Active fractal networks with stochastic force monopoles and force dipoles: Application to subdiffusion of chromosomal loci

Active matter is attracting increasing interest and spans diverse systems.^1–6 Much effort has been recently devoted to the phase diagram of active systems, with emphasis on “motility-induced phase separation.” ^3,7–11 Living systems, where the chemically stored energy is transformed to mechanical energy, can be regarded as a type of active matter, e.g., moving micro-organisms, in vivo intracellular dynamics, or in vitro active cytoskeleton dynamics.^12–18 Earlier studies of such bio-systems aimed to obtain steady-state dynamics, in particular, its manifestation in the mean square displacement (MSD) of a probe monomer or particle.^19–26 

In one such recent in vivo experiment, the motion of different chromosomal loci in bacteria and yeast cells was probed both in normal cells, in which active (“ac”) forces dominate, and in cells depleted of adenosine triphosphate (ATP), where fluctuations are governed by thermal (“th”) forces.^25,26 It was found that in both cases, chromosomal loci perform anomalous sub-diffusion, MSD $≈B t ν$ (⁠ $ν<1$⁠), with identical exponents $ν a c≃ ν t h≃0.4$⁠, yet different amplitudes $B$ (from which an apparent diffusion coefficient is defined in Ref. 26 as $B=2d D app$⁠, where $d$ is the dimensionality); the ATP-depleted cells yield a lower amplitude. Telomeres in chromatin of mammalian cells also exhibit subdiffusion with a similar exponent, $ν a c≃0.4$⁠.^23,24 Yet, the reason behind the equality of the two exponents reported in Ref. 26 is unclear.²⁷ While passive (thermal) and active Rouse dynamics of linear chains predict identical subdiffusion exponents, their values are $ν a c= ν t h=1/2$⁠,^28,29 larger than the observed values.

To explain these discrepancies, we take advantage of the fractal structure of chromatin (discussed next). Fractals, either disordered or deterministic, are characterized by a few broken dimensions:^30–32 (i) the mass fractal (Hausdorff) dimension $d f$ governing the scaling $M(r)∼ r d f$ of the mass $M(r)$ enclosed in concentric spheres of radius $r$⁠, (ii) the spectral dimension $d s$ governing the scaling $g(ω)∼ ω d s − 1$ of the vibrational density of states (DOS) $g(ω)$ with frequency $ω$⁠,^33,34 and (iii) the topological dimension $d l$⁠, which governs the scaling $M(l)∼ l d l$ of the mass $M(l)$ enclosed in concentric “spheres” of radius $l$ in the topological (or “manifold”/“chemical”) space. One may also define, instead of $d l$⁠, the chemical length (or the minimal path) dimension $d min= d f/ d l$⁠, which relates the real space distance $r$ between two points on the fractal to the minimal path distance $l$ between these points along the fractal network links, $l∼ r d min$⁠.^32,35,36 To clarify the concept of topological space and $d l$⁠, consider a linear chain—defining a 1D topological space–folded self-similarly in the 3D space without forming cross-links, thereby setting $d l=1$ independent of $d f$ (e.g., $d f=2$ and $5/3$ for Gaussian and self-avoiding chains, respectively), while cross-links may increase $d l$ above 1. Similarly, a 2D sheet may be crumpled self-similarly in the 3D space without internal cross-links, corresponding to $d l=2$ irrespective of $d f$⁠. The three broken dimensions, $d s$⁠, $d l$⁠, and $d f$⁠, obey the inequalities: $1≤ d s≤ d l≤ d f≤d$⁠, where $d$ is the Euclidean embedding space dimension.

In recent years, significant insight has been gained in understanding chromatin structure during the interphase stage using various experimental methods.^37–45 These experiments reveal a fractal-like density distribution and estimate the values for the (mass) fractal dimension $d f$ (see Refs. 44 and 46 for summaries of values). In particular, Hi–C experiments,⁴⁰ textural analysis,³⁷ and neutron scattering^38,39 yielded values of $d f≃2.8−2.9$⁠. The latter values of $d f$ are nearly those predicted by the fractal-globule model,^47–49 which builds on a linear polymer folded compactly in 3D space, thereby fractal dimensions $d l= d s=1$ and $d f=3$⁠. Given the abundance of lamin A proteins making internal cross-links,²⁴ we may expect that both $d l$ and $d s$ are somewhat larger than 1.

What are the anticipated effects of active forces acting on chromatin within living cells? These forces can arise from various sources.^50–53 Here, we focus on two main categories: force monopoles and force dipoles. Force monopoles are external forces that cannot emerge from within the network, i.e., they are exerted by the cytoplasm and membrane (in prokaryotic cells) or by the cytoskeleton (in eukaryotic cells).⁵² On the other hand, force dipoles are internal forces predominantly found in the cytoskeleton, such as the actomyosin system. They might also be present in chromatin, potentially through various enzymes that actively interact with DNA.⁵³ Higher-order multipoles, such as force quadrupoles, cannot be excluded completely when considering complicated enzyme or motor protein action but are likely rarer and weaker than the monopoles and dipoles.

Explanations have been proposed for the observed anomalous diffusion exponents in chromatin, $ν a c≃ ν t h≃0.4$⁠; yet they do not account for the behaviors of both normal and ATP-depleted cells on the same footing. One approach, based on a linear chain topology, has been to associate $ν t h<1/2$ in the presence of a viscoelastic solvent, whose frequency-dependent complex modulus behaves as a power-law, $G ∗(ω)∼ ( i ω ) α$⁠, yielding $ν t h=α/2$⁠.^25,54–56 However, a different exponent (also discussed briefly in  Appendix C and Sec. II A 1), $ν a c=3α/2−1$⁠, is predicted for an equivalent active system,⁵⁵ contradicting the experimental results. Another explanation was based on Rouse dynamics (of a folded linear chain) yielding $ν t h=2/(2+ d f)$⁠.⁵⁷ Alternatively, it has been shown that this exponent is always $ν t h=1− d s/2$⁠, irrespective of the fractal dimension;^58–61 hence, $ν t h=1/2$ is recovered for $d s=1$⁠. These two results for $ν t h$ become identical for Gaussian fractal networks (i.e., in the absence of self-avoidance) for which $d f=2 d s/(2− d s)$⁠.^62,63 This implies that $d s=6/5$ corresponds to $d f=3$⁠, leading to $ν t h=0.4$⁠, consistent with the experimental results for ATP-depleted cells. From a different perspective, a single particle model, which uses the generalized Langevin equation with a delta-correlated active force that does not obey the fluctuation–dissipation theorem (FDT), yields $ν a c=2 ν t h−1$ for $ν t h≥1/2$ and $ν a c=0$ for $ν t h<1/2$ such that identical passive and active exponents are obtained only for normal diffusion, $ν=1$⁠.^20,27,55,64 Active dynamics of critical percolation clusters—forming disordered fractals—of a 2D triangular network of springs have been studied previously; yet, the individual node MSD was not reported.⁶⁵ Here, we aim to generalize such models to arbitrary fractals with potential applications to chromatin.

This paper is structured as follows: In Sec. II, we begin with an overview of the model, the general analytical theory that solves it (detailed in  Appendix A) that applies to any fractal, and the primary results for MSD. Next, in Sec. III, we present simulation results for the Sierpinski gasket, a well-known deterministic fractal, thereby validating our analytical results and discovering new findings for the case of force dipoles. In Sec. IV, we extend the present general theory of the active fractal network to chromatin relying on its fractal nature and show general agreement with experimental findings. We conclude in Sec. V by highlighting the role of active forces in active fractal dynamics and the significance of the nature of active forces in chromatin dynamics.

It is of interest to extend the subdiffusion exponents to the case of a viscoelastic solvent, whose frequency-dependent complex modulus behaves as a power-law, $G ∗(ω)∼ ( i ω ) α$⁠, as discussed in Sec. I. This is presented in  Appendix C, where we obtain $ν t h=α ( 1 − d s 2 )$ and $ν a c=α ( 2 − d s 2 )−1$⁠, which reduce to the known results for a linear chain, $d s=1$⁠.^25,54–56 Hence, for any fractal in viscoelastic solvent, the two exponents are equal only for $α=1$⁠, the viscous case. Interestingly, for $α< ( 2 − d s 2 ) − 1$⁠, the active subdiffusion regime disappears.

For force dipoles—since for typical fractals $1− d s/2− d s/ d l<0$—an anomalous subdiffusion regime is absent, the MSD crosses over from the early ballistic motion $∼ t 2$ to saturation at $2⟨ r → 2 ⟩ ac$⁠. As now the latter diverges [see Eq. (A51)] with the upper integration limit $ω 0$⁠, it becomes essentially independent of $N$⁠. This implies that, surprisingly, for the combination of thermal and active forces, $⟨ r → 2 ⟩ th$ may dominate the total static MSD for large enough systems due to the Landau–Peierls instability. Moreover, if force monopoles are also present, they are likely to dominate the evolution (as discussed in Sec. IV).

These analytical results are based on the scalar elasticity Hamiltonian [Eq. (4)], assuming a large system, $N→∞$⁠. To verify the sensitivity of our results to various assumptions, we performed Langevin dynamics (LD) simulations of a bead-spring Sierpinski gasket (Fig. 11), using different generations where the $n$th generation is denoted by S $n$⁠, with $n=1,2,…$⁠. We solve Eq. (5) with $H= 1 2m ω 0 2 ∑ ⟨ i j ⟩ ( r → i − r → j − b r ^ i j ) 2$ for the Hamiltonian of harmonic springs having an equilibrium distance $b$⁠, where $r ^ i j$ is the unit vector of the distance $r → i− r → j$ between beads $i$ and $j$⁠. For $b=0$⁠, it reduces to the scalar elasticity Hamiltonian discussed above. We follow the position of an internal bead (see Fig. 12) in a gasket and calculate its time-averaged MSD using the standard method for stationary noise. Throughout the simulation, we worked with dimensionless units, where distance is in units of $b$⁠, force in units of $m ω 0 2b$⁠, and times are in units of $τ 0=γ/(m ω 0 2)$⁠, with $k BT/m ω 0 2 b 2=1$ (see details in  Appendix B).

Consider first the dynamics of a thermal (passive) network (previously studied^58–61). We compare the MSD obtained from simulations, for both vanishing and non-vanishing $b$⁠, with the numerical evaluation of Eq. (11a) for S $9$⁠, $N=9843$⁠. In Fig. 1(a), anomalous subdiffusion is observed spanning over three decades with exponents agreeing very well with the predicted theoretical value $ν th=1− d s/2=0.317$ for the Sierpinski gasket with $d s=1.365$⁠.^33,78 The MSD amplitude of the analytical model, which is based on the scalar elasticity Hamiltonian, matches well with the LD simulation results for $b=0$⁠, while it differs from the MSD for $b=1$⁠. The short and long-time behavior represents regular diffusion, $r 2∼t$⁠, such that the short time is the independent bead diffusion, while the long represents the CM diffusion. For MSD of internal and peripheral beads, we find almost identical exponents but somewhat different amplitudes [see Fig. 1(b)], showing the faster motion of the peripheral bead than that of the internal bead.

Turning to active fractal networks with force monopoles, to each bead, we assign an active force of random orientation. In Figs. 2(a) and 2(c) (drift velocity eliminated, see below), we depict the MSD of an internal bead for different values of active force parameters and arrested thermal motion (⁠ $T=0$⁠). We observe a crossover at $t∼τ$⁠, from short-time ballistic motion, MSD $∼ t 2$⁠, to subdiffusion at intermediate times. In Fig. 2(a), a crossover to ballistic motion appears at longer times, indicating the presence of a CM drift velocity. The latter emerges from the incomplete cancellation of the total force due to the system’s finite size. For $N→∞$⁠, $∑ i ≠ j f ^ i⋅ f ^ j=0$⁠; however, in a finite system, this sum may have a residual (random) value, resulting in drift velocity $v drift∼ ϕ f 0 p γ N$ (Appendix B 1), which will nevertheless vanish if we average over many realizations of the force field.

Calculating the MSD about the drift position using Eq. (B4) (Appendix B 1), we show in Fig. 2(c) that the intermediate-time subdiffusion regime becomes longer, $MSD∼ t ν ac$ with $ν ac≃1− d s/2≃0.32$ as analytically predicted, and for $t≫ τ N$⁠, the long-time ballistic motion is replaced with $∼t$ behavior associated with CM active diffusion. We can also observe the effects of the correlation time $τ$ and force amplitude $f 0$ on the MSD ballistic and subdiffusion amplitudes confirming our prediction $B a c s∝ϕ f 0 2 τ − 0.682$ (⁠ $t≪τ$⁠) and $B a c∝ϕ f 0 2τ$ (⁠ $t≫τ$⁠). Like the passive case, for $b=0$⁠, theory and simulations match exactly, see Fig. 2(b); for $b=1$⁠, the subdiffusion amplitude somewhat differs; yet, the exponent is unaltered.

In Fig. 2(d), we show the combined effect of active and thermal forces for a few sets of parameters and compare them with the thermal MSD. When thermal forces are present, at short times $t< τ 0$⁠, the thermal contribution dominates, resulting in diffusive behavior. At intermediate times, subdiffusion regimes with $ν≃1− d s/2≃0.32$ can be observed for all cases; in one combination, having a long $τ$ (⁠ $=100 τ 0$⁠) and small $f 0$ (⁠ $=m ω 0 2b$⁠), two consecutive such regimes appear, where the first is dominated by thermal motion and is followed by a second one, dominated by activity and having a much larger amplitude.

Next, we simulate force dipoles randomly assigned to a fraction $ϕ=20%$ of total bonds (unless otherwise specified), where we either prohibit or do not prohibit dipoles having a common bead. In biological networks, the choice of distinct beads may arise due to the excluded volume interaction between motor proteins. During the simulations, we update each dipole orientation to remain parallel to $r → i j$⁠.

We estimate the MSD for gasket S $9$ (Fig. 3) for different force parameters. For times $t≪τ$⁠, the MSD shows a ballistic behavior, $∼ t 2$⁠, which saturates to a constant value at $t∼τ$⁠. Both the existence of the early ballistic motion and the absence of the intermediate subdiffusion regime for the Sirepinski gasket where $1− d s/2− d s/ d l=−0.5464<0$ are in accord with the analytical predictions.

In Fig. 3, at longer times, a ballistic-like rise in MSD for higher $f 0 2τ$ is observed, which requires further analysis. It is followed by oscillations that can be picked for smaller gaskets; see Fig. 4(a) for S $4$⁠. In the movies (snapshots displayed in Figs. 16–18 in  Appendix C, Multimedia available online) showing trajectories of a few gasket generations, we can observe persistent rotational motion of the network, either clockwise or anti-clockwise, that may explain the MSD oscillatory behavior. To confirm this, we define a mean square angular displacement (MSAD) associated with the pure rotational motion of an arbitrary bead $i$⁠, located at a mean square distance $⟨ R 2⟩$ from the CM, $⟨ R 2⟩⟨ ( n ^ ( t ) − n ^ ( 0 ) ) 2⟩$⁠, where $n ^(t)$ is the unit vector of $R →$⁠. In Fig. 4(a), we plot the MSAD together with the complete MSD; the inset (linear scale) emphasizes the motion at long times. The overlap of the two oscillatory curves is almost perfect. Importantly, even though force dipoles cannot generate a net torque, they can induce a mean rotational velocity (see discussion in Appendix B 2).

To investigate the “persistent” rotational motion of the entire network, we compute the angular velocity $Ω → i$ of individual beads and subsequently determine the mean angular velocity (averaged over beads) $Ω → ¯$ [Eq. (B15)]. The color map [Fig. 5(b)] displays the temporal fluctuation of angular velocity $Ω → i$ for individual beads. Similarly, fluctuation of $Ω → ¯$ over time is shown in Fig. 5(c). However, the time average of mean angular velocity, $⟨ Ω → ¯ ⟩ T W$⁠, for different time-averaging windows, $T W$ [see Fig. 5(d)] is found to be non-vanishing and converges to a constant value for long values of $T W$⁠, which depends on the (quenched) realization of the dipolar forces. Given the local fluctuation in $Ω → ¯$⁠, the non-zero value of $⟨ Ω → ¯ ⟩ T W$ results in a “crawling” (non-coherent) rotation of a whole object, which appears as persistent rotational motion at long-time windows.

Condition of net zero torque requires that mean radius-square-weighted angular velocity is zero, i.e., $Ω → | R → | 2 ¯=0$ [Eq. (B14)], which does not necessarily imply, $Ω → ¯=0$⁠. Here, $R → i$ is the position of the $i$th bead in the CM frame of reference. The time-dependent fluctuations of $Ω → | R → | 2 ¯$ presented in Fig. 5(c) are found to be two orders of magnitude smaller than that of $Ω → ¯$⁠, despite that for all beads, $| R → i | 2>1$⁠. The fluctuations seen in $Ω → | R → | 2 ¯$ reflect the minor simulation errors, which improve with a smaller simulation time step, $Δt$⁠. When the simulation time step is tenfold decreased, these fluctuations get significantly smaller, shown in Fig. 5(e). Nevertheless, in Fig. 5(f), we show that the difference in $⟨ Ω → ¯ ⟩ T W$ between the two simulations (of different time steps) is minor and effectively vanishes when the time-averaging window gets long, demonstrating that the “crawling” rotation of the whole object is real.

We further argue that this rotational motion, whose direction and frequency randomly vary between realizations of the dipole spatial distribution [Fig. 13(a) in Appendix B 2], arises from a small residual anisotropy caused by the asymmetric distribution of forces associated with finite-size effects. As shown in Fig. 13(c), the onset of oscillations is delayed to longer times as $N$ increases, reflecting a reduction in anisotropy. Furthermore, simulating a gasket where force dipoles are assigned to all bonds completely eliminates the long-term MSD oscillations, as shown in Fig. 4(b).

Interestingly, in triangular micro-swimmers that are subject to force dipoles and hydrodynamic interaction, rotational motion emerges too.^79,80 In Fig. 15 (see  Appendix B), we show that even Rouse dynamics (i.e., without hydrodynamic interaction) of a triangular bead-spring, with anisotropy of force dipole strength (and random-telegraph fluctuations), is enough to produce rotational motion.

In fact, it has been suggested in general that, despite the vanishing total torque, a “crawling” rotational motion—associated with a given sequence of deformations—could be expected for a deformable body, once the rotational symmetry of (internal) forces, i.e., the force dipoles, is broken,^81–84 similar to translational swimming in low Reynolds’ numbers. The broken symmetry due to a force dipole distribution and the system’s geometry can create localized deformations that can spread through the system in a coordinated manner. This sequence of deformations (shape changes) can cause the body to “crawl” or exhibit a slow, persistent, and directional (albeit non-torque-driven) rotational motion.

The above behavior for force dipoles does not apply above a critical force amplitude $f 0 c≈m ω 0 2b$⁠. For $f 0> f 0 c$⁠, the gasket collapses to a random shape maintaining dynamical fluctuations, but with rotational motion effectively arrested (see Fig. 14). The radius of gyration $R g$ for contractile force dipoles against $f 0$ is shown in Fig. 6(a) for S $4$ and in Fig. 7(a) for generations $S 5−$S $7$⁠. The transition persists and sharpens for larger (higher generation) gaskets, implying a true, activity-induced, first-order phase transition.

The collapse may be rationalized as a result of the dynamical “persistence length” ^85,86—generalized to account for the spring resistance—rising above $b$⁠, leading to $f 0 c≃m ω 0 2b/ ( 1 − e − τ / τ 0 )$ (see Appendix B 3), in agreement with our results for $τ≳ τ 0$⁠, Fig. 7(b). Contrary to contractile force dipoles [Fig. 6(a)], for extensile force dipoles $R g$ rises again at larger forces, Fig. 6(b).

To connect with chromatin dynamics studies described in Ref. 26, we need to establish the parameter regime associated with chromatin. We hypothesize that the chemical distance between cross-links in chromatin is equal to the persistence length $L p ≃ 250$ nm,^87,88 and that $τ$ and $f o$ are associated with cytoplasmic motor protein processivity times and forces. Cytoplasmic typical values are in the range $f 0=1−10pN$⁠, $τ= 10 − 3−10s$⁠.^89–92 To obtain the dimensionless parameter range applicable to chromatin dynamics, we assume that the network segment between two beads behaves as a Gaussian 2D thermal spring with spring constant $m ω 0 2= 90 k B T L p 2 b 4$ for semi-flexible polymers,^93,94 and the bond length between two beads is $b= L p$⁠. Taking the solvent viscosity (water) $η=1× 10 − 3$ Pa s, we estimate $m ω 0 2b= 90 k B T b=1.5$ pN and $τ 0= γ m ω 0 2= 3 π η b 3 90 k B T=4× 10 − 4$ s. Hence, for chromatin, the dimensionless values of force amplitude and force correlation time are in the range $f 0/(m ω 0 2b)=0.5$–7 and $τ/ τ 0=1$– $10 4$⁠, respectively. These parameters yield MSDs of passive systems and purely active (i.e., $T=0$⁠) force monopole systems, exhibiting a subdiffusion regime with identical exponents, yet different amplitudes, as shown in Figs. 2(b)–2(c).

To describe realistically untreated cells, we should consider the interplay of thermal forces (as white noise) with both active monopolar and dipolar forces. Moreover, with $ν th= ν ac=1− d s/2≃0.4$ (the value observed in Ref. 26), we can predict $d s≃1.2$⁠, associated with a slightly cross-linked chain. This deviates from the original fractal globule model⁴⁷ that ignores the presence of such cross-links and corresponds to the case where $d s=1$⁠. Indeed, Rouse dynamics of passive and force-monopole-active linear Gaussian chains do yield $ν t h= ν a c=1/2$⁠, consistent with our result with $d s=1$⁠.^28,29 Our deduction of $d s≃1.2$ does agree with experimental findings showing slight deviations from the fractal-globule model predictions for the static correlations and the fractal dimension.⁴¹ Furthermore, Bronstein et al. demonstrated the role as cross-linkers of lamin A proteins in eukaryotic cell chromatin.²⁴ 

In Fig. 8, we present the combination of thermal, monopolar, and dipolar forces (with comparison to the individual force source cases) for a fractal network (⁠ $N≃ 10 4$⁠) with spectral dimension $d s=1.2$⁠. In Fig. 8(a), we estimate the MSD from the numerical integration of Eq. (A37), which does not include the CM motion, using $ω min= ω 0 N − 1 / d s$ for the lower limit and the spectral dimension $d s=1.2$ as we suggest for chromatin. In Fig. 8(b), we add the CM motion to Eq. (A37) according to Eqs. (A33)–(A34). We can still observe a significant subdiffusion regime described by an exponent $ν≃1− d s/2=0.4$⁠, which emerges since the dipolar sources alone lead only to a moderate MSD saturation value [denoted as $2⟨ r → 2 ⟩ ac$ in Eq. (A52)] allowing the combination of thermal and monopolar forces to take over. For comparison, in Fig. 9, we show the equivalent MSD results for a Sierpinski gasket (⁠ $N=9843$⁠) from LD simulations. While the early evolution is similar to Fig. 8 (except for a different subdiffusive exponent), for force dipoles alone, we also observe the rise from the plateau at long times (associated with the slow rotation), which is not captured by the analytic theory. However, it is clear that the dipolar forces do not contribute to the subdiffusive motion induced by the active monopolar and thermal forces. Hence, we can assess that monopolar and thermal forces are the cause of the subdiffusive motion in chromatin.

To examine further the predicted MSD amplitudes (or “apparent diffusion coefficients”) in comparison with the experimental results,²⁶ we consider in Fig. 10 the combination of thermal forces and active force monopoles in comparison with the purely thermal case (with dipolar forces ignored) for a chromatin-like fractal network where $d s=1.2$⁠. For small values of $ϕ f 0 2τ$ where $ϕ f 0 2τp(1−p)/(dγ k B)≪T$⁠, i.e., $T eff≃T$⁠, the thermal contribution to the MSD dominates over the active contribution. For higher values of $ϕ f 0 2τ$ where $T eff≳T$⁠, we observe the appearance of either a single subdiffusion regime dominated by active motion (for relatively small $τ$⁠) or a thermally dominated subdiffusive regime crossing over to an actively dominated subdiffusive regime with the same exponent $ν$⁠, albeit with a higher amplitude $B eff$⁠. Importantly, this is quite similar to the observation in chromatin reported in Ref. 26 (see, e.g., Fig. 1 therein). The experimentally measured MSD amplitude ratios, $B th/ B eff$⁠, of ATP-depleted to normal cells²⁶ (Table 1 therein) are theoretically reproduced depending on the combination $ϕ f 0 2τ$⁠. For example, for Saccharomyces cerevisiae, we need $ϕ f 0 2τ≃100$ to match the theoretical ratio with the experimental one, $B th/ B eff≃0.12$⁠, and for E. coli, we require $ϕ f 0 2τ≃13$ to match the measurements.

Some other recent MSD measurements of chromosomal locus in normal eukaryotic cells (i.e., no ATP depletion), either single-point MSD or two-point MSD,^95–97 do show subdiffusion exponent, $ν$ values closer to $1/2$⁠, and fractal-globule packing with $d f=3$⁠.⁹⁵ Importantly, these results are in perfect accord with our general approach. Provided that $d s=1$⁠, as is the case for a fractal globule, the anomalous diffusion exponent $ν=1− d s/2$ will always be 1/2, regardless of the fractal dimension $d f$⁠. It remains unknown whether ATP-depleted cells in such systems also exhibit the same exponent, a significant observation as we discussed. In case they would, for example, our general approach would enable us to infer that $d s=1$ in these cells, as in the original fractal-globule model. To draw such a conclusion, it is imperative to have MSDs of both ATP-depleted and normal cells, where the values of $ν$ are identical. This highlights how MSD measurements provide insights into the abundance of cross-links in chromatin.

Recent active, dipolar-force-based models for chromatin dynamics also appear to yield identical exponents, yet with $ν t h≃ ν a c≃0.5$⁠.^98,99 In Ref. 98, the active system is studied at finite temperatures, and the subdiffusion is found only for weak forces such that the system is thermally dominated.¹⁰⁰ When the system becomes dominated by stronger forces,¹⁰¹ the MSD shows a crossover from super-diffusion at times shorter than the correlation time, to a constant value at time longer than the correlation time. Notably, a similar behavior is obtained in the present work only for dipolar forces.

In Ref. 99, the time window where anomalous subdiffusion was found numerically in the pure active case, $0.05<t<0.30$⁠,¹⁰² is less than one decade and is likely picking the crossover behavior from the $∼ t 2$ to a constant value. This crossover from $t 2$ to a constant value is qualitatively identical to our findings for dipolar forces (e.g., Fig. 4). In contrast, our sub-diffusion regime for force-monopole ranges over a few decades (Fig. 2), demonstrating a true anomalous regime. Reference 99 claims to derive analytically $ν=1/2$ for force dipoles (to support the numerical study) and $ν=3/2$ for force monopoles, both for times shorter than the force correlation time. In contrast, our study yields $ν=2$ for both force monopoles and dipoles in this time range.

To conclude this section, we comment again on the inability of the “power-law fluid” interpretation, $G ∗(ω)∼ ( i ω ) α$⁠, to explain the observed equality of the active and thermal exponents on the same footing.^25,26 Our extended exponents $ν t h=α ( 1 − d s 2 )$ and $ν a c=α ( 2 − d s 2 )−1$ imply that $ν a c≠ ν t h$ for any $d s$⁠, unless $α=1$ (viscous fluid). Taking²⁵  $d s=1$ and $α$ in the range 0.7–0.8 implies $ν t h=0.35−0.4$ and $ν a c=0.05−0.2$⁠, which could be well discriminated experimentally. Furthermore, for $d s=1$ and $α<2/3$⁠, one obtains $ν a c=0$⁠, i.e., the active subdiffusion is absent, and the MSD saturates to a constant value or crosses over to the CM diffusion.

We studied analytically arbitrary fractal networks that obey harmonic scalar elasticity and Rouse-type dynamics and are subject to active force monopoles and force dipoles. We also performed Langevin dynamics simulations of the Sierpinski gasket considering a nonzero equilibrium spring length for which the elasticity is not perfectly scalar. Our results show the absence of a true power-law subdiffusion regime when only force dipoles are at play within an ideal fractal network. However, when thermal forces, active force monopoles, or both are introduced into the system, their impact on MSD surpasses that of the dipoles, with the latter contributing only a constant saturation value at intermediate times. The resulting subdiffusion exponent is always $ν=1− d s/2$ irrespective of whether thermal forces, active force monopoles, or a combination of both predominate the dynamics. The LD simulations confirm the theory with high accuracy and show that the non-vanishing spring length does not alter this exponent. The value of the exponent $ν$ depends solely on the spectral dimension $d s$ and is independent of the fractal dimension $d f$ of the self-similar structure. In particular, for a linear chain that folds in space without forming internal cross-links, we always have $d s=1$ and $ν=1/2$⁠, regardless of whether $d f=2$ for Gaussian chains, $d f=5/3$ for self-avoiding chains, or $d f=3$ for compact folding.

Therefore, we explain here the a priori surprising result of an identical subdiffusion exponent for both normal and ATP-depleted cells. Moreover, we showed that within the range of force strengths (⁠ $f 0$⁠), force correlation times (⁠ $τ$⁠), and force density (⁠ $ϕ$⁠), yielding different combinations of $ϕ f 0 2τ$⁠, we obtain amplitude ratios, $B th/ B eff$⁠, of ATP-depleted to normal cells that are similar to those experimentally observed.²⁶ We conclude that the observed normal cell chromosomal-loci subdiffusion is caused by the combination of active force monopoles and thermal forces. However, we do not claim that dipole forces are absent; they simply do not contribute to this observable. We stress that our conclusions do not rely on the type of self-similarity for chromatin packing, in particular, whether it obeys the fractal-globule model,^47,48 the loop-extrusion model,¹⁰³ or a combination of the fractal-globule model with added cross-links.^49,104

For force dipoles, we also discovered two other distinct behaviors that we consider fundamentally significant. First, we observed the collapse of the fractal to a compact object when the force amplitude $f 0$ exceeds a critical value, for which the dynamical persistence length, generalized to account for a harmonic potential, exceeds the equilibrium distance between beads. This is reminiscent of the previously studied “motility-induced phase transition” and other dynamical transitions in active systems.^3,7–11,105 However, it should be noted that the structure in the collapsed state will be modified if an excluded volume interaction is included, presenting a low cutoff for the collapsed state size. We, therefore, did not delve into a detailed study of its structure and dynamics.

Second, below the critical force mentioned above, we observed a slow rotational, “crawling” motion of the whole fractal network, whose direction is random and chosen by the symmetry (isotropy) breaking associated with the random realization of the dipolar force locations. We have shown that this behavior is very similar to the one observed in triangular microswimmers for which the asymmetry is broken by choice.^79,80 Even though the torque that each dipole exerts is vanishing, and so is the total torque, such a “crawling” rotational motion is possible, in general, for a deformable body^81–84 but not for a rigid body. Thus, force dipoles, which are expected to exist in chromatin,⁵³ should cause slow rotational motion of the chromosome (as in Fig. 4), provided that chromatin is free to rotate, e.g., when it is not anchored to the nuclear envelope in eukaryotic cells by lamin A (i.e., for lamin A depletion) or other linkers. Interestingly, a large-scale coherent motion has been experimentally observed in chromatin (possibly presenting local rotational motion of chromosomal territories) and backed up by simulations that include dipolar forces.¹⁰⁶ Rotational motion has also been observed in cytoplasmic flow and proposed—conceptually in accord with our simulations—to be caused by actomyosin force dipoles.^107,108