# Chromatin remodeling due to transient-link-and-pass activity enhances
subnuclear dynamics

Rakesh Das1,∗,#, Takahiro Sakaue2, G. V. Shivashankar3,4, Jacques Prost1,5,†
and Tetsuya Hiraiwa1,6,⋆ 1 Mechanobiology Institute, National University of
Singapore, Singapore 117411 2 Department of Physical Sciences, Aoyama Gakuin
University, Kanagawa 252-5258, Japan 3 Department of Health Sciences and
Technology (D-HEST), ETH Zurich, Villigen 8092, Switzerland 4 Division of
Biology and Chemistry, Paul Scherrer Institute, Villigen 5232, Switzerland 5
Laboratoire Physico Chimie Curie, Institut Curie, Paris Science et Lettres
Research University, 75005 Paris, France 6 Institute of Physics, Academia
Sinica, Taipei City 115201, Taiwan ∗<EMAIL_ADDRESS>†
<EMAIL_ADDRESS>⋆<EMAIL_ADDRESS># Current address: Max
Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38,
01187 Dresden, Germany.

###### Abstract

Spatiotemporal coordination of chromatin and subnuclear compartments is
crucial for cells. Numerous enzymes act inside nucleus—some of those
transiently link and pass two chromatin segments. Here we study how such an
active perturbation affects fluctuating dynamics of an inclusion in the
chromatic medium. Using numerical simulations and a versatile effective model,
we categorize inclusion dynamics into three distinct modes. The transient-
link-and-pass activity speeds up inclusion dynamics by affecting a slow mode
related to chromatin remodeling, viz., size and shape of the chromatin meshes.

Genetic information of a eukaryotic cell is stored in its chromatin, a $\sim
2$ m long polymeric entity comprising DNA and histone proteins, which is
packed inside the nucleus typically of size $7$-$10$ $\mu$m. In addition to
the chromatin, the nucleus contains a diverse variety of subnuclear
compartments (SNCs) like nucleoli, speckles, Cajal bodies, promyelocytic
leukemia bodies, transcription factories etc., ranging $\sim 50$ nm - $1$
$\mu$m in sizes, all dispersed in a viscous fluid medium called nucleoplasm
ZidovskaBiophysRev2020 ; ShivaBook2010 ; KosakBiochimBiophysActa2014 ;
SpectorTrendsGenet2011 ; BrangwynneCurrOpinCellBiol2015 . A spatiotemporal
coordination among these SNCs and the chromatin is necessary for the healthy
functionality of the cell ZidovskaBiophysRev2020 ; ShivaBook2010 ;
KosakBiochimBiophysActa2014 ; ShabanGenomeBiol2020 ; MisteliCurrOpinGenetDev ;
BrangwynneCurrOpinCellBiol2015 ; SuterTrendsCellBiol2020  lack of which
correlates with several diseases SabariDevCell2020 ;
SleemanCurrOpinCellBiol2015 .

Recent studies have attributed such spatiotemporal coordination of SNCs and
chromatin to the mechanical state of chromatin. Ref. BrangwynneNatPhys2021
shows that coalescence kinetics of inert liquid droplets dispersed inside a
nucleus depends on their dynamics dictated by the mechanics of the chromatic
environment, whereas Ref. WingreenPRL2021  shows how the number, size, and
localization of such subnuclear condensates are dictated by chromatin
mechanics. We found in our earlier study RakeshELife2022  that, transient-
link-and-pass activity (TLPA) associated with ATP-dependent actions of some
classes of enzymes, like Topoisomerase-II KouzineTranscription2013 ;
HsiehAnnuRevBiochem2013 ; RocaNuclAcidRes2009 ; NitissNatRevCancer2009 , can
affect the microphase-separation structure of hetero-/eu-chromatin—this
enables us to speculate that, even in a homogeneous medium of chromatin, e.g.,
even when looking into only the euchromatin part, enzymatic activities could
affect local mechanical states of chromatin. Such change in the local
mechanical state of chromatin could eventually affect dynamics of finite-size
inclusions such as SNCs. Indeed, it is known that the dynamics of the
chromatin and other SNCs are usually ATP-dependent ShivaBiophysJ2008 ;
ShivaPlosOne2012 ; SwedlowNatCellBiol2002 ; GorischPNAS2004 ; ZidovskaPNAS2013
; ZidovskaPNAS2018 ; DiPierroPNAS2018 ; ShiNatComm2018 , and ATP-dependent
remodeling of the chromatic environment is recognized as one of the mechanisms
for this dependency ShivaBiophysJ2012 ; GorischPNAS2004 ; ShivaPlosOne2012 .
Despite its possible relevance for biological functions, up to present, no
studies have clarified if an active perturbation like TLPA actually plays a
role in inclusion dynamics and what kind of chromatic remodeling can
contribute to it.

In this Letter, we study the effects of an active perturbation on the
chromatic medium and fluctuating dynamics of an inclusion, and investigate the
underlying physical mechanisms. For this purpose, we consider a single polymer
chain subjected to TLPA as a chromatin model, and a finite-size bead
disconnected from the polymer chain as an inclusion, inside a spherical cavity
(FIG. 1a). To implement TLPA, we follow our earlier work in which a model
enzymatic activity was constructed imagining Topoisomerase-II RakeshELife2022
. Through numerical simulations of our model, we first show that TLPA indeed
affects the inclusion dynamics. After that, we investigate what kind of
chromatin remodeling is associated with this effect. Finally, we construct an
effective model which is a fluctuating free particle model but keeps the
essence to reproduce the TLPA-dependency of the inclusion dynamics observed in
our simulation. Using the effective model, we identify the three major
dynamical modes in the system.

So far, experimentally-observed features of SNC dynamics has been explained
quantitatively by a combination of its diffusion within the chromatic-
interspace region, plus the translocation of that region due to chromatin
diffusion (which we refer as fast and normal diffusive modes, respectively,
later in the text) GorischPNAS2004 . However, in general, chromatic interspace
itself can remodel, i.e., its size and shape can change, and thus affect SNC
dynamics ShivaPlosOne2012 . In particular, mechanical actions of enzymes like
TLPA may directly drive such chromatin remodeling. The effective model
proposed here indeed demonstrates the relevance of a slow dynamical mode
linked to chromatin remodeling for the TLPA-dependent inclusion dynamics.

Model—We develop a self-avoiding linear homopolymer model for the chromatin
confined within a spherical cavity of diameter $D$ (FIG. 1a). The homopolymer
comprises $N$ soft-core beads of diameter $d_{B}$ consecutively connected by
finitely-extensible-nonlinear-elastic springs. A hard-core spherical inclusion
of diameter $d_{I}$ is placed at the center of the cavity. This inclusion
experiences steric forces due to the surrounding polymer.

The dynamics of the positions of the beads (${\bm{x}}_{B}$) and the inclusion
(${\bm{x}}_{I}$) are approximated by Brownian dynamics; see Supplemental
Material SM  and Ref. RakeshELife2022  for details. Stokes’ relation has been
assumed to mimic the frictional drag due to implicitly considered nucleoplasm.
Setting the thermal energy and the nucleoplasmic viscosity to unity, and
$D=12\,\ell$, we obtain the simulation units (s.u.) of length ($\ell$), time
($\tau$), and energy ($e$).

TLPA is implemented in the following sequential manner, which we call catch-
and-release mechanism. In the normal state, any pair of spatially proximal
beads experiences steric repulsion due to each other (self-avoidance potential
$h_{vex}>0$, see Supplemental Material SM ). The enzyme can catch that pair of
beads with a Poisson rate $\lambda_{ra}$ (FIG. 1b), and upon that, the beads
attract each other ($h_{vex}<0$). Next, the attraction between those two beads
is turned off with another Poisson rate $\lambda_{an}$, and the beads stay
there for a while without any steric interaction among themselves
($h_{vex}=0$). Eventually, the enzyme unbinds from the beads with a Poisson
rate $\lambda_{nr}$, and the beads return to their normal state with steric
repulsion between themselves. Therefore, in our model, enzymatic activity is
realized by the following sequence of Poisson transitions of the steric
interaction between a pair of proximal beads:
$\text{state}\left(h_{vex}>0\right)\xrightarrow{\lambda_{ra}}\text{state}\left(h_{vex}<0\right)\xrightarrow{\lambda_{an}}\text{state}\left(h_{vex}=0\right)\xrightarrow{\lambda_{nr}}\text{state}\left(h_{vex}>0\right)$.
These steps allow the beads to pass across each other stochastically and
thereby perturb the medium RakeshELife2022 .

We parameterize TLPA as
$\Lambda=\lambda_{ra}\left(1/\lambda_{an}+1/\lambda_{nr}\right)$, which can be
tuned in experiments by controlling ATP concentration WangJBiolChem1993 . We
choose $\lambda_{an}=16.7\,\tau^{-1}$ and $\lambda_{nr}=500\,\tau^{-1}$ and
tune $\lambda_{ra}$ to control the activity. For a given $\Lambda$, we
integrate the equations of motion of ${\bm{x}}_{B}$ and ${\bm{x}}_{I}$ using
Euler discretization method, and thereby we simulate the dynamics of the
polymer and the inclusion. This choice of the rates sets a typical timescale
$t_{TLPA}=\left(1/\lambda_{an}+1/\lambda_{nr}\right)\simeq 0.062\,\tau$ for
which an enzyme catches a pair of beads. We have checked that by this
timescale, a bead typically moves over only $\mathcal{O}(d_{B})$.

Each realization of the simulation starts from a steady-state configuration of
a self-avoiding polymer packed inside the cavity together with the inclusion
at the center of the cavity. The polymer and the inclusion follow the Brownian
dynamics as described above. We note that the inclusion eventually touches the
cavity boundary; so, we simulate our active polymer model (APM) until the
inclusion touches the cavity boundary for the first time. The results reported
below are obtained from $n$ number of independent realizations. (See
Supplemental Material SM  for further details.)

Figure 1:  (a) Bead-and-spring homopolymer model of chromatin packed inside a
spherical cavity together with an inclusion at the center; a schematic and a
typical simulation snapshot are shown. (b) Catch-and-release mechanism of
Topoisomerase-II’s enzymatic activity. There is no steric repulsion between
the pair of the beads when bound to the enzyme. (c) Mean $\pm$ s.e.m. of
$\Delta x_{I,\,0.1\tau}$-distribution are shown. Only the positive half is
shown for better representation. (d) MSDs of the inclusion are shown for
several $\Lambda$. Dashed lines are to guide the early-time and the late-time
diffusion. (e) Main—FPT distribution of the inclusion to the radius
$R=5\,\ell$ of the cavity starting from $R=0$ ($n=71$).
Inset—$\Lambda$-dependency of the mean $\pm$ s.e.m. of FPT. SFIG. 1
supplements FIG. 1c–e with curves for four $\Lambda$-values.

Inclusion dynamics—We first investigate how inclusion dynamics is affected by
TLPA by looking into its (i) one-dimensional displacement over a given time
duration, (ii) mean-square-displacement (MSD), and (iii) first passage time
(FPT) in the chromatic medium. Here, we consider an inclusion of size
comparable to that of the bead size ($d_{I}=0.40\,\ell$, $d_{B}\simeq
0.43\,\ell$).

First, we compare the distribution $P\left(\Delta x_{I,\,0.1\tau}\right)$ of
one-dimensional displacement of the inclusion over a time duration $\Delta
t=0.1\,\tau>t_{TLPA}$ for several $\Lambda$ (FIG. 1c). We note that
$P\left(\Delta x_{I,\,0.1\tau}\right)$ follows a Gaussian distribution and
widens with $\Lambda$. The distributions appeared to be symmetric around zero,
although with some fluctuations. However, we did not find any drift in the
inclusion dynamics (SFIG. 2).

Next, we calculate MSD of the inclusion as $\langle\Delta
r_{I}^{2}\left(\Delta
t\right)\rangle=\langle\lvert\bm{x}_{I}\left(t_{0}+\Delta
t\right)-\bm{x}_{I}\left(t_{0}\right)\rvert^{2}\rangle$, where
$\langle\cdot\rangle$ represents average over several $t_{0}$ and
realizations. We note (A) a short early-time diffusive regime, (B) an
intermediate subdiffusive regime, and (C) a late-time diffusive regime.
Following this regime-(C), we note a significant slowing down of the inclusion
dynamics (data not shown). We check that by this time, the inclusion reaches
close to the cavity boundary, and although the inclusion does not interact
with the boundary, its dynamics is indirectly affected by the polymeric
density at that locality which differs from the bulk region. The effect of
TLPA at that locality has a different physics than that in the bulk region,
and therefore, it will be discussed in the future. In this Letter, we focus on
the bulk region (up to radius $R=5\,\ell$ instead of the whole cavity of
radius $R=6\,\ell$). We show the regimes (A)-(C) of the inclusion MSD in FIG.
1d. Note that the crossover time from the early-time regime to the
intermediate regime increases with $\Lambda$.

FPT represents the time that the inclusion takes to reach $R=5\,\ell$ for the
first time starting its dynamics from $R=0$ at time $t=0$. We prepare a
histogram of the FPTs noted for several realizations of simulation and note
that $\Lambda$ affects it (FIG. 1e–main). Starting from a high value of the
mean FPT (MFPT) for $\Lambda=0$, it decreases monotonically up to a moderate
$\Lambda$ (FIG. 1e–inset). Taking the displacement distributions, MSDs, and
the FPT-data together, we conclude that TLPA enhances the inclusion dynamics.

Chromatic environment—The enhancement of the inclusion dynamics should be
attributed to some change in the chromatic environment due to TLPA. To
investigate it in detail, we compare the polymeric configurations with and
without TLPA. We note that the polymeric environment becomes more
heterogeneous with $\Lambda$ (FIG. 2a, Supplemental Videos). The mean density
of the beads, $n_{B}$, increases with $\Lambda$ (SFIG. 3). Along with that,
the spatiotemporal fluctuation in $n_{B}$, as manifested by its standard
deviation $\sigma_{B}$, also increases with $\Lambda$ (FIG. 2b), suggesting an
increase in heterogeneity of chromatin medium with TLPA.

To further elaborate on this observed heterogeneity, we calculate the
separation $s_{min}$ of any arbitrary point in the bulk region of the cavity
to its nearest bead and prepare its distribution $P(s_{min})$. As a corollary
to the increasing heterogeneity in chromatic environment, we note that the
tail of $P(s_{min})$ becomes heavier with $\Lambda$ (FIG. 2c). The largest
$s_{min}(\Lambda)$ with non-zero $P(s_{min})$ minus $d_{B}/2$ could be
interpreted as half of the meshsize in the corresponding chromatic
environment. It is evident from FIG. 2c–main that the typical meshsize
increases with $\Lambda$. Considering the $\langle\Delta
r_{I}^{2}\rangle$-data, it is straightforward to understand that the inclusion
performs early-time diffusion until when it feels the mechanical hindrance due
to its chromatic neighborhood beyond which it shows subdiffusion (FIG.
2c–schematic).

Figure 2:  (a) Cropped slices ($6\times 6$ in s.u.) of typical simulation
snapshots are shown for $\Lambda=0$ (left) and $0.6188$ (right). (b)
Fluctuation $\sigma_{B}$ in local density $n_{B}$ of beads increases with
$\Lambda$ ($n=71$). (c) Main—Distribution of the separation between an
arbitrary point and its nearest polymeric-bead. $\Lambda$-dependent increase
in the largest $s_{min}$ with $P(s_{min})>0$ indicates increase in chromatic
meshsize. Data for $s_{min}<0.2$ are not shown as the arbitrary points fall on
the beads ($d_{B}\simeq 0.43$ s.u.) in that range. Inset—Schematic of the
inclusion inside a chromatic mesh.

Effective model—Next, we develop an integrative understanding of chromatic
environment-mediated effect of TLPA on inclusion dynamics. We hypothesize that
inclusion dynamics in our APM could be mimicked by its Brownian dynamics with
colored noise defined over a coarse-grained timescale $\delta t$. Therefore,
we conceive a coarse-grained effective model—the inclusion is considered
alone, and its dynamics is given by

$\displaystyle\partial_{t}{\bm{x}}_{I}=v_{EM}\,{\bm{\zeta}}_{EM},$ (1)

where the right-hand-side is an effective noise whose characteristics can be
defined from the $\Lambda$-dependent behavior of the displacement vector
$\Delta{\bm{x}}_{I,\,\delta t}$ defined over $\delta t$. More specifically,
$v_{EM}^{2}=\langle\lvert\Delta{\bm{x}}_{I,\,\delta
t}\rvert^{2}\rangle/(\delta t)^{2}$, and ${\bm{\zeta}}_{EM}$ is a Gaussian
noise with zero-mean and autocorrelation
$\langle{\bm{\zeta}}_{EM}(0)\cdot{\bm{\zeta}}_{EM}(\Delta
t)\rangle=C_{\Delta{\bm{x}}_{I,\,\delta t}}\left(\Delta
t\right)=\langle\sum_{t_{0}}\left[\Delta{\bm{x}}_{I,\,\delta
t}\left(t_{0}\right)\cdot\Delta{\bm{x}}_{I,\,\delta t}\left(t_{0}+\Delta
t\right)\right]/$ $\sum_{t_{0}}\lvert\Delta{\bm{x}}_{I,\,\delta
t}\left(t_{0}\right)\rvert^{2}\rangle$ ($\langle\cdot\rangle$ indicating
average over several realizations). Hereafter, we consider $\delta
t=0.003\,\tau$ over which the inclusion dynamics should be affected by the
dynamics of its chromatic neighborhood (FIG. 1d; also see SFIG. 4).

We obtain $C_{\Delta{\bm{x}}_{I,\,\delta t}}\left(\Delta t\right)$ from our
simulation data for several $\Lambda$. A negative autocorrelation is noted for
$\Delta t\geq\delta t$ (FIG. 3a–inset) as expected for our current model
system—(visco-)elasticity of polymeric medium surrounding the inclusion may
tend to reverse the direction of inclusion motion TakahiroPRE2015 . The
negative part of the autocorrelation shows good fit to double exponential
function (FIG. 3a–main). Therefore, we write $C_{\Delta{\bm{x}}_{I,\delta
t}}(\Delta t)=A\delta(\Delta t)-g(\Delta t)$ with $g(\Delta t)=a_{f}e^{-\Delta
t/t_{f}}+a_{s}e^{-\Delta t/t_{s}}$, where the fitting parameters $a_{f,\,s}$
and $t_{f,\,s}$ depend on $\Lambda$. The parameter $A$ takes care of the
autocorrelation for $\Delta t<\delta t$.

Figure 3:  (a) Autocorrelation $C_{\Delta{\bm{x}}_{I,\delta t}}(\Delta t)$ of
the inclusion’s displacement vector $\Delta{\bm{x}}_{I,\,\delta t}$ over
$\delta t=0.003\,\tau$ for several $\Lambda$ (symbols). Main, semi-log plot to
focus on the negative part of $C_{\Delta{\bm{x}}_{I,\delta t}}$; inset, linear
plot. Dashed-lines are fits to $-C_{\Delta{\bm{x}}_{I,\delta t}}(\Delta t>0)$
with the double exponential function, $g(\Delta t)$. (b) Main—Parts of
$\langle\Delta r_{I,\,EM}^{2}\rangle$ corresponding to the three dynamical
modes (n, f, and s) for several $\Lambda$ (see FIG. 3a for legends). Top
inset—Diffusivity corresponding to s-mode increases with $\Lambda$. Bottom
inset—Schematic of each mode. Golden: chromatin media. Blue: inclusion.

Using the effective model, we analytically obtain MSD of the inclusion as

$\displaystyle\langle\Delta r_{I,\,EM}^{2}\left(\Delta
t\right)\rangle=\sum_{m\equiv f,s}2D_{m}t_{m}\left(1-e^{-\frac{\Delta
t}{t_{m}}}\right)+2D_{n}\Delta t,\;\;\;\;\,$ (2)

where $D_{f,s}=v_{EM}^{2}a_{f,s}t_{f,s}$ and
$D_{n}=v_{EM}^{2}\left(A/2-a_{f}t_{f}-a_{s}t_{s}\right)$ (Supplemental
Material SM ). Treating $A$ as a $\Lambda$-dependent fitting parameter, we
find good agreement between $\langle\Delta r_{I}^{2}\rangle$ and
$\langle\Delta r_{I,\,EM}^{2}\rangle$ (SFIGs. 5 and 6). Thus, our simulation
results are successfully reproduced by the effective model, where the
chromatic environment-mediated effect of TLPA on the inclusion dynamics is
captured by the $\Lambda$-dependency in the features of the coarse-grained
noise. Below we show that the construction of the effective model allows us to
unveil the physical mechanism how TLPA affects the inclusion dynamics.

We identify the MSD-terms corresponding to $m\equiv f,s$ with that obtained
for a particle subjected to a harmonic potential following an overdamped
Langevin equation (Supplemental Material SM ). This gives us a diffusivity
$D_{m}$ and a characteristic time $t_{m}$ for the mode-$m$. Thus, the
inclusion dynamics can be understood as that dictated by a fast (f) and a slow
(s) mode (with $t_{f}\sim 0.002\;\tau$ and $t_{s}\sim 0.02\;\tau$), plus a
normal (n) diffusive mode. The n-mode originates from the delta-correlated
forces that the inclusion feels due to the thermal noise, plus the polymeric
neighborhood coarse-grained over $\delta t$. The origin of the fast and slow
modes must underlie in the fluctuation in the potential that the inclusion
feels due to its polymeric neighborhood; we indeed found that the fluctuation
in the number of the beads interacting with the inclusion has the
characteristic timescale comparable to that of the fast mode, whereas
remodeling of the polymeric mesh around the inclusion has the timescale
comparable with the slow mode (SFIG. 7, Supplemental Material SM ). One can
visualize the corresponding scenario as following—let us consider a polymeric
mesh comprising some number of beads around the inclusion. At the scale of
$\sim 0.002\;\tau$, that number fluctuates because of the individual bead’s
thermal fluctuation (see SFIG. 8 for bead’s MSD), and therefore the potential
fluctuates. This leads to the fast mode-contribution to inclusion dynamics.
However, at a longer timescale $\sim 0.02\;\tau$, the size and shape of the
mesh are reconfigured leading to another source of the potential-fluctuation.
This dictates the slow mode-contribution to the inclusion dynamics.

Lastly, we investigate which mode mainly contributes to the speeding of the
inclusion dynamics with TLPA. FIG. 3b–main plots the parts of MSD,
$\langle\Delta r_{I,\,EM}^{2}\rangle$, separately derived from all the three
modes. We find that the $\Lambda$-dependency in MSD at the intermediate time
scale ($\sim 0.001<\Delta t<\sim 0.02$) is dominated by that of the s-mode. We
note a significant increase in $D_{s}$ with $\Lambda$ (FIG. 3b–top inset;
SFIG. 9). These results suggest that significant initial speeding up of the
inclusion dynamics (around $\Delta t=0.001$ in FIG. 1d) is induced through
s-mode, which is associated with the chromatin remodeling. In conclusion,
TLPA-assisted remodeling of the chromatic neighborhood plays a major role in
enhancing the inclusion dynamics. (See Supplemental Material SM  for
discussion on $\Lambda$-dependency of all three diffusivities.)

Discussion—In summary, through numerical simulations of APM and construction
and analysis of the effective model, we investigated the effect of enzymatic
action-induced TLPA on the inclusion dynamics in the chromatin environment. We
showed that the inclusion dynamics in this complex medium comprises three
modes—a fast mode dynamics within the local polymeric mesh, a slow mode
dynamics associated with the polymeric reconfiguration, and a normal diffusive
mode. TLPA speeds up the inclusion dynamics by significantly facilitating
chromatin remodeling. This finding is in line with Ref. ShivaPlosOne2012
where the authors emphasized on the importance of ATP-dependent chromatin
remodeling to explain their experimental observation. In Ref. GorischPNAS2004
, as briefly mentioned above, an effective model combining the fast and the
normal diffusive modes has been proposed to explain the inclusion dynamics.
Our results further showed the necessity of the slow mode associated with
chromatin remodeling to fully understand the inclusion dynamics in active
chromatic media (schematics in FIG. 3b–bottom inset). In Ref.
ShivaBiophysJ2012 , significant slowing down of transcription compartments’
dynamics upon depression of temperature from $37^{\circ}\text{C}$ to
$25^{\circ}\text{C}$ has been attributed to temperature-dependent active
processes. The chromatin remodeling-associated dynamical mode reported here
can be responsible for that experimental observation. Our current study also
implicates how a global active perturbation may regulate site-specific target
searching of SNCs by enhancing their encounter-frequency
SuterTrendsCellBiol2020 ; HansenNatChemBiol2020 ; CortiniNatComm2018 .
Furthermore, as Topoisomerase-II activity is known to correlate with cell
cycle BergerGenes2019 , aging MullerMutationRes1989 , nucleoplasmic ATP
content YanoSciRep2021 , our study may elucidate such cell state-dependent
subnuclear dynamics.

Despite apparent similarity in the inclusion dynamics for various model-
scenarios in terms of a coarse-grained quantity like MFPT, we find our
effective model useful in distinguishing the underlying differences in terms
of the three dynamical modes (Supplemental Material SM ) RD_in_preperation .

$d_{I}$ | $0.2\,\ell$ | $0.4\,\ell$ | $1.2\,\ell$
---|---|---|---
MFPTΛ=0.3094/MFPTΛ=0 | $1.04\pm 0.08$ | $0.90\pm 0.11$ | $0.57\pm 0.21$
Table 1: Mean $\pm$ s.e.m. propagated from both the MFPTs.

Dynamics of an inclusion in a polymeric media is expected to depend on the
inclusion size RubinsteinMacrmolecules2011 . We find that the effect of TLPA
becomes significant for large inclusions (Table 1, Supplemental Material SM )
RD_in_preperation .

Our model provides a framework which may be helpful for understanding various
active gels LiverpoolPhilTransRSocA2006 ; JacquesNatPhys2015 ;
MacKintoshScience2007 ; WeitzCell2014 , usually the long-time stochasticity
stemming from activity and the short-time modes being thermal. For instance,
Refs. MacKintoshScience2007 ; WeitzCell2014  reported time-dependence of
inclusion MSDs, which may be related to a reminiscent structural remodeling-
associated mode.

###### Acknowledgements.

R.D. acknowledges useful discussions he had with Raphaël Voituriez and David
Weitz. R.D. and T.H. appreciate Yuting Lou and Akinori Miyamoto for valuable
discussions. Authors acknowledge anonymous referees for their insightful
comments. This research was supported by Seed fund of Mechanobiology Institute
(to J.P., T.H.) and Singapore Ministry of Education Tier 3 grant,
MOET32020-0001 (to G.V.S., J.P., T.H.) and JSPS KAKENHI No. JP18H05529 and
JP21H05759 from MEXT, Japan (to T.S.).

## References

  * (1) Y. S. Mao, B. Zhang, and D. L. Spector, Biogenesis and function of nuclear bodies, Trends Genet. 27, 295 (2011).
  * (2) A. Zidovska, The rich inner life of the cell nucleus: dynamic organization, active flows, and emergent rheology, Biophys. Rev. 12, 1093 (2020).
  * (3) G. V. Shivashankar, Nuclear mechanics and genome Regulation (Academic Press, Amsterdam, 2010).
  * (4) A. M. Wood, A. G. Garza-Gongora, and S. T. Kosak, A Crowd sourced nucleus: Understanding nuclear organization in terms of dynamically networked protein function, Biochim. Biophys. Acta 1839, 178 (2014).
  * (5) L. Zhu and C. P. Brangwynne, Nuclear bodies: the emerging biophysics of nucleoplasmic phases, Curr. Opin. Cell Biol. 34, 23 (2015).
  * (6) H. A. Shaban, R. Barth, and K. Bystricky, Navigating the crowd: visualizing coordination between genome dynamics, structure, and transcription, Genome Biology 21, 278 (2020).
  * (7) E. Soutoglou and T. Misteli, Mobility and immobility of chromatin in transcription and genome stability, Curr. Opin. Genet. Dev. 17, 435 (2007).
  * (8) D. M. Suter, Transcription factors and DNA play hide and seek, Trends in Cell Biol. 30, 491 (2020).
  * (9) B. R. Sabari, Biomolecular condensates and gene activation in development and disease, Dev. Cell 55, 84 (2020).
  * (10) J. E. Sleeman and L. Trinkle-Mulcahy, Nuclear bodies: new insights into assembly/dynamics and disease relevance, Curr. Opin. Cell Biol. 28, 76 (2014).
  * (11) D. S. W. Lee, N. S. Wingreen, and C. P. Brangwynne, Chromatin mechanics dictates subdiffusion and coarsening dynamics of embedded condensates, Nat. Phys. 17, 531 (2021).
  * (12) Y. Zhang, D. S. W. Lee, Y. Meir, C. P. Brangwynne, and N. S. Wingreen, Mechanical Frustration of Phase Separation in the Cell Nucleus by Chromatin, Phys. Rev. Lett. 126, 258102 (2021).
  * (13) R. Das, T. Sakaue, G. V. Shivashankar, J. Prost, and T. Hiraiwa, How enzymatic activity is involved in chromatin organization, eLife 11, e79901 (2022).
  * (14) L. Baranello, F. Kouzine, and D. Levens, DNA Topoisomerases beyond the standard role, Transcription 4, 232 (2013).
  * (15) S. H. Chen, N.-L. Chan, and T.-s. Hsieh, New mechanistic and functional insights into DNA topoisomerases, Annu. Rev. Biochem. 82, 139 (2013).
  * (16) J. Roca, Topoisomerase II: a fitted mechanism for the chromatin landscape, Nucleic Acids Res. 37, 721 (2009).
  * (17) J. L. Nitiss, DNA topoisomerase II and its growing repertoire of biological functions, Nat. Rev. Cancer 9, 327 (2009).
  * (18) D. K. Sinha, B. Banerjee, S. Maharana, and G. V. Shivashankar, Probing the dynamic organization of transcription compartments and gene loci within the nucleus of living cells, Biophys. J. 95, 5432 (2008).
  * (19) M. Platani, I. Goldberg, A. I. Lamond, and J. R. Swedlow, Cajal body dynamics and association with chromatin are ATP-dependent, Nat. Cell Biol. 4, 502 (2002).
  * (20) A. Zidovska, D. A. Weitz, and T. J. Mitchison, Micron-scale coherence in interphase chromatin dynamics, Proc. Natl. Acad. Sci. 110, 15555 (2013).
  * (21) D. Saintillan, M. J. Shelley, and A. Zidovska, Extensile motor activity drives coherent motions in a model of interphase chromatin, Proc. Natl. Acad. Sci. 115, 11442 (2018).
  * (22) M. Di Pierro, D. A. Potoyan, P. G. Wolynesa, and J. N. Onuchic, Anomalous diffusion, spatial coherence, and viscoelasticity from the energy landscape of human chromosomes, Proc. Natl. Acad. Sci. 115, 7753 (2018).
  * (23) G. Shi, L. Liu, C. Hyeon, D. Thirumalai, Interphase human chromosome exhibits out of equilibrium glassy dynamics, Nat. Comm. 9, 3161 (2018).
  * (24) F. M. Hameed, M. Rao, and G. V. Shivashankar, Dynamics of passive and active particles in the cell nucleus, PLOS ONE 7, e45843 (2012).
  * (25) S. M. Görisch, M. Wachsmuth, C. Ittrich, C. P. Bacher, K. Rippe, and P. Lichter, Nuclear body movement is determined by chromatin accessibility and dynamics, Proc. Natl. Acad. Sci. 101, 13221 (2004).
  * (26) S. Maharana, D. Sharma, X. Shi, and G. V. Shivashankar, Dynamic organization of transcription compartments is dependent on functional nuclear architecture, Biophys. J. 103, 851 (2012).
  * (27) See Supplemental Material at [url] for further details about the numerical model, the effective model, and the supporting discussion. It includes Refs. RakeshELife2022 ; JusufiJPCM2001 ; CamargoJCP2009 ; VoituriezNatChem2012 ; VoituriezNature2016 ; TakahiroPRR2023 .
  * (28) A. Jusufi, J. Dzubiella, C. N. Likos, C. von Ferber, and H. Löwen, Effective interactions between star polymers and colloidal particles, J. Phys.: Condens. Matter 13, 6177 (2001).
  * (29) M. Camargo and C. N. Likos, Phase separation in star-linear polymer mixtures, J. Chem. Phys. 130, 204904 (2009).
  * (30) T. Guérin, O. Bénichou, and R. Voituriez, Non-Markovian polymer reaction kinetics, Nat. Chem. 4, 568 (2012).
  * (31) T. Guérin, N. Levernier, O. Bénichou, and R. Voituriez, Mean first-passage times of non-Markovian random walkers in confinement, Nature 534, 356 (2016).
  * (32) Y. Sakamoto and T. Sakaue, First passage time statistics of non-Markovian random walker: Dynamical response approach, accepted in Phys. Rev. Res. (2023).
  * (33) J. E. Lindsley and J. C. Wang, On the coupling between ATP usage and DNA transport by yeast DNA topoisomerase II, J. Biol. Chem. 268, 8096 (1993).
  * (34) T. Saito and T. Sakaue, Driven anomalous diffusion: An example from polymer stretching, Phys. Rev. E 92, 012601 (2015).
  * (35) A. S. Hansen, A. Amitai, C. Cattoglio, R. Tjian, and X. Darzacq, Guided nuclear exploration increases CTCF target search efficiency, Nat. Chem. Biol. 16, 257 (2020).
  * (36) R. Cortini and G. J. Filion, Theoretical principles of transcription factor traffic on folded chromatin, Nat. Comm. 9, 1740 (2018).
  * (37) J. H. Lee and J. M. Berger, Cell cycle-dependent control and roles of DNA topoisomerase II, Genes 10, 859 (2019).
  * (38) H. C. Schröder, R. Steffen, R. Wenger, D. Ugarković, and W. E.G. Müller, Age-dependent increase of DNA topoisomerase II activity in quail oviduct; modulation of the nuclear matrix-associated enzyme activity by protein phosphorylation and poly(ADP-ribosyl)ation, Mutation Res. 219, 283 (1989).
  * (39) K. Morotomi‐Yano and K.‐i. Yano, Nucleolar translocation of human DNA topoisomerase II by ATP depletion and its disruption by the RNA polymerase I inhibitor BMH‐21, Sci. Rep. 11, 21533 (2021).
  * (40) R. Das, T. Sakaue, G. V. Shivashankar, J. Prost, and T. Hiraiwa, Manuscript in preparation.
  * (41) L.-H. Cai, S. Panyukov, and M. Rubinstein, Mobility of nonsticky nanoparticles in polymer liquids, Macromolecules 44, 7853 (2011).
  * (42) T. B. Liverpool, Active gels: where polymer physics meets cytoskeletal dynamics, Phil. Trans. R. Soc. A 364, 3335 (2006).
  * (43) J. Prost, F. Jülicher, and J. F. Joanny, Active gel physics, Nat. Phys. 11, 111 (2015).
  * (44) D. Mizuno, C. Tardin, C. F. Schmidt, and F. C. MacKintosh, Nonequilibrium Mechanics of Active Cytoskeletal Networks, Science 315, 370 (2007).
  * (45) M. Guo et al., Probing the Stochastic, Motor-Driven Properties of the Cytoplasm Using Force Spectrum Microscopy, Cell 158, 822 (2014).

Supplemental Material to “Chromatin remodeling due to transient-link-and-pass
activity enhances subnuclear dynamics”

Rakesh Das1,∗,#, Takahiro Sakaue2, G. V. Shivashankar3,4, Jacques Prost1,5,†
and
Tetsuya Hiraiwa1,6,⋆
1 Mechanobiology Institute, National University of Singapore, Singapore 117411
2 Department of Physical Sciences, Aoyama Gakuin University, Kanagawa
252-5258, Japan
3 Department of Health Sciences and Technology (D-HEST), ETH Zurich, Villigen
8092, Switzerland
4 Division of Biology and Chemistry, Paul Scherrer Institute, Villigen 5232,
Switzerland
5 Laboratoire Physico Chimie Curie, Institut Curie, Paris Science et Lettres
Research University, 75005 Paris, France
6 Institute of Physics, Academia Sinica, Taipei City 115201, Taiwan
∗<EMAIL_ADDRESS>†<EMAIL_ADDRESS>⋆<EMAIL_ADDRESS>
# Current address: Max Planck Institute for the Physics of Complex Systems,
Nöthnitzer Strasse 38, 01187 Dresden, Germany.

###### Contents

  1. .1 Details of the simulation setting
  2. .2 Brownian dynamics
  3. .3 Preparation of initial configurations
  4. .4 Analysis of inclusion’s drift dynamics
  5. .5 Density of polymer beads $n_{B}$
  6. .6 Supplemental Videos
  7. .7 Coarse-grained effective model
  8. .8 Identification of the fast- and slow-modes
  9. .9 $\Lambda$-dependency of the diffusivities corresponding to the three dynamical modes
  10. .10 Power of the proposed effective model
  11. .11 Inclusion dynamics for several inclusion sizes

##  .1 Details of the simulation setting

We consider a spherical cavity of diameter $D=1.4112$ $\mu$m containing a
$50.8070$ Mega base pair (Mbp) long chromatin, yielding a typical diploid
genome density of human somatic cells. The chromatin is mimicked by a bead-
and-spring model comprising $N=20992$ beads, each one representing a $2.4203$
kbp coarse-grained genomic content. Assuming nucleosomes as spheres of
diameter $22$ nm containing $200$ bp of genomic content and close compaction
of nucleosomes within coarse-grained beads, we obtain diameter of the beads as
$d_{B}=22\times(\text{number of nucleosomes per bead})^{1/3}=50.5086$ nm.

We set $D=12\,\ell$ which gives us the simulation unit (s.u.) of length
$\ell=117.6$ nm. Our model has been simulated at a physiological temperature
$T=310$ K, and we consider energy unit as $e=1\,k_{B}T=4.28$ pN$\cdot$nm. The
frictional drag on the beads due to the implicitly considered nucleoplasm is
approximated by Stokes’ law, and the corresponding viscosity is represented by
$\eta$. Considering the nucleoplasmic viscosity as unity in simulations, we
get the simulation unit of time $\tau=\eta\ell^{3}/e$.

##  .2 Brownian dynamics

We consider the following Brownian dynamics equations to represent the
position ${\bm{x}}_{B}$ of the $B^{th}$ bead and ${\bm{x}}_{I}$ of the
inclusion:

$\displaystyle\frac{\partial{\bm{x}}_{B}}{\partial t}$
$\displaystyle=-\frac{1}{6\pi\eta(d_{B}/2)}\frac{\partial
H_{B}}{\partial{\bm{x}}}+\sqrt{\frac{2k_{B}T}{6\pi\eta(d_{B}/2)}}{\bm{\zeta}}_{B},$
(S1) $\displaystyle\frac{\partial{\bm{x}}_{I}}{\partial t}$
$\displaystyle=-\frac{1}{6\pi\eta(d_{I}/2)}\frac{\partial
H_{I}}{\partial{\bm{x}}}+\sqrt{\frac{2k_{B}T}{6\pi\eta(d_{I}/2)}}{\bm{\zeta}}_{I}.$
(S2)

Here, ${\bm{\zeta}}$’s represent univariate white Gaussian noises with zero
mean. We consider the potential energies $H_{B}$, that realized by the
$B^{th}$ bead, and $H_{I}$, that realized by the inclusion, as

$\displaystyle H_{B}$ $\displaystyle=$ $\displaystyle
H_{spring}+H_{vex}+h_{BI}+H_{confinement}$ (S3)

and

$\displaystyle H_{I}$ $\displaystyle=$ $\displaystyle H_{BI}.$ (S4)

We explain the terms on the right-hand side of eqs. (S3) and (S4) sequentially
in the following paragraphs. To numerically integrate eqs. (S1) and (S2) with
eqs. (S3) and (S4), time is discretized into steps as usual, and the positions
${\bm{x}}_{B}$ of all the beads and ${\bm{x}}_{I}$ of the inclusion are
updated sequentially over steps.

The potential energy associated with the spring connecting two consecutive
beads along the polymer is given by
$h_{spring}=-\frac{1}{2}kr_{0}^{2}\ln\left[1-\left(r_{ij}/r_{0}\right)^{2}\right]$,
where $r_{ij}$ is the distance between the $i^{th}$ and $j^{th}$ beads. We
consider $H_{spring}=\sum_{j(\in n.i)}h_{spring}$, where the summation
$\sum_{j(\in n.i)}$ runs over the beads next to $i$ (i.e., $j$ takes $i-1$ and
$i+1$ if the $i^{th}$ bead is not located at one of the polymer ends, whereas
it takes only either of them if $i^{th}$ bead is located at an end). Here $k$
is the spring constant of a finitely extensible nonlinear elastic spring whose
stretch $r_{ij}\leq r_{0}$.

When a pair of beads separated by $r_{ij}\leq\ell$ is not bound to a
Topoisomerase-II (Topo-II) enzyme, each bead realizes steric repulsion due to
one another. We consider that interaction potential between those two beads as

$\displaystyle h_{vex}$
$\displaystyle=\epsilon_{vex}\exp\left(-\alpha_{vex}r_{ij}^{2}\right),\text{
when bead-pair ($ij$) not bound to Topo-II},$ (S5)

and the total steric potential realized by the $i^{th}$ bead is
$H_{vex}=\sum_{j\in r_{ij}\leq\ell}h_{vex}$. Here $\epsilon_{vex}$ is the
amplitude of the Gaussian interaction potential chosen, and $\alpha_{vex}$
determines its variance. We set size of the beads in our simulation model by
setting a criterion that the minimum of $h_{spring}+h_{vex}$ appears at
$r_{m}=\sum_{i\in\text{connected beads}}d_{B}/2$. This sets a constraint over
the choice of the parameters in $h_{spring}$ and $h_{vex}$ as
$kr_{m}^{2}/\left(1-r_{m}^{2}/r_{0}^{2}\right)=2\epsilon_{vex}\alpha_{vex}r_{m}^{2}\exp\left(-\alpha_{vex}r_{m}^{2}\right)$.

As per our active polymer model described in the main text, if a pair of beads
($ij$) separated by $r_{ij}\leq\ell$ is bound to a Topo-II enzyme, the beads
either realize attraction for each-other, or there is no steric interaction
among those beads. We implement this model scheme by tuning $h_{vex}$ as
following:

$\displaystyle h_{vex}$
$\displaystyle=-\epsilon_{vex}\exp\left(-\alpha_{vex}r_{ij}^{2}\right),\text{
when bead-pair ($ij$) bound to Topo-II and in the attraction state,}$ (S7)
$\displaystyle=0,\text{when bead-pair ($ij$) bound to Topo-II}\text{ and in
the no-interaction state}.$

The interaction potential $h_{vex}$ between a pair of beads ($ij$) separated
by $r_{ij}\leq\ell$ starts the following series of Poissonian transitions:
$\text{state}\left(h_{vex}>0\right)\xrightarrow{\lambda_{ra}}\text{state}\left(h_{vex}<0\right)\xrightarrow{\lambda_{an}}\text{state}\left(h_{vex}=0\right)\xrightarrow{\lambda_{nr}}\text{state}\left(h_{vex}>0\right)$,
where each transition step can take place in between any consecutive two steps
of ${\bm{x}}_{B}$-dynamics (following eq. (S1) and eq. (S3) discretized in
time) and is implemented stochastically according to the Poisson process with
the rate $\lambda_{XX}$ ($XX\equiv$ ra, an, nr). When the bead-pair ($ij$) get
separated by $r_{ij}\geq\ell$ during this process, those pair of beads
temporarily stop proceeding with the stochastic steps of the transitions
described above.

To confine the polymer inside the cavity, we use a potential $H_{confinement}$
that acts upon a bead at a separation $r$ from the boundary of the cavity. We
consider this potential energy as that between a wall and a star polymer with
functionality $s=2$ (therefore, a linear polymer) SMJusufiJPCM2001 ;
SMCamargoJCP2009 :

$\displaystyle H_{confinement}$ $\displaystyle=$ $\displaystyle
ps^{3/2}\left[-\ln\left(\frac{r}{R_{s}}\right)-\left(\frac{r^{2}}{R_{s}^{2}}-1\right)\left(\frac{1}{1+2\kappa^{2}R_{s}^{2}}-\frac{1}{2}\right)+\gamma\right],\text{
for }r\leq R_{s},$ (S8) $\displaystyle=$ $\displaystyle
ps^{3/2}\,\gamma\,{\operatorname{erfc}\left(\kappa
r\right)}/{\operatorname{erfc}\left(\kappa R_{s}\right)}\text{, for }r>R_{s}.$

Here we have modeled individual bead as a star polymer with radius of gyration
$R_{g}=d_{B}/2$. In eq. (S8), $p$ is a free parameter, $R_{s}=0.65\,R_{g}$
stands for radius of corona of the polymer, the parameter $\kappa$ should be
$\mathcal{O}\left(R_{g}^{-1}\right)$,
$\gamma=\sqrt{\pi}\operatorname{erfc}\left(\kappa
R_{s}\right)\exp\left(\kappa^{2}R_{s}^{2}\right)/\left[\kappa
R_{s}\left(1+2\kappa^{2}R_{s}^{2}\right)\right]$, and $\operatorname{erfc}$
stands for complementary error function.

The steric repulsion between the $B^{th}$ bead and the inclusion, separated by
$r_{BI}$, is approximated by WCA potential,

$\displaystyle h_{BI}$ $\displaystyle=$ $\displaystyle
4\epsilon_{BI}\left[\left(\frac{\sigma_{BI}}{r_{BI}}\right)^{12}-\left(\frac{\sigma_{BI}}{r_{BI}}\right)^{6}\right]+\epsilon_{BI},\text{
for }r_{BI}\leq\frac{d_{B}+d_{I}}{2},$ (S9) $\displaystyle=$ $\displaystyle
0,\text{ otherwise}.$

Therefore, the total potential energy realized by the inclusion due to
neighboring polymeric beads is
$H_{BI}=\sum_{r_{BI}\leq(d_{B}+d_{I})/2}h_{BI}$.

We summarize our choice of model parameters in Table 2.

Potential | Parameters
---|---
$H_{spring}$ |  $k=22\,e\ell^{-2}$; $r_{0}=2d_{B}$.
$H_{vex}$ |  $\epsilon_{vex}=8\,e$; $\alpha_{vex}=7.9585\,\ell^{-2}$.
$H_{confinement}$ |  $p=4\,e$; $s=2$; $R_{s}=0.65R_{g}$; $\kappa=1/R_{g}$; $R_{g}=d_{B}/2$.
$H_{BI}$ |  $\epsilon_{BI}=e$; $2^{1/6}\sigma_{BI}=(d_{B}+d_{I})/2$.
Table 2: Choice of model parameters.

Equations (S1) and (S2) are integrated over time in Euler method, where we use
a discretized time step $10^{-4}\,\tau$. The simulations are done using lab-
developed code where we use CUDA to exploit GPU acceleration SMRakeshELife2022
. We start the simulations from specially prepared equilibrated configurations
(described in Sec. .3) and register the dynamics henceforth.

##  .3 Preparation of initial configurations

We prepared the initial configurations for our main simulations in two steps.
We started with a polymer arbitrarily packed inside the spherical cavity,
together with the spherical inclusion of diameter $0.8\,\ell$. We assumed the
polymer to be an ideal chain (i.e., $h_{vex}=0$ for any pair of the beads),
although the interaction between the beads and the inclusion was considered
the same as described in Sec. .2. We performed Brownian dynamics simulations
to evolve the polymeric system for a duration of $4020\,\tau$ (by which mean-
squared-displacement of individual bead goes beyond $2D$). During this
simulations, the inclusion was kept fixed at the center of the cavity. The
last configurations obtained from such $12$ set of simulations were considered
as the initial configurations for the next step of initial configuration
preparation. In the next step, we considered individual configurations
obtained from the first step and replace the previous inclusion by an
inclusion particle of desired diameter $d_{I}$. This time we considered the
polymer to be self-avoiding (i.e., $h_{vex}>0$) and allowed the system to
evolve for another duration of $4020\,\tau$ (by which the mean-squared-
displacement of individual bead goes beyond $D$). The inclusion was still kept
fixed at the center. For each set of $12$ initial configurations, we picked
$3$ different simulation configurations and consider those $36$ configurations
as the initial configurations for our main simulations for a given $d_{I}$.
During our main simulations, we have used different sequence of random numbers
to generate more than $36$ realizations.

##  .4 Analysis of inclusion’s drift dynamics

We checked whether there is any emergence of drift motion of the inclusion due
to the enzymatic activity. Note that we considered an isotropic geometry of
the cavity. So, by analyzing the distribution of the one-dimensional
displacements in one of the three axes of the simulation-reference-frame
(SRF), we may not be able to conclude unambiguously the existence of drift.
So, we did the following. For a given realization of simulation, we know the
displacement vector ${\bm{d}}_{t}={\bm{x}}_{I}(t+0.1\,\tau)-{\bm{x}}_{I}(t)$
of the inclusion at time $t$ on the SRF. We obtained a transformed vector
${\bm{d}}_{transformed,\,t}=\mathcal{R}{\bm{x}}_{I}(t+0.1\,\tau)-\mathcal{R}{\bm{x}}_{I}(t)\equiv(\Delta
x_{t,\,transformed},\Delta y_{t,\,transformed},\Delta z_{t,\,transformed})$
such that $\mathcal{R}{\bm{x}}_{I}(t)\equiv(0,0,z_{transformed})$,
$\mathcal{R}$ representing necessary sequences of rotations. We defined
$\Delta z_{transformed}=\langle\Delta z_{t,\,transformed}\rangle_{t\in\text{
course of the simulation}}$ in radial grids (used $6$ concentric radial grids,
each of width $\ell$), and similarly $\Delta x_{transformed}$ and $\Delta
y_{transformed}$. We calculated $(\Delta z_{transformed},\Delta
x_{transformed},\Delta y_{transformed})$’s for $71$ realizations for each
$\Lambda$ (small open symbols in SFIG. 2), and thereafter we calculated the
mean over realizations (big closed symbols in SFIG. 2). While $\Delta
x_{transformed}$ and $\Delta y_{transformed}$ should be zero as per our
definitions, a non-zero value of $\Delta z_{transformed}$ would imply a drift.
However, we do not see any significant non-zero value of $\Delta
z_{transformed}$.

SFIG. 1:  Supplementary to Fig. 1(c–e) showing data for four $\Lambda$ values.
SFIG. 2:  We show $(\Delta z_{transformed},\Delta x_{transformed},\Delta
y_{transformed})$’s for $71$ realizations for each $\Lambda$ by small open
symbols. The mean $\pm$ std. dev. over realizations are shown by big closed
symbols. We do not see any significant non-zero value of $\Delta
z_{transformed}$ inferring no drift motion of the inclusion.

##  .5 Density of polymer beads $n_{B}$

The bulk region of the cavity has been segmented into $5^{3}$ cubic grids of
linear dimension $\ell$, and the number of the polymeric beads in individual
grids are counted for several hundreds of snapshots for individual
realization. First we calculate the mean density of the beads and the
corresponding fluctuation for individual realizations, and then taking
averages over $71$ realizations, we obtain $n_{B}$ (SFIG. 3) and $\sigma_{B}$
(FIG. 2b in the main text).

SFIG. 3:  Density of polymeric beads, $n_{B}$, increases with $\Lambda$. Mean
$\pm$ s.e.m. over realizations are shown (n=71).

##  .6 Supplemental Videos

Two supplemental videos are shown for $\Lambda=0$ (video 1) and $0.6188$
(video 2). In each of the frames, only a slice of breadth $1$ s.u. (around a
great circle $y^{\prime}=0$) of the whole simulation system is shown. Each
slice (each great circle) is chosen such that it contains the inclusion (blue,
$d_{I}=0.4$ s.u.) for that frame, and the real simulation coordinates
$(x,y,z)$ are transformed accordingly to $(x^{\prime},y^{\prime},z^{\prime})$.
The black square has linear dimension $12$ s.u.

SFIG. 4:  Non-monotonic behavior of $C_{\Delta{\bm{x}}_{I,\,\delta t}}(\Delta
t=\delta t)$ with $\delta t$ attaining its optimal about
$\mathcal{O}(0.001\,\tau)$.

##  .7 Coarse-grained effective model

The autocorrelation function of $\Delta{\bm{x}}_{I,\,\delta t}$ is defined as

$C_{\Delta{\bm{x}}_{I,\,\delta t}}\left(\Delta
t\right)=\frac{\langle\sum_{t_{0}}\left[\Delta{\bm{x}}_{I,\,\delta
t}\left(t_{0}\right)\cdot\Delta{\bm{x}}_{I,\,\delta t}\left(t_{0}+\Delta
t\right)\right]\rangle}{\langle\sum_{t_{0}}\lvert\Delta{\bm{x}}_{I,\,\delta
t}\left(t_{0}\right)\rvert^{2}\rangle}.$

We note a non-monotonic behavior of $C_{\Delta{\bm{x}}_{I,\,\delta t}}(\Delta
t=\delta t)$ with $\delta t$ (SFIG. 4).

The effective model has been defined in the main text by eq. (1) with the
characteristics of the noise $v_{EM}\,{\bm{\zeta}}_{EM}$ mentioned therein.
Since $\langle{\bm{\zeta}}_{EM}\rangle=0$, the first moment of displacement of
the inclusion, $\langle\Delta r_{I,\,EM}(\Delta t)\rangle$ over a time
duration $\Delta t$ is zero. The second moment of the displacement over the
duration $\Delta t$ can be written as

$\displaystyle\langle\Delta r_{I,\,EM}^{2}(\Delta t)\rangle$ $\displaystyle=$
$\displaystyle v_{EM}^{2}\int_{t}^{t+\Delta t}ds\int_{t}^{t+\Delta
t}ds^{\prime}\langle{\bm{\zeta}}_{EM}(s)\cdot{\bm{\zeta}}_{EM}(s^{\prime})\rangle$
(S10) $\displaystyle=$ $\displaystyle v_{EM}^{2}\int_{t}^{t+\Delta
t}ds\int_{t}^{t+\Delta t}ds^{\prime}[\,A\delta(\lvert
s-s^{\prime}\rvert)-a_{f}\exp(-\lvert
s-s^{\prime}\rvert/t_{f})-a_{s}\exp(-\lvert s-s^{\prime}\rvert/t_{s})\,].$

A simplification of the above expression leads us to eq. (2) in the main text.
Treating $A$ as a $\Lambda$-dependent fitting parameter, we find nice
agreement between $\langle\Delta r_{I}^{2}(\Delta t)\rangle$ and
$\langle\Delta r_{I,\,EM}^{2}(\Delta t)\rangle$ (SFIG. 5). The fitted and the
used parameters in the effective model are shown in SFIG. 6.

SFIG. 5: The MSD of the inclusion obtained from the simulation model
($\langle\Delta r_{I}^{2}\rangle$, symbols) show excellent match with that
obtained using the effective model ($\langle\Delta r_{I,\,EM}^{2}\rangle$,
dashed-lines). SFIG. 6: Parameters used in the effective model.

##  .8 Identification of the fast- and slow-modes

Let us consider a particle subjected to a harmonic potential performing
Brownian motion. Its dynamics is given by the following overdamped Langevin
equation—

$\displaystyle\frac{\partial{\bm{x}}_{p}}{\partial
t}=-\frac{1}{\gamma_{p}}k{\bm{x}_{p}}+\sqrt{2D_{p}}{\bm{\zeta}}.$ (S11)

${\bm{\zeta}}$ represents a Gaussian white noise with
$\langle{\bm{\zeta}}\rangle=0$ and
$\langle{\bm{\zeta}}(t_{1})\cdot{\bm{\zeta}}(t_{2})\rangle=\delta_{\alpha\beta}\delta(t_{1}-t_{2})$;
$\alpha,\,\beta\in$ Cartesian components. The diffusivity is set by the
friction coefficient $\gamma_{p}$ and the thermal energy $k_{B}T$:
$D_{p}=k_{B}T/\gamma_{p}$. The spring constant $k$ and friction coefficient
$\gamma_{p}$ set a characteristic timescale $t_{c}=\gamma_{p}/k$.

The solution to eq. S11 is given by

$\displaystyle{\bm{x}}_{p}(t)=\sqrt{2D_{p}}\int_{-\infty}^{t}ds\;\exp\left(-\frac{t-s}{t_{c}}\right)\;{\bm{\zeta}}(s).$
(S12)

It is straightforward to calculate the MSD of the particle—

$\displaystyle\text{MSD}_{p}(\Delta t)=2D_{p}t_{c}\left[1-\exp\left(-\Delta
t/t_{c}\right)\right].$ (S13)

This expression is analogous to the fast- and the slow-mode contributions to
the inclusion MSD, as discussed in the main text; see eq. (2).

SFIG. 7:  Extraction of the fast and the slow timescales from autocovariance
(ACV) of several quantities. (Top) ACV of the number of the polymeric beads
interacting with the inclusion are fitted with exponential functions at the
early lag regime. The corresponding characteristic timescales compare well
with the fast timescales ($t_{f}(\Lambda)$). ACV of the radius of gyration
(Bottom-left) and the shape anisotropy (Bottom-right) of polymeric mesh around
the inclusion are fitted with exponential functions at the moderate lag
regime. The corresponding characteristic timescales compare well with the slow
timescale ($t_{s}(\Lambda)$). Data shown for $d_{I}=0.40$ s.u., averaged over
six realizations.

To investigate the fast and the slow modes, we tracked the beads interacting
with the inclusion for all the timeframes in our simulation. Therefore, we
could count the number of the beads interacting with the inclusion and analyze
the size and the shape of the polymeric mesh around it. In SFIG. 7, we show
the autocovariance (ACV) of (Top) the number of beads interacting with the
inclusion, (Bottom-left) the radius of gyration (i.e., size) of the mesh, and
(Bottom-right) shape anisotropy of the mesh. The reason behind these choices
of the quantities is that any fluctuation in these quantities should induce
fluctuation in the interaction potential between the inclusion and the beads
surrounding it. We find good exponential fitting of the ACV of the number of
beads interacting with the inclusion for early-time regime, giving us
characteristic times for several $\Lambda$. Interestingly, the corresponding
timescales compare well with the fast timescales obtained previously.
Therefore, we conclude that the fast mode corresponds to the fluctuation in
the number of polymeric beads interacting with the inclusion. We check the MSD
of the polymeric beads and note that over this short timescale $\sim t_{f}$,
the individual beads show normal diffusion because they are yet to realize the
constraints due to their neighbors along the polymer chain (SFIG. 8). This
suggests that $t_{f}\sim\mathcal{O}(\text{fluctuation at individual bead
level})$. Thus, we argue that the fluctuation in the number of the beads
interacting with the inclusion at such a short timescale is introduced by
random inclusion or exclusion of individual bead from the the polymeric mesh
around the inclusion.

We construct gyration tensor for each timeframe using the coordinates of the
beads interacting with the inclusion (i.e., the constituents of the mesh) and,
by diagonalizing that tensor, we find the radius of gyration and the shape
anisotropy of the mesh. We show ACV of these quantities in SFIG. 7–Bottom and
fit those curves with exponential functions for the moderate time regime. We
could find considerably good fitting to those curves leading to characteristic
timescales that compare well with the slow timescales ($t_{s}(\Lambda)$).
Since, these quantities are related with the configuration of the mesh around
the inclusion, we argue that slow mode corresponds to the reconfiguration
aspect of the polymeric mesh.

SFIG. 8:  MSD of polymeric beads. Individual bead diffuses normally (dynamic
exponent $1$) in an early-time regime until it feels its neighbor along the
polymer chain. Beyond this, the bead needs to move along with its neighbor
which slows down the dynamics of the individual bead. The dynamic exponent in
this regime is known to be $0.5$.

##  .9 $\Lambda$-dependency of the diffusivities corresponding to the three
dynamical modes

SFIG. 9:  $\Lambda$-dependency of the diffusivities corresponding to the
normal (blue plus symbols), fast (brown triangles), and slow modes (red
circles).

In this section, we discuss how the three modes proposed in this article alter
the inclusion dynamics with TLPA. We show the parts of $\langle\Delta
r_{I,\,EM}^{2}\rangle$ derived from all the three modes in FIG. 3b–main and
the changes in the corresponding diffusivities with $\Lambda$ in SFIG. 9. It
is intuitive to understand that the fluctuation in the beads increases with
$\Lambda$ (also see SFIG. 8 to compare bead’s MSD for several $\Lambda$). At
short timescale $\sim t_{f}(\Lambda)$, this enhanced fluctuation should be
randomly reverting the inclusion’s motion within the mesh and thereby slowing
down the inclusion dynamics. This justifies the decrease in $D_{f}$ with
$\Lambda$. However, since the fast mode-part becomes irrelevant after a short
timescale $t_{f}$, we do not see any significant $\Lambda$-dependency of this
contribution to $\langle\Delta r_{I,\,EM}^{2}\rangle$. On the contrary to the
fast mode, we note a significant increase in $D_{s}$ with $\Lambda$ resulting
in significant speeding up of the inclusion dynamics. This infers that TLPA-
assisted remodeling of the chromatic neighborhood plays a major role in
enhancing the inclusion dynamics. Also, $D_{n}$ increases with $\Lambda$,
which is rather intuitive to understand, as the TLPA is expected to enhance
fluctuation in the chromatic medium which is encapsulated within $D_{n}$. Note
that this also enhances inclusion dynamics by a considerable extent.

SFIG. 10:  Effective model has been employed for the RAR and RNR models.

##  .10 Power of the proposed effective model

Here we illustrate the capacity of the proposed phenomenological effective
model in recognizing the dynamical contributions associated with the distinct
modes. To do so, we use the following two types of variants of our original
active model:

  * •

RAR model—the steric interaction between a pair of proximal beads are
subjected to the following sequence of Poisson transitions:
$\text{state}\left(h_{vex}>0\right)\xrightarrow{\lambda_{1}}\text{state}\left(h_{vex}<0\right)\xrightarrow{\lambda_{2}}\text{state}\left(h_{vex}>0\right)$.

  * •

RNR model—the steric interaction between a pair of proximal beads are
subjected to the following sequence of Poisson transitions:
$\text{state}\left(h_{vex}>0\right)\xrightarrow{\lambda_{1}}\text{state}\left(h_{vex}=0\right)\xrightarrow{\lambda_{2}}\text{state}\left(h_{vex}>0\right)$.

The activity parameters are defined for both these models as
$\Lambda=\lambda_{1}/\lambda_{2}$. Recall that in our original active model,
the enzymatic activity is realized by the following sequence of Poisson
transitions of the steric interaction between a pair of proximal beads:
$\text{state}\left(h_{vex}>0\right)\xrightarrow{\lambda_{ra}}\text{state}\left(h_{vex}<0\right)\xrightarrow{\lambda_{an}}\text{state}\left(h_{vex}=0\right)\xrightarrow{\lambda_{nr}}\text{state}\left(h_{vex}>0\right)$,
and the activity parameter is defined as
$\Lambda=\lambda_{ra}\left(1/\lambda_{an}+1/\lambda_{nr}\right)$. For the
above two variants of the original active model, we set
$1/\lambda_{2}=\left(1/\lambda_{an}+1/\lambda_{nr}\right)$ and tune
$\lambda_{1}$ to set the corresponding activity parameter. Rest of the details
of these variant models are kept exactly the same as in the original active
model.

We simulate inclusion dynamics ($d_{I}=0.4\,\ell$) for both the RAR and RNR
models with $\Lambda=0.3094$ and employ the phenomenological effective model
to analyze the simulation data in the same fashion as described in the main
text. We find that our effective model successfully reproduces the simulation
MSD data of the inclusion (SFIG. 10). We use the corresponding dynamical
parameters obtained by the effective model-scheme and determine the
contributions of the distinctive dynamical modes to the total MSD of the
inclusion (SFIG. 11a). We also calculate diffusivities corresponding to the
three dynamical modes and compare those with the results obtained from our
original model (SFIG. 11b). Thus, we realize that our proposed effective model
is significantly powerful in (i) representing the dynamics of the complex
systems like those presented here in terms of a comparatively easier model and
(ii) identifying the dynamical parameters associated with different physical
origins (viz., dynamics within a polymeric mesh, dynamics due to mesh
reconfiguration, and overall normal diffusive dynamics), which may differ from
each-other across different model-scenarios. We note that a coarse-grained
physical quantity like MFPT is unable to distinguish the system dynamics
between the cases of the original model with $\Lambda=0$ (MFPT = $64.1364\pm
5.2413$), RAR model (MFPT = $65.5864\pm 4.8428$), and RNR model (MFPT =
$63.6359\pm 4.9206$). This observation further decorates the utility of our
phenomenological effective model in differentiating apparently similar model-
dynamics in terms of their details. It will be a fascinating task to
analytically formulate FPT distribution using our effective model (which
represents a non-Markovian process) SMVoituriezNatChem2012 ;
SMVoituriezNature2016 ; SMTakahiroPRR2023 , which may explain the apparent
similarity of these different model-scenarios in terms of MFPT.

(a)
(b)

SFIG. 11:  Comparison between a variety of models in terms of their dynamical
modes.

##  .11 Inclusion dynamics for several inclusion sizes

We simulate the original APM for $d_{I}\in[0.2\,\ell,1.2\,\ell]$. The MSD
profiles for all the $d_{I}$’s investigated look similar to that we reported
in the main text for $d_{I}=0.4\,\ell$ (SFIG. 12a). However, we note that the
exponent of the subdiffusive regime approaches $\sim 0.5$ with increasing
$d_{I}$ (exponents are $\sim 0.85$ and $\sim 0.5$ for $d_{I}=0.2\,\ell$ and
$1.2\,\ell$, respectively). We further note that the MSD of the polymeric
beads in the corresponding time regime has an exponent $0.5$ (SFIG. 8). These
two observations mutually corroborates as the constrained-dynamics (i.e., the
subdiffusive behavior) of an inclusion bigger than the chromatic meshsize must
be dictated by the dynamics of the neighboring beads. Similar to what we
reported in the main text, the crossover between the early-time diffusion and
the intermediate subdiffusion is delayed by $\Lambda$ for all the $d_{I}$’s.

We also analyze the FPT statistics of the inclusion for all the $d_{I}$’s
mentioned above. We note that the effect of TLPA is insignificant for the
smallest $d_{I}$; however, it becomes significant for larger inclusions (SFIG.
12b). For the largest $d_{I}$ investigated, for some $\Lambda>0$, the MFPT is
reduced to almost the half its value without TLPA.

SFIG. 12:  (a) MSD of the inclusion of different sizes (indicated on the right
margin; typical meshsize $d_{mesh}\simeq
2\,max\left(s_{min}|P(s_{min})>0\right)-d_{B}\simeq 0.37\,\ell$ (FIG. 2c)).
The downward shift of the curves with increasing $d_{I}$ is due to the
corresponding mobility parameter set by Stokes’ law. (b) Ratio of the MFPT for
$\Lambda=0.3094$ to that for $\Lambda=0$ are shown for several inclusion
sizes. The activity-dependent remodeling of the chromatic environment becomes
more significant for bigger inclusion. $n=91,\,71,\,36,\,71,\,20$ realizations
are used for $d_{I}=0.2,\,0.4,\,0.6,\,0.8,\,1.2\,\ell$, respectively. The
error bars are obtained by propagating s.e.m. of the numerators and the
denominators.

## References

  * (1) R. Das, T. Sakaue, G. V. Shivashankar, J. Prost, and T. Hiraiwa, How enzymatic activity is involved in chromatin organization, eLife 11, e79901 (2022).
  * (2) A. Jusufi, J. Dzubiella, C. N. Likos, C. von Ferber, and H. Löwen, Effective interactions between star polymers and colloidal particles, J. Phys.: Condens. Matter 13, 6177 (2001).
  * (3) M. Camargo and C. N. Likos, Phase separation in star-linear polymer mixtures, J. Chem. Phys. 130, 204904 (2009).
  * (4) T. Guérin, O. Bénichou, and R. Voituriez, Non-Markovian polymer reaction kinetics, Nat. Chem. 4, 568 (2012).
  * (5) T. Guérin, N. Levernier, O. Bénichou, and R. Voituriez, Mean first-passage times of non-Markovian random walkers in confinement, Nature 534, 356 (2016).
  * (6) Y. Sakamoto and T. Sakaue, First passage time statistics of non-Markovian random walker: Dynamical response approach, accepted in Phys. Rev. Res. (2023).