# Directed Motion and Spatial Coherence in the Cell Nucleus

M. Hidalgo-Soria  Department of Physics and Institute of Nanotechnology and
Advanced Materials, Bar-Ilan University, Ramat-Gan 5290002, Israel  Department
of Physics, University of California, San Diego, La Jolla, CA 92093, USA  Y.
Haddad  Department of Physics and Institute of Nanotechnology and Advanced
Materials, Bar-Ilan University, Ramat-Gan 5290002, Israel  E. Barkai
Department of Physics and Institute of Nanotechnology and Advanced Materials,
Bar-Ilan University, Ramat-Gan 5290002, Israel  Y. Garini  Faculty of
Biomedical Engineering, Technion, Israel Institute of Technology, Haifa
3200003, Israel  S. Burov  Department of Physics and Institute of
Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 5290002,
Israel

###### Abstract

Investigating the dynamics of chromatin and the factors that are affecting it,
has provided valuable insights into the organization and functionality of the
genome in the cell nucleus. We control the expression of Lamin-A, an important
organizer protein of the chromatin and nucleus structure. By simultaneously
tracking tens of chromosomal loci (telomeres) in each nucleus, we find that
the motion of chromosomal loci in Lamin-A depleted cells is both faster and
more directed on a scale of a few micrometers, which coincides with the size
of chromosome territories. Moreover, in the absence of Lamin-A we reveal the
existence of correlations among neighboring telomeres. We show how these
pairwise correlations are linked with the intermittent and persistent
character of telomere trajectories, underscoring the importance of Lamin-A
protein in chromosomal organization.

##  I INTRODUCTION

The eukaryotic cell’s nucleus contains numerous genes arranged on chromosomal
DNA. This compact environment houses dense genetic material and various
nuclear bodies. The control and processing of genetic information rely on
proteins’ ability to locate specific DNA sequences within the crowded nucleus,
which is influenced by its structure and dynamics. Previous studies have
extensively examined the dynamics of single nucleus entities Chuang  _et al._
(2006); Bronstein  _et al._ (2009); Kepten  _et al._ (2015); Bronshtein  _et
al._ (2015); Stadler and Weiss (2017); Vivante  _et al._ (2019). These studies
have revealed the ultra-slow movement caused by crowding. Typically,
measurements of single particle dynamics in the nucleus involve either time or
ensemble averages, which overlook important information and correlations. This
research aims to uncover these correlations using multiple methods,
particularly by simultaneously tracking many individual particles. This
approach allows us to gain insights into directional correlations within a
path, as well as pair-wise correlation functions. Additionaly, this
investigation sheds light on patchy dynamics within the nucleus, where motion
is correlated within specific patches.

Zidovska and colleagues Zidovska  _et al._ (2013) developed a potent technique
called displacement correlation spectroscopy (DCS) to map the dynamics of
large-scale correlated chromatin movements. They found that the motion of
chromatin regions ($4-5\,\ \mu$m) was coherent for several seconds. In our
study, we focus on investigating the motion of individual telomeres. The
coherent motion of chromatin territories reveals how different nucleus regions
communicate and respond to DNA damage Bronshtein  _et al._ (2015).
Nonetheless, it remains unclear how this large-scale coherent motion affects
individual molecules.

Telomeres are specialized DNA sequences present at the cap of chromosomes in
eukaryotic cells that prevent their degradation and fusion with other
chromosomes. In this study, we analyze the movement of individual telomeres to
elucidate the dynamics of chromatin loci and their interrelations. We focus on
two types of correlations: self-correlation along the telomere path and
correlation between pairs of telomeres. How do these correlations manifest in
the behavior of telomeres, and how can they be manipulated in experimental
settings? Our study shows that Lamin-A, a critical protein for nuclear
organization, controls the extent and nature of these crucial correlations.

Studies on the dynamics of single molecules in the cellular environment have
shown that their behavior is primarily governed by caging Bronstein  _et al._
(2009); Bronshtein  _et al._ (2015); Vivante  _et al._ (2020), leading to
anti-persistent motion, although active forces can induce persistent or
directional motion Chuang  _et al._ (2006). Our work reveals that both types
of behavior occur in telomeres within the cell nucleus and that Lamin-A plays
a role in their regulation. Despite previous progress, the presence of
spatially coherent structures implies that correlations may exist between the
dynamics of pairs of particles. Our study addresses this challenge by showing
that chromatin loci such as telomeres must be treated collectively, as many-
body correlations are present.

This investigation involves studying the dynamics of telomeres in two
different types of cells: wild-type cells (WT) that express Lamin A and cells
where Lamin A is knocked out and is not expressed (KO). Our findings show that
telomeres in KO cells exhibit persistent motion, which is absent in WT cells
with intact Lamin-A. We also discover that the pairwise correlation between
particles is associated with the persistent motion of individual telomeres.
These results emphasize the importance of Lamin-A in regulating chromatin
dynamics and reveal the collective behavior of telomeres in the nucleus.

Lamin-A is a crucial component of the nuclear lamina, providing essential
structural support to the nucleus and various factors, such as age, stress,
and disease, can influence its expression De Vos  _et al._ (2010); Melcer  _et
al._ (2012); Mahen  _et al._ (2013). Dysregulation of Lamin-A expression has
been linked to a range of pathological conditions, collectively called
laminopathies Gilchrist  _et al._ (2004). Our research demonstrates that in KO
cells, the magnitude of the telomere dynamics is significantly greater than in
WT cells. In particular, we observe that in the absence of Lamin-A, telomeres
can move persistently over considerable distances, up to 20% of the nucleus
diameter. It indicates that low levels of Lamin-A can disrupt the dynamics in
the whole volume of the nucleus by considerably modifying the active dynamics.

## Results

###  I.1 Experimental setup

Measurements were carried out in cell lines of mouse embryonic fibroblasts
(MEFs), one that is KO and their WT Sullivan  _et al._ (1999). They were
kindly provided by Prof. Susana Gonzalo from Saint Louis University School of
Medicine, St Louis, MO, USA. The cells were maintained in Dulbecco’s high-
glucose modified Eagle’s medium containing 10% fetal bovine serum, 1%
penicillin and streptomycin antibiotics and 1% L-glutamine, all materials are
from Biological Industries, Beit HaEmek, Israel.

For telomere tracking, we used transient labeling of the shelterin subunit
TRF2 fused to green fluorescent protein (GFP), a kind gift from Prof. Sabine
Mai (Manitoba Institute of Cell Biology, Winnipeg University, Canada). The
imaging system includes an inverted Olympus IX-81 fluorescence microscope
coupled to an FV-1000 confocal set-up (Olympus, Tokyo, Japan), and a UPLSAPO
X60 objective lens (NA=1.35). Cells were placed in a $37^{\circ}$C incubator
(Tokai, Shizuoka-ken, Japan) with a 5% $CO_{2}$ level. The measurements were
performed in three dimensions to correct for cell linear and rotational drift,
although the actual dynamic analysis was performed only on the planar $xy$
motion, see details below. Normally 35 equally spaced planes were measured
with a lag time of 18.5 seconds, a voxel size of 108 X 108 X 350 $nm^{3}$ and
50 or 100 time-points. The planar resolution of our measurements is 170 nm,
and the spatial precision is better than 10 nm (localization error), based on
both calculation and measurement of a fixed sample. Finally, Quantitative
single particle tracking (SPT) analysis of time-lapse image sequences was
performed.

###  I.2  Correction of linear and rotational drift.

As mentioned above we employed confocal microscopy to obtain 3D data sets of
telomere’s trajectories. As the cells are alive, they tend to move during the
data acquisition. In order to describe the motion of telomeres with respect to
the nucleus, the nucleus motion has to be subtracted from the telomere’s
trajectories. Indeed, the observation of telomere dynamics before and after
corrections reveals that it is necessary to rectify the translational drift,
and also but less importantly the nucleus rotation (Supplementary Material
Fig. S1).

We now explain how to correct telomere trajectories from cell motion, this is
particularly important as we wish to explore the correlations between pairs of
telomeres. To see how the drift of each different cell behaves with time
($t_{i}$), we computed the drift vector
$\vec{R}(t_{i})=(R_{x}(t_{i}),R_{y}(t_{i}))$, where each component is the
respective average center of mass:
$R_{x}(t_{i})=\sum_{j=1}^{M}x^{j}(t_{i})/M$, and
$R_{y}(t_{i})=\sum_{j=1}^{M}y^{j}(t_{i})/M$, (where $j=1,\ldots,M$ and $M$ is
the number of tracked telomeres in the nucleus). Fig. 1 a-b) shows the
relative magnitude of the drift vector $|\vec{R}(t)|-|\vec{R}(t_{0})|$ as a
function of time, with $|\vec{R}(t_{i})|=\sqrt{R_{x}^{2}+R_{y}^{2}}$, for few
cells in both WT and KO cells. The relative drift vector indicates the
translational drift of the nucleus. In all cases the relative magnitude of the
drift vector follows a smooth nearly linear behavior with time. More
precisely, for twelve different cells displayed (8 WT and 4 KO), we observe
that seven of them (60%) has almost zero drift and the others cells exhibit a
quasi-monotonous trend with slope close to $\sim 0.1$, i.e. $\sim\pm 0.005\mu
m/s$.

Figure 1:  Translational and rotational drift versus time for telomeres in WT
and KO cells, before detrending procedure. a) For WT cells and total
measurement time $t=50\ast\Delta$, relative magnitude of the drift vector
$|\vec{R}(t)|-|\vec{R}(t_{0})|$ versus time. b) The same as in a) for KO cells
and $t=100\ast\Delta$. c-d) Corresponding rotational drift in degrees versus
time for WT and KO cells respectively. The time is measured as
$t=n\ast\Delta$, with $n=[1,50]$ or $n=[1,100]$.

To inspect the behavior of the rotation of the cell, for all the trajectories
inside a given nucleus, we computed at each time frame the average rotation:
$\phi(t_{i})=\sum_{j=1}^{M}\theta^{j}(t_{i},\Delta t)/M$, with
$\theta^{j}(t_{i},\Delta t)$ the relative angle between successive vectors
defined by Eq. (I.5). Here $\phi(t_{i})$ indicates at each time frame the
average rotation around the center of mass of all telomeres in the cell. Fig.
1 c-d) shows the rotation angle along the trajectory for few cells in both WT
and KO cells. As one can see, the rotation is very small in most of the cases.

Given these rather smooth/stable drift and small rotation angles, the drift
subtraction and rotation correction procedures that we used are completely
justified as one can find for other similar experimental set-ups Zhang  _et
al._ (2022). A complete detail of our techniques are described in the
Materials and Methods section and in the SM.

###  I.3 Tracking telomeres in the nucleus cell

We tracked simultaneously fluorescently tagged telomeres in WT and KO MEF cell
lines, for a total measurement time of either $T=925$ sec or $1850$ sec, see
Materials and Methods V for details. Typical telomere trajectories are shown
in Fig. 2 for WT and KO cells. As mentioned above, while in WT cells telomeres
are mainly confined in regions of typical length $\sim 100-200$ nm, in KO
cells telomeres cover significantly larger distances up to $\sim 3\mu$m.
Strikingly, we note that in KO cells, the diffusion not only spans a
significantly larger volume, but one can notice that there are time-windows in
which the motion is highly directional (Fig. 2 b). This variation in the
directional dynamics, either caging with no preferential direction in WT cells
or unidirectional motion in KO cells, can be attributed to the respective
differences in the nuclear structure of each type of cells.

Figure 2: Trajectories of telomeres in the nucleus. a) Wild type (WT) cells.
b) Cells depleted of lamin A (KO). Telomeres moving in cells depleted of
Lamin-A cover larger distances compared with telomeres diffusing in WT cells,
and directional motion often occurs. The color gradient represents time
measured as $t=n\ast\Delta$ with $n=[1,100]$. See Fig. S4 in SM for a detailed
description of telomere tracks in WT cells.

It was lately found that the dynamics of gene loci in WT cells is sub-
diffusive, and it can turn into normal-like diffusion in KO cells Bronshtein
_et al._ (2015). It was suggested Bronshtein  _et al._ (2015) that Lamin-A
forms intra and inter-chromosomal cross-links of DNA, thereby turning the free
long chromosome polymer into a gel-like structure, which naturally exhibits
highly restricted dynamics in space and time. Moreover, it was shown that by
transfecting Lamin-A back to these cells, the sub-diffusive nature of the
chromatin dynamics is reestablished. It demonstrates that Lamin-A is of major
importance for maintaining a constrained diffusion. Hereinafter, we explain
how the presence/absence of Lamin-A protein renders a different behavior in
the spreading of telomere trajectories, directed single particle dynamics, and
correlation between pairs of particles.

###  I.4 Laplace-like PDF of the telomeres displacement

After detrending telomeres trajectories in the cell nucleus, first we analyze
the dynamics of telomeres in WT and KO cells by calculating the distribution
of the traced displacements over different time intervals $\Delta t$. The
distribution of displacements for WT telomeres was previously studied in
Stadler and Weiss (2017). Here we ask how does Lamin-A depletion modify this
distribution and show that it has a dramatic effect on the scale of the
motion. However the distribution of displacements, as a stand alone does not
yield insights on the coherent motion and correlations amongst telomeres,
features that will be treated later.

The displacements along the $x$ and $y$ axes are defined as $\Delta
X=X(t+\Delta t)-X(t)$ and $\Delta Y=Y(t+\Delta t)-Y(t)$; with $t=n\ast\Delta$
and $\Delta\xi=\\{\Delta X,\Delta Y\\}$. Fig. 3 shows the positional PDF in
semi-log scale for telomeres in WT and KO cells with $\Delta t=10\Delta$, see
panels a and b respectively. For both cell types we found that the positional
PDF collapses to the same plot fitted by a model composed of a linear
combination of Gaussian and Laplace distributions (solid lines), which is
defined as

$\displaystyle
P(\Delta\xi)=(1-w)\frac{e^{-\frac{\Delta\xi^{2}}{4\sigma^{2}}}}{\sqrt{4\pi\sigma^{2}}}+w\frac{e^{-\frac{|\Delta\xi|}{\lambda}.}}{2\lambda},$
(1)

here $\lambda$ is the decay length of the exponential tails, $\sigma^{2}$ the
variance of the Gaussian central part and $w$ the weight of the Laplace
distribution. The analysis of residuals shown in Fig. S6 in SM corroborates
that Eq. (1) is a good model for describing telomere dynamics. We also tried
to fit the data to a linear combination of two Gaussian curves. We found that
the linear combination of a Gaussian and Laplace distributions gives
significantly better results, see SM for details.

There are several noteworthy observations about the distributions of telomere
displacement (Fig. 3). First, we note that there are few displacements in Fig.
3a) at large positive range. They correspond to only 0.15% of the total set of
points and can be considered as outliers. Secondly, note that there is a
slight difference in the x and y coordinates in Fig. 3b). It results from only
3 out of 7 cells that are measured for KO cells (see Fig. S8 in SM). The
nucleus of the cell is not always isotropic, so these results can occur
naturally. Additionally, since cells and nuclei are non-identical objects, it
is unsurprising that some of the cells are more anisotropic than others.

Packets with exponential decaying tails have been thoroughly investigated both
theoretically Chaudhuri  _et al._ (2007); Hapca  _et al._ (2009); Chechkin
_et al._ (2017); Barkai and Burov (2020); Chubynsky and Slater (2014); Sposini
_et al._ (2018); Lanoiselée and Grebenkov (2018); Wang  _et al._ (2020a);
Hidalgo-Soria  _et al._ (2021) and experimentally Leptos  _et al._ (2009);
Wang  _et al._ (2009, 2012); Lampo  _et al._ (2017); Janczura  _et al._
(2021); Sabri  _et al._ (2020); Åberg and Poolman (2021); Corci  _et al._
(2023); Hu  _et al._ (2023). Fig. 3 c-d show the distributions of the
normalized displacements for $\Delta t=\\{\Delta,10\Delta,20\Delta\\}$. The
normalization is performed by dividing the displacements $\Delta\xi$ by the
standard deviation, defined as $\sigma_{\Delta\xi}=\sqrt{Var(\Delta\xi)}$. As
one can see in both cell types and for all $\Delta t$, the PDF is well
described by Eq. (1) (black solid line). For comparison in each case the
Gaussian distribution is shown as a black dashed line, further details of the
fitting parameters are shown in the SM. The striking features are the
exponential tails and the larger spread of the KO case, compared to the WT
case (Fig. 3 a -b).

Figure 3: Distribution of telomere displacements $\Delta\xi=\\{\Delta X,\Delta
Y\\}$ (red triangles and blue squares in correspondence) WT a) and KO b)
cells, with $\Delta t=10\Delta$. c-d) Same data for normalized displacements
$\Delta\xi/\sigma_{\Delta\xi}$, for WT and KO cells respectively for $\Delta
t=\\{\Delta,10\Delta,20\Delta\\}$, and X-Y directions (colored triangles
x-coordinate and colored squares y-coordinate). Here $\sigma_{\Delta\xi}$ is
the respective standard deviation. For comparison in each case Gaussian
statistics is shown in black dashed lines. In all cases for any time scale,
$\Delta\xi$ and cell type the positional PDF collapses into the same plot
given by the model of a superposition of a Gaussian center and exponential
tails Eq. (1) (solid lines). Nonetheless, the spreading of the positional PDF
is larger for telomeres in KO cells compared with WT cells.

Independently of the level of Lamin-A in the nucleus, the short displacements
of the telomeres are described by Gaussian statistics and for large
displacements the distribution has exponential tails, which coincides with the
behavior of the Laplace distribution. A strong effect on the spreading of the
displacements of telomeres was found when Lamin-A is completely depleted. From
the fitting of Eq. (1) to experimental data, the decay length obtained by the
corresponding fitting, was $\lambda\sim 0.4$ $\mu m$ for telomeres in KO cells
when $\Delta t=20\Delta$, which is four times larger compared with WT cells,
$\lambda\sim 0.1$ $\mu m$. The average total displacement
$\langle|\Delta\xi|\rangle=\langle|\xi(T)-\xi(0)|\rangle$, behaves similarly
with $\langle|\Delta X|\rangle\sim 0.5$ $\mu m$ and $\langle|\Delta
Y|\rangle\sim 0.6$ $\mu m$ for telomeres in KO cells compared to
$\langle|\Delta X|\rangle\sim 0.15$ $\mu m$ and $\langle|\Delta Y|\rangle\sim
0.15$ $\mu m$ for telomeres in WT cells. Despite this difference in the
spreading, for both cases we found that the decay length $\lambda$ in Eq. (1)
grows as the square root of $\Delta t$, i.e. $\lambda\propto\sqrt{\Delta t}$,
which is consistent with a linear MSD (see Fig. S5 and Fig. S9 in SM). This
dependence of $\lambda$ was also found in other heterogeneous systems Wang
_et al._ (2009); Guan  _et al._ (2014); Miotto  _et al._ (2021); Pastore  _et
al._ (2022).

The information gained from Fig. 3 is naturally an averaged quantity, namely
one cannot pinpoint the origin of the exponential decaying tails, i.e. Laplace
diffusion Laplace (1986). Does it appear because different telomeres exhibit
distinct types of motion? Or is each trajectory heterogeneous, which means
that there is an interplay of fast and slow motion? Fig. 3 suggests that the
Gaussian part of the PDF is associated with a slow dynamics phase (for short
distances), while the Laplace exponential decay is associated with fast motion
(for long distances). In the section Model, we show how these slow and fast
phases are related to caging, pure diffusion and directed motion.

###  I.5 Caged and Directed motion of telomeres inside the nucleus

The positional PDF shown above emphasizes the spatial and temporal scales of
telomeres in the nuclei of WT and KO cells, but the directional trajectories
that only appear in short time-windows are averaged out. In order to expose
its nature, we use a statistical tool that calculates the relative angle
between each two points along the particle trajectory separated by $\Delta t$.
Namely the absolute value of the turning angle which is denoted by
$\theta(t;\Delta t)$ Burov  _et al._ (2013); Bos  _et al._ (2015); Kadoch  _et
al._ (2017); Fang  _et al._ (2022); Balcerek  _et al._ (2024), see inset in
Fig. 4 a. The relative angle for any $\Delta t$ is defined as:

$\displaystyle\cos\theta(t,\Delta t)=\frac{\Delta\vec{X}(t;\Delta
t)\cdot\Delta\vec{X}(t+\Delta t;\Delta t)}{||\Delta\vec{X}(t;\Delta t)||\,\,\
||\Delta\vec{X}(t+\Delta t;\Delta t)||},$

with the vector displacements $\Delta\vec{X}(t;\Delta t)=\Big{(}X(t+\Delta
t)-X(t),Y(t+\Delta t)-Y(t)\Big{)}$ and $||\cdot||$ the Euclidean norm.

We compute for each $t$ the relative angle between successive displacements
within different lapse-times $\Delta t=\\{\Delta,5\Delta,10\Delta,20\Delta\\}$
and calculate the respective distribution of the relative angle. In persistent
trajectories a particle tends to move in the same direction, exhibiting small
values of $\theta(t;\Delta t)$. On the other hand, in anti-persistent tracks a
particle flips its direction of movement, remaining caged, and
$\theta(t;\Delta t)$ has values close to $\pi$ Burov  _et al._ (2013); Weber
_et al._ (2012a); Sadegh  _et al._ (2017).

Fig. 4 a shows the relative angle distribution for telomeres in WT cells, as a
function of the normalized variable $\theta/\pi$. We see that for any time
scale, the relative angle distribution has a peak at $\pi$ and hence the
displacements are negatively correlated, see the autocorrelation function of
the displacements in Fig. S10 a-b in SM. Therefore the process is anti-
persistent, implying caged dynamics Weber  _et al._ (2012a); Burov  _et al._
(2013); Sadegh  _et al._ (2017).

In contrast, for KO cells $P(\theta)$ has a maximum at $\theta=0$ for and
$\Delta t$ studied, see Fig. 4 b. This implies that the motion is now directed
for $\Delta t\leq 20\Delta$. This persistent behavior in telomeres moving in
KO cells is translated into trajectories like those displayed in Fig. 2 b,
where telomeres tend to continue in the same direction for a significant time
window.

There is a striking difference between the behavior of $P(\theta)$ for both WT
and KO cells (Fig. 4) and the case of regular Brownian motion, e.g. a random
walk. For Brownian dynamics the distribution of the relative angle is uniform
in the whole range of $\theta$, irrespective of $\Delta t$ Burov  _et al._
(2013); Sadegh  _et al._ (2017). Indeed, every movement is uncorrelated with
the previous one, so every direction is equally probable (for any time scale).
The non-uniform profiles of $P(\theta)$ for the WT and KO configurations
reveal the temporal structure of the motion. For the KO case within time
scales of $t\lesssim 300$ sec, the motion is directed as the particle has a
preference to travel in the previously explored direction. On the other hand,
WT dynamics shows the presence of caging at any time scale (a peak for the
relative angle distribution at $\theta=\pi$, see Fig. 4 a).

Figure 4:  Probability distribution of the deflection angle along telomeres
trajectories as a function of the normalized angle $\theta/\pi$ for few
different lag-times $\Delta t=\\{\Delta,5\Delta,10\Delta,20\Delta\\}$. a) WT
cells, the distribution mode is at $\theta=\pi$, indicating anti-persistent
diffusion. Inset: examples of relative angles $\theta_{i},\theta_{j}$ defined
by Eq.(I.5). b) KO cells, the distribution mode is at $\theta=0$ indicating
the presence of directed motion. The black dashed line, represents the
corresponding angular distribution for a random walk (a uniform distribution).

The difference between caging and directed motion, becomes even clearer when
we observe the evolution of the average relative angle as a function of the
sampling time (Fig. 5). For trajectories of telomeres in WT cells, the
ensemble average relative angle ($\langle\theta(t,\Delta t)\rangle/\pi$) and
the ensemble time average of the relative angle
($\langle\overline{\theta}(\Delta t)\rangle/\pi$) are displayed in red circles
and blue circles respectively. For tracks in KO cells, the respective averages
are shown in cyan and magenta circles, the black dashed line indicates the
average relative angle corresponding to a random walk, i.e.
$\langle\theta(t,\Delta t)\rangle/\pi=0.5$ . Following results shown in Fig.
4, in WT cells we observe caging dynamics for all measurement times. We
observe a reduction of the average relative angle of 10% from the short-time
gap value to the long-time gap. For KO cells the average relative angles are
always less compared with the random walk reference value achieving a
reduction of 8%, implying an increase in persistent motion along trajectories.
The minimum of the average relative angle is attained at $\Delta
t=15\Delta\sim 270$ sec.

The relative angle statistics provides strong evidence for the existence of
mechanisms that trigger both caged and directed motion inside the cell
nucleus, and this difference in the dynamics depends on the presence/absence
of Lamin-A protein. Next we study how these directional dynamics are
correlated on pair of telomeres in WT and KO cells.

Figure 5: Ensemble average relative angle ($\langle\theta(t,\Delta
t)\rangle/\pi$) and ensemble time average relative angle per trajectory
$\langle\overline{\theta}(t,\Delta t)\rangle/\pi$ as a function of $\Delta
t=n\Delta$ with $\Delta=18.5$s and $n=[1,20]$. For WT cells the ensemble
average (red circles) and the ensemble time average (blue circles) for all
$\Delta t$ are above the corresponding values of a random walk, i.e. 0.5
(black dashed line), indicating dominance of anti-persistent motion or caging.
For KO cells the ensemble average (cyan circles) and the ensemble time average
per trajectories (magenta circles) for any $\Delta t$ are below 0.5, which
implies presence of persistent behavior for any time scale $\Delta
t<20\Delta$.

###  Pair correlation for telomere dynamics

We now focus on the correlated motion of pairs of telomeres, and show that the
motion cannot be considered as mutually independent. In other words we search
for the spatial correlation length; a scale on which dynamics is correlated.
We compute the pair correlation of telomeres as follows: for a pair of
telomere tracks labeled as telomere $A$ and telomere $B$, in the same nucleus,
we measure the initial separating distance
$d_{AB}=\sqrt{(x^{A}(t_{1})-x^{B}(t_{1}))^{2}+(y^{A}(t_{1})-y^{B}(t_{1}))^{2}}$.
We then compute the correlation function:

$\displaystyle\beta_{A,B}(\Delta
t,d_{AB})=\frac{1}{N}\displaystyle\sum\limits_{n=1}^{N}\hat{\Delta
X}_{A}(t_{n},\Delta t)\cdot\hat{\Delta X}_{B}(t_{n},\Delta t),$

with $N$ the number of sampling points in each trajectory, $t_{n}=n*\Delta$,
$\hat{\Delta X}_{A}(t_{n},\Delta t)=\overrightarrow{\Delta X}_{A}(t_{n},\Delta
t)/||\overrightarrow{\Delta X}_{A}(t_{n},\Delta t)||$, where
$\overrightarrow{\Delta X}_{A}(t_{n},\Delta
t)=\overrightarrow{X}_{A}(t_{n}+\Delta t)-\overrightarrow{X}_{A}(t_{n})$ and
$\overrightarrow{X}_{A}(t_{n})=(x^{A}(t_{n}),y^{A}(t_{n}))$. We compute the
ensemble average $\langle\cdot\rangle$ of Eq. (Pair correlation for telomere
dynamics) per cell type and bins for different values of $d_{AB}$ (see details
in SM). Then, for a fixed value of $d_{AB}$, the average per cell type (WT or
KO) is performed. For uncorrelated random walkers the pair correlation has
zero values for any $d_{AB}$ and $\Delta t$, as shown in Fig. S12 in SM.

Fig. 6 presents the pair-correlation of telomeres for different time-scales
(Eq. (Pair correlation for telomere dynamics)) as a function of the initial
spreading distance between them. As one can see, in Fig. 6 for both cell types
when telomeres are initially close to one another, the pair correlation is
positive which means that they move coherently in the same direction. This
effect is clearly more pronounced for KO cells, where the pair correlation
exhibited significant positive values, for which the maximum attained is at
0.38. Particularly telomeres in WT cells exhibit regions of $\sim 1\mu$m in
extent where the pair correlations are positive, see Fig. 6 a. For KO cells
the extent of positive correlations is $\sim 4\mu$m, see Fig. 6 b.
Interestingly, in both cell types when telomeres are far away from each other
inside the same cell, they move coherently in the opposite direction.
Remarkably we found that irrespectively the presence of Lamin-A, these pair
correlations exhibit the same shape in both cell types, but the correlation
level is different. Given the above observations, we developed a physical
model for telomeres dynamics that can explain these results, paying specific
attention to the caging and directed motion dynamics.

Figure 6: Pair correlation of telomeres as a function of their initial
separating distance. a): For telomeres diffusing in WT cells, corresponding
ensemble average of the pair correlation of telomeres defined by Eq. (Pair
correlation for telomere dynamics), for different measurement times $\Delta
t=\\{\Delta,5\Delta,10\Delta,20\Delta\\}$ (color triangles). b): The same as
above but for telomeres in KO cells. For WT cells there are very small
positive correlations of $\sim 0.05$ at a separation distance of $\sim 1.4\mu
m$ (for all time-scales) and a small negative correlation of $\sim 0.2$ for
large distances. For KO cells the pair correlation exhibits large positive
values (maximum of 0.38) at short telomere distances of $\sim 1.4\mu m$, and a
significant negative correlation of $-0.25$ at larger distances. This
indicates that for both cell types the same correlation pattern appears.
Nevertheless, it is significantly more pronounced in KO cells relative to WT
cells.

###  I.6  Model

Based on the experimental data of telomere dynamics in WT and KO cells, we
developed a physical model that fits the main features exhibited by
experimental data, like the exponential decay in the distribution of
displacements and the relative angle distributions. All the parameters
employed in our model are obtained from experimental data except for a few
free parameters dealing with the width of the fluctuations of the angular
increments.

Figure 7: Representative time series, with predominantly small jumps a), large
jumps b) and switching between small and large jumps c) of telomeres in WT
cells. In d)-f) we show the corresponding two state series representation of
the length size jumps $L(t,\Delta t)=\\{0,1\\}$ (cyan solid lines). For
telomeres in KO cells we show representative tracks with predominantly non-
persistent dynamics g)-h) and persistent one i). The corresponding two state
series representation of the level of persistence ${\cal P}(t,\Delta
t)=\\{0,1\\}$ are shown in j)-l) (magenta solid lines). The color gradient
represents time as $t=n\ast\Delta$ with $n=[1,50]$. We employed for the two
state analysis $\Delta=10\Delta$ for WT cells and $\Delta=5\Delta$ for KO
ones.

As reported in many single particle tracking experiments we see a wide
fluctuation in behaviors of individual molecules Bénichou  _et al._ (2011);
Weron  _et al._ (2017). For instance for telomeres in WT cells we found
intermittency between caged and diffusive intervals, see Fig. 7 a-f). This was
deduced by computing the local convex hull (LCH) at each time frame and given
a specific time window, see Fig.8 and Material and Methods. Specifically at
each time frame for the respective LCH we obtain the average largest diameter
labeled as $\langle\mathcal{D}_{max}\rangle$. Then we define a criterion for
discriminating between the “large jumps” or diffusive phase and “small jumps”
or caged phase as following: If $\mathcal{D}_{max}(t,\Delta
t)>\langle\mathcal{D}_{max}\rangle$ the LCH contains “large jumps” (and its
labeled as $L(t,\Delta t)=1$ ). In other case, $\mathcal{D}_{max}(t,\Delta
t)<\langle\mathcal{D}_{max}\rangle$, the LCH conatins “small jumps” (calling
$L(t,\Delta t)=0$). Thus we generate a dichotomous process $L(t,\Delta
t)=\\{0,1\\}$ see cyan lines in Fig. 7 d-f). For telomeres in KO cells using a
critical value for the relative angle at each time frame and within a specific
time interval, we developed a similar decomposition of trajectories between
persistent (${\cal P}(t,\Delta t)=1$) and non-persistent intervals (${\cal
P}(t,\Delta t)=0$), see magenta lines in Fig. 7 j-l) and detalis in Materials
and Methods.

Figure 8:  Schematic representation of the LCH (red polygons) for a typical
telomere trajectory (blue line). Typical LCH areas and maximum diameters are
shown in red color and black dashed line respectively.

Therefore, for the dynamics of telomeres in WT cells, we suggest a modified
Pearson random walk model. Recall that for a Pearson random walk Pearson
(1905) at each time step, a random angle is chosen on the 2D plane and then
the walker advances a distance $r$ on a straight line. Accordingly we assume a
two-state model of caging and pure diffusion, where intermittent switching
occurs. During caging, the Pearson random walk is assumed in a circular cage
of radius $r_{c}$ with reflecting boundary conditions. For the pure diffusive
phase $r_{c}\rightarrow\infty$. We also employ different displacement lengths
during different phases, namely $r_{S}<r_{L}$, with $r_{S}$ the displacement
during caging and $r_{L}$ during pure diffusion. The duration of each of the
two phases is randomly selected from an exponential distributions that are
determined from the experimental data, see Fig. S13 of the SM. The complete
algorithm is presented in Materials and Methods VII.

Fig. 9 demonstrates a comparison of the experimental (Figs.3, 4) and our
simulation results (black triangles and gray squares). Fig. 9 a,b shows the
comparison the displacements and relative angle in WT cells. The agreement
between theory and experiment is excellent for all the observed data. The
simulation data precisely follows the Gaussian distribution at short
displacement and Laplace tails for larger ones Eq. (1). It also nicely follows
the angular distribution data for all lag-times. Remaining statistics are
displayed in Fig. S15 in SM.

As explained before, previous works demonstrated the lack of chromatin cross
links, which leads to directed dynamics for $\Delta t\leq 20\Delta$. And as we
show in Fig. 7, in KO cells we found that telomere trajectories display a
switching between non-persistent and persistent intervals. Given this our
model assumes intermittency between pure diffusion and directed dynamics.
Therefore in the KO case the theory presents a two state model with a regular
Pearson random walk and varying persistent motion. Our model works as follows.
Initially, for each track we draw a random value for $\theta_{int}$ from a
uniform distribution $U(0,2\pi)$. We then build the trajectory with the first
step equal to $(x_{1},y_{1})=(r\cos\theta_{int},r\sin\theta_{int})$ and each
following step equal to
$(x_{j},y_{j})=(x_{j-1}+r\cos\tilde{\theta}_{j},y_{j-1}+r\sin\tilde{\theta}_{j})$.
Here $\tilde{\theta}_{j}=\tilde{\theta}_{j-1}+\tilde{\Delta\theta}_{j}$, where
$\Delta\theta_{j}$ defines the level of persistence, and it is drawn at each
step from a uniform distribution $\tilde{\Delta\theta}_{j}\sim
U(-\theta_{c},\theta_{c})$. Finally, a new $\theta_{int}$ is drawn again after
$20\Delta$ sec, see further details in Materials and Methods VII. By
introducing $\theta_{int}$ at the beginning of each persistent phase, we
correlate the direction of successive segments of the trajectory. This is done
during time scales less than $20\Delta$, after this time a new $\theta_{int}$
is drawn and it is correlated in the successive segments during the same lapse
of time.

We have the representative segments of trajectories, for instance: by
correlating directed stages as above, $\theta_{c}\rightarrow 0$ implies
alternating between straight lines and random walk paths, rendering persistent
segments during a time scale of $20\Delta$. And on the other hand, when
$\theta_{c}\geq\pi/2$ we have random walk behavior along the corresponding
segments of the trajectories. The distribution of displacements and one of the
relative angles obtained from simulated trajectories are shown in grey squares
and black triangles in Fig. 9 c and d respectively. We find a good agreement
with those obtained from experimental data. A Gaussian bulk with exponential
decaying tails for $P(\Delta\xi)$ and a zero peaked angular distribution are
recovered, see further statistics in Fig. S17 in SM.

Figure 9: Comparison between experimental data of telomeres in WT and KO cells
with simulations. a) For telomeres in WT cells and simulation data of a two
state model with caging and diffusive motion, distribution of displacements
$\Delta\xi=\\{\Delta X,\Delta Y\\}$ with $\Delta t=10$. Experimental data is
shown in red triangles and blue squares and simulation data in black triangles
and grey squares respectively. b) For telomeres in WT cells and simulation
data of a two state model with caging and pure diffusive motion, relative
angle distribution experimental data in color lines and simulation data in
grey squares ($\Delta t=\Delta$) and black triangles ($\Delta t=10\Delta$). c)
and d) the same as in a) and b) but for telomeres in KO cells and simulation
data from a two state model with Pearson and varying persistence random walks.
The parameters used in the simulations are specified in Materials and Methods
VII.

##  II  DISCUSSION

By following the details of the dynamics along single telomere trajectories,
major insights were exposed. First, we identified a two-state behavior in both
cell types WT and KO. In WT cells, the trajectories switch between caged and
Brownian normal diffusion, while in KO cells they switch between pure
diffusion and directed motion. Second, we found that in both cell types the
dynamics lead to Gaussian statistics for small displacements and Laplace
(exponential tails) statistics at large displacements. Our analysis revealed
that the Gaussian statistics result from caging in WT cells and free diffusion
in KO cells. Additionally, the Laplace tails results from the large
displacements of the free diffusion in the WT cells and by the directed motion
in KO cells. Thirdly, we note that when the dynamics of WT and KO cells is
compared, it is clear that the directional motion observed in KO cells is
suppressed by the effect of Lamin-A in WT cells. It also explains the
difference in the large spatial spread of the motion in KO cells versus small
spread in WT cells.

Interestingly for trajectories in KO cells, we found persistent dynamics for
times scales $\Delta t<20\Delta\sim 370$ sec. For this case, if after each
switching between the phases the telomere “forgets” the directionality of the
previously visited directed stage, the observed directionality is expected to
disappear, in the sense that a uniform distribution of the relative angle
describes such dynamics. Instead, the observed behavior of the relative angle
PDF shows that the direction is persistent. Furthermore, we revealed that this
persistent motion is connected to pair wise correlations of telomeres, Fig. 6
b. This suggests, that the directionality field in the nucleus of the cell
when Lamin-A is absent, is patchy, in the sense that different micro regions
impose quenched directionality of the motion. We may speculate that the
correlated directed motion we found, is related to chromatin territory Cremer
and Cremer (2001); Meaburn and Misteli (2007); Lieberman-Aiden  _et al._
(2009); Cremer and Cremer (2010); Li  _et al._ (2012); Vivante  _et al._
(2019), or to the vicinity of the chromatin loci to the boundaries of the
cell, or possibly to spatial domains with high concentration of activity,
possibly ATP driven Weber  _et al._ (2012b), issues that must be studied
later.

We speculate that the observed structure of directional correlations (Fig. 6)
may be the product of the contraction or expansion of the nucleus during the
different stages of the cell cycle. A similar idea was proposed in Kepten  _et
al._ (2015), in the context of long-range anisotropy-enhanced parallel
diffusion of centromeres and telomeres. The short distances scale dynamics is
most likely related to the coherent motion of chromatin loci found in Zidovska
_et al._ (2013) using DCS technique. The two works show that active forces are
correlated on scales of micrometers.

The exponential decay of the PDF in the displacements of telomeres that was
observed in this study and in Stadler and Weiss (2017), has also been observed
in a several number of systems from glassy systems Chaudhuri  _et al._ (2007)
to diffusion of molecules within living cells Leptos  _et al._ (2009); Wang
_et al._ (2009, 2012); Lampo  _et al._ (2017); Janczura  _et al._ (2021);
Sabri  _et al._ (2020); Åberg and Poolman (2021); Corci  _et al._ (2023) and
tracers in colloidal suspensions Weeks  _et al._ (2000); Kegel and van
Blaaderen (2000); Weeks and Weitz (2002); Chakraborty and Roichman (2020);
Lavaud  _et al._ (2021). Laplace diffusion has been theoretically modelled, by
diffusing diffusivity models Hapca  _et al._ (2009); Chubynsky and Slater
(2014); Chechkin  _et al._ (2017); Sposini  _et al._ (2018); Lanoiselée and
Grebenkov (2018); Wang  _et al._ (2020a); Hidalgo-Soria  _et al._ (2021),
spatial quenched disorder Luo and Yi (2018, 2019); Postnikov  _et al._ (2020);
Pacheco-Pozo and Sokolov (2021), and interacting particle approaches Baldovin
_et al._ (2019); Hidalgo-Soria and Barkai (2020); Nampoothiri  _et al._ (2021,
2022). It was shown that Laplace exponential statistics for the decay of
positional PDF attains similar universal features as the Gaussian behavior of
the PDF for the central part of the distribution Barkai and Burov (2020); Wang
_et al._ (2020b). In this study, we showed for KO cells (via numerical
simulations) that a correlated run and tumble model also leads to Laplace
exponential decay. Nevertheless, the usual run and tumble models Thiel  _et
al._ (2012); Taktikos  _et al._ (2014); Fedotov  _et al._ (2022), as they
traditionally have been developed for explaining the dynamics of bacteria,
lack the long time correlations along trajectories that we have employed.

Directional dynamics appears not only for run and tumble models Thiel  _et
al._ (2012); Taktikos  _et al._ (2014); Fedotov  _et al._ (2022), but also for
active Brownian particles Basu  _et al._ (2018); Majumdar and Meerson (2020);
Malakar  _et al._ (2020) and Lévy walks Zaburdaev  _et al._ (2015); Kanazawa
_et al._ (2020); Wang  _et al._ (2020c); Han  _et al._ (2021); Mukherjee  _et
al._ (2021). The majority of these models do not establish intrinsic
directionality as a mechanism of active motion, as we suggest here for
telomeres in KO cells. Another popular approach to model active matter
utilizes Langevin equations with active noises Bechinger  _et al._ (2016);
Jeanneret  _et al._ (2016). The absence of spatial correlations (for the
noise) in these models make them insufficient as a framework for our
experimental system. Interestingly in Ro  _et al._ (2021) the emergence of
long-ranged active dynamics has been linked with the presence of disordered
landscapes, where the concentration of active matter is non-uniform, in some
cases fractal. Our results for KO cells are in agreement with the latter idea
since we saw that the inherent directionality of telomeres tracks is quenched,
and temporally correlated for hundreds of seconds.

##  III  SUMMARY

Our study focuses on several aspects of the dynamics of telomeres, for WT and
KO cells. In both cases we observe that the spreading of packets of particles
is not described by Gaussian statistics, but rather by Laplace-like
exponential tails. As well known exponential tails decay slowly compared to
Gaussian tails, however this observation as a stand alone does not lead to a
direct insight with respect to coherence and persistence. A dramatic increase
in the magnitude of displacements is found for the KO cells. This is related
to the fact that KO cells malfunction. The motion of single particles exhibits
a combination of Brownian and trapping/directed motion. The directed motion in
KO cells is a clear indication for persistence and correlation between micro
displacements along the paths of individual particles. This is accompanied
with pair wise correlations between pair of particles, indicating coherent
motion of single molecules  in the cell nucleus, which as far as we know is
the first observation of such a phenomenon. The level of directional motion
crucially depends on Lamin-A’s presence. In contrast, the Laplace spreading of
the telomeres dynamics and the overall structure of the pairwise correlation
function does not depend on Lamin-A.

Unexpectedly, when we study the directional correlations of pairs of telomeres
inside the same cell, we observe that there is a general structure for such
correlations in WT and KO cells, i.e. positive correlations for short
distances between telomeres and negative values for large distances (Fig. 6).
The latter is important because independently of the presence of Lamin-A, it
suggests the existence of long-ranged forces that create this coherent motion
in the nucleus. Determining the origin of such forces is set up as an open
question.

##  IV MATERIALS AND METHODS

##  V Analysis of Trajectories

We perform SPT analysis of fluorescently labeled telomeres in WT cells, where
363 trajectories of telomeres were analyzed, 270 tracks with $\Delta
t=50\Delta=925$ sec and 93 with $\Delta t=100\Delta=1850$ sec. In the case of
telomeres in KO cells, 128 trajectories were analyzed, 59 tracks with $\Delta
t=50\Delta=925$ sec and 69 with $\Delta t=100\Delta=1850$ sec. Particularly
for WT cells and measurement time $T=50\Delta$, we employed 8 different cells
and on each one we tracked the following numbers of trajectories: cell #1 25
tracks, cell #2 35 tracks, cell #3 38 tracks, cell #4 25 tracks, cell #5 35
tracks, cell #6 48 tracks, cell #7 25 tracks, and cell #8 39 tracks ( with a
total of 270 trajectories). For WT cells and $T=100\Delta$, we employed 5
different cells for which we trace: cell #1 21 tracks, cell #2 24 tracks, cell
#3 20 tracks, cell #4 13 tracks, and cell #5 15 tracks (with a total of 93
trajectories). In the case of KO cells and measurement time $T=50\Delta$, we
used 3 different cells each one following: cell #1 19 tracks, cell #2 17
tracks, cell #3 23 tracks (with a total of 59 trajectories) . And finally for
KO cells and $T=100\Delta$, we employed 4 different cells with: cell #1 18
tracks, cell #2 26 tracks, cell #3 10 tracks, and cell #4 15 tracks (with a
total of 69 trajectories).

In all cases a lag time of $\Delta=18.5$ sec was employed, and after
performing the respective SPT, a nucleus drift and rotation correction were
applied. For the correction of the nucleus drift, per cell at each multiple of
the lag time $t_{i}=i\ast\Delta$, the center of mass of each coordinate was
computed: $\langle x(t_{i})\rangle=\sum_{j=1}^{M}x^{j}(t_{i})/M$, $\langle
y(t_{i})\rangle=\sum_{j=1}^{M}y^{j}(t_{i})/M$, $\langle
z(t_{i})\rangle=\sum_{j=1}^{M}z^{j}(t_{i})/M$ (where M is the number of
telomeres in the nucleus). Thereafter each center of mass was subtracted
forward in time (for $t_{k}>t_{i}$ with $k>i$) to the corresponding
coordinate: $x^{j\ast}(t_{k})=x^{j}(t_{k})-\langle x(t_{i})\rangle$,
$y^{j\ast}(t_{k})=y^{j}(t_{k})-\langle y(t_{i})\rangle$ and
$z^{j\ast}(t_{k})=z^{j}(t_{k})-\langle z(t_{i})\rangle$. Next the nucleus
rotation correction was performed as follows: per cell and for all the tracks
inside at each frame of time we follow: when $i=1$ (the first frame of time)
we compute $r_{d}=\sqrt{(x^{j\ast}(t_{1}))^{2}+(y^{j\ast}(t_{1}))^{2}}$ and
$\theta^{j}(t_{1})=acos(x^{j\ast}(t_{1})/r_{d})$, if $y^{j\ast}(t_{1})<0$, we
make $\theta^{j}(t_{1})=-\theta^{j}(t_{1})$. Then we compute the average as
$\langle\theta(t_{1})\rangle=\sum_{j=1}^{M}\theta^{j}(t_{1})/M$. When $i>1$,
we compute $r_{1}=\sqrt{(x^{j\ast}(t_{i}))^{2}+(y^{j\ast}(t_{i}))^{2}}$ and
$r_{2}=\sqrt{(x^{j\ast}(t_{i-1}))^{2}+(y^{j\ast}(t_{i-1}))^{2}}$. With
$\theta^{j}(t_{i})=acos((x^{j\ast}(t_{i})\ast
x^{j\ast}(t_{i-1})+y^{j\ast}(t_{i})\ast y^{j\ast}(t_{i-1}))/(r_{1}r_{2}))$ and
if $asin((-x^{j\ast}(t_{i})\ast y^{j\ast}(t_{i-1})+y^{j\ast}(t_{i})\ast
x^{j\ast}(t_{i-1}))/(r_{1}r_{2}))<0$, we make
$\theta^{j}(t_{i})=-\theta^{j}(t_{i})$. Then the average is computed just when
$r_{2}>0.5$ (neglecting trajectories close to the origin), as
$\langle\theta(t_{i})\rangle=\sum_{j=1}^{M}\theta^{j}(t_{i})/\tilde{M}$, with
${M}$ the number of track which satisfy this criterion. Then as above for
$t_{k}>t_{i}$ with $k>i$, the inverse rotation matrix was applied as:
$x^{j\prime}(t_{k})=x^{j\ast}(t_{k})\cos(\langle\theta^{j}(t_{i})\rangle)+y^{j\ast}(t_{k})\sin(\langle\theta^{j}(t_{i})\rangle)$
and
$y^{j\prime}(t_{k})=-x^{j\ast}(t_{k})\sin(\langle\theta^{j}(t_{i})\rangle)+y^{j\ast}(t_{k})\cos(\langle\theta^{j}(t_{i})\rangle)$.
In this way the final coordinates $(x^{j\prime}(t_{k}),y^{j\prime}(t_{k}))$
have no trend or bias in rotation. The full algorithm is shown in SM.

##  Characterizing the time series

Using the relative angle time series. To differentiate between periods of
persistent and non-persistent motion on a single track in KO cells, we
employed the relative angle defined by (I.5) at an intermediate span of time
$\Delta t=5\Delta$ and $\Delta t=10\Delta$. Thus from the time series of the
telomeres without Lamin-A $\\{x(t),y(t)\\}$ with $t=n\ast\Delta$, first we
obtain the time series on the angle $\\{\theta(t,\Delta t)\\}$. For each time
step we compare $\theta(t,\Delta t)$ with a critical value named $C$. In this
case $C$ defines a threshold of the angles, letting the separation between the
persistent and non-persistent motions, we define the critical relative angle
as $C=30^{\circ}$ Mukherjee  _et al._ (2021).

So the criterion for distinguish between persistent and non-persistent motion
is the following: If $\theta(t,\Delta t)<C$ (there is persistent motion) and
we label it as ${\cal P}(t,\Delta t)=1$. Else if $\theta(t,\Delta t)\geq C$
(the dynamics in non-persistent) and it is labeled as ${\cal P}(t,\Delta
t)=0$. In this way we generate a two state time series which we call ${\cal
P}(t,\Delta t)=\\{0,1\\}$ (see magenta lines in Fig. 7).

Using local convex hull analysis. A typical statistical tool for detecting the
existence of different phases of transport with various jump sizes is the
local convex hull (LCH) analysis Lanoiselée and Grebenkov (2017); Sabri  _et
al._ (2020). LCH technique is described as follows. For a data set with points
$\\{\vec{X}_{1},\vec{X}_{2},\ldots,\vec{X}_{m}\\}$ its convex hull ($CH$) is
the set of all convex combinations which can be made with the data points,

$\displaystyle
CH=\Bigg{\\{}\displaystyle\sum\limits_{k=1}^{m}\alpha_{k}\vec{X}_{k}\Bigg{|}\displaystyle\sum\limits_{k=1}^{m}\alpha_{k}=1\Bigg{\\}}.$
(4)

In the 2$D$ case $\vec{X}\in\mathbb{R}^{2}$, $CH$ is the minimal convex
polygon that encloses all the data points
$\\{\vec{X}_{1},\vec{X}_{2},\ldots,\vec{X}_{m}\\}$. For employing CH in the
time series of the telomeres, we used time windows $\Delta t=5\Delta,$ and
$\Delta t=10\Delta$, such that at each $t=n\ast\Delta$ we define the
corresponding LCH given by (4) on the data points
$\\{\vec{X}_{t},\vec{X}_{t+\Delta},\vec{X}_{t+2\Delta},\ldots,\vec{X}_{t+\Delta
t}\\}$. Then for each LCH we compute the maximum diameter
$\mathcal{D}_{max}(t,\Delta t)$, largest distance between the points
$\\{\vec{X}_{t},\vec{X}_{t+\Delta},\vec{X}_{t+2\Delta},\ldots,\vec{X}_{t+\Delta
t}\\}$, of the corresponding polygon, see Fig. 8. In this figure we show two
typical LCH polygons (using Eq. (4) in the main text and within $\Delta
t=5\Delta$) in red color solid line. Also in the same figure we show the Area
$A$ (displayed in red shaded color) and the maximum diameter (black dashed
lines) of the respective LCH polygon.

##  VI  Estimation of the waiting times

Representative examples of the two state decomposition of trajectories in
terms of the level of persistence (${\cal P}(t,\Delta t)$) for KO cells and
the relative jump size ($L(t,\Delta t)$) for WT cells, is shown in Fig. 7.

In both cells types using the time series of ${\cal P}(t,\Delta t)$ (magenta
lines) and $L(t,\Delta t)$ (cyan lines), we extracted the waiting times
corresponding to the persistent phase and non-persistent phase, labeled as
$\tau_{{\cal P}}$ and $\tau_{N{\cal P}}$ respectively. And the waiting times
$\tau_{L}$ and $\tau_{S}$ for the long jump phase and the small jump one. In
all cases the corresponding distributions follow a discrete exponential
distribution, well fitted by the geometric distribution with mass distribution
$P(\tau)=(1-p)^{\tau-1}p$, with $0<p\leq 1$ and $\tau=1,2,3,\ldots$, (see Fig.
S13 in SM). Via this classification of trajectories we obtained the mean
waiting times for trajectories in WT cells,
$\langle\tau_{L}\rangle=5.8\Delta<\langle\tau_{S}\rangle=9.5\Delta$ using
$\Delta t=5\Delta$, and
$\langle\tau_{L}\rangle=9.2\Delta<\langle\tau_{S}\rangle=14.9\Delta$ employing
$\Delta t=10\Delta$. For trajectories in KO cells, $\langle\tau_{{\cal
P}}\rangle=1.6\Delta<\langle\tau_{N{\cal P}}\rangle=6.7\Delta$ using $\Delta
t=5\Delta$ and $\langle\tau_{{\cal P}}\rangle=2\Delta<\langle\tau_{{N{\cal
P}}}\rangle=7.4\Delta$.

##  VII  Simulation models

Two state model with caging and pure diffusion. In this case we define two
waiting times, one for the caged with small length jumps called $\tau_{S}$ and
other for the pure diffusive phase with larger length jumps named $\tau_{L}$.
Both waiting times are sampled from the corresponding empirical distribution
$P(\tau_{S})$ and $P(\tau_{L})$, see Fig. S13 b. These distributions were
obtained from the two state decomposition in terms of the jump size length of
the trajectories and represented by the time series of $L(t,\Delta t)$ (see
Fig. 7). And both were fitted with the geometric distribution with mean values
$\langle\tau_{L}\rangle=9\Delta<\langle\tau_{S}\rangle=14.9\Delta$, see SM.

We consider equilibrium initial conditions, namely with probability
$\langle\tau_{S}\rangle/[\langle\tau_{S}\rangle+\langle\tau_{L}\rangle]$ the
particle starts from the caging phase and with probability
$\langle\tau_{L}\rangle/[\langle\tau_{S}\rangle+\langle\tau_{L}\rangle]$ it
starts from the pure diffusive one. Then the trajectory alternates between the
two phases until a final time $T\ast N$, with $N$ the sampling time.

Particularly each phase is defined as follows. For the caging phase during a
random time $\tau_{S}\ast N$ for each sub-step we implement a Pearson random
walk, with $\tilde{\theta}_{i}\sim U(0,2\pi)$ and
$(x_{i},y_{i})=(x_{i-1}+r_{S}*\cos\tilde{\theta}_{i},y_{i-1}+r_{S}*\sin\tilde{\theta}_{i})$,
and reflecting boundary conditions within a circular cage with radius
$r_{c}=0.01\mu$m. For the pure diffusive phase, $r_{c}\longrightarrow\infty$,
during a random time $\tau_{L}\ast N$, we sample $\tilde{\theta}_{i}\sim
U(0,2\pi)$ and
$(x_{i},y_{i})=(x_{i-1}+r_{L}*\cos\tilde{\theta}_{i},y_{i-1}+r_{L}*\sin\tilde{\theta}_{i})$.
Notice that the displacement length depends on the phase where the particle
is, they satisfy $r_{S}<r_{L}.$

We sample the positions along the corresponding trajectory each $N$ sub-steps.
The simulations were done using parameters close to the experimental ones. A
final time of $T=50$, sampling time of $N=20$, displacement length in the
caging phase $r_{S}=0.005\mu$m, $r_{L}=0.02\mu$m in the pure diffusive phase
and $\langle\tau_{S}\rangle=9\Delta<\langle\tau_{L}\rangle=15\Delta$. We
employ an ensemble of 3000 different particles.

Two state model with Pearson and varying persistence random walks. Similarly
as for the model defined above we define a non-persistent Pearson random walk
phase with waiting times $\tau_{N{\cal P}}$ and a varying persistence phase
with $\tau_{{\cal P}}$. The corresponding waiting times are sampled from the
distributions obtained from the two state analysis of the level of
persistence, see Fig. S13 d in SM.

For each trajectory we draw an angle $\theta_{c}\sim
N(\bar{\theta},\sigma^{2}_{\bar{\theta}})$, with $\bar{\theta}(t,\Delta t)$
the average relative angle and $\sigma^{2}_{\bar{\theta}}$ its respective
variance. Also at the beginning $\theta_{int}\sim U[0,2\pi]$ is drawn. We
consider equilibrium initial conditions, namely with probability
$\langle\tau_{N{\cal P}}\rangle/[\langle\tau_{N{\cal
P}}\rangle+\langle\tau_{{\cal P}}\rangle]$ the particle starts from the
Pearson random walk phase and with probability $\langle\tau_{{\cal
P}}\rangle/[\langle\tau_{N{\cal P}}\rangle+\langle\tau_{{\cal P}}\rangle]$ it
starts from the varying persistent one. Then the trajectory alternates between
the two phases until a final time $T\ast N$, with $N$ the sampling time.

Thus for the Pearson random walk phase during a random time $\tau_{N{\cal
P}}\ast N$ for each sub-step we do (without restrictions of motion)
$\tilde{\theta}_{i}\sim U(0,2\pi)$ and
$(x_{i},y_{i})=(x_{i-1}+r*\cos\tilde{\theta}_{i},y_{i-1}+r*\sin\tilde{\theta}_{i})$.
For the varying persistent phase we proceed as follows. During a random time
$\tau_{{\cal P}}\ast N$, for the first sup-step we make
$\tilde{\theta}_{i}=\theta_{int}$ and we update the positions as
$(x_{i},y_{i})=(r*\cos\theta_{int},r*\sin\theta_{int})$. Then for sub-steps
$j>i$, we draw $\tilde{\Delta\theta}_{j}\sim U(-\theta_{c},\theta_{c})$, we
make $\tilde{\theta}_{j}=\tilde{\theta}_{j-1}+\tilde{\Delta\theta}_{j}$ and
$(x_{j},y_{j})=(x_{j-1}+r*\cos\tilde{\theta}_{j},y_{j-1}+r*\sin\tilde{\theta}_{j})$.
In order to avoid correlations in the varying persistence phases for time
scales larger that $20\Delta$, at each $20\ast\Delta$, $\theta_{int}$ is drawn
again.

Sampling the positions along the corresponding trajectory each $N$ sub-steps.
The respective parameters employed were $T=50$, $N=20$, $r=0.015\mu$m,
$\langle\tau_{{\cal P}}\rangle=2\Delta<\langle\tau_{N{\cal
P}}\rangle=7\Delta$, $\bar{\theta}=1.48$, $\sigma^{2}_{\bar{\theta}}=0.86$ and
an ensemble of 3000 different particles.

#### Code availability

Numerical simulations of our model, computation of the relative angles, and
autocorrelation functions were done employing $C$ language programs. The
computation of displacements, correlation coefficient, and LCH analysis were
done with Octave/Matlab scripts. The fitting and computation of normalized
histograms were obtained with gnuplot generic routines. All scripts are
available at the GitHub repository:
https://github.com/mariohidalgosoria/Telomeres_Dynamics.

#### Data availability

All the data sets obtained experimentally, by data analysis or by numerical
simulations are available from the corresponding authors upon request.

## Author Contributions

EB, YG and SB designed the research; YG and YH performed the experiments; MHS
and SB performed the data analysis; MHS performed the numerical simulations;
MHS, SB, EB, and YG wrote the paper.

## Conflicts of interest

The authors declare no competing interests.

## Acknowledgements

The support of Israel Science Foundation Grants 1614/21 (EB and MHS) and
2796/20 (SB) is acknowledged as well as the Israel Science Foundation Grants
1902/12, 1219/17 and 2624/22 (YG and YH) and the S. Grosskopf grant for
Generalized dynamic measurements in live cells at Bar Ilan University.

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