Active diffusion of model chromosomal loci driven by athermal noise

Abstract

Active diffusion, i.e., fluctuating dynamics driven by athermal noise, is found in various out-of-equilibrium systems. Here we discuss the nature of the active diffusion of tagged monomers in a flexible polymer. A scaling argument based on the notion of tension propagation clarifies how the polymeric effect is reflected in the anomalous diffusion exponent, which may be of relevance to the dynamics of chromosomal loci in living cells.

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I Introduction

It is now well acknowledged that chromosomes in living cells are highly dynamic entities.^1,2 Clarifying their dynamic structure in various length and timescales is one of the major challenges in the current biological science. In this respect, it is remarkable that recent experiments are starting to reveal the rich fluctuating dynamics of chromosomal loci in prokaryotic cells and eukaryotic nuclei.^3–5 As a rule, the motion of loci looks apparently random, but there are indications from the measurement of the mean square displacement (MSD) 〈Δr(t)²〉 ∼ t^α that its nature is different from ordinary Brownian motion, i.e., α ≠ 1.⁶ Several reasons can be conceived: (i) chromosomes exist in a highly crowded environment in cells, which may endow the viscoelastic memory effect for the motion. (ii) The metabolic activity in living cells drives the system out-of-equilibrium. Several experiments evidenced that the in vivo motion of loci is ATP-dependent, thus non-thermal.^2,3,5,7 (iii) Being long DNA molecules complexed with structure proteins, chromosomes behave as flexible polymers in length scale larger than ca. 30 nm. Any chromosomal locus is, thus, a part of a long polymer chain (Fig. 1).

Fig. 1 Schematic of the model chromosome chain in a nuclear-like crowded, active environment. The boldest curve shows only a part of the chain with the remaining part of the same chain and other chains being drawn in thinner curves. Other crowding agents are drawn as small objects, some of which act as the active noise generators consuming chemical fuels (ATP).

The motion of chromosomal loci in cells is, in a sense, similar to the sub-diffusional motion, i.e., 〈Δr₁(t)²〉 ∼ t^α₁ with α₁ < 1, of micron-sized non-polymeric, say colloidal particles in complex medium (here the subscript “1” is introduced to mark non-polymeric particles, which will be referred to as simple probes hereafter).^8–12 However, the measured exponent α for the loci, though dependent on various factors (species, cell cycle, the drug treatment, etc.), is often substantially smaller than that for simple probes, i.e., α < α₁.^4,13 Given the point (iii) in the preceding paragraph, one asks if it is a general consequence of the polymeric nature of chromosomes.

In ref. 13, the Rouse model supplemented with a cytoplasmic viscoelasticity has been suggested to account for the sub-diffusional motion of loci in bacterial cells, where the motion of an n-th monomer (n ∈ (0,N)) reads in the continuum limit

where Γ(t) = (γ₀/τ₀)(2 − p)(1 − p)|t/τ₀|^−p (0 < p < 1) is a memory kernel. The spring constant k = 3k_BT/a² (k_BT: the thermal energy, a: the size of monomers) for the chain connectivity and the monomeric friction coefficient γ₀ define the microscopic timescale τ₀ = γ₀/k. In addition, [small xi, Greek, vector]_th(n,t) is the thermal noise with zero mean whose second moment obeys the fluctuation-dissipation theorem (FDT) 〈[small xi, Greek, vector]_th(n,t)[small xi, Greek, vector]_th(m,s)〉 = k_BTΓ(t − s)δ(n − m)I with I being 3 × 3 identity matrix. The main outcome from the model is that, while the sub-diffusion of the simple probes subjected to the same kernel Γ(t) is characterized by the MSD exponent α₁ = p, the exponent of tagged monomers, i.e., chromosomal loci, halves α = p/2, see eqn (5) below. More recently, the model has been extended to include the effect of the activity in living cells, in which the noise term in eqn (1) is modified as [small xi, Greek, vector]_th(t) → [small xi, Greek, vector](t) = [small xi, Greek, vector]_th(t) + [small xi, Greek, vector]_A(t).¹⁴ The newly added active noise [small xi, Greek, vector]_A(t) with zero mean was supposed to have the correlation of the form

〈[small xi, Greek, vector]_A(n,t)[small xi, Greek, vector]_A(m,s)〉 = Ae^{−|t−s|/τ_A}δ(n − m)I (2)

where A and τ_A characterize the strength and the persistence time of the active noise. The analysis of the model indicates a much richer scenario including a super-diffusion behavior in a certain time interval, which is connected to the out-of-equilibrium nature of the system manifested by the fact that the noise [small xi, Greek, vector](t) does not obey FDT.

Our purpose here is to try to extract a key physics in the behaviors of polymers in the viscoelastic active medium from a simple scaling argument based on the notion of tension propagation. This is particularly interesting due to its potential relevance to the problem of chromosomal loci in cells. As a main result, we pinpoint how the motion of chromosomal loci, modeled as tagged monomers in a long polymer, differs from that of simple probes in various situations (viscous, viscoelastic fluids with or without active noise). We also discuss the stationary conformation of polymers in active media; while the large scale behavior is characterized by an exponent, which may (or may not) be different from conventional Flory's, there is a certain subtlety in the model regarding the short scale behavior. Another merit of our argument lies in its flexibility for various generalizations, for instance, to include the effect of the self-avoidance, hydrodynamic interactions, non-linear molecular architecture, and possibly the convective flow which may be present in the medium.

II Tagged-monomer diffusion viewed from tension propagation dynamics

Even without entanglements, the dynamics seen in polymeric systems differs in many aspects from those in other non-polymeric counterparts. The anomalies in the former often trace back to the viscoelastic response associated with the dynamics of tension propagation. This notion of tension propagation is getting increasingly popular in recent studies on non-equilibrium dynamics of single polymers,^15–18 in particular, in the context of polymer translocation.^19–23 The term “non-equilibrium” here indicates the transient process, in which the large driving force induces the conformational distortion. Not only in the transient process, however, the tension propagation is equally important as a decisive factor in the dynamics and rheology of polymers in the stationary state. In fact, the dynamics of tagged monomers in a long polymer chain can be most easily described in terms of the time-dependent friction effected by the tension propagation. We first illustrate it using the Rouse polymer in viscous media, and then proceed to look into the viscoelasticity and/or active noise effects. In Section 3, we discuss various generalizations.

In viscous media

A sub-diffusion of tagged monomers is a generic property in long polymer molecules, which can be found already in the Rouse model, the simplest model for polymer dynamics.†²⁴ Here we provide two arguments for it. Eqn (1) reduces to a classical Rouse model by replacing the left-hand side with γ₀dr⃑(n,t)/dt (or equivalently by taking the limit p → 1). One can see that the deterministic part of the Rouse equation of motion takes a form of the diffusion equation with the diffusion coefficient k/γ₀. This implies that a disturbance at particular sites inside the chain will diffusively propagate along the chain, i.e., the number n(t) of monomers that can adjust themselves to the disturbance grows as (t/τ₀)^1/2, which defines a terminal timescale τ_N ≃ τ₀N² at which the tension propagates up to the chain end. In the first argument (I), we appreciate that n(t) corresponds to the number of monomers that move coherently, leading to the time-dependent friction γ*(t) = γ₀n(t) ≃ γ₀(t/τ₀)^1/2 for the motion of the tagged monomer. We thus find the MSD scaling

〈Δr(t)²〉 ≃ D*(t)t ≃ a²(t/τ₀)^1/2 (3)

where D*(t) = k_BT/γ*(t) is the effective diffusion coefficient according to the Einstein relation. The second argument (II) is based on the observation that the section of polymers with n monomers can equilibrate itself during the time interval τ_n ≃ τ₀n². This implies that the amplitude of the tagged-monomers' positional fluctuation in the time interval coincides with the equilibrium size an^1/2 of the section. Assuming the sub-diffusion scaling 〈Δr(t)²〉 ∼ t^α for t < τ_N, we thus write 〈Δr(τ_n)²〉 ≃ n^2α ≃ n¹, from which we again find α = 1/2. One may first wonder whether these two arguments are redundant. However, a closer inspection reveals that while the argument (II) invokes the equilibrium conformational properties of the polymer, the argument (I) does not. The equivalent result is reached, however, through the Einstein relation (FDT). This point manifests later, when we discuss the dynamics in the active media. Finally, we note that the exponent α = 1/2 can be compared to the normal diffusion α₁ = 1 for simple probes in viscous fluids.

In viscoelastic media

A power-low memory kernel Γ(t) ∼ |t|^−p implies the time-dependent friction γ₁*(t) ∼ t^1−p for the motion of simple probes, hence, the MSD 〈Δr₁(t)²〉 ≃ D₁*(t)t with D₁*(t) = k_BT/γ₁*(t), i.e., a sub-diffusion with exponent α₁ = p. To see how the medium viscoelasticity affects the motion of tagged monomers, we point out that the dynamics of tension propagation is governed by the following fractional diffusion equation;where the operator represents the fractional derivative of order p with respect to time.‡ For the “spatial” derivative along the chain (n-coordinate), the order is q = 2 for Rouse polymers.

Eqn (4) indicates that the tension propagates along the chain as n(t) ∼ (t/τ₀)^p/2. Then, the effective friction grows as γ*(t) ∼ γ₁*(t)n(t) ∼ t^1−(p/2), where we have taken into account the time dependence of the effective monomeric friction γ₁*(t) due to medium viscoelasticity. This leads to the MSD 〈Δr(t)²〉 ≃ D*(t)t with D*(t) = k_BT/γ*(t), hence

α = p/2. (5)

Perhaps more illuminating is to express the tagged monomer MSD aswhere γ_pol/γ₀ represents the friction enhancement due to the polymeric effect, which coincides with the number n(t) of associated monomers in the Rouse model. This form explicitly indicates the contribution of memory from the tension dynamics to otherwise sub-diffusing simple probes with the exponent α₁.

In the above, we have followed the line of argument (I). One can easily re-derive the same result from the argument (II), by noting the terminal time scaling τ_N ≃ τ₀N^2/p from the tension propagation dynamics.

In viscoelastic active media

Here again, one can follow the same step: first identify the behaviors of simple probes, then, superimpose the effect of tension propagation. One should note, however, that the steady-state conformation of the polymers may no longer be a simple random walk. Indeed, the additive nature of the active and thermal noises and their mutual statistical independence imply

R_n ≃ an^1/2 + bn^ν_A (7)

where R_n is the stationary spatial size of polymers with the number n of constituting monomers, and b is related to the length scale (effective segment size) introduced by the active dynamics. Below we focus on the situation, in which the active noise dominates over the thermal noise R_n ∼ n^ν_A, and determine how the exponent ν_A is related to the dynamics.

Assuming that the medium viscoelasticity is still described by power-law kernel with the exponent p,§ the MSD of simple probes is calculated as 〈Δr₁(t)²〉 ≃ D₁*(t)t, where the effective diffusion coefficient depends on the timescale

The MSD exponent is thus α_1,A = 2p at t ≪ τ_A and α_1,A = 2p − 1 at t ≫ τ_A. Such behaviors arise from the combined effects of medium caused time-dependent friction γ₁*(t) ∼ t^1−p and the persistent (random) action of the active noise in the short (long) timescale, and have been observed in the dynamics of engulfed microspheres in a living eukaryotic cell.²⁵ Notice that unlike passive systems driven by the thermal noise, the active fluctuation is free from the Einstein relation.

We now turn our attention to the dynamics of tagged monomers in the timescale t ≫ τ_A. The dynamics of tension propagation is governed by eqn (4), thus, n(t) ≃ (t/τ₀)^p/2. Applying the argument (I), this leads to the MSD of tagged monomers 〈Δr(t)²〉 ∼ (1/n(t)) × D₁*(t)t ∼ (1/n(t)) × t^α₁,_A, thus

α_A = α_1,A − (p/2) = (3p/2) − 1 (t ≫ τ_A). (9)

The stationary size of a polymer can be found from the condition imposed from the argument (II); 〈Δr(t)²〉 ∼ t^α_A ∼ n(t)^2ν_A, hence

α_A = pν_A. (10)

Combining it with eqn (9), we find

ν_A = (3/2) − p⁻¹ (t ≫ τ_A), (11)

in agreement with rigorous calculations in ref. 14.

In shorter timescale t ≪ τ_A, the dynamics of simple probes is different (eqn (8)), which implies α, thus, ν_A as well may be different from those at t ≫ τ_A. The simple probe behavior (eqn (8) upper line) combined with tension propagation dynamics suggests

α_A = 3p/2 (t ≪ τ_A), (12)

again in agreement with ref. 14. There is, however, a certain subtlety in the short scale properties of the model. To see it, let us define the characteristic number of monomers n_A ≃ (τ_A/τ₀)^p/2. For a smaller section of chain with n < n_A monomers, the action of the active noise may be too strong to over-swell it. Indeed, applying the argument (II) in the timescale t < τ_A, one again expects the scaling relation of eqn (10), which combined with eqn (12) leads to ν_A = 3/2 > 1.¶ A related issue is associated with the bond stretching. In Rouse and many other flexible polymer models, the connectivity is ensured by the entropic spring constant k ≃ k_BT/a², and its ratio to the monomeric friction coefficient determines the microscopic timescale τ₀ = γ₀/k. In active media, the strength of active noise affects the entropic nature of the spring. In particular, in the timescale t ≫ τ_A, the active noise enters the “equi-partition” relation k_Aa_A²/k_BT = 3[1 + (T_A/T)], where we introduce the effective temperature T_A = Aτ_A/(k_Bγ₀),¹⁴ and thus, the spring constant k_A and/or the bond length a_A, and the monomeric timescale τ_0A as well may be altered. Two extreme cases are the followings: (i) the fixed spring constant k_A = k implies the bond stretching a_A ≃ a(T_A/T)^1/2 and τ_0A = τ₀; (ii) fixed bond length a_A = a, which results in the stiffening k_A ≃ kT_A/T and the reduction in the microscopic timescale τ_0A ≃ τ₀T/T_A ≃ (γ₀a)²/(Aτ_A).

III Generalization

There are situations where the monomers' self-avoidance and/or the solvent mediated hydrodynamic interactions become apparent. One may be also interested in behaviours of objects with a fractal connectivity, i.e., polymeric fractals, for which we need to distinguish the total number N of monomers from the number M of monomers in a given direction along the chemical structure. To deal with these aspects, we introduce several exponents ν, z and d_s. The first relates the equilibrium (stationary) spatial size R_n of the polymer with the number n of constituting monomers R_n ≃ an^ν_eq; in active media, this relation is replaced by R_n ≃ an^ν_eq + bn^ν_A, see eqn (7). Hereafter, we write R_n ∼ n^ν with ν = ν_eq or ν_A in thermal or active media (T_A ≫ T), respectively; if necessary, we shall adopt the same notation for other exponents as well. The second exponent specifies the ratio of the friction coefficient of the polymer to that of monomers as γ_pol/γ₀ = (R_n/a)^z−2, which is thus related to the dissipation mechanism. For free-draining polymers z = 2 + ν⁻¹, the ratio is equal to the number of monomers γ_pol/γ₀ = n irrespective of the conformation (often called Rouse dynamics). In dilute solutions, however, the hydrodynamic interactions between monomers are usually relevant to alter the exponent to z = 3; then the ratio is proportional to the spatial size of the polymer γ_pol/γ₀ = (R_n/a), reminiscent to the Stokes formula (often called Zimm dynamics).^26,28 The last one is used for the relation N ≃ M^d_s, where the spectral dimension d_s characterizes the topology, or the chemical formula of the object (describing linear sequences and branch points), but is independent of the spatial conformation of the object.²⁷

Crucial to our discussion is the dynamics of tension propagation described by eqn (4). Now the self-avoidance and/or hydrodynamic interactions introduce long-range spatial memories, making the order q non-integer in general cases. Since the tension propagates along the path of connectivity as m(t) ≃ (t/τ₀)^p/q, the number of monomers which contributes to the effective friction grows as n(t) ≃ (t/τ₀)^pd_s/q. The terminal time is thus τ ≃ τ₀M^q/p = τ₀N^q/(d_sp). We now apply the argument (I) to calculate the MSD of tagged monomers 〈Δr(t)²〉 ≃ (a/R_n(t))^z−2D₁*(t)t, where the time-dependent effective diffusion coefficient D₁*(t) for simple probes is already given (before eqn (4) for thermal media, and eqn (8) for active media, respectively). With R_n(t) ∼ n(t)^ν ∼ m(t)^d_sν ∼ t^pd_sν/q, this leads to the MSD exponent

α = α₁ − pd_sν(z − 2)q⁻¹ (13)

On the other hand, the argument (II) suggests the relation 〈Δr(t)² 〉 ∼ t^α ∼ n(t)^2ν, from which we obtain the generalized version of eqn (10)||

α = 2d_sνp/q (14)

Consistency between two arguments requires α₁ = pd_sνz/q, thusand These results deserve some appreciation.

In thermal media

The expression of q_eq is seen to be the generalization of the form suggested from the Gaussian approximation, where the equation of motion is linearized by means of pre-averaging with the equilibrium monomer distribution function.^18,28 Since ν_eq is known as an equilibrium property, once the dissipation mechanism (z_eq) is specified, the MSD exponent can be determined.

In active media (t ≫ τ_A)

In this long timescale, the active noise acts as non-correlated random kick reminiscent of the white noise. This suggests ν_A → ν_eq in the case of viscous monomer dynamics p → 1, since in this limit, the noise and the friction terms satisfy the proportionality relation, analogous to the FDT, albeit with the effective temperature T_A. Inserting p = 1 in eqn (15: middle), we thus find q_A = d_sν_eqz_eq(=q_eq), which allows us to obtainand, from eqn (16: middle), For Rouse polymers, their defining properties q = 2 and z = 2 + ν⁻¹ make it possible to rewrite the middles of eqn (15) and (16) as ν_A = (2p − 1)/(pd_s) − 0.5 and α_A = 2p − 1 − (pd_s)/2, which reduce to eqn (11) and (9), respectively, already obtained for linear polymers d_s = 1.

In active media (t ≪ τ_A)

In this scale, the spectrum of active noise does not match that of friction kernel for any exponent p, so a strong out-of-equilibrium feature is expected. To get some insight into the short scale behaviors, let us again find the exponents for Rouse polymers; with q = 2 and z = 2 + ν⁻¹, the bottoms of eqn (15) and (16) lead to ν_A = (2/d_s) − 0.5 and α_A = (4 − d_s)p/2, respectively, which reduce to the results already obtained for linear polymers d_s = 1. The pathological over-swelling behaviour ν_A > 1 for Rouse polymers with d_s = 1 may be avoided by the non-local spatial interactions. From eqn (15: bottom), we have ν_A = (q_A/d_s) − 0.5 for free draining polymers. Thus, for linear polymers d_s = 1, the upper limit ν_A = 1, i.e., elongation, can be realized by q_A = 3/2. This modifies the MSD exponent at a short time from eqn (12) to α_A = 4p/3. Another way to look at the effect is the following. Polymers synthesized from the mixture of bi- and tri-functional units develop tree-like structures with random branching, which may be characterized by the fractional spectral dimension d_s ≃ 4/3. The fractal dimension of such objects without self-avoidance is known to be 4 in thermal equilibrium.²⁴ This is consistent with ν_eq = 1/4 derived from eqn (15: top) with Rouse polymer characteristics q_eq = 2 and z_eq = 2 + ν_eq⁻¹. Such over-collapsing behavior (ν_eq⁻¹ > space dimension) indicates the significant effect of the self-avoidance. In the active media, however, the active noise inhibits the shrinkage, and leads to the stretching in the short length scale. For the branched polymers with a quenched connectivity, the upper limit of elongation is ν_A = 1/d_s. This again implies the necessity of the non-local correlation with q_A = (2 + d_s)/2 ≃ 5/3. Numerical simulations of model polymers with bond length restriction, e.g. finitely-extensible bonds, may be interesting to resolve the issue. Notice that the analogue of the length scale n_A introduced below eqn (12) has been observed in a recent simulation of a semiflexible filament in the active media, where the undulation modes from bending energy control the tension dynamics.²⁹

IV Discussion

To what extent is the model analyzed relevant to real chromosomal dynamics? While we are confident in revealing fundamental polymeric aspects of active dynamics, we feel that more elaboration is needed to answer the above question. The following remarks and questions may be pertinent to the further development of the model.

Medium viscoelasticity

The present model assumes the viscoelasticity of the medium in a phenomenological way through the non-trivial memory kernel exponent p ≠ 1 for the monomer motion. This has been motivated in ref. 13 from the observation that the fluorescently labeled RNA molecules (with size ∼100 nm supposed to be simple probes in the present terminology) in live E.coli cells exhibit sub-diffusion with α₁ ≃ 0.7.³⁰ By setting p = α₁, the authors have claimed that eqn (1) describes the observed sub-diffusion of chromosomal loci with α ≃ 0.4 reasonably well; cf.eqn (5). So the question to be addressed is the molecular origin of the memory kernel.

Such power-law memory kernels are ubiquitously found in complex fluids, which characterize their mechanical properties in the scale larger than the mesh size.³¹ For instance, in a standard microrheology set-up, one monitors the motion of simple probes in dense polymer solution. Here, the motion of the probe, whose size is larger than the mesh size of the polymer solution becomes sub-diffusive due to the viscoelastic response of the polymers.¹⁰ The medium where chromosomes are packed in cells is highly crowded, so certain viscoelastic properties would be conceivable. Microrheological measurements have reported a sort of hopping dynamics of probe particles with size 100 nm.³² The motion of larger probes with size 1 μm is caged over long time, but the external force larger than ∼10² pN enables the probes to hop between cages.³³ These studies provide an estimate for the intra-nuclear mesh size 200–300 nm, which would be larger than the effective monomer size of a chromosomal polymer. This might suggest that the chromosomal monomers feel the local (interstitial) viscosity, and if so, the limit p → 1 in eqn (1) and (4), hence the usual viscous dynamics would be appropriate for the dynamics in a monomeric scale, which results in α_eq = α_A,t≫τA = 2ν_eq/(2ν_eq + 1). The possible alternative mechanism for the non-Rouse exponent α ≠ 1/2 could be related to the topological constraints, which may either restrict the motion of monomers on fractal support (random walk trajectory) or may induce a non-trivial fractal conformation;³⁴ see the following remarks.

Topological constraints

The motion of tagged monomers in entangled polymer solutions is a simple example of (anomalous) diffusion on fractal. There is an intermediate time region with the MSD exponent of tagged monomers α ≃ 1/4 as a consequence of the Rouse dynamics eqn (3) inside the entanglement tube which makes random walk spatial configuration.^24,28 This has indeed been suggested as a possible mechanism to explain the sub-diffusion of telomeres with α ≃ 0.3.⁴ The apparently same result is obtained with p = 1/2 in thermal media (eqn (5)), but the underlying physics leading to the result is different.

Note that the entanglement in the linear polymers invoked above is a matter of timescale. Growing evidence indicates that the conformation of long chromosomes in vivo is distinct from the entangled Gaussian conformation characteristic of linear polymers in their equilibrium melt;³⁴ see ref. 35 for the estimates for the reptation time of chromosomal chains in various species in comparison to the characteristic time of cellular processes. The loci dynamics in the non-trivial chromosome conformation would be different from that in the conventional reptation picture in the entangled melt.** A numerical simulation on the tagged monomer diffusion in crumple globule, i.e., a knot-free self-similar compact structure with ν = 1/3,³⁶ has reported the exponent α ≃ 0.38.³⁷ In our frame in Section 3, this corresponds to the passive dynamics with d_s = 1 and z_eq = 2 + ν_eq⁻¹ = 5, thus, eqn (16: top) yields α_eq = 0.4. In addition, q_eq = 5/3(<2) from eqn (15: top) effects the long-range spatial correlation along the chain, which effectively ensures the compact conformation compared to the ideal conformation with q = 2. A similar scaling argument for α exponent can be found in ref. 18 and 37 as well. Finally, due to the reason of topological constraints mentioned above, the melt of unlinked and unknotted rings has attracted considerable interest as a model system of the interphase chromosome,³⁸ for which a recent theory predicts the exponent α = 2/7.³⁹

Active noise

Inside cells and nuclei, there are various ATP-driven active processes taking place along chromosomes.⁴⁰ Examples include nucleosome repositioning by chromatin remodeling enzymes, transcription by RNA polymerases and local condensation/decondensation controlled by various histone modifications. Recent experiments have evidenced that the effect of metabolic activity is at work on the chromosomal dynamics in living cells.^2,3,5,7,33 For instance, ref. 5 reported that ATP depletion leads to the reduction in the apparent diffusion coefficient of loci, while the sub-diffusive exponent α ≃ 0.4 remains unchanged.†† There are also reports on the ATP-dependent jump motion³ and the super-diffusion of loci.⁴¹

In the present paper, the effect of metabolic activity is incorporated via the active noise with the property (2). This is one of the simplest ways, and indeed seems to explain the observed active dynamics of simple probes in cells,²⁵ but there may be other possibilities to model the active effect on the chromosome dynamics^42,43,44 such as the convective flow induced by the conformational change of molecular machines in the medium or the chromatin itself. We stress, however, that the present argument is rather generic; representative formula (6) is a consequence of the collective dynamics of the polymer determined by its mode spectra,¹⁸ and does not depend on the details of the noise. Hence, the idea should be applicable to the semiflexible filaments, too.^29,45 It would be also interesting to look at the dynamics of the polymer driven by self-propelled particles from the present viewpoint.‡‡^46,47 Furthermore, aside from the situation, where the active noise dominates over the thermal noise T_A ≫ T, a further study should clarify how the competition between these two noises leads to various crossover dynamical behaviors. We hope that its simplicity helps us understand the complex dynamics of chromosomal loci, not only the anomalous fluctuation driven by noise but also the directional motion driven by external bias,¹⁸e.g. chromosome segregation, and other related macromolecular dynamics including, for instance, the translocation across a nanopore^19–23 and the rotational dynamics⁴⁸ in- and out-of-equilibrium situations.

Acknowledgements

The present analysis was motivated by the discussion with C. Vanderzande, H. Vandebroek and E. Carlon, which was done when one of us (T. Sakaue) stayed at KU Leuven. We thank T. Sugawara, S. Shinkai and A. Mikhailov for discussion on chromosome and active dynamics in living cells. This work is supported by KAKENHI (No. 16H00804, “Fluctuation and Structure”) from MEXT, Japan.

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Footnotes

† Among various classes of stochastic processes for anomalous diffusion,⁴⁹ the tagged monomer dynamics in free space discussed here corresponds to the fractional Brownian motion. See ref. 18 and 50 for microscopic derivation of the generalized Langevin equation with power-law memory kernel.

‡ Eqn (4) with q = 2 is the deterministic part of eqn (1). One can easily check that the left-hand sides of these equations are equivalent in a scaling sense. A concise explanation can be found in the ESI of ref. 14.

§ Note that the active noise may disturb the media so that the exponent p may be altered from the thermal system.

¶ This analysis suggests, unlike the large scale behavior, the short scale conformation has no p dependence. The calculation of Rouse modes in viscous solution under the condition n ≪ n_A ⇔ t ≪ τ_A shows 〈R_n²〉 ≃ (k_BT/k_A)n + Ak_A⁻²n³, where k_A is the spring constant of the polymer in active media (see the main text).

|| Notice the correspondence to the problem of random walk on fractal objects, in which the MSD of random walker is 〈Δr(t)²〉 ∼ t^d_s/d_f.²⁷ The random walk (thus, p = 1 and q = 2) along the path and the walker's displacement in real space correspond to the tension dynamics (described by eqn (4)) and the amplitude of the monomer's positional fluctuation, respectively, in our problem.

** There is a recent report that the MSD of loci in budding yeast follows a scaling consistent with the Rouse model, i.e., α = 1/2.⁵¹ This is presumably because the entanglement effect would be very weak in budding yeast as estimated from the length and density of yeast chromosomes in nucleus³⁸.

†† Within the viscoelastic Rouse model (p ≠ 1, q = 2), the lack of variation of α on ATP was interpreted as a sign that the ATP-driven active noise has a frequency dependence similar to that of thermal noise in viscoelastic media.^5,7 Our analysis indicates an alternative possible scenario p = 1, q ≠ 2 with the active noise which acts as random kick at t ≫ τ_A; compare eqn (16: top) and (18).

‡‡ Indeed, the large scale exponent ν_A = ν_eq, cf.eqn (17) with p = 1, and the non-trivial expansion and stiffening observed in the simulation of short polymers⁴⁶ are in line with our predictions.

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