Molecular Crowding Increases Knots Abundance in Linear Polymers

Article pubs.acs.org/Macromolecules

Molecular Crowding Increases Knots Abundance in Linear Polymers Giuseppe D’Adamo* and Cristian Micheletti SISSA, International School for Advanced Studies, via Bonomea 265, I-34136 Trieste, Italy

Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on September 7, 2015 | http://pubs.acs.org Publication Date (Web): August 17, 2015 | doi: 10.1021/acs.macromol.5b01323

S Supporting Information *

ABSTRACT: Stochastic simulations are used to study the effects of molecular crowding on the self-entanglement of linear polymers. We consider flexible chains of beads of up to 1000 monomers and examine how their knotting properties vary in the presence spherical crowders that are 4 times smaller than the chains themselves and which occupy 35% of the solution volume. We find that crowding boosts the incidence of physical knots by more than an order of magnitude for all considered chain lengths. Furthermore, most crowding-induced knots are found to be significantly longer than in the free case. We show that the observed properties follow from the screening of excluded volume interactions mediated by the crowders at length scales larger than their size.

P

crowders changes the overall abundance, type, length, and chain location of spontaneously occurring physical knots. The comparative analysis is carried out for various chain lengths while keeping fixed both the volume fraction occupied by crowding agents, ϕc, and the polymer-to-crowders size ratio, q. These parameters, which are realistically set equal to ϕc = 0.35 and q = 4, are known to be the key determinants of the metric and thermodynamic properties of asymptotically long chains in good solvent.56−59 Accordingly, by maintaining them fixed, one can best expose the effects of crowding on chains that, despite their different contour lengths, experience analogous levels of crowding-induced compactification. By using Langevin dynamics simulations and Monte Carlo sampling, we profile the properties of chains of up to 1000 monomers and find that crowders systematically boost the incidence of physical knots by more than an order of magnitude. This major effect, besides being remarkable per se, is accompanied by a further notable change in the distribution of knot lengths. For no crowders, in fact, the latter features a peak at about 150 monomers for all chain lengths; with crowders, however, the peak is systematically and noticeably shifted to larger and larger values as chain length increases. This effect, along with the overall enhancement of knots incidence, can be rationalized in terms of the screening of excluded volume interactions mediated by crowders for length scales larger than their size. This screening effect, in fact, promotes the compactification, and hence the self-entanglement and knotting, of relatively long portions of the chains.

hysical knots can be trapped only temporarily in linear chains and yet can significantly affect their physical properties such as relaxation dynamics, translocation through narrow pores, and resistance to mechanical tension.1−17 Because of these implications, several efforts are being spent to characterize the spontaneous formation of knots in open chains16,18 and how their overall incidence in thermodynamic equilibrium can be controlled via a suitable choice of the system properties. One relevant parameter, for instance, is clearly the chain contour length. For isolated chains in equilibrium, in fact, the probability to be unknotted decreases rapidly with chain length, 4,18 analogously to what is known to hold for equilibrated, but topologically unconstrained, rings.19−27 A second relevant parameter is the degree of compactness of the chain which can be conveniently changed by varying the quality of the solvent4,28,29 or, more directly, through spatial confinement. In fact, the knotting probability of a chain can be dramatically affected by its degree of compactification inside channels, slits, or cavities.9−11,16,19,30−36 A further general and practical way for modulating a polymer’s compactness is through molecular crowding.37−49,51−54 In fact, the excluded volume interaction of the polymer with crowding agents introduces an entropic attraction between the constitutive monomers which generally favors the compactification of the chain. By analogy to the mentioned cases of spatial confinement,55 one may envisage that the crowding-induced compactification ought to significantly affect the degree of self-entanglement of the chain and particularly the occurrence of physical knots. To clarify this effect, which to our knowledge has not yet been explored, we investigate here the impact of molecular crowding on the equilibrium knotting properties of fully flexible self-avoiding chains. Specifically we examine how going from an isolated chain to one immersed in a solution of spherical © XXXX American Chemical Society

Received: June 18, 2015 Revised: August 1, 2015

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DOI: 10.1021/acs.macromol.5b01323 Macromolecules XXXX, XXX, XXX−XXX

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C. Crowding Effects. The impact of crowding was studied by setting the volume fraction of the crowding agents to ϕc = 0.35 and the polymer-to-crowder size ratio to q = 2Rg*/σcc = 4, which are realistic values for these parameters.48,61 The corresponding values of Rg* and σcc for the considered lengths are given in Table 1.

I. METHODS A. Coarse-Grained Model of a Polymer in Crowded Environment. We model the polymer as a fully flexible chain of beads and the crowding agents as monodispersed spherical particles. The implicit-solvent potential energy = for a model system comprising a chain of Nm monomers and Nc crowders is written as Nm − 1

==

∑


i=1

Nm

vb +

∑ i = 1, j > i

Nc

vmm +

∑ i = 1, j > 1

Nm

vcc +

Table 1. Polymer Radius of Gyration and Crowder Effective Radius for Different Values of Nma

Nc

∑ ∑ vcm i=1 j=1

(1)

Nm Rg* σcc/2

where the m and c subscripts stand for monomers and crowders, respectively. The terms vmm, vcc, and vcm capture respectively the monomer−monomer, crowder−crowder, and crowder−monomer excluded volume interactions which are enforced via a truncated and shifted Lennard-Jones (LJ) potential:

vAB

a

300 14.611(7) 3.653

500 19.882(5) 4.971

1000 30.11(1) 7.528

All the quantities are expressed in units of the bead diameter σmm.

The properties of the crowded systems were mostly characterized with the following pair distribution functions:

⎧ ⎡⎛ σ ⎞12 ⎛ σ ⎞6 1⎤ AB ⎟ ⎪ − ⎜ AB ⎟ + ⎥ for d < 21/6σAB ⎪ 4ϵ⎢⎝⎜ ⎝ d ⎠ 4⎦ =⎨ ⎣ d ⎠ ⎪ ⎪0 for d ≥ 21/6σAB ⎩

gAB(r) =

1 NANB ρA ρB

NA

NB

∑ ∑′ δ(r − r iA + r Bj ) i=1 j=1

(4)

where A and B denote the particle type, ρA is the number density of particles A, and the primed summation denotes that the terms i = j are omitted if A = B. Because the system is homogenous and isotropic, in what follows we consider the spherical averages of gAB(r), ω(r) and S(k), which we denote simply as gAB(r), ω(r) and S(k). Langevin Dynamics Simulations with Explicit Crowders. We first used Langevin dynamics simulations to sample the conformational space of the polymer and the dispersed crowders in a periodic cubic simulation box of side equal to 8Rg* which accommodates Nc = 2737 crowders at each value of Nm to meet the ϕc and q constraints. As discussed in the Results section, the box is found a posteriori to be large enough to avoid chain self-interactions across the periodic boundaries and to yield the bulk behavior of the crowding agents. The dynamics was integrated numerically with the LAMMPS simulation package.62 The elementary integration step was set equal to δt = 0.012τLJ, where τLJ = (mmσm2/ϵ)1/2 is the characteristic LJ time involving the monomer mass mm. We used standard values for both the monomer mass, mm = 1, and the monomer friction coefficient, ξmm = 2τLJ,63 whereas the analogous quantity for the crowders is rescaled by the ratio of the crowder-to-monomer size, i.e., ξcc = ξmmσcc/σmm. This rescaling holds under the assumption that mm = mca choice which does not affect the equilibrium static properties of the system. The initial state of the system was prepared by placing in the box a relaxed, Monte Carlo generated configuration for the chain and then by attempting subsequent random insertions of the crowding particles until a configuration free of overlaps is obtained. To promote the system to equilibrium, a short preliminary run was performed. The duration of the simulations ranged from ≈2.4 × 108τLJ for Nm = 100 to ≈9.4 × 108τLJ for Nm = 1000 and sufficed to gather about 3× 104 (for Nm = 1000) to 1.2× 105 (for Nm = 100) configurations that are nominally independent, i.e., with a time separation larger than the relaxation time of the end-to-end distance vector (Rouse time64). Monte Carlo Sampling with Implicit Crowders. To improve the sampling efficiency of Langevin simulations, which is limited by the chain relaxation time which grows rapidly with Nm, we complemented them with the Monte Carlo

(2)

where A and B denote the m or c particle types, ϵ is the characteristic LJ energy scale, and σAB is the sum of the nominal radii of particle types A and B. The argument of vAB is the distance of particles i and j, d = |rAi − rBj |, where rA is the position of the center of the ith particle of type A. The chain connectivity is enforced by the FENE bonding potential vb acting on all pairs of consecutive monomers i and i +1 ⎛ d ⎞2 ⎤ ⎛ b0 ⎞2 ⎡ ⎢ vb = −15ϵ⎜ ⎟ log 1 − ⎜ ⎟ ⎥ ⎢⎣ ⎝ σmm ⎠ ⎝ b0 ⎠ ⎥⎦

100 7.459(1) 1.865

(3)

where b0 = 1.5σmm is the maximum bond elongation and the argument of vb is the distance between the centers of monomers i and i + 1, d = |rmi − rmi+1|. As it is customary, we take the monomer diameter as the unit length of the system, σmm = 1, while the energy unit is given by the LJ potential amplitude which is also taken equal to the thermal energy of the system in canonical equilibrium at a temperature T, ϵ = kB T = 1. B. No-Crowders Case. The chain metric and knotting properties in the absence of crowders were established by using Metropolis Monte Carlo (MC) simulations of a single isolated polymer in canonical equilibrium. Starting from a straight configuration, we evolved the polymer by means of a suitable combination of single monomer displacement, pivot, and generalized crankshaft moves.60 We considered chains of length Nm = 100, 300, 500, and 1000. For each case, we collected ∼106−107 uncorrelated configurations which were used to assess the occurrence of knots and to compute various metric properties. Specifically, we considered the average gyration tensor, whose Cartesian component is S*α,β = ∑iΛ2* >Λ3*, the rootmean-square radius of gyration R*g = (Λ*1 + Λ*2 + Λ*3 )1/2, the intramolecular pair distribution function ω*(r) = (1/ N Nm)⟨(∑i≠jmδ(r − rmi + rmj )⟩, and the polymer structure factor S*(k) = ω̂ *(k) + 1, with ω̂ *(k) denoting the Fourier transform of ω*(r). B


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target radial distribution function is reproduced as accurately as possible, we proceed by iterative improvements of the crowders induced potential, u0(r) → u1(r) → u2(r) → .... At the beginning of the iteration, u0 is computed from eq 6 using as input the computed ω(r), hcc(r), and hcm(r) for the system with explicit crowders: u0(r) = uHNC |ω (r). Next, u0 is used within a Monte Carlo sampling scheme carried out exclusively on the polymer (no crowders) from which the corresponding radial distribution function, ω0, is computed. Typically, ω0 will differ from the input, explicit-crowders one. To improve on that, the potential is subsequently updated according to the following recurrence relationship:

simulations of an equivalent model where crowding effects are treated implicitly (see Figure 1).


HNC ui + 1(r ) = ui(r ) + λ[u|HNC ωi (r ) − u|ω (r )]

where λ is a mixing parameter i u(rmm ij ). Specifically, for vanishingly small polymer volume fraction the PRISM relationships can be recasted in Fourier space as65,68,69

II. RESULTS AND DISCUSSION We used stochastic sampling techniques to investigate how the knotting properties of fully flexible self-avoiding linear chains change when going from the free (no crowders) case to a solution of spherical crowders. We considered chains of Nm= 100, 300, 500, and 1000 monomers of diameter σmm. For each case, we set the crowders’ volume concentration equal to ϕc = 0.35 and the polymer-to-crowders size ratio equal to q = 4. The latter condition implies that the crowders’ diameter, σcc, is varied concertedly with the polymer length according to σcc = R*g /2, where R*g is the root-mean-square gyration radius of the free chain. The ϕc and q parameters were kept fixed because they are known to be the key determinants, at fixed solvent quality, of the metric properties of asymptotically long polymers in the presence of hard spherical crowders, provided that these are much larger than the chain monomers, σcc ≫ σmm. At the same time, to have an appreciable effect on the chain internal organization, the crowders should arguably not exceed the bulk polymer size, i.e., σcc < 2R*g . The choice q = 4, which has also been used in other studies,48,59 respects these two bounds at all considered chain lengths, including the shortest one, Nm = 100, where the two bounds are most stringent (see Table 1). A. Structural Properties of a Polymer in a Model Crowded Environment. We start by presenting the Langevin simulation results for the relative positioning of the two basic

hcĉ (k) = ccĉ (k) + ρc ccĉ (k)hcĉ (k) ̂ (k) = S(k)c ̂ (k)[1 + ρ h ̂ (k)] hcm cm c cc

(7)

(5)

where ρc = ϕc6/(πσcc ) is the crowder number density, ĥαβ(k) is defined as the Fourier transform of gαβ(r) − 1, with gαβ(r) the radial distribution function of particle types α and β, and ĉαβ(k) is the direct correlation function in the reciprocal space.70 The first equation in eq 5 is the Orstein−Zernike relationship for crowders in the absence of polymers relating ccc to the usual radial distribution function hcc which, upon substitution in the second equation, yields the definition for ccm. The latter is finally used to obtain the desired potential via the so-called HNC closure71,72 1 u HNC ̂ (k) = − ρc ccm ̂ (k)[1 + ρc hcĉ (k)]cmc ̂ (k ) β (6) 3

where β = 1/kB T = 1 in simulation units. The final total nonbonded interaction is given by adding the inverse Fourier transform of eq 6 to the bare pairwise potential of eq 2 acting only on the pairs of nonsubsequent beads. To ensure that the C


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observables: the structure factor, S(k), and the gyration radius tensor. The former is shown in Figure 3a in the form of a Kratky plot where S(k) is multiplied by (kR*g )2/Nm. This multiplicative


types of particles in the system, namely the crowders and the polymer monomers. Their radial distribution functions, g(r), are presented Figure 2a as a function of the particles’ reduced

Figure 2. (a) Crowders−crowders correlation function, g(r)cc; the same correlation function computed without the chain (i.e., bulk fluid of LJ particles) is shown for reference. (b) Crowders−monomers correlation function, g(r)cm. (c) Crowders−chain center of the mass correlation function, g(r)c−pol.

distance r/Rg*. It is noticed that the crowder−crowder functions, g(r)cc, for systems with chains of different length are essentially superposed onto each other and coincide with the analogous distribution of an equivalent fluid of LennardJones particles. This shows a posteriori that the simulation box is large enough that crowders retain, on average, their bulk behavior. More interestingly, one observes that the crowder−monomer functions, gcm, are well superposed, too (see Figure 2b). This fact confirms that at fixed q and ϕc Rg* is the natural length scale to use for highlighting properties that are independent of the polymer length, Nm, and hence universal. The effective width of the depletion region between monomers and crowders centers, which is obtained by suitably integrating gcm(r),74 is equal to ≈0.4R*g . Therefore, the depleted region extends significantly beyond the sum of the particles’ bare radii, σm + σc ≈ 0.25R*g . The observed depletion width is nevertheless smaller than the one expected in the limit of vanishingly small crowders concentration, ϕc → 0, which is equal to ≈0.65R*g (see Supporting Information). This clearly reflects the relevance of excluded volume interactions between the crowders themselves at the considered concentration, ϕc = 0.35. The concentrated crowders can nevertheless effectively coordinate and permeate the polymer. This is shown in panel c which portrays the radial distribution function of the crowders and the polymer center of mass, gc−pol(r), and its comparison with the analogous quantity for ϕc → 0. In fact, even at the smallest distances, gc−pol(r) is non-negligible and ranges from 0.25 to 0.4, depending on Nm, and these values are even larger than the ones found for ϕc → 0. This clarifies that the pressure exerted by the finite crowders concentration reduces significantly the depletion effects between crowders and monomers. To complete the characterization of the effect of crowding on the chain spatial organization, we consider two additional

Figure 3. (a) Kratky plot of the structure factor S(k) for chains of chains of different length, Nm, with and without crowders. For visual clarity some curves have been rescaled as indicated in the figure. Results for both the explicit and implicit crowders simulations are shown. The value of k associated with the monomer size, kbond, are marked with arrows. (b) Crowding induced reduction of the gyration radius, Rg, and of the root-mean-square gyration tensor eigenvalues, Λ1,2,31/2. Filled and open symbols refer respectively to explicit and implicit crowders simulation. Lines are intended as a guide for the eye.

term is customarily introduced so to balance out the Debye scaling regime observed in ideal chains (i.e., S(k) ∝ k−2 for kR*g > 1) and hence expose the difference or consistency with this prototypical reference system.75 The comparison of the Kratky curves of Figure 3a for chains with and without crowders is best discussed starting from the regime of large k’s or, equivalently, of short wavelengths. When k is larger than the value associated with the monomer size σmm (marked with arrows in the plot), the curves have an oscillatory behavior. The perfect superposition of the curves for the free and crowding cases agrees with the fact that no significant differences are expected at the submonomer length scales, given the much larger size of the dispersed particles. Moving to progressively smaller k’s, i.e., longer length scales, the two sets of curves become clearly distinguishable when kR*g ∼ 10, independently of Nm. In particular, when 1 ≲ kRg* ≲ 10, the curves for the crowding cases exhibit a plateau. This indicates that the structure factor has the same functional D


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Macromolecules dependence of ideal chains, S(k) ∝ k−2. This fact agrees with the expectation that the excluded-volume effects on the chain are screened by the effective monomer−monomer attraction induced by the crowders (depletion interaction). Such screening occurs for kR*g ≲ 10, implying that the free-chain behavior should be progressively recovered for length scales ≲0.6R*g . This length scale about coincides with the nominal diameter of the crowders, σcc = 0.5Rg*, as well as with the typical transverse size of the accessible interstitial cavities between the crowders [(8R*g )3/Nc − 4π/3(σcc/2)3]1/3 ∼ 0.5R*g , which are both intuitively important length scales in the problem. Finally, moving to kR*g ≲0.6, one finds that the free and crowding cases become progressively indistinguishable as one approaches the long wavelengths Guinier regime76 where S(k)(kR*g )2/Nm ≈ (kR*g )2 − k4R2g R*2g /3. The insight offered by the Kratky plots is aptly complemented by considering the principal components of the gyration tensor which captures the overall compactness and anisotropy of the chains with and without crowders. To this end, the radius of gyration and the root-mean-square gyration tensor eigenvalues, Λ11/2 > Λ21/2 > Λ31/2, normalized to the nocrowders case are shown in Figure 3b for various chain lengths. All quantities are smaller than one and hence give a consistent indication that crowding causes an overall reduction of the chain size, consistently with the aforementioned screening effects. It is also seen that the relative reduction of the polymer radius of gyration grows with chain length and, upon extrapolation, appears to approach a limiting value of about ≈0.7 (see Supporting Information). Interestingly, the polymer compactification is not isotropic because the reduction of Λ1 is much larger than for Λ3. Crowding thus causes an overall reduction of the chain asphericity with respect to the free case.77 The data of Figure 3a,b, which were obtained from Langevin dynamics simulations, exclusively pertain to the internal properties of the chain. They provide a valuable term of comparison for validating the implicit-crowders model for which the same quantities can be calculated to a much higher precision for the same computational cost. In particular, in Figure 3a the polymer structure factors of the implicit-crowders model are shown with filled data points and are perfectly superposed to those of the explicit-crowders case which are shown with solid lines. This superposition is a direct consequence of the scheme used to derive the effective interactions of the implicit model which, in fact, is based on matching the two structure factors. More interesting is the comparison shown, again, in Figure 3b for the radius of gyration and the principal components of the gyration tensor. In this case, too, the implicit model data points are in very good agreement with the explicit-crowders ones. As a matter of fact, the observed deviations for corresponding data points are 0.7% on average and never larger than 2%. Building on this remarkable agreement, in the following section we shall often complement the Langevin simulations results with the Monte Carlo ones of the implicit model. The latter, in fact, being more amenable to extensive sampling, offers a better control of the ensemble statistics. These results are in line with the intuitive expectation that crowding effect should bring about an overall size reduction for flexible chains. In this regard, it is however worth mentioning that crowding-induced swelling has also been reported in specific situations. These include the case of crowders with

shape polydispersity50 or with size comparable to51 or smaller than the chain monomers.52,53 B. Knotting Properties. For each considered chain length the incidence of physical knots was assessed over a set of at least 104 uncorrelated configurations obtained from Langevin simulation with explicit crowders. Instead, for both the free (no-crowders) case and for the implicit-crowders model, we used a much larger set of 106 uncorrelated conformation generated via a Monte Carlo scheme. The corresponding knotting properties are given in Table 2 and shown in Figure 4. The absolute incidence of knots is detailed in the Supporting Information. Table 2. Knotting Probabilities, Pk(ϕc)%, for Chains of Different Length with and without Crowders and the Knotting-Probability Enhancement Pk(ϕc)/P*k case free chain no crowders

explicit crowders

implicit crowders

Nm 100 300 500 1000 100 300 500 1000 100 300 500 1000

Pk(ϕc)/P*k

Pk(ϕc)% 2.0(5) 6.5(3) 2.202(6) 6.13(8) 5(1) 1.1(1) 2.7(1) 6.7(8) 4.9(2) 1.23(4) 2.94(5) 8.15(7)

× × × × × × × × × × × ×

−4

10 10−3 10−2 10−2 10−3 10−1 10−1 10−1 10−3 10−1 10−1 10−1

1 1 1 1 ≈25 17(2) 13.4(9) 11(2) ≈25 19(2) 14.5(7) 13.4(9)

Figure 4. Knotting probability Pk as a function of the chain length Nm with and without crowders. Solid lines are intended as a guide for the eye.

For free chains one has that the knotting probability grows rapidly with Nm, going from 2.0(5) × 10−4% to 6.13(8) × 10−2% as Nm increases from 100 to 1000. The relatively small incidence of knots at these chain lengths is of the same order of earlier results for equivalent or related polymer models at comparable chain lengths. For instance, for a freely jointed chain of Nm = 1024 segments Tubiana et al.18 found Pk(1024) = 8 × 10−2%, while for a pearl-necklace model Dai et al.78 found Pk(1000) = 9.4 × 10−2%. Apart from differences in the model definition, the smaller value reported here at Nm = 1000 E


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Macromolecules arguably reflects the use of a more conservative criterion for selecting genuine physical knots (see Supporting Information). We find that the introduction of crowding agents enhances dramatically the incidence of physical knots. In fact, the data in Figure 4 show that the knotting probability increases by more than 1 order of magnitude at each Nm. More precisely, the relative knots enhancement, Pk(ϕc)/Pk*, is about equal to 12 for the longest chains, whereas for Nm = 100 and 300 it is close to a factor 20. The same enhancement is observed for the implicitcrowders model whose knotting probabilities are also shown in Figure 4. Such data points appear to be very close to the explicit-crowders ones. In fact, they fall within their 95% confidence interval, thus confirming the viability of the implicit model. Both with and without crowders, the knot spectrum is dominated by the simplest topology, that is, the 31 or trefoil knot. In fact, at each Nm, this knot type accounts for more than ≈90% of the observed knots both with and without crowders. The remaining knots population consists about entirely of 41 or figure-eight knots. The fact that the increased abundance of knots is not accompanied by a substantial increase of their complexity is intriguing because it is different from what has been observed in simulations of polymers that are compactified by three-dimensional spatial confinement or under poorsolvency conditions.28,79 In such cases, in fact, the knotting enhancement is paralleled by a complication of the knot spectrum. An equal footing quantitative comparison of these cases with the presently discussed crowding effects cannot be made because of the differences either in the nature of the model chains, their lengths, and degree of compactness. However, for the case of ideal polymer rings in a spherical confinement, the scaling relationship conjectured by Michels and Wiegel suggests that for fixed relative reduction of the chain calliper size the enhancement of knots should increase with chain length.19 This is not what is observed here and this suggests that the knotting effects of crowding, at least within the considered setup, differ from those expected from subjecting a free chain to an equivalent degree of spatial compactification. Crowders Effect on Chain Knotting. Although the knots repertoire is practically monodisperse, the various instances of observed knots can differ substantially for their contour length. This poses the question of whether the significantly increased incidence of physical knots introduced by molecular crowding is uniform across the various knot lengths or not. We accordingly computed the normalized probability distributions of lk with and without the effects of crowding. Because such characterization requires large samples for statistical accuracy, it was carried out exclusively for the implicit crowders model. The lk distributions for the two cases are shown in Figure 5 for the three largest chain lengths. The comparison reveals remarkable qualitative differences for both the distribution breadth and the behavior of the modal value of lk as a function of the chain contour length. For the free case, the modal value of lk is practically constant at all chain lengths (and about equal to 150), in agreement with previous studies of circular and linear knotted chains in a variety of different conditions but without crowders.15,18,36,80 In such contexts, too, in fact, the probability distribution features a peak whose location was found to be largely independent of the chain length, a phenomenon referred to as “metastability”33 followed by an approximate power-law slow decay which

Figure 5. Knot length distribution P(lk) for chains of different length, Nm, with crowders (implicit model, solid lines) and without crowders (dashed lines). The vertical lines correspond to the average knot lengths.

ultimately determines how the average knot length scales with Nm.79,81−83 Notably, when crowding effects are taken into account, the peak location is shifted to larger and larger values of lk as Nm increases and the distribution breadth grows concomitantly, too (see Figure 5). In particular, for the longest chain, Nm = 1000, the average knot length is about 3 times larger than the free chain case. A qualitatively similar enhancement of knot size is observed in compact rings82 where, however, the knot complexity is dramatically enhanced by chain compactness,84 unlike what is observed here. The observed effects of knot length suggests that the internal organization of the chain is affected by additional length scales introduced by crowding effects. In this regard we recall that it is the relative crowders-topolymer size, q, and not the absolute crowders size, that is held fixed across the considered range of Nm. One implication of the enhancement of longs knots regards the position along the chain of a knot’s midpoint. Its probability distribution is shown in Figure 6. The introduction of crowders changes the distribution more and more as Nm is increased. In particular, compared to the free case, the distribution becomes more peaked at the chain center and depleted at the termini. The connection with the effects discussed for Figure 5 emerges by considering that the midpoint of a knot of length lk can be accommodated only in the [lk/2; Nm − lk/2] interval along the chain contour and by assuming the midpoint occupies with about equal probability any point in such interval. These two premises suffice to derive a simple theoretical reference for the midpoint position probability density by suitably convolving the lk distributions in Figure 5. The resulting estimates, which are shown in Figure 5, are comparable to those based on actual observations and hence clarify that the two above-mentioned considerations provide a viable starting point for describing the positioning of knots along a chain contour. Next, to rationalize the differences emerging from Figure 5, we turned to consider the compactness of the chain, which is usually a key promoter of entanglement, and examined on what length scales is affected by crowding. Specifically, we considered the gyration radius, rg, of increasingly long subportions of free chains (which are mostly unknotted). These same quantities were next separately computed for knotted portions r*g,k of F


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contour lengths and grows systematically for longer ones. As a practical means of establishing the length scale at which the size reduction sets in, we considered the arclength l for which the trends of rg(l) starts to deviate from free case counterparts, r*g (see Supporting Information). This threshold length, l ̅, is highlighted in Figure 7 and corresponds to a gyration radius of about ≈0.3Rg*. Equivalently, l ̅ corresponds to an average monomer−monomer distance of 0.45R*g , which is comparable to the excluded-volume screening length, i.e., ≈0.6Rg*, estimated from the structure factors of Figure 3a. This observation, which holds for other chains lengths too as shown in the Supporting Information, allows for a seamless interpretation of the crowding-induced modifications to the overall incidence of knots and to the knot length distribution. The striking enhancement of the knots’ abundance arguably reflects the fact the l ̅ is substantially smaller than the chain contour length for all considered values of Nm. Indeed, a pervasive increase of compactness, and hence self-entanglement, is introduced systematically in the chain ensemble over portions longer than l ̅. The impact on the knot length distribution can be understood from the fact that the occurrence of knots longer than l ̅ should be significantly enhanced compared to shorter one, which ought to be unaffected by the crowding-induced compactification. In fact, for free chains of 300 beads only few knots are shorter than lN̅ m=300 = 60, and consequently the crowding-induced knot enhancement operates uniformly over the whole range of knots length. This can be appreciated in Figure 5 by noticing how the normalized probability distribution in the presence of crowders, P(lk), is practically indistinguishable from its free-case counterpart. Conversely, for the longest considered chains of 1000 beads, knots shorter than lN̅ m=1000 = 150 are an appreciable fraction of the total, and hence, in the presence of crowders, their relative incidence is suppressed in favor of longer knots (see Figure 5). Finally, we emphasize that the fact that the fraction of knots shorter than lN̅ m grows with Nm (see also Supporting Information) is due to the constraints of fixing the crowding parameters ϕc and q at all values of Nm. Clearly, this sophisticated interplay of different length scales affecting knotting properties could be suitably altered with alternative setups, e.g., by keeping the length of the chain fixed, and the crowders relative size, q, is varied at fixed ϕc.

Figure 6. Probability distribution of the knots midpoint location along the chain contour for polymers of different length with crowders (implicit model, solid lines) and without crowders (dashed line). The thin lines show a theoretical reference distribution based on the argument provided in the main text.

various length, lk. The results are displayed in Figure 7 for the longest chains, Nm = 1000, and are reported in the Supporting Information for Nm = 300 and 500.

Figure 7. Gyration radius of portions of length l extracted from chains of Nm = 1000 beads without and with crowders (implicit model). The gyration radius of the sole knotted portions is displayed, too, and its running average is shown with a dashed line.

III. SUMMARY AND CONCLUSIONS We considered flexible self-avoiding chains consisting of Nm monomers, with Nm ranging from 100 to 1000 and examined how their metric and entanglement properties change when spherical crowders, considerably larger than the monomer size, are added in solution. In crowding conditions, the salient properties of asymptotically long chains are controlled by only few parameters, such as the volume fraction occupied by the crowders, ϕc, and the polymer-to-crowders size ratio, q. For definiteness, these parameters are here fixed at the physically viable values ϕc = 0.35 and q = 4. The conformational space of the chains with and without crowders is explored by means of stochastic simulations, specifically Langevin dynamics and Monte Carlo sampling techniques. The main findings are two. First, for all considered chain lengths, it is seen that the introduction of crowders enhances the spontaneous knotting probability by more than an order of magnitude at each considered chain length. Such increase of

Consistently with intuition, the data show that knotted regions are substantially more compact than average. In fact, r*g,k is typically smaller than r*g by a factor of ∼√2, and the same holds when crowding effects are included. Interestingly, this reduction factor is about the same that is expected going from linear to circular chains. Indeed, we found that the termini of the knotted region are usually rather close to each other independently of the knot length, so that to a first approximation they behave similarly to rings (see Supporting Information). Besides quantifying the different compactness of knotted and unknotted chain portions, the comparison of the curves in Figure 7 is useful for clarifying that the compactification induced by crowding sets is negligible at sufficiently small G


Article


Macromolecules overall knots incidence is about equivalent to the one expected for the free chains (no crowders) upon increasing their contour length by more than a factor of 2. However, the second notable feature of crowding-induced knotting is that the observed knots have a very different contour length distribution compared to the free case. In the latter situation, the most probably knot length is largely independent of Nm. For the crowders case, instead, the modal knot length grows concomitantly with the chain length; as a matter of fact, the whole knot length distribution becomes progressively skewed toward larger knot lengths as Nm is increased for the crowders case. To clarify crowding-induced effects that underpin the observed qualitative changes of the knotting phenomenology, we examined several metric observables that capture the internal chain organization at different length scales. The analysis shows that both the boost of the knotting probability and the bias toward forming longer knots have common, intuitive origins. In fact, on one hand, the crowders induce an effective depletion-like attraction of the chain monomers which promotes the overall compactification of the chain, and this, in turn, favors the occurrence of physical knots. On the other hand, the screening of the chain self-avoiding effects operated by the induced attraction occur only above a length scale that is a fraction of the free chain size. As a consequence, the chain self-knotting is preferentially promoted over long contour lengths, hence the observed dramatic enhancement of long knots. The findings, besides being interesting from a general polymer physics point of view, for they shed light on a previously unreported aspect of the interplay of metric and topological properties in systems with polymers and crowders, might have practical ramifications, too. In fact, the observed sensitive dependence of abundance and length of physical knots suggests that these key geometrical and topological properties of flexible polymers could be externally tuned via a suitable choice of the crowding conditions. In this respect, we believe that further interesting ramifications could emerge by examining the effects on chain entanglement by the combined introduction of molecular crowding and spatial confinement.

Figure 8. Crowding induced potential between the monomers for chains of different length Nm. The potential is rescaled by the absolute value of its minimum, u0, and is plotted as a function of the rescaled distance x = (r − σmm)/Rg*. The inset shows u0 and Rg* for the indicated values of Nm.

(r − σmm)/Rg*, and by rescaling the potential amplitude by |u0| a nice collapse of the different curves is observed for Nm > 100. As a matter of fact, this seems to suggest that at least in range of Nm considered in this work the potential can be reasonably parametrized as −3 (2) udep (r ; Nm) = u0R g* f (r = xR g* + σmm)

(A1)

where f(x) is a scaling function, nearly independent of Nm. From a theoretical point view, such a scaling form provides a simple microscopic interpretation of the numerical evidence that by keeping fixed q and ϕc the overall physical behavior exhibited for different polymer lengths remains approximately the same. Indeed, as is customary for polymer solutions, the deviations from the good solvency condition can be expressed by measuring the contribution to the monomeric second virial coefficient Bdep 2 due to the presence of the crowders, which can be approximated as

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APPENDIX. EFFECTIVE POTENTIAL DESCRIPTION In this Appendix we briefly discuss the major features of the crowding induced potential between the monomers. The numerical outcomes of the inversion procedure described in section I.C for the various polymer contour lengths Nm considered in this work are portrayed in Figure 8. On qualitative grounds, we find that for all the cases the potential is purely attractive for distances ≲σcc in agreement with the theoretical expectations based on the Asakura−Oosawa−Vrij theory for the depletion interaction.85−87 For larger distances a modest repulsive contribution appears before a dumped oscillatory tail in the potential takes place with a spatial modulation reminiscent of the liquid-like order of the crowders. These effects clearly originates from the nonideality of the crowding agents. Moreover, the potential becomes progressively more long-ranged but less attractive as Nm increases, with a characteristic amplitude at contact which is numerically found to scale approximately as σcc−3 or equivalently as R*g−3. In particular, by fitting the available data for the minimum of u(r), u0, as a function of R*g we found that it is reasonably described by the relation u0 = −28.99R*g−3 (see the inset of Figure 8). Furthermore, we find that by plotting the potentials for different values of Nm as a function of the reduced distance x =

B2dep ≈ 2π

∫σ

∞

dr r 2(1 − e−βu(r ; Nm))

mm

(A2)

By substituting the scaling form of eq A1 into the previous expression and by approximating 1 − e−βu(r;Nm) ≈ βu(r;Nm), we find B2dep ≈ 2π

∫σ

∞

dr r 2βu(xR g* + σmm)

mm

= 2πβu0

∫0

∞

dx x 2f (x)

(A3)

which turns out to be independent of Nm. In other terms, at fixed q and ϕc but different Nm, the solvent quality remain approximately the same as the crowded induced potential becomes progressively more long-ranged but less attractive in a such a way that the two effects seem to be compensated. Such an effect is well verified numerically. Indeed, by using eq A1 and −3 assuming a cutoff for u(r) equal to 1.3R*g , we found Bdep = 2 σmm −1.00, −1.00, −1.01, and −1.00 for Nm = 100, 300, 500, and 1000, respectively. Finally, we note that as the bare excluded volume contribution to the total virial coefficient B2 is approximately 4π/6σmm3 ≈ 2σmm3, B2 is decreased compared H


Article

Macromolecules

(21) Janse van Rensburg, E. J.; Whittington, S. G. J. Phys. A: Math. Gen. 1990, 23, 3573. (22) Koniaris, M. M. K. Phys. Rev. Lett. 1991, 66, 2211. (23) Koniaris, M. M. K. J. Chem. Phys. 1991, 95, 2873. (24) Deguchi, T.; Tsurusaki, K. J. Phys. Soc. Jpn. 1993, 62, 1411. (25) Shimamura, M. K.; Deguchi, T. Phys. Lett. A 2000, 274, 184. (26) Katritch, V.; Olson, W. K.; Vologodskii, A.; Dubochet, J.; Stasiak, A. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2000, 61, 5545. (27) Baiesi, M.; Orlandini, E. Phys. Rev. E 2012, 86, 031805. (28) Baiesi, M.; Orlandini, E.; Stella, A. L. Phys. Rev. Lett. 2007, 99, 058301. (29) Virnau, P.; Rieger, F. C.; Reith, D. Biochem. Soc. Trans. 2013, 41, 528. (30) Micheletti, C.; Marenduzzo, D.; Orlandini, E.; Sumners, D. J. Chem. Phys. 2006, 124, 064903. (31) Micheletti, C.; Marenduzzo, D.; Orlandini, E.; Sumners, D. W. Biophys. J. 2008, 95, 3591. (32) Reith, D.; Virnau, P. Comput. Phys. Commun. 2011, 182, 1945. (33) Grosberg, A. Y.; Rabin, Y. Phys. Rev. Lett. 2007, 99, 217801. (34) Orlandini, E.; Micheletti, C. J. Biol. Phys. 2013, 39, 267. (35) Lin, P. k.; Hsieh, C.-C.; Chen, Y.-L.; Chou, C.-F. Macromolecules 2012, 45, 2920. (36) Dai, L.; Renner, C. B.; Doyle, P. S. Macromolecules 2015, 48, 2812. (37) Shaw, M. R.; Thirumalai, D. Phys. Rev. A: At., Mol., Opt. Phys. 1991, 44, R4797. (38) van der Schoot, P. Macromolecules 1998, 31, 4635. (39) Sear, R. P. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1998, 58, 724. (40) Odijk, T. Biophys. Chem. 1998, 73, 23. (41) Castelnovo, M.; Gelbart, W. Macromolecules 2004, 37, 3510. (42) Kramer, T.; Schweins, R.; Huber, K. Macromolecules 2005, 38, 9783. (43) Le Coeur, C.; Demé, B.; Longeville, S. Phys. Rev. E 2009, 79, 031910. (44) Le Coeur, C.; Teixeira, J.; Busch, P.; Longeville, S. Phys. Rev. E 2010, 81, 061914. (45) Jones, J. J.; van der Maarel, J. R. C.; Doyle, P. S. Nano Lett. 2011, 11, 5047. (46) Kudlay, A.; Cheung, M. S.; Thirumalai, D. J. Phys. Chem. B 2012, 116, 8513. (47) Denton, A. R. In New Models of the Cell Nucleus: Crowding, Entropic Forces, Phase Separation, and Fractals; Hancock, R., Jeon, K. W., Eds.; Academic Press: New York, 2014; Vol. 307, pp 27−71. (48) Kang, H.; Pincus, P. A.; Hyeon, C.; Thirumalai, D. Phys. Rev. Lett. 2015, 114, 068303. (49) Shin, J.; Cherstvy, A. G.; Metzler, R. ACS Macro Lett. 2015, 4, 202. (50) Kang, H.; Toan, N. M.; Hyeon, C.; Thirumalai, D. arXiv:1505.01623, 2015. (51) Reddy, G.; Yethiraj, A. Macromolecules 2006, 39, 8536. (52) Kim, J. S.; Szleifer, I. J. Phys. Chem. C 2010, 114, 20864. (53) Kim, J. S.; Backam, V.; Szleifer, I. Phys. Rev. Lett. 2011, 106, 168102. (54) Liao, G.-J.; Chien, F.-T.; Luzhbin, D.; Chen, Y.-L. J. Chem. Phys. 2015, 142, 174904. (55) Micheletti, C.; Marenduzzo, D.; Orlandini, E. Phys. Rep. 2011, 504, 1. (56) Fuchs, M.; Schweizer, K. S. Europhys. Lett. 2000, 51, 621. (57) Mutch, K. J.; van Duijneveldt, J. S.; Eastoe, J. Soft Matter 2007, 3, 155. (58) D’Adamo, G.; Pelissetto, A.; Pierleoni, C. Mol. Phys. 2013, 111, 3372. (59) D’Adamo, G.; Pelissetto, A.; Pierleoni, C. J. Chem. Phys. 2014, 141, 244905. (60) Sokal, A. D. In Monte Carlo and Molecular Dynamics Simulations in Polymer Science; Binder, K., Ed.; Oxford University Press: New York, 1995; pp 47−124.

to the no-crowders case but still positive, mimicking the behavior observed in solvents of intermediate quality between the Θ and the good-solvent limit.

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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.5b01323. Additional details for the implicit crowders model; radial distribution functions in the zero-density limit and the depletion thickness; crowding-induced chain compactification, extrapolation for Nm → ∞; typical relaxation times for a chain of beads with/without crowders; selection of genuine physical knots; size of knotted and unknotted subchains; absolute incidence of knots: number of sampled configurations and knotted states (PDF)

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AUTHOR INFORMATION

Corresponding Author

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

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are indebted to Enzo Orlandini, Luca Tubiana and Alexander Grosberg for valuable discussions. We acknowledge support from the Italian Ministry of Education grant PRIN 2010HXAW77.

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REFERENCES

(1) Arai, Y.; Yasuda, R.; Akashi, K.; Harada, Y.; Miyata, H.; Kinosita, T.; Itoh, H. Nature 1999, 399, 446. (2) Saitta, A. M.; Soper, P. D.; Wasserman, E.; Klein, M. L. Nature 1999, 399, 46. (3) Bao, X.; Lee, H.; Quake, S. Phys. Rev. Lett. 2003, 91, 265506. (4) Virnau, P.; Kantor, Y.; Kardar, M. J. Am. Chem. Soc. 2005, 127, 15102. (5) Huang, L.; Makarov, D. E. J. Chem. Phys. 2008, 129, 15. (6) Grosberg, A. Polym. Sci., Ser. A 2008, 51, 70. (7) Kivotides, D.; Wilkin, S. L.; Theofanous, T. G. Phys. Rev. E 2009, 80, 041808. (8) Matthews, R.; Louis, A. A.; Yeomans, J. M. Europhys. Lett. 2010, 89, 20001. (9) Micheletti, C.; Orlandini, E. Macromolecules 2012, 45, 2113. (10) Micheletti, C.; Orlandini, E. Soft Matter 2012, 8, 10959. (11) Dai, L.; van der Maarel, J. R. C.; Doyle, P. S. ACS Macro Lett. 2012, 1, 732. (12) Reisner, W.; Larsen, N. B.; Silahtaroglu, A.; Kristensen, A. Proc. Natl. Acad. Sci. U. S. A. 2010, 107, 13294. (13) Rosa, A.; Di Ventra, M.; Micheletti, C. Phys. Rev. Lett. 2012, 109, 118301. (14) Szymczak, P. Eur. Phys. J.: Spec. Top. 2014, 223, 1805. (15) Dai, L.; Renner, C. B.; Doyle, P. S. Macromolecules 2014, 47, 6135. (16) Micheletti, C.; Orlandini, E. ACS Macro Lett. 2014, 3, 876. (17) Mansfield, M. L. Macromolecules 1998, 31, 4030. (18) Tubiana, L.; Rosa, A.; Fragiacomo, F.; Micheletti, C. Macromolecules 2013, 46, 3669. (19) Michels, J.; Wiegel, F. Proc. R. Soc. London, Ser. A 1986, 403, 269. (20) Sumners, D. W.; Whittington, S. G. J. Phys. A: Math. Gen. 1988, 21, 1689. I


Article


Macromolecules (61) Ellis, R. J.; Minton, A. P. Nature 2003, 425, 27. (62) Plimpton, S. J. Comput. Phys. 1995, 117, 1. (63) Kremer, K.; Grest, G. J. Chem. Phys. 1990, 92, 5057. (64) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, 1986. (65) Curro, J. G.; Schweizer, K. S. Macromolecules 1987, 20, 1928. (66) Schweizer, K. S.; Curro, J. G. Phys. Rev. Lett. 1987, 58, 246. (67) Schweizer, K. S.; Curro, J. G. Adv. Chem. Phys. 1997, 98, 1. (68) Khalatur, P.; Zherenkova, L.; Khokhlov, A. J. Phys. II 1997, 7, 543. (69) Khalatur, P.; Zherenkova, L.; Khokhlov, A. Phys. A 1997, 247, 205. (70) Hansen, J.; McDonald, I. Theory of Simple Liquids: with Applications to Soft Matter; Elsevier Science: Amsterdam, 2013. (71) Chandler, D.; Singh, Y.; Richardson, D. M. J. Chem. Phys. 1984, 81, 1975. (72) Grayce, C. J.; Schweizer, K. S. J. Chem. Phys. 1994, 100, 6846. (73) Tubiana, L.; Orlandini, E.; Micheletti, C. Prog. Theor. Phys. 2011, 191, 192. (74) Lekkerkerker, H. N.; Tuinier, R. Colloids and the Depletion Interaction; Springer Science: Berlin, 2011; Vol. 833. (75) Müller, M.; Binder, K.; Schäfer, L. Macromolecules 2000, 33, 4568. (76) Doi, M.; See, H. Introduction to Polymer Physics; Clarendon Press: Oxford, 1996. (77) Zifferer, G. Macromol. Theory Simul. 1999, 8, 433. (78) Dai, L.; Renner, C. B.; Doyle, P. S. Phys. Rev. Lett. 2015, 114, 037801. (79) Baiesi, M.; Orlandini, E.; Stella, A. L. Macromolecules 2014, 47, 8466. (80) Tubiana, L. Phys. Rev. E 2014, 89, 052602. (81) Farago, O.; Kantor, Y.; Kardar, M. Europhys. Lett. 2002, 60, 53. (82) Marcone, B.; Orlandini, E.; Stella, A. L.; Zonta, F. J. Phys. A: Math. Gen. 2005, 38, L15. (83) Orlandini, E.; Stella, A. L.; Vanderzande, C. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2003, 68, 031804. (84) Tubiana, L.; Orlandini, E.; Micheletti, C. Phys. Rev. Lett. 2011, 107, 188302. (85) Asakura, S.; Oosawa, F. J. Chem. Phys. 1954, 22, 1255. (86) Asakura, S.; Oosawa, F. J. Polym. Sci. 1958, 33, 183. (87) Vrij, A. Pure Appl. Chem. 1976, 48, 471.

J


Molecular Crowding Increases Knots Abundance in Linear Polymers

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