span (top) and longitudinal velocity of the PE center of mass (bottom) at E = 0.6ϵ/qσ. For each of the main stages of the elongation process a representative snapshots of the chain and bound ions is shown. Panel (b) shows analogous plots, but for stalled configurations, where the elongation stages are four, see main text. (c) Chain indices of the monomers leading the chain motion of nonstalled or stalled trajectories at the beginning of stage II. See Figure S6 for the persistence of the leading bead during stage II of stalled trajectories.

nonstalled trajectories have very similar x(t) traces but are distinguished by the very different magnitude of the longitudinal velocity, which is significantly lower for nonstalled states. As a matter of fact, chain conformations from the two stalled and nonstalled classes present clear differences already at the transition between stages I and II. This crossover occurs when the field action eventually pulls out a portion of the globular chain from the “pole” opposite the field vector (because the polyion is negatively charged). The characteristics of this firstextracted portion are crucial for the later development of the elongation process and, in fact, define the jamming properties. Smoothly evolving trajectories develop from first-extracted portions that are located at a small (chemical) distance from one of the termini, typically within 150 beads for E = 0.6ϵ/qσ (see Figure 3c). The dragging action of the field causes the extracted portion to bend into a U-shaped loop and to grow in length at the expenses of the globule. The loop growth is stopped shortly when the proximal terminus is disengaged from the globular polyion. At this point, the entire terminal region extracted from the globule extends under the action of the field and becomes uniformly elongated, as shown in section II of Figure 3a. This leading stretch then drives the subsequently smooth unravelling of the globule until the steady-state extension III is reached. By contrast, stalled trajectories occur when the first-extracted portion is close to the chain midpoint (i.e. further than about 150 beads from either chain end at E = 0.6ϵ/qσ; see Figure 3c). In this case, the growth of the extracted loop is not arrested early by the disengaging of a nearby terminus. The U-shaped loop can thus develop into extended, needle-like backfolded conformations such as the one shown for stage II* of Figure 3b. This stage corresponds to stalled states. In fact, the needle-like conformations have no analogue in smoothly evolving trajectories. The backfolded states are bridged by trivalent counterions and are different, both for shape and for the originating mechanism, from U-shaped states that transiently emerge in gel electrophoresis when polyelectrolytes are mechanically pinned (in this case the U-bend is trailing and not leading motion).57,58 They also differ from the intriguing tadpole states adopted by ring polymers at high sedimentation forces.55 In these, in fact, the trailing arms are not bridged as in our linear PE, but are

kept nearby by the closed topology of the ring and, especially, the leading part is globular while in our case is not. As a matter of fact, a distinctive property of backfolded states is the droplet-like shape of their leading tip, the “needle’s eye”, as seen in Figure 3b. The shape of the needle’s eye remains constant during dynamics, but it depends on the field strength. Specifically, its span and width increase with E (see Figure S7). A further notable feature is that the longitudinal velocity of backfolded states II* is significantly higher than the steady-state elongated ones III. For instance, by comparing the bottom panels Figure 3a,b, one observes that at E = 0.6ϵ/qσ the average center-of-mass velocity of needle-like states is |v|II* ∼ 0.27σ/τLJ and drops to the steady-state value, |v|III ∼ 0.20σ/τLJ, when the chain eventually unfolds and reaches the asymptotic elongated state. As we discuss later, these two intriguing and seemingly unrelated effects, droplet shape of the tip and enhanced longitudinal velocity, are closely connected. Counterion Distribution. The complex phenomenology can be rationalized by profiling the average number trivalent counterions, n, bound to each monomer of the PE chain (see Figure 4 and for monovalent ions see Figure S8). At steady state, the trivalent ions cover the extended conformations III fairly uniformly, n ∼ 0.37, except near (within 75 beads at E = 0.6ϵ/qσ) the termini due to the interaction with the ionic plug flow. Specifically, the trailing end has a slight excess of bound counterions, while the leading one is mostly free of them. It is the coupling of this unscreened leading end with the external field that dictates the longitudinal motion of the chain. In fact, the observed increase of the longitudinal velocity with the field is directly related to the concomitant lengthening of the unscreened tip. The analogous profile of bound counterions for backfolded, stalled conformations is shown in Figure 4b. These observations explain the effects noted in the previous section. First, the fact that the unscreened tip of backfolded states II* is about twice as long as in extended ones III clearly reflects in their different coupling with the external field and hence explains the higher longitudinal velocity of the states II*. Second, it is now clear that the droplet-like shape of the needle’s eye results from the self-repulsion of the unscreened backfolded tip competing with the bridging action of multivalent counterions. This mechanisms is an out-ofD

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relative sliding is also accompanied by a stochastic longitudinal rotation (see Figure S9). An insightful description of the lifetimes of backfolded states is obtained by considering the average waiting time, tw, needed by a backfolded state with longitudinal span x to reach the steady-state span, x,̅ and is provided in Figure 5a. Note that the data points are for the strongest field, E = 0.6ϵ/qσ, where within a computationally affordable time all jammed states eventually reach the steady state, which is a prerequisite for being able to compute tw. The lifetime curve in Figure 5a can be quantitatively accounted for by the following minimalistic model of the sliding kinetics. Consider a backfolded conformation with span x. Clearly, the span of the longest arm is itself equal to x, while the shortest arm has span x̅ − x (see Figure 5b). Next, consider the pulling action of the field on each of the two arms taken separately, i.e., as if they were not connect at the tip of the needle’s eye. As a first approximation, we assume that each arm moves rigidly under the pulling force and that the latter arises exclusively from field acting on the unscreened side of the needle’s eye, i.e. we neglect the contribution from the residual charge along the rest of the strand. In the dissipative system, the motion of the strands results from the opposite action of the pulling force, Fe, which is statistically the same for both strands because of the needle’s eye symmetry, and the frictional drag that instead is different for the two arms because it is proportional to their span. Accordingly, the translational velocity of the shortest arm is (Fe/γ̃)(x̅ − x) and that of the longest one is (Fe/γ̃)x, where γ̃ is a parameter that defines the effective friction coefficient, γ̃ σ. Over a small time interval, Δt, the unbalance of pulling and frictional effects would create a mismatch in the longitudinal displacement of the two arms. Specifically, the tip of the short arm (experiencing less friction) should advance the other by a quantity equal to FeΔt(1/(x̅ − x) − 1/x)/γ̃. In the actual system, where the arms’ tips are connected, this advancement would become equally distributed in the two arms, yielding the following equation for the time evolution of the length of the longest arm:

Figure 4. (a) Average number of trivalent ions, n, binding each PE monomer as a function the monomer (normalized) index. The profile is averaged over independent steady-state (elongated) configurations at E = 0.6ϵ/qσ. The counterion-depleted region corresponding to the tip (see representative configuration above the plot) is highlighted with a yellow background. Panel (b) shows the same quantities but computed for back-folded conformations with arms of approximately equal length. The highlighted depleted region now corresponds to the “eye” of the needle-like conformation.

equilibrium counterpart of the static competition between bending energy and self-attraction that stabilizes raquet-like conformations in semiflexible DNA filaments adsorbed on a surface.59−63 We believe that the models introduced in such contexts ought to be transferable to the case of PE needle-like states and might hence help future quantitative characterization of the needle’s eye shape and its size dependence on the external field. Lifetime of Backfolded States. We now turn to discuss the origin of the jammed elongation kinetics of backfolded states. The inspection of the trajectories shows that their exceedingly slow conversion to extended state proceeds via the relative sliding of the two juxtaposed arms, with the long strand becoming longer at the expenses of the short one. This slow

F ⎛ 1 dx 1⎞ = e⎜ − ⎟ ⎝ dt 2γ ̃ x ̅ − x x⎠

(1)

Figure 5. (a) Lifetime, tw, of backfolded states as a function of their longitudinal span. The lifetime is defined as the waiting time to reach the extended state. The continuous line is the best fit of the data points based on the theoretical relationship of eq 2. The fitting parameter is γ̃/2Fe = 0.35τLJ/σ2. The steady-state longitudinal span, x,̅ and its half value are indicated for reference. (b) Schematic representation of the chain, with the needle unscreened part in yellow, occupying a finite portion of the chain, and the electric force effectively acting only on this chain portion. E

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Figure 6. (a) Time evolution of the longitudinal span, x, of a chain that is in a needle-like state at t = 0 (previously formed in a trivalent salt) and is then evolved in the presence of monovalent, divalent, and trivalent counterions at E = 0.6ϵ/qσ. Snapshots of the initial and later states are shown in panel (b).

like states opened up very rapidly, within 103τLJ, and adopted a U-shaped conformations (see Figure 6b). The loss of the bridging mediated by trivalent ions also makes the relative motion of the arms in the U-shaped states less hindered, which explains the faster unfolding dynamics (see Figure 6a and Figure S10). As a complement of the above analysis we also explored the effects of varying chain length by increasing N from 1000 to 1500. The results, again for E = 0.6ϵ/qσ, are illustrated in Figure 7a and are relevant in several respects. First, the lifetime of needle-like states shows a considerable increase with chain length, as at N = 1500 a visibly greater

Through variable separation and integration in the range [x, x]̅ for the span and [tw, 0] for t (note that a span equal to x̅ corresponds to zero waiting time), we obtain the following expression for the waiting time: tw =

⎞ ⎛ x̅ ⎞ γ ̃ ⎛ x̅ 2 2 ⎟ + x − xx ⎟ ⎜ log⎜ ̅ ⎝ ⎠ 2Fe ⎝ 2 2x − x ̅ ⎠

(2)

As it is shown in Figure 5a the actual data points are described very well by this equation, which has a single fitting parameter, the prefactor γ ̃ . Incidentally, the latter is expected to have an 2Fe

implicit dependence on N because it necessarily subsumes the length-dependent approximations of neglecting the internal elasticity of the two arms, their residual charge, and their sliding hindrance caused by the bridging of the counterions. Note that expression 2 also predicts that the lifetimes should decrease for larger fields, a fact reflective of the differences found for E = 0.4, 0.6, and 0.8ϵ/qσ in Figure 2 and Figure S4. Above all, the deterministic model shows that the backfolded state lifetime diverges and hence exceeds any characteristic time scale of the system when x → x/2. In the actual stochastic ̅ system, this situation is precisely realized by the longest-lived jammed states which, we recall from Figure 2, are characterized by two balanced, equally long arms with longitudinal span equal to half the steady-state one. This distinctive limiting signature of the crossover between stages I and II* now finds a posteriori an intuitive rationale. The dramatic slowing down of the elongation process in such cases is illustrated in the SI movie (see also section X of the Supporting Information) that provides an equal time comparison against the globule-toextended transition of nonjammed states. Effect of Changing Counterion Valence and Chain Length. To clarify whether trivalent counterions are key to observe needle-like states, we proceeded as follows. We considered a set of stalled trajectories at E = 0.6ϵ/qσ and extracted the newly formed needle-like conformations at the crossover between stages II and II*. Next, we replaced all trivalent counterions with divalent (or monovalent) ones and then added additional divalent (or monovalent) counterions to achieve the overall charge neutrality of the system. We next compared the evolution of the system, shown in Figure 6a, to the original one of Figure 2b. We found that neither monovalent nor divalent counterions could keep the two strands bridged together. The initial needle-

Figure 7. (a) Time evolution of the longitudinal span, x, for PE chains of 1500 beads after a longitudinal electric field, E = 0.6ϵ/qσ, is suddenly switched on at t = 0. The curves are for 20 independent trajectories and are color-coded based on the rapid (blue) or slow (red) evolution toward the steady-state elongation. In one of the trajectories, a practically simultaneous extraction of two loops was observed. Various snapshots from such trajectory are shown in panel (b) along with a schematic representation of their structure. F

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Macromolecules fraction of these trajectories are stalled at t ∼ 106τLJ, a property clearly reflecting the longer bridgings. The size of the needle’s eye is instead about equivalent to that of the N = 1000 case at the same field strength. The larger collapsed globule also offers more chances for extracting a deep loop, the seed of the needle’s eye, and consistent with this we observed two practically simultaneous loop extraction events in one of the trajectories (see Figure 7b). Because of their different depth, the corresponding needles’ eyes unfolded at different times and formed elaborate, serpentine conformations with multiple backfolds before becoming fully extended. Detection of Backfolded States with AC Fields. The long-lived backfolded states predicted here ought to be detectable with current single-molecule experiments using DNA or polystyrenesulfonate as possible polyelectrolytes, while possible trivalent ions could be spermidine or the inorganic cation Co(NH3)63+.29,64−66 For instance, one could employ microfluidic setups similar to those of ref 38, where AC fields were used to elongate DNA molecules while keeping them at a stagnation point, i.e., with no net drift. Backfolded states might then be revealed by periodically clearing out the chains and funnelling them through a solid state nanopore. Then, similarly to the recent breakthrough detection of DNA knots,1 the magnitude of the ionic blockade during pore translocation should be revealing about linear or backfolded geometries. A possible alternative to nanopore translocation could be to use fluorescence kymographs. A proof-of-concept demonstration is explored in Figure 8, which shows the kymographs of the

The differences of the kymographs of linear and backfolded states are readily picked up; in particular, one notes the plateau at half the asymptotic elongation for the needle-like state. Comparing Figures 8 and 2 also shows that the lifetimes in AC fields are reduced severalfold compared to DC fields of the same strength. This is consistent with the known different behavior of PE systems in DC/AC fields due to the pulling action occurring at only one or both ends.36,40

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CONCLUSIONS We presented a first systematic study of the out-of-equilibrium elongation process of long flexible PE filaments in trivalent salt that are suddenly exposed to strong electric fields. We found that in about half of the simulated trajectories the transition from the initial (equilibrium) compact state to the elongated steady state becomes stalled midway. Stalled trajectories are associated with the spontaneous emergence of needle-like backfolded states. As we clarify with a quantitative interpretative model, the long lifetime of these states, which exceeds by orders of magnitude other characteristic time scales in the system, originates from the slow relative sliding of the two arms. The incidence of jammed states, as well as their lifetimes, is controlled by the interplay of the chain contour length and the length of the looped tip, the “needle’s eye”, which can be varied with the field strength. The predicted long-lived backfolded states ought to be detectable with current microfluidic experimental setups.

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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.8b00019. Additional details of the model, the relaxation and elongation response at fields and counterion valence different from those in the main text, the spontaneous emergence and effects of physical knots, and the elongation curves in AC fields (PDF) Movie comparing the elongation response of jammed and nonjammed states (MOV)

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Figure 8. Kymograph of the PE longitudinal span in an AC field of strength E = 0.6ϵ/qσ and period 103τLJ. Panels (a) and (b) are for nonstalled and stalled (backfolded) trajectories, respectively; see also section XII of the Supporting Information.

AUTHOR INFORMATION

Corresponding Author

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

Antonio Suma: 0000-0002-5049-9255 Marco Di Stefano: 0000-0001-6195-4754 Cristian Micheletti: 0000-0002-1022-1638

longitudinal footprint of linear and needle-like states in a sinusoidal AC field of strength equal to E = 0.6ϵ/qσ, as in the static case. The AC period was set by balancing two competing requirements. The period should be small enough that the cyclic longitudinal offsets are much smaller than the chain extension, but it should not be too small, as otherwise the system behavior would depart from the static field case. We accordingly chose an AC period of 103τLJ, which is comparable to the intrinsic chain response time at the considered field strengths (∼4 × 103−1.5 × 104τLJ), and still it introduces periodic longitudinal displacements that are small compared to the chain extension (see Figure S11). In Figure 8 these offsets, in fact, appear as a sawtooth modulation of amplitude ∼40σ that is superposed to the widening projected span of the elongating chains.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Alexander R. Klotz, Roland Netz, Rudi Podgornik, and Walter Reisner for useful discussions. We acknowledge financial support from the Italian Ministry of Education.

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REFERENCES

(1) Plesa, C.; Verschueren, D.; Pud, S.; van der Torre, J.; Ruitenberg, J. W.; Witteveen, M. J.; Jonsson, M. P.; Grosberg, A. Y.; Rabin, Y.; Dekker, C. Direct observation of DNA knots using a solid-state nanopore. Nat. Nanotechnol. 2016, 11, 1093−1097.

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DOI: 10.1021/acs.macromol.8b00019 Macromolecules XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.macromol.8b00019 Macromolecules XXXX, XXX, XXX−XXX