# The effect of loops on the mean square displacement of Rouse-model chromatin

Tianyu Yuan  Integrated Graduate Program in Physical and Engineering Biology,
Yale University, New Haven, Connecticut 06520, USA  Department of Physics,
Yale University, New Haven, Connecticut 06520, USA  Hao Yan  Integrated
Graduate Program in Physical and Engineering Biology, Yale University, New
Haven, Connecticut 06520, USA  Department of Physics, Yale University, New
Haven, Connecticut 06520, USA  Mary Lou P. Bailey  Integrated Graduate Program
in Physical and Engineering Biology, Yale University, New Haven, Connecticut
06520, USA  Department of Applied Physics, Yale University, New Haven,
Connecticut 06520, USA  Jessica F. Williams  Department of Cell Biology, Yale
School of Medicine, New Haven, Connecticut 06520, USA  Ivan Surovtsev
Department of Physics, Yale University, New Haven, Connecticut 06520, USA
Department of Cell Biology, Yale School of Medicine, New Haven, Connecticut
06520, USA  Megan C. King  Integrated Graduate Program in Physical and
Engineering Biology, Yale University, New Haven, Connecticut 06520, USA
Department of Cell Biology, Yale School of Medicine, New Haven, Connecticut
06520, USA  Department of Molecular, Cell and Developmental Biology, Yale
University, New Haven, Connecticut 06511, USA  Simon G. J. Mochrie  Integrated
Graduate Program in Physical and Engineering Biology, Yale University, New
Haven, Connecticut 06520, USA  Department of Physics, Yale University, New
Haven, Connecticut 06520, USA  Department of Applied Physics, Yale University,
New Haven, Connecticut 06520, USA<EMAIL_ADDRESS>
(April 21, 2023)

###### Abstract

Chromatin exhibits polymer properties and its dynamics are now commonly
described using the classical Rouse model. The subsequent discovery, however,
of intermediate scale chromatin organization known as topologically
associating domains (TADs) in experimental Hi-C contact maps for chromosomes
across the tree of life, together with the proposed loop extrusion factor
(LEF) model that aims to explain TAD formation, motivates efforts to
understand the effect of loops and loop extrusion on chromatin dynamics. This
paper seeks to fulfill this need by combining LEF-model simulations with
extended Rouse-model polymer simulations to investigate the dynamics of
chromatin with loops and dynamic loop extrusion. Specifically, we extend the
classical Rouse model by modifying the polymer’s dynamical matrix to
incorporate extra springs that represent loop bases. We also theoretically
generalize the friction coefficient matrix so that the Rouse beads with non-
uniform friction coefficients are compatible with our Rouse model simulation
method. We show that loops significantly suppress the averaged mean square
displacement (MSD) of a gene locus, consistent with recent experiments that
track fluorescently-labelled chromatin loci. We also find that loops reduce
the MSD’s stretching exponent from the classical Rouse-model value of $1/2$ to
a loop-density-dependent value in the 0.45-0.40 range. Remarkably, stretching
exponent values in this range have also been observed in recent experiments
[S. C. Weber, A. J. Spakowitz, and J. A. Theriot, Phys. Rev. Lett. 104, 238102
(2010) and M. L. P. Bailey, I. Surovtsev, J. F. Williams, H. Yan, T. Yuan, S.
G. Mochrie, and M. C. King, Mol. Biol. Cell (in press)]. We also show that for
a wide range of plausible loop-extrusion parameters, the dynamics of loop
extrusion itself negligibly affects chromatin mobility. By studying a static
“rosette” loop configuration, we also demonstrate that chromatin MSDs and
stretching exponents depend on the location of the locus in question relative
to the position of the loops. Finally, we show that non-uniformity in friction
coefficients for different parts of a chromatin polymer does not significantly
modify its averaged dynamics. However, unsurprisingly, individual loci with
distinctly high friction coefficients exhibit correspondingly reduced MSDs.

††preprint: APS/123-QED

##  I Introduction

The classical Rouse model for the dynamics of a polymer in a viscous fluid
depicts the polymer as an array of overdamped beads, connected together by
nearest-neighbor springs [1]. Each bead actually represents a sub-polymer,
whose end-to-end distance follows a Gaussian distribution, which determines
the Rouse model spring stiffness. Despite its simplicity, in cases where
interaction between different polymers restores $N^{\frac{1}{2}}$-scaling of
the polymer end-to-end distance [2] ($N$ is the number of monomers in the
polymer in question), and where hydrodynamic interactions between different
submolecules are screened by other polymers [3], and where the polymers are
shorter than their entanglement length [4], the Rouse model’s predictions
match experimental measurements on many polymer melts [5, 6, 7] and solutions
[8, 9, 10].

Beyond synthetic polymers, the possibility that the Rouse model might also
provide an appropriate description of the dynamics of chromatin in living
cells has emerged from experiments which quantitatively characterize the
motion of a fluorescently-labelled gene locus via its mean-square displacement
(MSD) [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. In many of
these experiments, the experimentally measured mean-square displacement of a
labelled gene locus behaves similarly to the Rouse-model prediction that the
MSD initially increases as $t^{\frac{1}{2}}$ with increasing time, $t$.
However, chromatin, which is comprised of DNA and its myriad associated
proteins, is now understood to possess a more elaborate spatial organization
than the simple random-walk polymer envisioned in the classical Rouse model.
It is unclear whether the classical Rouse model predictions should be expected
to apply to such a polymer.

Until recently, chromatin organization was well established only at the two
extremes of the genome scale - with DNA wrapped around histones in a
nucleosome at the molecular scale (hundreds of DNA base pairs) [25, 26, 27],
and with each chromosome largely occupying its own space in the nucleus in
chromosome territories (millions of base pairs) [28, 29, 30]. However, the
advent of chromatin configuration capture (Hi-C) methods has now unveiled an
inhomogeneous, hierarchical, domains-within-domains, organization at
intermediate scales ($10^{4}-10^{6}$ bps) [31, 32, 33, 34, 35, 36, 37, 38, 39,
40]. Gene loci within the same domain (termed topologically associated domains
or TADs) have a much higher probability to come into contact, even if they are
genomically distant, than do loci from different TADs [41]. High contact
probability between two distant loci supports the idea that the two loci in
question are likely to be at the base of a chromatin loop, consistent with the
long-standing hypothesis that loops are a fundamental organizing principle for
chromatin [42, 43, 44].

The loop extrusion factor (LEF) model has emerged as the preferred candidate
mechanism for TAD formation [45, 46, 47, 48, 49, 50, 51, 52, 53]. In this
model, LEFs bind to chromatin and then initiate dynamic loop extrusion, until
they either stall when they encounter another LEF or at specific boundary
elements (generally identified as CCCTC-binding factor or CTCF binding sites),
which mark TAD boundaries, or until they unbind. Thus, a population of LEFs
establishes a dynamic steady-state, which largely recapitulates contact
probabilities determined by experimental Hi-C maps [54]. On the experimental
side, the structural maintenance of chromosomes (SMC) complexes, cohesin and
condensin, have been identified as possible LEFs, with TADs disappearing from
Hi-C maps in the absence of cohesin [55, 56]. At larger, sub-chromosomal
scales, phase separation of different regions has been proposed to further
organize chromatin into chromatin “compartments” [30, 57, 58, 59, 60].

Although a number of modifications to the classical Rouse model have been
proposed to better describe chromatin dynamics [61, 23, 24], to-date how
polymer loops might affect the predictions of the Rouse model have not been
described, to our knowledge. In this paper, motivated by the LEF model, we
consider a modified version of the Rouse model that includes loops in order to
investigate how loops may affect chromatin MSDs. Specifically, we augment the
classical nearest-neighbor Rouse model by adding an additional spring (of the
same spring constant) between the pair of monomers at the base of each loop.
The number, sizes, and locations of these additional springs evolve according
to LEF model simulations. Because the additional springs lead to far-from-
diagonal terms in the Rouse-model dynamical matrix, our modified Rouse model
is no longer analytically tractable. Nevertheless, it is straightforward to
simulate in an exact manner as follows. First, we diagonalize the modified
dynamical matrix numerically, based on the current loop configuration, and
find its eigenvalues and eigenvectors, which define the coordinate
transformation to normal coordinates. The time evolution of each independent
normal coordinate is then simulated, using a version of the method described
in Ref. [62], assuming equipartition with an effective temperature. Next, the
bead positions versus time are recovered by the inverse coordinate
transformation. The above steps are repeated for each different loop
configuration to obtain the time series of bead positions. Finally, the MSD of
each bead, representing the MSD of the corresponding gene locus, is determined
from the time series of bead positions.

This method applied to the classical Rouse model reproduces the well-known
analytical results for the behavior of the MSD: At early times, the 1-D MSD
varies as $2Dt^{\alpha}$ with the stretching exponent $\alpha$, taking on the
predicted value of $\alpha=1/2$ and the amplitude, $D$, taking on the
predicted value of $D=k_{B}T/\sqrt{\pi\kappa\zeta}$, where $\kappa$ is the
effective spring constant between subpolymers and $\zeta$ is the friction
coefficient for each subpolymer; at long times the MSD achieves a (boundary-
condition dependent) limiting value. By contrast, for the Rouse model with
loops, the MSD is significantly reduced. In addition, it shows a noticeably
reduced and subtly time-dependent stretching exponent with values that clearly
fall below one-half. Interestingly, these results are reminiscent of recent
experimental results, that examine the dynamics of gene loci in living cells
[13, 14, 15, 63].

Beyond the LEF model, our approach is applicable to any polymer configuration
involving loops, by appropriately picking the locations of the additional
springs in the Rouse-model dynamical matrix. Another loop configuration that
has attracted attention is a “rosette”, in which similarly-sized loops emanate
from an organizing center. For example, rosettes are believed to be relevant
to the E. coli nucleoid [64, 65]. Examining a fixed-loop-configuration rosette
facilitates investigation of the MSDs for several distinct genomic locations,
relative to the loops. Thus, we find that the tip of a loop exhibits the
largest mobility, at early times appearing unconstrained by loops, while the
mobility at the base of a loop is constrained the most. The mobility of loci
on the polymer backbone, between loops, resembles that of the tips at early
times but approaches the mobility of the loop bases at longer times.

Our approach also allows us to assign different friction coefficients to
individual beads, granting us the ability to investigate polymer dynamics in
environments with inhomogeneous viscosity. Extending the Rouse model to
embrace non-uniform friction environments is motivated by the observation that
chromatin itself is locally heterogeneous. We assign each bead a fixed
friction coefficient drawn from a log-normal distribution with mean $\mu_{0}$
and standard deviation of $\sigma_{0}=\mu_{0}$ and calculate the MSDs and
stretching exponents for each bead. The averaged MSD and stretching exponent
over 60 uniformly-spaced beads are calculated as well. We find that although
the MSD and stretching exponent for each individual bead vary in a wide
interval from bead to bead, the 60-bead-averaged MSD and stretching exponent
shows insignificant deviation from the averaged MSD and stretching exponent
for the polymer with uniform friction coefficient of $\mu_{0}$. So-called
replication factories may provide another example, where inhomogeneous
friction is important. In a replication factory, chromatin is hypothesized to
be folded into a rosette-like structure, at whose center transcription factors
and SMCs form a phase-separated droplet and assist the simultaneous
replication of the rosette loops [66, 67, 68, 69, 70, 71, 72, 73]. Plausibly,
the phase separation gives rise to a locally-high viscosity. For this
scenario, unsurprisingly, we find that the mobility of chromatin loci within
the high-friction central cluster is significantly reduced compared to the
case when the friction is the same for all beads, suggesting a potential role
for locus dynamics in elucidating phase separation within the nucleus [74, 75,
76].

This paper is organized as follows. In Sec. II.1, we examine theoretically the
dynamics of $N$-coupled beads, subject to a random force, for general,
symmetric, dynamical and friction coefficient matrices in terms of the
appropriate normal coordinates. In Sec. II.2, we use the equipartition theorem
of statistical mechanics to express the mean-square amplitude of each normal
coordinate in terms of an effective temperature and an eigenvalue of the
dynamical matrix. Sec. II.3 applies the results of Sec. II.1 and II.2 to the
classical Rouse model with free ends (free boundary conditions), reproducing
the well-known analytic results for the Rouse-model MSD. Sec. II.5 reviews the
scaling of $\kappa$, $\zeta$, and $D$ with the size of the subpolymers and
explains how experimental measurements of $D$ and literature estimates of the
chromatin persistance length and of the number of DNA base pairs per unit
chromatin contour length allow us to estimate appropriate values of $\kappa$
and $\zeta$ for our polymer simulations. Sec. III.1 describes two slightly
different versions of the LEF model, which we call the random loop model and
the CTCF model, respectively, and which are the basis of our loop extrusion
factor (LEF) simulations. Sec. III.2 explains how we carry out Rouse model
simulations, incorporating loops and loop extrusion. In Sec. IV, we present
our results. Sec. IV.1 examines the simulated dynamics of the chromatin
polymer with dynamic loops, driven by the LEF model. Comparison between
polymers with and without loops reveals that loops significantly reduce
chromatin mobility, as measured by the mean-square displacement (MSD). Loops
also reduce the stretching exponent $\alpha$ from a plateau value of 0.5
without loops to a plateau value lying between about $0.45$ and $0.4$,
depending on the density of loops. By examining the behavior of polymers with
loops but without loop extrusion, i.e. polymers with static loops, Sec. IV.1
also reveals that the dynamics of loop extrusion itself has a minimal effect
on the dynamics of the chromatin polymer, as measured by both the MSD and
stretching exponent, suggesting that although LEFs actively extrude loops,
their dynamics do not dictate how chromatin polymers move. Sec. IV.2 examines
the simulated dynamics of static loops in a rosette, revealing that mobility
is reduced near the base of a loop in comparison to the mobility at the tip of
a loop. In Sec. IV.3, we examine Rouse-model polymers without loops, when the
beads’ friction coefficients are randomly distributed according to a log-
normal distribution. We calculate the MSDs and stretching exponents for
individual beads as well as the averaged MSD and stretching exponent for 60 of
these beads, uniformly distributed along the polymer. Comparing these averaged
results to those for a polymer with uniform friction, we find that there are
insignificant differences for both MSD and stretching exponent between the
non-uniform-friction and the uniform-friction case. Sec. IV.4 shows that
polymers with rosette structures subject to locally high friction at the
central cluster, mimicing replication factories, exhibit significantly reduced
MSDs in the high-friction region (rosette bases) but exhibit similar MSDs at
the tips of loops, where the friction environment is unchanged. Finally, we
conclude in Sec. V.

##  II Theoretical background

###  II.1 Dynamics of $N$-coupled beads

The Rouse model describes a polymer as $N$ subpolymers connected together into
a chain. Each subpolymer is conceived to be sufficiently long that its end-to-
end distance is a Gaussian random variable. Consequently, each subpolymer acts
as a Hooke’s-law spring with a spring constant equal to
$\kappa=dk_{B}T/\langle R_{EE}^{2}\rangle$, where $k_{B}$ is the Boltzmann
constant, $T$ is the effective temperature, $d$ is the number of spatial
dimensions, and $\langle R_{EE}^{2}\rangle$ is the mean square end-to-end
distance of a subpolymer [77]. It is often convenient to conceive the center
of mass of each subpolymer as a bead. This picture leads to a chain comprised
of $N$ beads, each with friction coefficient, $\zeta$, and connected to its
neighbors via springs of spring constant, $\kappa$.

The equations of motion for $N$ beads, coupled together by springs, in the
low-Reynolds-number, overdamped regime, may be expressed as the matrix
equation

$Z\dot{X}=-KX+F,$ (1)

where $X$ is the vector of bead coordinates, $Z$ is the friction matrix, $K$
is the dynamical matrix, and $F$ is a vector of stochastic forces. To ensure
that our discussion is applicable to a Rouse polymer with loops, we envision
$K$ and $Z$ to be as general as possible. Nevertheless, Newton’s Third Law
requires that $K$ and $Z$ are both symmetric; in addition, $Z$ must be
positive definite in order to ensure dissipation, and $K$ must be positive
semi-definite to ensure that the minimum potential energy corresponds to zero
displacements relative to the center of mass. It is convenient to introduce
dimensionless versions of $Z$ and $K$, namely $J$ and $A$, respectively, via
$Z=\zeta J$ and $K=\kappa A$. In terms of $J$ and $A$, Eq. 1 reads

$\frac{dX}{dt}=-\frac{\kappa}{\zeta}J^{-1}AX+\frac{1}{\zeta}J^{-1}F.$ (2)

To proceed, we diagonalize Eq. 2 (assuming that $J^{-1}A$ is diagonalizable).
Supposing that $S$ is the matrix that diagonalizes $J^{-1}A$, i.e.
$S^{-1}J^{-1}AS=D_{1}$, where $D_{1}$ is a diagonal matrix, then Eq. (2)
becomes

$\frac{dX}{dt}=-\frac{\kappa}{\zeta}SD_{1}S^{-1}X+\frac{1}{\zeta}J^{-1}F.$ (3)

Introducing the normal coordinates, $\mathcal{X}=S^{-1}X$, we obtain

$\frac{d\mathcal{X}}{dt}=-\frac{\kappa}{\zeta}D_{1}\mathcal{X}+\frac{1}{\zeta}\widetilde{F},$
(4)

where $\widetilde{F}=S^{-1}J^{-1}F$. Thus, $N$ coupled equations of motion
decouple into $N$ independent equations of normal coordinates,
$\mathcal{X}_{m}$, which satisfies, for each $m$,

$\frac{d\mathcal{X}_{m}}{dt}=-\frac{\kappa}{\zeta}\Lambda_{m}\mathcal{X}_{m}+\frac{1}{\zeta}\widetilde{F}_{m},$
(5)

where $\Lambda_{m}$ is the $m$-th eigenvalue of $J^{-1}A$. Eq. 5 leads to the
following results:

$\left<\mathcal{X}_{m}(t\to\infty)\right>=\left<\mathcal{X}_{m}\right>=0,$ (6)
$\left<\mathcal{X}^{2}_{m}(t\to\infty)\right>=\sigma^{2}_{\mathcal{X}_{m}},$
(7)

and

$\left<\mathcal{X}_{m}(t+s)\mathcal{X}_{m}(s)\right>_{s\to\infty}=\sigma^{2}_{\mathcal{X}_{m}}e^{-\frac{\kappa}{\zeta}\Lambda_{m}t},$
(8)

where $\left<....\right>$ denotes an ensemble average, and
$\sigma^{2}_{\mathcal{X}_{m}}$ and $\langle\mathcal{X}_{m}\rangle$ are the
variance and mean of eigenmode $m$ at steady state [62].

###  II.2 Equipartition and theoretical MSD for $N$-coupled beads

To calculate $\sigma^{2}_{\mathcal{X}_{m}}$, we turn to statistical mechanics,
assuming an effective temperature, $T$. To this end, first, we express the
$N$-bead potential energy, $U$, in terms of the normal coordinates:

$\displaystyle U$ $\displaystyle=\frac{1}{2}X^{T}KX$ (9)
$\displaystyle=\frac{\kappa}{2}\mathcal{X}^{T}S^{T}AS\mathcal{X}$
$\displaystyle=\frac{\kappa}{2}\mathcal{X}^{T}(S^{T}AS)\mathcal{X}.$

Evidently, the $N$-bead potential energy is a quadratic form of the normal
coordinates. Because $S$ was chosen to diagonalize $J^{-1}A$, not $A$, it is
not obvious from Eq. 9 that the potential energy is a function of
${\mathcal{X}_{m}}^{2}$-terms only. However, if we can prove $S^{T}AS$ is
diagonal, then the potential energy is guaranteed to decouple in the normal
coordinates $\mathcal{X}$, and we can straightforwardly calculate
$\sigma^{2}_{\mathcal{X}_{m}}$, using the equipartition theorem. Because $A$
is positive semi-definite and symmetric, we may write

$S^{T}AS=(A^{\frac{1}{2}}S)^{T}A^{\frac{1}{2}}S.$ (10)

Eq. 10 informs us that for $S^{T}AS$ to be diagonal requires that
$\\{A^{\frac{1}{2}}v_{m}\\}$ are orthogonal vectors, where $\\{v_{m}\\}$ are
the eigenvectors of $J^{-1}A$ and constitute the columns of $S$. To show that
$\\{A^{\frac{1}{2}}v_{m}\\}$ are orthogonal, we start with the eigenvalue
equation:

$J^{-1}Av_{m}=\Lambda_{m}v_{m}.$ (11)

Multiplying both sides of Eq. 11 by $A^{\frac{1}{2}}$, we find

$A^{\frac{1}{2}}J^{-1}A^{\frac{1}{2}}(A^{\frac{1}{2}}v_{m})=\Lambda_{m}(A^{\frac{1}{2}}v_{m}).$
(12)

Thus, we see that $A^{\frac{1}{2}}v_{m}$ is an eigenvector of the matrix,
$A^{\frac{1}{2}}J^{-1}A^{\frac{1}{2}}$, with eigenvalue, $\Lambda_{m}$. The
matrix, $A^{\frac{1}{2}}J^{-1}A^{\frac{1}{2}}$, is symmetric because $A$ and
$J$ are symmetric and raising a matrix to a power commutes with transposing
the matrix. Because a symmetric matrix is guaranteed to possess orthogonal
eigenvectors, it follows that $\\{A^{\frac{1}{2}}v_{m}\\}$ are indeed
orthogonal. (In the case of two equal eigenvalues, it is always possible to
pick orthogonal eigenvectors in that eigen-subspace.) Defining the diagonal
matrix $D_{2}=S^{T}AS$, we thus have

$U=\frac{\kappa}{2}\mathcal{X}^{T}D_{2}\mathcal{X}=\frac{\kappa}{2}\sum_{m}{\lambda}_{m}\mathcal{X}_{m}^{2}.$
(13)

where $\lambda_{m}$ is the $m$-th eigenvalue (diagonal entry) of
$D_{2}=S^{T}AS$. Thus, for non-zero $\lambda_{m}$’s, the equipartition theorem
gives

$\langle\mathcal{X}_{m}^{2}\rangle=\sigma^{2}_{\mathcal{X}_{m}}=\frac{k_{B}T}{\kappa\lambda_{m}},$
(14)

and Eq. 7 and 8 become

$\left<\mathcal{X}_{m}^{2}(t)\right>=\frac{k_{B}T}{\kappa\lambda_{m}},$ (15)

and

$\left<\mathcal{X}_{m}(t)\mathcal{X}_{m}(0)\right>=\frac{k_{B}T}{\kappa\lambda_{m}}e^{-\frac{\kappa}{\zeta}\Lambda_{m}t},$
(16)

respectively. For $\lambda_{m}=0$, we set the corresponding normal coordinate,
$\mathcal{X}_{m}$, which is proportional to the polymer’s center of mass, to
zero, which eliminates the overall diffusion, mimicking chromatin’s
confinement within the nucleus.

$X(t)$ may be expressed in terms of $\mathcal{X}(t)$:

$X_{n}(t)=\sum_{m=1}^{N}S_{nm}\mathcal{X}_{m}(t).$ (17)

Using the fact that different normal coordinates are uncorrelated, it
immediately follows that

$\displaystyle\left<X_{n}^{2}(t)\right>$
$\displaystyle=\sum_{m=1}^{N}\sum_{p=1}^{N}S_{nm}S_{np}\left<\mathcal{X}_{m}(t)\mathcal{X}_{p}(t)\right>$
$\displaystyle=\sum_{m=1}^{N}\ S^{2}_{nm}\frac{k_{B}T}{\kappa\lambda_{m}},$
(18)

and

$\displaystyle\left<X_{n}(t)X_{m}(0)\right>$
$\displaystyle=\sum_{m=1}^{N}\sum_{p=1}^{N}S_{nm}S_{np}\left<\mathcal{X}_{m}(t)\mathcal{X}_{p}(0)\right>$
$\displaystyle=\sum_{m=1}^{N}S^{2}_{nm}\left<\mathcal{X}_{m}(t)\mathcal{X}_{m}(0)\right>$
$\displaystyle=\sum_{m=1}^{N}S^{2}_{nm}\frac{k_{B}T}{\kappa\lambda_{m}}e^{-\frac{\kappa}{\zeta}\Lambda_{m}t}.$
(19)

The MSD of bead $n$ is, therefore,

$\displaystyle\left<[X_{n}(t)-X_{n}(0)]^{2}\right>$
$\displaystyle=2\left<X^{2}_{n}(0)\right>-2\left<X_{n}(t)X_{n}(0)\right>$
$\displaystyle=\frac{2k_{B}T}{\kappa}\sum_{m=1}^{N}\
\frac{S^{2}_{nm}}{\lambda_{m}}(1-e^{-\frac{\kappa}{\zeta}\Lambda_{m}t}).$ (20)

These results are applicable in general when $K$ and $Z$ (or $A$ and $J$) are
both symmetric, $K$ (or $A$) is positive semi-definite and $Z$ (or $J$) is
positive-definite. We also require that $J^{-1}A$ is diagonalizable. An
important special case is when $J$ is the identity matrix. In this instance,
since $D_{2}=S^{T}AS=S^{T}SS^{-1}J^{-1}AS=S^{T}SD_{1}=D_{1}$, $\lambda_{m}$
reduces to $\Lambda_{m}$.

###  II.3 Classical Rouse model

For the classical nearest-neighbor Rouse model of $N$-beads with free ends,
the dynamical matrix, $K$, is given by,

$K=\kappa A=\kappa\begin{bmatrix}1&-1&0&\cdots&\cdots&0\\\
-1&2&-1&\cdots&\cdots&0\\\ 0&-1&2&\cdots&\cdots&0\\\
\vdots&\vdots&\vdots&\ddots&\ddots&\vdots\\\
\vdots&\vdots&\vdots&\ddots&2&-1\\\ 0&0&0&\cdots&-1&1\end{bmatrix},$ (21)

and the friction matrix, $Z$, is diagonal,

$Z=\zeta J=\zeta\begin{bmatrix}1&0&0&\cdots&\cdots&0\\\
0&1&0&\cdots&\cdots&0\\\ 0&0&1&\cdots&\cdots&0\\\
\vdots&\vdots&\vdots&\ddots&\ddots&\vdots\\\
\vdots&\vdots&\vdots&\ddots&1&0\\\ 0&0&0&\cdots&0&1\end{bmatrix}.$ (22)

In this case, $K$, $A$, and $J^{-1}A$ are diagonalized by the orthogonal
matrix, $S$, given by:

$S=\sqrt{\frac{2}{N}}\begin{bmatrix}1/\sqrt{2}&\cos\frac{1\pi/2}{N}&\cos\frac{1\pi}{N}&\cdots\\\
1/\sqrt{2}&\cos\frac{3\pi/2}{N}&\cos\frac{3\pi}{N}&\cdots\\\
1/\sqrt{2}&\cos\frac{5\pi/2}{N}&\cos\frac{5\pi}{N}&\cdots\\\
\vdots&\vdots&\vdots&\vdots\\\ \vdots&\vdots&\vdots&\vdots\\\ \end{bmatrix},$
(23)

i.e.

$S_{nm}=\begin{cases}\sqrt{\frac{2}{N}}\cos\left[\frac{\pi\left(n+\frac{1}{2}\right)m}{N}\right]\quad&m\neq
0\\\ \sqrt{\frac{1}{N}}\quad&m=0,\end{cases}$ (24)

given the matrix index range as $\\{n,m\\}\in\\{0,1,2,\dots,N-1\\}$, and $N$
as the number of beads [78, 79]. Matrix $S$ diagonalizes $A$, i.e.
$SAS=D_{1}$. The resultant diagonal matrix $D_{1}$ contains the eigenvalues of
$A$ on its diagonal:

$D_{1}=\begin{bmatrix}0&0&0&\cdots\\\ 0&2-2\cos\frac{\pi}{N}&0&\cdots\\\
0&0&2-2\cos\frac{2\pi}{N}&\cdots\\\ \vdots&\vdots&\vdots&\ddots\\\
\end{bmatrix},$ (25)

i.e.

$\Lambda_{m}=2-2\cos\frac{m\pi}{N}.$ (26)

Using these results (and setting the normal coordinate with zero eigenvalue to
zero), Eq. 18 and Eq. 20 become

$\displaystyle\langle X_{n}^{2}(0)\rangle$ $\displaystyle=\langle
X_{n}^{2}(t)\rangle$ $\displaystyle=\frac{k_{B}T}{\kappa
N}\sum_{m=1}^{N-1}\cos^{2}\left[\frac{\pi(n+\frac{1}{2})m}{N}\right]\frac{1}{\left[1-\cos\left(\frac{m\pi}{N}\right)\right]}$
$\displaystyle=\frac{k_{B}T}{\kappa
N}\left[\left(n-\frac{1}{2}(N-1)\right)^{2}+\frac{1}{12}(N^{2}-1)\right],$
(27)

and

$\displaystyle\langle$
$\displaystyle\left[X_{n}(t)-X_{n}(0)\right]^{2}\rangle=\text{MSD}_{n}(t)=$
$\displaystyle\frac{2k_{B}T}{\kappa
N}\sum_{m=1}^{N-1}\cos^{2}\left[\frac{\pi(n+\frac{1}{2})m}{N}\right]\frac{1-e^{-\frac{2\kappa}{\zeta}\left(1-\cos(\frac{m\pi}{N})\right)\left\lvert
t\right\rvert}}{1-\cos\left(\frac{m\pi}{N}\right)},$ (28)

respectively. The index of the first bead starts from $n=0$. At early times,
Eq. 28 may be shown to be well-described by

$\text{MSD}_{n}(t)=\left<[X_{n}(t)-X_{n}(0)]^{2}\right>=2Dt^{\frac{1}{2}},$
(29)

with

$D=\frac{k_{B}T}{\sqrt{\pi\zeta\kappa}},$ (30)

where $D$ plays a role similar to a diffusion coefficient [80].
Conventionally, one defines the polymer time-scale to be the quantity,
$\tau_{p}=\zeta/(4\kappa)$, which is proportional to the inverse of the
largest eigenvalue and proportional to the shortest relaxation time. The one-
dimensional MSD thus can also be expressed as

$\text{MSD}_{n}(t)=\left<[X_{n}(t)-X_{n}(0)]^{2}\right>=\frac{k_{B}T}{\sqrt{\pi}\kappa}\sqrt{\frac{t}{\tau_{p}}}.$
(31)

The form of Eq. 28 depends on the boundary conditions (see Eq. 56 and Eq. 63
for periodic and fixed boundary conditions). By contrast, Eq. 31 turns out to
be independent of boundary conditions.

###  II.4 Rouse model with loops

To incorporate chromatin loops into our Rouse simulation, we model each loop
base as an additional spring that connects the two loci to which it is bound.
For simplicity, we choose the additional springs to have spring constants
equal to those of the usual nearest-neighbor springs. We then modify the
dynamical matrix, $K$, accordingly, as illustrated in Fig. 1. Because $K$
remains symmetric upon adding loops, the theory in Sec. II.1 remains
applicable.

Figure 1:  Simplified illustration of a (free-boundary-condition) Rouse
polymer with LEFs and its corresponding dynamical matrix $A$. The two LEFs are
modeled as springs and color-coded as red and green, and their effects on the
dynamical matrix are color-coded accordingly. The subpolymers are numbered
from left to right.

###  II.5 Parameter scaling

There is considerable laxity in how to pick the subpolymers (and, therefore,
$N$). However, the Rouse model parameters are constrained by the following
scaling: If the size of the subpolymer is chosen a factor $f$ larger, then the
number of subpolymers becomes $\frac{N}{f}$, the spring constant of the new
subpolymer becomes $\frac{\kappa}{f}$, the friction coefficient of the new
subpolymer becomes $f\zeta$, and the characteristic polymer time of the Rouse
polymer becomes $f^{2}\tau_{p}$ [80]. Under this scaling $D$ is invariant (see
Eq. 30), as must be the case for a measurable quantity.

We chose to model 6 Mb regions of the mouse genome using $N=600$ beads,
corresponding to subpolymers of size $N_{1}=10$ kb. The spring constant
between these beads may be expressed as

$\kappa=d\frac{k_{B}T}{(N_{1}/C)l_{k}},$ (32)

where $C$ is the number of base pairs per unit chromatin contour length (so
that $\frac{N_{1}}{C}$ is the contour length of a subpolymer) and $l_{k}$ is
the chromatin Kuhn length. The chromatin is considered as a freely-jointed
chain of Kuhn segments of length, $l_{k}$. Combining Eq. 30 and Eq. 32 yields

$\zeta=\frac{\frac{N_{1}}{C}l_{k}k_{B}T}{d\pi D^{2}}.$ (33)

Ref. [81] estimates $C=50$ bp nm-1 and $l_{k}=138$ nm, while $D$ is measured
to be roughly $D\simeq 4.1\times 10^{-3}~{}\mu$m2 s${}^{-\frac{1}{2}}$ for
$d=2$ [63]. These results lead to the following numerical estimates for a
chromatin Rouse polymer built from subpolymers comprising 10 kb of DNA:
$\frac{\kappa}{k_{B}T}\simeq 7.25\times 10^{-5}$ nm-2,
$\frac{\zeta}{k_{B}T}\simeq 2.61\times 10^{-4}$ nm-2 s, and
$\tau_{p}=\zeta/(4\kappa)=0.9$ s. We used these numerical values in the
simulations shown below in Sec. IV.

##  III Methods

###  III.1 LEF model simulations

To generate dynamic loop configurations, we carried out simulations of the
loop extrusion factor (LEF) model, closely following Refs. [45, 46, 47, 48,
49], where the LEF model is introduced and described in detail. In our
implementation, the chromatin polymer is conceived as a linear array of LEF
binding sites. Each LEF possesses two legs, and binds at random, initially
occupying two empty, neighboring binding sites. Subsequently, the LEF starts
to extrude a loop. Each loop extrusion event moves a site that was previously
a backbone site into a loop. Only outward steps, that grow the loop, are
permitted, with growth to the left or to the right occurring randomly and
independently with equal probability. A LEF cannot occupy a site that is
already occupied by another LEF. Therefore, LEF binding and loop extrusion are
blocked by other LEFs. A LEF can also unbind, dissipating its loop. In our
simulations, a LEF immediately rebinds after dissociation, so that the overall
number of bound LEFs remains fixed throughout our simulations.

We implemented two versions of the LEF model. One is just as described in the
previous paragraph, which we call the random loop model. The second, we call
the CTCF model. In the CTCF model, loop extrusion is partially blocked at
specific locations along the genome, where CCCTC-binding factors (CTCFs) bind
to specific DNA sequences. As shown in the simulations of Ref. [47], this
model is able to reproduce several of the key features seen in chromosome
conformation capture (Hi-C) experiments, including, in particular, TAD
boundaries, separating different genomic regions of high self contact
probability. By construction, the random loop model does not exhibit TAD
boundaries.

To incorporate CTCF-related TAD boundaries in simulations, following Ref.
[47], we binned the experimental mouse CTCF ChIP-Seq coverage data from Ref.
[54], in 10-kb bins, and summed over the coverage data in each bin. In our
simulations, each such bin is located between neighboring LEF binding sites,
so that CTCF abundance scales down the loop extrusion rate from one LEF
binding sites to the next by multiplying a factor

$p(x)=\frac{1}{1+e^{x/x_{0}-\mu}},$ (34)

where $x$ is the experimental CTCF ChIP-Seq coverage of the bin, $x_{0}=20$,
and $\mu=3$, based on values given in Ref. [47]. In this paper, we chose to
focus on three regions of the mouse genome, namely 32-38 Mb on Chromosome 12,
4.8-10.8 Mb on Chromosome 13, and 52-58 Mb on Chromosome 18. These three
genomic regions have 58, 41, and 62 CTCF peaks with varying strengths,
respectively, based on the peak-calling protocol described in Ref. [54].

We simulated these LEF models via a Gillespie-type algorithm [82], implemented
in MATLAB. For each possible next event, say event $k$, a random variable is
generated representing the time $t_{k}$, at which event $k$ would occur. In
each case, $t_{k}$ is drawn from an exponential distribution characterized by
the appropriate rate for the event in question, given the current loop
configuration. Which of these possible next events is actually realized is the
one with the smallest value of $t_{k}$, say, $t_{\tilde{k}}$, advancing the
simulation by a time $t_{\tilde{k}}$. This process is repeated for the
duration of the simulation. The parameters of our simulations are presented in
Table 1. We picked the loop extrusion velocity in either left or right
direction to be 60 bp/s, giving a total loop extrusion velocity of 120 bp/s,
which is equal or comparable to the velocities of 100 bp/s estimated in Ref.
[51] and 125 bp/s in Ref. [83], respectively. We picked the number of LEFs and
the loop dissociation rate to make corresponding processivity and loop density
to lie near the center of the ranges that are given in Ref. [47].

LEF simulation parameter |  Typical value
---|---
number of LEFs |  24, 48, 72
simulated chromatin length |  6 Mb (600 sites)
loop extrusion rate |  120 bps/s (0.012 sites/s)
loop dissociation rate |  0.0005 s-1
LEF processivity |  240 kb
mean LEF separation |  250kb, 125kb, 83kb
Table 1: LEF simulation parameters. Each lattice site represents 10 kb. The
loop extrusion rate given here is for both ends of a LEF combined; for each
end of a LEF, the extrusion rate is 60 bps/s. The LEF density values given
here is defined by the ratio of total simulated genomic length over total
number of LEFs.  Figure 2:  Time evolution of polymer compaction by loops.
Compaction is defined as the ratio of mean squared radius of gyration $\langle
R_{G}^{2}\rangle$ (Eq. 35) of a looped polymer to that of a Gaussian polymer
without loop. The colored lines represent CTCF loop extrusion simulations run
on three different genomic regions using: red, 52-58 Mb of Chr 18; green,
4.8-10.8 Mb of Chr 13; blue, 32-38 Mb of Chr 12. The parameters used in the
CTCF LEF simulations are given in Table 1, and each polymer has 48 LEFs bound
to it. The black line represents the loop extrusion simulation using the
random loop model. The light thin lines represent independent simulation runs.
The dark thick lines are the averages of 5 independent simulation runs. The
dashed lines are exponential fit to the mean curves, which give steady-state
$\langle R_{G}^{2}\rangle$ values of 0.43, 0.40, 0.44, and 0.28 for red,
green, blue and black, respectively. The fitted decay times are: red, 718 sec;
green, 1030 sec; blue, 1126 sec; black, 876 sec.

A key question for any simulation is the time needed to reach a steady-state.
To assess how long it takes our LEF simulations to reach steady-state, we
exploited the observation that treating chromatin as a Gaussian polymer
permits us to determine the polymer’s mean squared radius of gyration,
$\langle R_{G}^{2}\rangle$, directly from the time-dependent loop
configuration (assuming that the polymer time is sufficiently small that each
loop configuration is sufficiently explored by the polymeric degrees of
freedom). Under these conditions, the mean squared radius of gyration between
sites A and B along the genome is given by

$\langle
R_{G}^{2}\rangle=\frac{l_{k}^{2}}{2N(N+1)}\sum_{i,j=A}^{B}(N_{eff})_{ij},$
(35)

where $(N_{eff})_{ij}$ is the effective genomic distance between location $i$
and $j$, which can be determined directly from LEF simulations. Appendix A
presents a derivation of Eq. 35 and a detailed explanation of
$(N_{eff})_{ij}$.

Figure 3:  Abstract representations, following Ref. [84], of three randomly-
chosen, steady-state loop configurations (a, b, and c) taken from three
independent CTCF LEF simulations of a 6 Mb region of mouse Chr 12 (32-38 Mb),
Chr 13 (4.8-10.8 Mb) and Chr 18 (52-58 Mb), respectively, and (d) taken from
the LEF simulation using random loop model. The loop extrusion simulations are
done on 600 lattice sites, each representing 10-kb of genomic length. The
black, horizontal line represents stretched-out chromatin; the semicircles
represent loops extruded on the chromatin polymer. There are 48 loops in each
loop configuration.

Fig. 2 shows the polymer compaction as a function of time as measured by the
relaxation of $\langle R_{G}^{2}\rangle$, normalized to the $\langle
R_{G}^{2}\rangle$ for a Gaussian polymer without loops, for four different LEF
simulations. Each simulation starts with 48 LEFs bound to neighboring lattice
cites and ready to undergo loop extrusion. The solid, black lines show the
relaxation of the normalized $\langle R_{G}^{2}\rangle$ for a chromatin
polymer with loops extruding according to the random loop model. The thin
black lines show the relaxation of the normalized $\langle R_{G}^{2}\rangle$
for five independent LEF simulations, while the thicker black line shows their
average. The red, green, and blue lines show the relaxation behaviors of the
normalized $\langle R_{G}^{2}\rangle$, according to the CTCF model, for three
different genomic regions, namely 52-58 Mb of Chr 18, 4.8-10.8 Mb of Chr 13,
and 32-38 Mb of Chr 12, respectively.

By fitting each mean relaxation curve to a single exponential function, shown
as the dashed lines in the figure, we find that the best-fit relaxation times
for the three genomic regions, 52-58 Mb of Chr 18, 4.8-10.8 Mb of Chr 13, and
32-38 Mb of Chr 12, are 718, 1030, and 1126 seconds, respectively, while the
relaxation time in the case of the random loop model is 876 seconds. The
corresponding steady-state compaction factors for the three different genomic
regions are 0.43, 0.40, and 0.44, respectively, while the steady-state
compaction factor in the case of the random loop model is 0.28. Evidently, in
every case, the simulations achieve a steady-state within a few thousand
seconds. Accordingly, because the loop extrusion simulations are
computationally inexpensive, to ensure a loop extrusion steady state prior to
starting data collection, we run all of our LEF simulations until the total
elapsed time exceeds $2\cdot${10}^{4}$$ seconds, prior to data collection.

Fig. 3 shows abstract representations of three randomly-chosen, steady-state
loop configurations from CTCF-model simulations of mouse chromosome 12, 13,
and 18, and one from the LEF simulation using random loop model, all using
parameters in Table 1. The loop configuration originated from the random loop
model shows a higher level of compaction by possessing denser and bigger loops
along the backbone. We ascribe the reduced steady-state radius of gyration in
the random loop model, in comparison to the CTCF model, to the absence of
limitations on loop extrusion, provided by CTCF boundary elements.

###  III.2 Rouse model simulations

For a random variable obeying Eq. 5, Gillespie [62] showed that

$\displaystyle\mathcal{X}_{m}(t$ $\displaystyle+\Delta t)$ $\displaystyle=~{}$
$\displaystyle\text{\bf{N}}\left(\mathcal{X}_{m}(t)e^{-\frac{\kappa}{\zeta}\Lambda_{m}\Delta
t},\sigma^{2}_{\mathcal{X}_{m}}(1-e^{-2\frac{\kappa}{\zeta}\Lambda_{m}\Delta
t})\right)$ $\displaystyle=~{}$
$\displaystyle\text{\bf{N}}\left(\mathcal{X}_{m}(t)e^{-\frac{\kappa}{\zeta}\Lambda_{m}\Delta
t},\frac{k_{B}T}{\kappa\lambda_{m}}(1-e^{-2\frac{\kappa}{\zeta}\Lambda_{m}\Delta
t})\right)$ $\displaystyle=~{}$
$\displaystyle\mathcal{X}_{m}(t)e^{-\frac{\kappa}{\zeta}\Lambda_{m}\Delta
t}+\text{\bf{N}}(0,1)\sqrt{\frac{k_{B}T}{\kappa{\lambda}_{m}}(1-e^{-\frac{2\kappa}{\zeta}\Lambda_{m}\Delta
t})}.$ (36)

where $\text{\bf{N}}(\mu,\sigma^{2})$ is the Gaussian distribution with mean
$\mu$ and variance $\sigma^{2}$, and where we used Eq. 14 after the second
equality. Eq. III.2 provides a prescription for how to simulate a Rouse-model
polymer, either with or without loops. We chose to simulate Rouse-model
polymers containing 600 beads, matching the number of beads to the number of
LEF binding sites in the LEF simulations. The parameters used in our Rouse-
model simulations are given in Table 2.

To progress a simulation of a Rouse-model polymer from time $t_{1}$, to time
$t_{2}$, we first determine the transformation matrix from bead positions to
normal coordinates, using the loop configuration at $t_{1}$. Using the bead
positions at $t_{1}$, $X(t_{1})$, we then use this $t_{1}$-transformation
matrix to determine the normal coordinates at $t_{1}$, $\mathcal{X}(t_{1})$.
As long as there is no change in the loop configuration between $t_{1}$ and
$t_{2}$, we evolve the normal coordinates according to Eq. III.2 to get normal
coordinates at $t_{2}$, $\mathcal{X}(t_{2})$. To then determine the bead
positions at $t_{2}$, $X(t_{2}$), we calculate the inverse of the
$t_{1}$-transformation matrix, and apply it to $\mathcal{X}(t_{2})$.
Alternatively, if there is a change in the loop configuration at time $t_{3}$,
intermediate between $t_{1}$ and $t_{2}$, we progress the simulation as just
described from $t_{1}$ to $t_{3}$. We then recalculate the
$t_{3}$-transformation matrix, based on the new loop configuration. Using this
new $t_{3}$-transformation matrix, we then progress the simulation from
$t_{3}$ to $t_{2}$.

During each simulation, the dynamical matrix is repeatedly updated to reflect
the current loop configuration, and a new corresponding set of eigenvalues and
normal coordinates are repeatedly calculated. Thus, the entire Rouse
simulation is separated by loop extrusion events, into sub-simulations, each
with its own stochastic update formula specified by the eigensystem of the
current loop configuration. For any two consecutive sub-simulations, the
former’s final conditions serve as the latter’s initial conditions, ensuring
continuity of the entire Rouse simulation. As noted in Ref. [62], this
procedure represents an “exact” simulation, with no small-time approximation
in the sense that there is no constraint on the size of time steps used
($t_{2}-t_{1}$ or $t_{3}-t_{1}$ or $t_{2}-t_{3}$), provided each different
loop configuration is properly included and accounted for.

The Rouse simulation is run for loop configurations that have already relaxed
to steady state (Fig. 2). Specifically, as discussed above, we used loop
configurations from beyond $2\cdot${10}^{4}$$ seconds into each loop extrusion
simulation. The starting bead positions in the Rouse simulation are
initialized by transforming the initial normal coordinates, which are randomly
drawn from the normal distribution given in Eq. III.2, with
$\mathcal{X}_{m}(t=0)=0$ and $\Delta t=\infty$. The initial bead positions
calculated in this way are guaranteed to follow the statistics at the
equilibrium, subject to a mean position of zero. We chose the update time step
in our Rouse simulations to be 1 s, comparable to the Rouse polymer time, and
we ran each Rouse-model simulation for ${10}^{4}$ s.

Rouse simulation parameter |  Typical value
---|---
friction coefficient, $\zeta$ |  $1.08\times 10^{-6}$ Ns/m
spring constant, $\kappa$ |  $3\times 10^{-7}$ N/m
temperature, T |  300 K
polymer time, $\tau_{p}$ |  0.9 sec
update time step |  1 sec
simulation length |  $\sim${10}^{4}$$ sec
Table 2: Rouse simulation parameters.

One possible caveat to our approach is that it is possible to envisage the
development of out-of-equilibrium values of the Rouse model spring potential
energy, if the increase in potential energy as a result of loop extrusion
outpaces Rouse-model energy relaxation and dissipation. To investigate this
possibility, in Fig. 4, we plot examples of the potential energy versus time
for several different situations. The red line represents the potential energy
from simulations of a classical, two-dimensional Rouse-model polymer without
loops, where we find that the potential energy fluctuates about a mean value
of 603 $k_{B}T$. In comparison, the theoretically-expected value, from the
equipartition theorem, is (600 beads)$\times$(2
dimensions)$\times(\frac{1}{2}k_{B}T)=600~{}k_{B}T$. The black line is the
potential energy time-series for the Rouse polymer with loops for 32-38 Mb of
mouse chromosome 12, corresponding to the simulation parameters given in
Tables 1 and 2. In this case, we see that the potential energy fluctuates
about a mean value of 598 $k_{B}T$, also near the theoretical expectation.
This observation indicates that for the loop extrusion and Rouse parameters
used in our simulations, Rouse-model polymers with loops remain in thermal
equilibrium at temperature, $T$. We also carried out simulations for which all
of the rates in the loop extrusion simulations were increased by a factor of
either ten or one hundred. The resulting potential energy versus time traces
are shown in blue and green, respectively. The corresponding mean potential
energies are 600 and 599 $k_{B}T$, respectively. Thus, we see that even for
loop extrusion that is one hundred times faster than estimated, the potential
energy does not noticeably exceed the value expected from equipartition.

Figure 4:  Time series of Rouse polymers’ potential energy for different loop
extrusion rates. The red line shows the simulated spring potential energy
versus time for a free Rouse polymer without loops, using the parameters given
in Table 2. The black line shows the simulated spring potential energy versus
time for a random-loop-model Rouse polymer, for 48 LEFs and the parameters
given in Table 1 and 2. The blue and green lines show the potential energy
versus time for random-loop-model polymers, populated by 48 LEFs, when all of
the loop extrusion rates given in Table 1 are increased by a factor of 10 and
100, respectively, while the polymer parameters give in Table 2 are held
fixed. The resolution of the line points in the plot is set to 70 seconds to
ensure a clear view.  Figure 5:  (a) Simulated MSDs of Rouse-model polymers
with free boundary conditions for the classical Rouse model with no loops,
shown in red, for the CTCF model with loops, shown in blue, and for the random
loop model, shown in green. (b) The corresponding simulated MSD stretching
exponent, $\alpha(t)$, calculated according to Eq. 37 for no loops (red), for
the CTCF model (blue), and for the random loop model (green). The dashed black
line in (a) shows the theoretical mean MSD given by Eq. 28, and then the
theoretical mean stretching exponent is calculated using Eq. 37 and shown in
(b). In both (a) and (b), the thin lines correspond to the results for 60
individual beads, uniformly distributed along the chromatin. The thick lines
correspond to the average across all 60 beads. Each thin line is an average
over 30 independent simulations with identical simulation parameters. The LEF
and Rouse simulation parameters used are given in Table 1 and 2.

##  IV results and discussions

###  IV.1 Chromatin mobility in the presence of loops

Fig. 5(a) compares the MSDs versus time of a Rouse-model polymer without loops
to the MSDs of Rouse-model polymers with loops, generated by loop extrusion,
all under free boundary condition. Shown in red are simulated MSDs for the
classical Rouse model without loops. The thin red lines show the MSDs,
averaged over thirty independent simulations, for sixty individual beads,
equally spaced along the polymer, while the thicker red line shows these MSDs
averaged over all sixty of these beads. In comparison, the dashed black line
in Fig. 5(a) shows the theoretical mean MSD (Eq. 28). Evidently, the
simulation closely matches the analytic theory. Shown in blue and green are
simulated MSDs for the CTCF model and the random loop model, respectively,
both with 48 LEFs, with free boundary conditions, and averaged over thirty
independent simulations. In these cases too, the thin lines show the MSDs for
60 individual beads, while the thicker lines show the MSDs averaged over these
beads. Importantly, in both cases, the MSD in the presence of loops is
significantly smaller than the MSD for the classical Rouse model without
loops, indicating that the presence of loops significantly constrains
chromatin mobility. The difference between the MSDs of the two different loop
extrusion models is small, although clearly the MSDs of the random loop model
are more reduced than the MSDs of the CTCF model.

MSDs versus time are often described using a “stretching exponent”. Fig. 5(b)
shows the time-dependent stretching exponent, $\alpha(t)$, corresponding to
the MSDs shown in Fig. 5(a), defined as

$\alpha(t)=\frac{\log\left[\text{MSD}(t+\Delta
t)\right]-\log\left[\text{MSD}(t)\right]}{\log(t+\Delta t)-\log(t)},$ (37)

where $\Delta t$ is the time step of the Rouse simulations. For ranges of time
over which $\alpha(t)$ is constant, the MSD approximates power law behavior
versus time. For the classical Rouse model without loops (red curves), with
increasing time, $\alpha(t)$ decreases from a value of 1 at the earliest times
(data not shown) to achieve a plateau value close to 0.5 for times greater
than a few tens of $\tau_{p}$. This is consistent with the theoretically
predicted $\alpha(t)$, shown as the black, dashed line in Fig. 5(b), which was
calculated from Eq. 28. For Rouse polymers with loops (blue and green curves),
interestingly the stretching exponent falls significantly below the value for
the Rouse polymer without loops. In the case of the CTCF model simulations,
for times beyond a few tens of $\tau_{p}$, $\alpha$ achieves a plateau value
close to $\alpha=0.45$. For the random loop model simulations, $\alpha$
appears to decrease continuously with time from a value near 0.45 at 30
$\tau_{p}$ to a value near 0.4 for $10^{3}~{}\tau_{p}$.

Figure 6:  (a) Comparison between the simulated MSDs of Rouse-model polymers
with free, periodic and fixed boundary conditions, shown as the solid, dashed,
and dotted lines, respectively, for the classical Rouse polymer with no loops,
shown in red, for the CTCF model with 48 LEFs, shown in blue, and for the
random loop model with 48 LEFs, shown in green. (b) Comparison between the
corresponding simulated MSD stretching exponent, $\alpha(t)$, calculated
according to Eq. 37 for no loops (red), for the CTCF model with 48 LEFs
(blue), and for the random loop model with 48 LEFs (green), each considered
separately with the three different boundary conditions. In both (a) and (b),
each line is averaged over 30 independent simulations and averaged over 60
individual beads, which are uniformly distributed along the chromatin.

The simulations of Fig. 5 correspond to polymers with free boundary
conditions. We also carried out comparable simulations for both closed polymer
rings (periodic boundary conditions) and polymers with fixed ends (fixed
boundary conditions). A comparison of the mean MSDs for each possible boundary
condition is shown in Fig. 6(a) for different loop models. It is clear from
this figure that the MSDs for different boundary conditions are very similar
to each other over the range of times studied. Fig. 6(b) shows the
corresponding mean stretching exponents, which are also very similar to each
other over the range of times studied. Solutions to the classical Rouse model,
subject to periodic and fixed boundary conditions, are given in Appendix B and
Appendix C, respectively.

Figure 7:  (a) Simulated MSDs of Rouse-model polymers with periodic boundary
conditions for the classical Rouse model with no loops, shown in red, and for
the CTCF model with different number of LEFs, shown in cyan, blue, magenta,
and green, corresponding to 24, 48, 72 and 144 LEFs, respectively. (b) The
corresponding simulated MSD stretching exponent, $\alpha(t)$, for no loops
(red), and for the CTCF models with 24 (cyan), 48 (blue) 72 (magenta), and 144
(green) LEFs. The dashed black line in (a) shows the theoretical mean MSD
given by Eq. 56. The dashed line in (b) is the corresponding theoretical mean
stretching exponent, calculated using Eq. 37, applied to Eq. 56. In both (a)
and (b), each line is averaged over 30 independent simulations and averaged
over 60 individual beads, which are uniformly distributed along the chromatin.

How MSD depends on the number of LEFs is illustrated in Fig. 7, which depicts
MSDs (Fig. 7(a)) and stretching exponents (Fig. 7(b)) for CTCF-model polymers
with 0 LEFs, shown as the red line, 24 LEFs, shown as the cyan line, 48 LEFs,
shown as the blue line, 72 LEFs, shown as the magenta line, and 144 LEFs,
shown as the green line. It is clear from this figure that the MSDs are
progressively repressed as the number of LEFs increases. Fig. 7(b) displays
the corresponding stretching exponents for different number of LEFs, revealing
that the stretching exponent also initially decreases progressively as the
number of LEFs increases as the number of LEFs increases. Interestingly, the
stretching exponent at intermediate times appears to show a limiting value of
0.43 for large number of LEFs.

Figure 8:  (a) Comparison between the MSDs versus time of polymers with 48
randomly-located static loops (orange) and polymers with dynamic loops,
evolving according to the CTCF model with 48 LEFs (blue). (b) The
corresponding time-dependent stretching exponents. The LEF and Rouse
simulation parameters used are given in Table 1 and 2. In both (a) and (b),
the thin lines correspond to results of 60 individual beads, uniformly
distributed along the polymer; the thicker lines are averages over the 60
beads. The result for each individual bead is averaged over 30 independent
simulations with identical simulation parameters.

LEFs actively extrude loops along the chromatin. Therefore, although chromatin
mobility is reduced overall by the introduction of loops, it is interesting to
investigate the extent to which the dynamics of loop extrusion might make a
positive contribution to chromatin mobility. To identify any possible
contribution to chromatin dynamics from LEF dynamics, we investigated Rouse
polymers with randomly-located static loops, for which there are no LEF
dynamics. Each static loop configuration studied corresponds to one randomly-
picked time point from the steady state of the CTCF model with 48 LEFs on
32-38 Mb of mouse Chr 12. For each such static loop configuration, a Rouse
simulation was then carried out to determine its MSDs and stretching
exponents. Finally, we averaged the MSDs and stretching exponents from 30
different static loop configurations. The resultant mean MSDs and stretching
exponents for looped polymers with random, static loops (orange) are compared
in Fig. 8 to the simulated MSDs and stretching exponents of looped polymers
with dynamic loops (blue), that evolve in time according to the CTCF model
with 48 LEFs in the same genomic region. The MSDs of individual beads, equally
separated along the polymer, shown as thin lines in Fig. 8(a), show similar
distributions and trends for both the static and dynamic loops. The stretching
exponents in the case of static and dynamic loops, shown in Fig. 8(b), also
behave similarly to each other. It is apparent from this comparison that the
MSDs and stretching exponents of polymers with dynamic, on one hand, and
static loops, on the other, show insignificant differences over the range of
times studied, suggesting in turn that loop extrusion dynamics do not
contribute significantly to chromatin dynamics, at least for the chosen
parameter values.

Our simulational results clearly demonstrate that the presence of loops
significantly reduces chromatin mobility. We believe this behavior follows
from the additional, positional constraints that loops impose, specifically
that the two beads at the base of each loop must necessarily lie close to each
other. Notably, the simulational results, presented in this paper, are
consistent with our recent experimental measurements of the MSDs of gene loci
in fission yeast, described in Ref. [63], where we found that fluorescently-
labelled gene loci in yeast strains with absent or functionally-compromised
versions of the putative LEFs, cohesin and condensin, exhibited larger MSDs
than the same gene loci in wild-type strains with properly functioning cohesin
and condensin. Thus, Ref. [63], in conjunction with the results of this paper,
provides clear experimental support for the idea that loops reduce chromatin
mobility, as characterized by the MSD of a gene locus.

A less expected result to emerge from our simulations is the observation that
the stretching exponent in polymers with loops is noticeably smaller than that
for the classical Rouse model without loops. Strikingly, however, in our own
measurements of the MSDs of fluorescently-labelled gene loci in fission yeast,
we also observed stretching exponents with values smaller than 0.5 (about
0.45), further connecting our simulations and experiments [63]. Our
simulational results also recall the experimental observation in Ref. [13]
that the MSD of fluorescently-labelled gene loci in Escherichia coli shows a
stretching exponent of about $\alpha=0.4$. In Ref. [13], this value was
explained by envisioning the bacterial genome to be an unlooped Rouse-model
polymer within a viscoelastic medium. The calculations of Ref. [61] showed
that such a viscoelastic background could reduce the classical Rouse model
stretching exponent from 0.5 to the observed value. However, a population of
loops across the E. coli genome would also contribute to a smaller stretching
exponent than 0.5 in this case too. Polymer simulations of chromosome regions
with high compaction show stretching exponents of around 0.3 [85, 86], which
may also partly originate from a population of loops.

Figure 9:  The rosette configuration investigated in Sec. IV.2. Shown in blue
is the chromatin polymer itself; loop bases are highlighted as the brown
circles. Each loop is 17 lattice sites long, and each backbone segment between
neighboring loops is 8 lattice sites long. Thus, the fraction of the chromatin
polymer inside loops is 0.68 in this configuration. There are 24 repetitions
of the loop-backbone structure within the periodic chromatin polymer, making a
total of 600 lattice sites, representing a genome of 6 Mb. The dashed blue
line represents repetitions of the identical loop-backbone structures.
Examples of each family of locus locations are labelled tip, base and
backbone.  Figure 10:  (a) Simulated MSDs of the Rouse-model polymer with the
rosette structure illustrated in Fig. 9 for three different families of
locations: tip (yellow), base (magenta), and backbone (cyan). (b) The
corresponding simulated MSD stretching exponent for different families of
locations on the rosette-structured polymer: tip (yellow), base (magenta), and
backbone (cyan). In both (a) and (b), each line shows the averaged result of
720 independent simulations for beads in each family. The black, dashed lines
in (a) and (b) show the theoretical MSD and stretching exponent, respectively,
of a Rouse-model polymer without loops, with periodic boundary condition, as
given in Eq. 56 and 37.

###  IV.2 Locus dynamics around a chromatin rosette

Our Rouse simulation method is applicable to any configuration of loops. As
noted in the introduction, genomic rosettes have been suggested to be an
important motif for E. coli nucleoids [64, 65] and replication factories [66,
67, 68, 69, 71, 72, 70]. Accordingly, we have investigated the MSDs and
stretching exponents for the static, rosette configuration, illustrated in
Fig. 9, in which 24 loops are periodically and equally spaced about a closed
chromatin polymer. For our simulations, each loop contains 17 beads, and is
separated from its neighboring loops by a segment of the backbone containing 8
beads. By symmetry, we can expect the mean dynamics within different repeating
units of the rosette to be identical. Thus, because we can average the
behavior of corresponding loci in different repeating units, the rosette
facilitates investigating the dynamics at the midpoint of a loop, at the base
of a loop, and at the midpoint of a backbone region, for example. Each family
of locations is illustrated and labelled as examples in Fig. 9.

In Fig. 10, shown in yellow, magenta, and cyan, respectively, are the MSDs and
stretching exponents for beads at the tip of the loops (yellow), base of the
loops (magenta), and midpoint of the backbone (cyan). By symmetry of the
rosette structure, beads belonging to the same family are statistically
equivalent. Therefore, the yellow, magenta, and cyan lines shown in Fig. 10
are averaged results of 720 independent simulations for the tip, base, and
backbone families, respectively, indifferent about the loop index. The
theoretical MSD and stretching exponent, calculated by Eq. 56 and 37, are
shown as the black, dashed lines in Fig. 10(a) and (b), respectively.

It is clear from Fig. 10(a) that loop bases (magenta) exhibit the smallest
mean MSD at all times studied, taking on a value that is approximately one-
half the theoretical MSD for periodic Rouse-polymer without loops, while loop
tips show the largest MSDs. At early times, the mean MSD for backbone beads
show similar behavior to the mean MSD for loop tip beads. However, with
increasing time, the mean MSD for backbone beads deviate (at about 50
$\tau_{p}$) to smaller values to eventually achieve (by about 1000 $\tau_{P}$)
similar behavior to the mean MSD for the bases beads. The mean MSD for loop
tips closely follow the theoretical MSD for a polymer without loops until
about 500 $\tau_{p}$, when it deviates below the theoretical MSD.

The corresponding stretching exponents, shown in Fig. 10(b), also reveal
elaborate behavior. The loop tips in the rosette structure exhibit the same
mean stretching exponent as the theoretical stretching exponent for polymers
without loops at early times, until about 200 $\tau_{p}$. During this early
time period, the mean stretching exponent for the loop tips decrease from
value of 1 at the earliest time (not shown), to a level of roughly 0.5. At
times larger than 200 $\tau_{p}$, the mean stretching exponent for the loop
tips starts to deviate to a lower level compared to the theory. The mean
stretching exponent for the backbone beads has similar values to that for the
loop tips and the theoretical value for polymers without loops; however, it
starts to deviate to a lower level much earlier, at about 10 $\tau_{p}$, and
decreases to a much reduced value of about 0.3 at a few hundred $\tau_{p}$, at
which it starts to recover and increase back to 0.4 at very late times. The
mean stretching exponent for the loop bases reside between 0.4 and 0.45 at
earlier times, before $t=100~{}\tau_{p}$, then increase beyond the 0.5 level
at around 500 $\tau_{p}$, and finally drop back to the level around 0.5,
similar to the theoretical stretching exponent for polymers without loops at
the very later times. It is clear from the results shown in this figure, that
the dynamics of a locus are dictated by the position of the locus in question
relative to the positions of loops. This behavior does not emerge in the case
of dynamic loops, because commonly in a loop extrusion steady state any
particular locus randomly alternates between being in the backbone, being at a
loop base, being at a loop tip, etc.

Figure 11:  (a) Comparison between the MSDs of periodic polymers without
loops, subject either to uniform friction, shown in red, or to log-normal-
distributed (non-uniform) friction, shown in green. For the polymer subject to
log-normal-distributed non-uniform friction, each bead has a fixed friction
coefficient, drawn from the scaled Lognormal(0, log(2)) with mean of $\zeta$,
given in Table 2. (b) The corresponding MSD stretching exponents, calculated
according to Eq. 37, for uniform friction case (red) and log-normal-
distributed (non-uniform) case (green). In both (a) and (b), thin lines
represent 60 individual beads uniformly distributed along the polymer, and
each thin line is an average of 30 independent simulations. Each thicker line
is an average of all corresponding thin lines.  Figure 12:  The hypothetical
replication factory configuration discussed in Sec. IV.4. The shaded region is
where the friction coefficient is 10 times higher than in the unshaded region.
Loci experiencing higher friction coefficients are all located at the bases of
the loops. The midpoints of each loop are labelled as “tip”, and the bases of
each loop are labeled as “base”. Total number of LEFs is 24. Each loop has
size of 24. The backbone length between neighboring loops is 1, which is the
minimal length a backbone can form given that the LEFs will block each other
at distance 1. The dashed line represents repetitions of the same loop
structure. Figure 13:  (a) Shown in red and magenta lines are the mean MSDs
for the loop tips and bases, respectively, of a rosette polymer in the non-
uniform-friction envionrment, as shown in Fig. 12. Shown in blue and green
lines are the mean MSDs for the loop tips and bases, respectively, of the same
rosette polymer, but in a uniform-friction envioronment, i.e. without the
region of ten-fold friction shown in Fig. 12. Each line is an average of 720
independent simulations for the tip beads or bases beads, indifferent about
the loop repetitions to which each bead belongs, as a result of symmetry. (b)
The corresponding averaged stretching exponents.

###  IV.3 Rouse polymer with non-uniform friction

The theory described in Sec. II.1 also allows us to study polymers with
friction coefficients that vary from bead to bead. Accordingly, we first
examined a linear Rouse polymer with each bead’s friction coefficients drawn
from a log-normal distribution. We use a log-normal distribution proportional
to Lognormal(0,log(2)), so that the ratio of the standard deviation and the
mean is 1, and scale it to have a mean of $\zeta$, which is the friction
coefficient in Table 2. Each bead is assigned a fixed friction coefficient
drawn from such a scaled log-normal distribution. Fig. 11(a) compares the MSDs
of a periodic Rouse polymer with friction coefficients distributed according
to a log-normal distribution to the MSDs of a periodic Rouse polymer with
uniform friction coefficient. There are no loops in both cases. The thin lines
show the MSDs for 60 individual beads, equally spaced along the polymer, while
the thicker lines show the averaged MSDs over all sixty of these beads. Each
thin line is also an average of 30 independent simulations. The mean MSDs for
the uniform friction coefficient and log-normal-distributed friction
coefficient are very similar across all time-scales considered here, with a
small discrepancy that the mean MSD for the non-uniform-friction case is
slightly higher than that for the uniform-friction case at all time scales
considered. Fig. 11(b) shows the corresponding stretching exponents, and the
mean stretching exponents for the two cases are, again, almost identical. Both
mean stretching exponents decreases from 1, at the earliest time (not shown
here), to around 0.5, at 100 $\tau_{p}$, with the small discrepancy between
the two cases such that the stretching exponent for the polymer with non-
uniform friction is slightly lower than the stretching exponent for the
uniform-friction polymer. After 100 $\tau_{p}$, the stretching exponent for
the non-uniform-friction polymer fluctuates around the 0.5 level while the
stretching exponent for the uniform-friction polymer continues to decrease to
a level below 0.45 at 100 $\tau_{p}$. Even though the mean MSDs and stretching
exponents are similar for polymers with uniform and non-uniform friction, the
MSDs and stretching exponents of individual beads in the non-uniform-friction
polymer show a greater spread than those of uniform friction case. This is
consistent with the simulation setup in which each bead experiences a
different but fixed friction coefficient drawn from the log-normal
distribution. The above results indicate that fluctuations in the friction
coefficient does not strongly affect the mean MSD and mean stretching
exponent, at least up to the spread of the beads’ friction coefficients
considered here.

###  IV.4 Dynamics of a rosette that models a replication factory

In a replication factory, chromatin is hypothesized to be folded into a
rosette-like structure, at whose center transcription factors and SMCs may
form a phase-separated droplet and assist the simultaneous replication of the
DNA within the loops [66, 67, 68, 69, 71, 72, 70]. A simple version of such an
organization is illustrated in Fig. 12. The blue lines shown represent the
chromatin polymer segment, and the brown circles represent the putative SMCs,
cohesin or condensin molecules, that establish the replication-factory
configuration. In our simulations, we model a replication factory as
consisting of 24 loops, each of which contains 24 beads. Plausibly, the phase-
separated droplet, shown as the grey region in this figure, which covers the
loop bases, may give rise to a high local viscosity. In this case, we model
the high viscosity region by using a ten-fold higher friction coefficient for
the loop bases than for the beads in the loops.

Fig. 13 compares the averaged (a) MSDs and (b) stretching exponents for the
loop tips and bases in the replication-factory configuration (Fig. 12) with a
uniform-friction or with a non-uniform-friction environment. Because each loop
repetition is statistically equivalent, the simulation is indifferent to the
loop repetition each loop tip or base belongs to. Therefore, for each line in
Fig. 13, we take an average of 720 independent simulations, regardless of the
loop index. The time axis in the plot is measured in units of $\tau_{p}$,
calculated using an unmodified friction coefficient ($1.08\times 10^{-6}$
Ns/m). In Fig. 13(a), the MSDs for the replication-factory polymer with
uniform friction are shown in red and magenta, for the loop tips and bases,
respectively. In comparison, the MSDs for the replication-factory polymer with
ten-fold higher friction at the loop bases are shown in blue and green, for
loop tips and loop bases, respectively. Unsurprisingly, the difference between
the mean MSDs at loop tips for the two cases is small, with the mean MSDs
nearly identical at early times ($\lesssim 400~{}\tau_{p}$). The loop tips in
the replication-factory polymer admitting non-uniform friction have slightly
lower mean MSD than those in a uniform-friction polymer, for times beyond
about $400~{}\tau_{p}$. Also unsurprisingly, the MSDs at loop bases are
significantly smaller when the loop bases experience enhanced friction, than
when they do not.

Fig. 13(b) shows the corresponding stretching exponents. The mean stretching
exponents for the loop tips in uniform (red) and non-uniform (blue) friction
environments show similar behaviors across the entire range of times studied.
From the earliest time (not shown) to about 10 $\tau_{p}$, both stretching
exponents decrease from 1 to the level around 0.5 and maintain that level of
0.5 until 100 $\tau_{p}$. From 100 $\tau_{p}$ to 1000 $\tau_{p}$, both start
to decrease, in a similar manner, from the level around 0.5 to the level
between 0.2 and 0.3. By constrast, the stretching exponent for the loop bases
appears very differently for the two cases: in the uniform friction case, the
loop bases display a stretching exponent that decreases from 0.4 to $\sim$0.25
as the time increases from a few $\tau_{p}$ to $\sim 100~{}\tau_{p}$. It then
appears to reverse this trend, recovering to 0.35 for times $\gtrsim
300\tau_{p}$. For loop bases with ten-fold higher friction, the stretching
exponent for the loop bases decreases from 0.85 at early times (a few
$\tau_{p}$) to reach 0.4 (for times $\gtrsim 300~{}\tau_{p}$) where it appears
to plateau. We can understand the initial decrease of the stretching exponent
from a large value in the enhanced friction case by first recognizing that in
actuality all stretching exponents start from 1 at the earliest times. This
behavior usually occurs at smaller values of the time than we plot. However,
because the polymer time of the loop bases is effectively increased by a
factor of 10, as a result of the ten-fold higher friction that they
experience, it follows that the evolution of the stretching exponent from 1 to
smaller values now occurs within the range of times covered by our
simulations. According to Eq. 30, the diffusion coefficient, and thus the MSD
amplitude, is proportional to $1/\sqrt{\zeta}$. Therefore, in theory, when the
friction coefficient is increased by a factor of 10, the MSD amplitude should
decrease by a factor of about 3. By examining the discrepancy between the mean
MSD for loop bases with normal friction coefficient $\zeta$ (magenta) and the
mean MSD for loop bases with friction coefficient of 10 $\zeta$ (green) in
Fig. 13(a), we see that at very early times, from 0 $\tau_{p}$ to around 50
$\tau_{p}$, the mean MSD for loop bases with friction $\zeta$ is about 3 times
greater than that with friction 10 $\zeta$, coinciding with the theoretical
prediction. However, the dependence of MSD amplitude on the friction
coefficient becomes less significant for times greater than 50 $\tau_{p}$, and
the two MSD amplitudes only differ by about 20% at 2000 $\tau_{p}$.

##  V Conclusions

We have incorporated loops and loop extrusion into the Rouse model of polymer
dynamics in a fashion that permits exact simulations of the resultant looped
polymer. By carrying out simulations for polymers whose loop configuration
evolves in time according to the loop extrusion factor (LEF) model, we have
demonstrated that chromatin loops in a dynamic steady state reduce chromatin
mobility, as measured by the time- and ensemble-averaged MSDs of gene loci. We
have also shown that this reduction in mobility increases with increasing LEF
density. Also, in contrast to the classical Rouse model stretching exponent,
$\alpha$, which admits the value of $1/2$ in the absence of loops, we have
shown that loops reduce the stretching exponent at early times. The reduction
in the stretching exponent also increases with increasing LEF density, but
achieves a value near 0.45 for best-estimate loop densities, estimated from an
analysis of Hi-C experiments. Remarkably, the simulated stretching exponent of
0.45 is consistent with the value measured in our own recent experiments that
study the MSDs of several fluorescently-labelled gene loci in living fission
yeast. Active loop extrusion via the LEF model automatically ensures that our
simulations sample across an ensemble of different loop configurations.
However, by finding indistinguishable results, when we explicitly average MSDs
over an ensemble of different static loop configurations, we can infer that
LEF dynamics themselves contribute negligibly to the polymer dynamics of
chromatin, given that the polymer time is sufficiently smaller than the time
scale of loop extrusion.

By studying a static rosette configuration (Fig. 9), we have shown that the
MSD for loop bases is significantly suppressed compared to that of loop tips,
which, at early times, behave akin to a polymer without loops. The chromatin
backbone resembles the loop tips at early times and resemble the loop bases at
later times, in terms of both MSD and stretching exponent.

Our exact simulation method also allows us to examine Rouse models with non-
uniform friction. By assigning each bead of an unlooped polymer a fixed
friction coefficient drawn from a scaled log-normal distribution, we have
shown that a non-uniform friction environment does not affect the averaged
dynamics predicted by the Rouse model. We have also shown that chromatin
mobility at the center of a replication factory is much reduced if a high-
friction environment is present at the center. More generally, our Rouse model
simulation can provide predictions and evidence for chromatin organization and
dynamics that are subject to loops or other genomic organizations and
functions.

###### Acknowledgements.

This research was supported by the NSF via EFRI CEE 1830904. M.L.P.B. was
supported by NIH Grant No. T32EB019941 and the NSF GRFP.

##  Appendix A Mean squared radius of gyration

The mean squared radius of gyration of a chromatin polymer segment, that
extends from genomic location A to genomic location B, given the equivalent
definition of radius of gyration from [87], can be represented as

$\langle
R_{G}^{2}\rangle=\frac{1}{2N(N+1)}\sum_{i=A}^{B}\sum_{j=A}^{B}\langle(\bm{r}_{i}-\bm{r}_{j})^{2}\rangle$
(38)

where $\bm{r}_{i}$ and $\bm{r}_{j}$ are the position vectors of genomic
location $i$ and $j$, respectively. At length scales longer than the Kuhn
length, $l_{k}$, we treat chromatin as a Gaussian polymer, so that the
probability distribution of separation vector between $i$ and $j$ in
$d$-dimension, namely $\bm{r}_{ij}$, is given by

$\mathcal{P}(\bm{r}_{ij})=(\frac{d}{2\pi
N_{ij}l_{k}^{2}})^{d/2}e^{-d\bm{r}_{ij}^{2}/2N_{ij}l_{k}^{2}},$ (39)

where $N_{ij}$ is the number of Kuhn segments between $i$ and $j$, and the
mean squared separation between $i$ and $j$ is given by

$\langle(\bm{r}_{i}-\bm{r}_{j})^{2}\rangle=N_{ij}l_{k}^{2}.$ (40)

Due to the existence of loops, and thus a change of chromatin polymer
topology, the separation between any two locations $i$ and $j$ could be
reduced by intervening loops such that their separation vector follows a
probability distribution given in Eq. 39 but with a smaller $N_{ij}$ value, or
the effective genomic distance $(N_{eff})_{ij}$. The effective genomic
distance can be calculated given only the loop configuration of the polymer
and is expressed in the unit of number of Kuhn segments here. Therefore, the
mean squared separation between $i$ and $j$ with loop configuration is given
by

$\langle(\bm{r}_{i}-\bm{r}_{j})^{2}\rangle=(N_{eff})_{ij}l_{k}^{2},$ (41)

and the mean squared radius of gyration in (38) reads as

$\langle
R_{G}^{2}\rangle=\frac{l_{k}^{2}}{2N(N+1)}\sum_{i=A}^{B}\sum_{j=A}^{B}(N_{eff})_{ij}.$
(42)

In the absence of loops, $(N_{eff})_{ij}$ reduces to $N_{ij}=|i-j|$.
Therefore, Eq. 42 becomes

$\displaystyle\langle R_{G}^{2}\rangle$
$\displaystyle=\frac{l_{k}^{2}}{2N(N+1)}\sum_{i=1}^{N}\sum_{j=1}^{N}|i-j|$
(43) $\displaystyle=\frac{l_{k}^{2}}{2N(N+1)}\frac{(N-1)N(N+1)}{3}$ (44)
$\displaystyle=\frac{(N-1)l_{k}^{2}}{6},$ (45)

which exactly recovers the mean radius of gyration of a Gaussian polymer with
$N$-beads ($N-1$ segments) and Kuhn length $l_{k}$.

##  Appendix B Rouse model for periodic boundary conditions

In this case, the two ends of the polymer are connected together via a
nearest-neighbor spring, and the matrix $A$ ($N$-by-$N$) becomes

$A_{periodic}=\begin{bmatrix}2&-1&0&\cdots&\cdots&-1\\\
-1&2&-1&\cdots&\cdots&0\\\ 0&-1&2&\cdots&\cdots&0\\\
\vdots&\vdots&\vdots&\ddots&\ddots&\vdots\\\
\vdots&\vdots&\vdots&\ddots&2&-1\\\ -1&0&0&\cdots&-1&2\end{bmatrix},$ (46)

which is a circulant matrix. The eigenvectors of circulant matrices contain
entries of solutions of $N$-th root of unity; more precisely, the complex
eigenvector matrix of (46) is given by

$U_{jk}=e^{\frac{2\pi i}{N}jk},\quad j,k\in\\{0,1,...,N-1\\}.$ (47)

Since (46) is real, the real and imaginary parts of $U$ each gives a set of
eigenvectors, although neither of them is complete, due to the parities of
sine and cosine. However, a linear combination of the real and imaginary parts
breaks the parity and thus forms a set of linearly independent and complete
eigenvectors. Here we choose a symmetric linear combination of cosine and
sine, which results in the real eigenvector matrix:

$\displaystyle U_{jk}$
$\displaystyle=\frac{N_{0}}{2}\left(\sqrt{2}\cos(\frac{2\pi
jk}{N})+\sqrt{2}\sin(\frac{2\pi jk}{N})\right)$ (48)
$\displaystyle=N_{0}\sin\left(\frac{2\pi jk}{N}+\frac{\pi}{4}\right)$ (49)
$\displaystyle=\sqrt{\frac{2}{N}}\sin\left(\frac{2\pi
jk}{N}+\frac{\pi}{4}\right)$ (50)

where the normalization constant $N_{0}=\sqrt{\frac{2}{N}}$ ensures unitarity
of $U$. The eigenvalue corresponding to $k$-th column of $U$ (the $k$-th
eigenvector) is given by

$\displaystyle\lambda_{k}$
$\displaystyle=\sum_{j=0}^{N-1}A_{0j}U_{jk}=\sum_{j=0}^{N-1}A_{0j}e^{\frac{2\pi
i}{N}jk}$ (51) $\displaystyle=A_{00}+A_{01}e^{\frac{2\pi
i}{N}k}+A_{0,N-1}e^{\frac{2\pi i}{N}k(N-1)}$ (52)
$\displaystyle=2-e^{\frac{2\pi i}{N}k}-e^{-\frac{2\pi i}{N}k}$ (53)
$\displaystyle=2-2\cos\left(\frac{2\pi k}{N}\right).$ (54)

Using the above results, Eq. 18 and Eq. 20 become

$\displaystyle\langle X_{n}^{2}(0)\rangle$ $\displaystyle=\langle
X_{n}^{2}(t)\rangle$ $\displaystyle=\frac{k_{B}T}{\kappa
N}\sum_{k=1}^{N-1}\sin^{2}\left(\frac{2\pi
nk}{N}+\frac{\pi}{4}\right)\frac{1}{\left[1-\cos\left(\frac{2\pi
k}{N}\right)\right]}$ $\displaystyle=\frac{k_{B}T}{12\kappa N}(N^{2}-1),$ (55)

and

$\displaystyle\langle$
$\displaystyle\left[X_{n}(t)-X_{n}(0)\right]^{2}\rangle=$
$\displaystyle\frac{2k_{B}T}{\kappa N}\sum_{k=1}^{N-1}\sin^{2}\left(\frac{2\pi
nk}{N}+\frac{\pi}{4}\right)\frac{1-e^{-\frac{2\kappa}{\zeta}(1-\cos(\frac{2\pi
k}{N}))\left\lvert t\right\rvert}}{\left[1-\cos\left(\frac{2\pi
k}{N}\right)\right]},$ (56)

respectively. The first bead starts with index $n=0$. Note that the mean
squared position $\langle X_{n}^{2}(0)\rangle$ (relative to Rouse polymer’s
center of mass) of the beads under periodic boundary condition does not depend
on the bead index, which makes intuitive sense since the position of one bead
relative to the others in the Rouse chain is identical for all beads.

Figure 14:  (a) Simulated MSDs of Rouse-model polymers with fixed boundary
conditions for the classical Rouse model with no loops, shown in red, for the
CTCF model with loops, shown in blue, and for the random loop model, shown in
green. (b) The corresponding simulated MSD stretching exponent, $\alpha(t)$,
calculated according to Eq. 37 for no loops (red), for the CTCF model (blue),
and for the random loop model (green). The dashed black line in (a) shows the
theoretical mean MSD given by Eq. 63, and then the theoretical mean stretching
exponent is calculated using Eq. 37 and shown in (b). In both (a) and (b), the
thin lines correspond to the results for 60 individual beads, uniformly
distributed along the chromatin. The thick lines correspond to the average
across all 60 beads. Each thin line is an average over 30 independent
simulations with identical simulation parameters. The LEF and Rouse simulation
parameters used are given in Table 1 and 2.

##  Appendix C Rouse model for fixed boundary condition

In this case, the two ends of the polymer are attached to two fixed location
via additional springs and the matrix $A$ ($N$-by-$N$) becomes

$A_{fixed}=\begin{bmatrix}2&-1&0&\cdots&\cdots&0\\\ -1&2&-1&\cdots&\cdots&0\\\
0&-1&2&\cdots&\cdots&0\\\ \vdots&\vdots&\vdots&\ddots&\ddots&\vdots\\\
\vdots&\vdots&\vdots&\ddots&2&-1\\\ 0&0&0&\cdots&-1&2\end{bmatrix}.$ (57)

The eigenvector matrix is given by

$S=\sqrt{\frac{2}{N+1}}\begin{bmatrix}\sin\frac{\pi}{N+1}&\sin\frac{2\pi}{N+1}&\sin\frac{3\pi}{N+1}&\cdots\\\
\sin\frac{2\pi}{N+1}&\sin\frac{4\pi}{N+1}&\sin\frac{6\pi}{N+1}&\cdots\\\
\sin\frac{3\pi}{N+1}&\sin\frac{6\pi}{N+1}&\sin\frac{9\pi}{N+1}&\cdots\\\
\vdots&\vdots&\vdots&\vdots\\\ \vdots&\vdots&\vdots&\vdots\\\ \end{bmatrix},$
(58)

i.e.

$S_{mn}=\sqrt{\frac{2}{N+1}}\sin\frac{mn\pi}{N+1}.$ (59)

$S$ diagonalizes $A$, i.e. $SAS=D_{1}$. The resultant diagonal matrix $D_{1}$
contains the eigenvalues of $A$ on its diagonal:

$\Lambda=\begin{bmatrix}2-2\cos\frac{\pi}{N+1}&0&0&\cdots\\\
0&2-2\cos\frac{2\pi}{N+1}&0&\cdots\\\ 0&0&\ddots&\cdots\\\
\vdots&\vdots&\vdots&\ddots\\\ \end{bmatrix},$ (60)

i.e.

$\Lambda_{m}=2-2\cos\frac{m\pi}{N+1}.$ (61)

Using these results, Eq. 18 and Eq. 20 become

$\displaystyle\langle X_{n}^{2}(0)\rangle$ $\displaystyle=\langle
X_{n}^{2}(t)\rangle$
$\displaystyle=\frac{k_{B}T}{(N+1)\kappa}\sum_{k=1}^{N}\sin^{2}\left(\frac{nk\pi}{N+1}\right)\frac{1}{\left[1-\cos\left(\frac{k\pi}{N+1}\right)\right]}$
$\displaystyle=\frac{k_{B}T}{\kappa}n\left(1-\frac{n}{N+1}\right),$ (62)

and

$\displaystyle\langle$
$\displaystyle\left[X_{n}(t)-X_{n}(0)\right]^{2}\rangle=$
$\displaystyle\frac{2k_{B}T}{(N+1)\kappa}\sum_{k=1}^{N}\sin^{2}\left(\frac{nk\pi}{N+1}\right)\frac{1-e^{-\frac{2\kappa}{\zeta}(1-\cos(\frac{k\pi}{N+1}))\left\lvert
t\right\rvert}}{\left[1-\cos\left(\frac{k\pi}{N+1}\right)\right]},$ (63)

respectively. Note that the bead indexing $n$ starts from 1 to N in this case.

Fig. 14 shows the simulated MSD and stretching exponent of a Rouse-model
polymer with loops and without loops, under fixed boundary condition. The
dashed black line gives the theoretical mean MSD and stretching exponent
calculated using Eq. 63 and Eq. 37, as a comparison to the simulation results
for classical Rouse polymer under fixed boundary condition.

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