Takahiro
Sakaue
*^{a} and
Takuya
Saito
^{b}
^{a}Department of Physics, Kyushu University, Fukuoka 819-0395, Japan. E-mail: sakaue@phys.kyushu-u.ac.jp; Fax: +81 92802 4107; Tel: +81 92802 4066
^{b}Fukui Institute for Fundamental Chemistry, Kyoto 606-8103, Japan

Received
31st March 2016
, Accepted 25th May 2016

First published on 25th May 2016

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.

The motion of chromosomal loci in cells is, in a sense, similar to the sub-diffusional motion, i.e., 〈Δr_{1}(t)^{2}〉 ∼ t^{α1} with α_{1} < 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 < α_{1}.^{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

(1) |

〈_{A}(n,t)_{A}(m,s)〉 = Ae^{−|t−s|/τA}δ(n − m)I | (2) |

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.

〈Δr(t)^{2}〉 ≃ D*(t)t ≃ a^{2}(t/τ_{0})^{1/2} | (3) |

(4) |

Eqn (4) indicates that the tension propagates along the chain as n(t) ∼ (t/τ_{0})^{p/2}. Then, the effective friction grows as γ*(t) ∼ γ_{1}*(t)n(t) ∼ t^{1−(p/2)}, where we have taken into account the time dependence of the effective monomeric friction γ_{1}*(t) due to medium viscoelasticity. This leads to the MSD 〈Δr(t)^{2}〉 ≃ D*(t)t with D*(t) = k_{B}T/γ*(t), hence

α = p/2. | (5) |

(6) |

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} ≃ τ_{0}N^{2/p} from the tension propagation dynamics.

R_{n} ≃ an^{1/2} + bn^{νA} | (7) |

Assuming that the medium viscoelasticity is still described by power-law kernel with the exponent p,§ the MSD of simple probes is calculated as 〈Δr_{1}(t)^{2}〉 ≃ D_{1}*(t)t, where the effective diffusion coefficient depends on the timescale

(8) |

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/τ_{0})^{p/2}. Applying the argument (I), this leads to the MSD of tagged monomers 〈Δr(t)^{2}〉 ∼ (1/n(t)) × D_{1}*(t)t ∼ (1/n(t)) × t^{α1,A}, thus

α_{A} = α_{1,A} − (p/2) = (3p/2) − 1 (t ≫ τ_{A}). | (9) |

α_{A} = pν_{A}. | (10) |

ν_{A} = (3/2) − p^{−1} (t ≫ τ_{A}), | (11) |

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) |

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/τ_{0})^{p/q}, the number of monomers which contributes to the effective friction grows as n(t) ≃ (t/τ_{0})^{pds/q}. The terminal time is thus τ ≃ τ_{0}M^{q/p} = τ_{0}N^{q/(dsp)}. We now apply the argument (I) to calculate the MSD of tagged monomers 〈Δr(t)^{2}〉 ≃ (a/R_{n(t)})^{z−2}D_{1}*(t)t, where the time-dependent effective diffusion coefficient D_{1}*(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)^{dsν} ∼ t^{pdsν/q}, this leads to the MSD exponent

α = α_{1} − pd_{s}ν(z − 2)q^{−1} | (13) |

α = 2d_{s}νp/q | (14) |

(15) |

(16) |

(17) |

(18) |

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 ∼10^{2} 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.

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}

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,^{18}e.g. chromosome segregation, and other related macromolecular dynamics including, for instance, the translocation across a nanopore^{19–23} and the rotational dynamics^{48} in- and out-of-equilibrium situations.

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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_{n}^{2}〉 ≃ (k_{B}T/k_{A})n + Ak_{A}^{−2}n^{3}, 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^{ds/df}.^{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 nucleus^{38}. |

†† 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 polymers^{46} are in line with our predictions. |

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