# Effect of active loop extrusion on the two-contact correlations in the
interphase chromosome

Dmitry Starkov1,2 and Sergey Belan1,3<EMAIL_ADDRESS>1Landau Institute for
Theoretical Physics, Russian Academy of Sciences, 1-A Akademika Semenova av.,
142432 Chernogolovka, Russia  2National Research University Higher School of
Economics, Faculty of Mathematics, Usacheva 6, 119048 Moscow, Russia
3National Research University Higher School of Economics, Faculty of Physics,
Myasnitskaya 20, 101000 Moscow, Russia

###### Abstract

The population-averaged contact maps generated by the chromosome conformation
capture technique provide important information about the average frequency of
contact between pairs of chromatin loci as a function of the genetic distance
between them. However, these datasets do not tell us anything about the joint
statistics of simultaneous contacts between genomic loci in individual cells.
This kind of statistical information can be extracted using the single-cell
Hi-C method, which is capable of detecting a large fraction of simultaneous
contacts within a single cell, as well as through modern methods of
fluorescent labeling and super-resolution imaging. Motivated by the prospect
of the imminent availability of relevant experimental data, in this work we
theoretically model the joint statistics of pairs of contacts located along a
line perpendicular to the main diagonal of the single-cell contact map. The
analysis is performed within the framework of an ideal polymer model with
quenched disorder of random loops, which, as previous studies have shown,
allows to take into account the influence of the loop extrusion process on the
conformational properties of interphase chromatin.

##  I Introduction

A series of recent experiments [1, 2, 3, 4, 5, 6, 7] have shown that the
structural maintenance of chromosomes (SMC) proteins, such as condensin and
cohesin, when binding to DNA can exhibit (ATP-dependent) motor activity
leading to progressive growth of DNA loops. Due to these works, the loop
extrusion process, originally proposed as a hypothetical molecular mechanism,
has received direct experimental confirmation [8, 9, 10].

The influence of loop extrusion machinery on the conformational properties of
chromatin can be modeled within the framework of ideal chain model with array
of random loops, see Fig. 1. In [11] we showed that this simple model
qualitatively correctly reproduces the specific experimental profile of
pairwise contact frequency in its dependence on the genomic distances at the
scales up to $1Mb$. In addition, we have previously used this model to extract
theoretical predictions regarding the conditional contact probability [12] and
the mean square of physical distance [13] between pairs of genomic loci.
Generalization of the model to the more realistic case of a non-ideal polymer
was discussed in [11] and [14].

Figure 1: Left: Interphase chromosome of higher eukaryotes can be modeled as
polymer chain with an array of sufficiently sparse random loops which are in
practice produced due to motor activity of cohesin proteins. Right: Modern
experimental techniques allow to construct single-cell contact map depicting
the pattern of simultaneous contacts - both random and cohesin-induced. The
size of the pixels is determined by the resolution of experimental technique
and can be also controlled at the stage of data analysis via coarse-graining.
We are interested in joint statistics of contacts belonging to the line
perpendicular to the main diagonal of the contact map, see e.g. pixels 1 and
2.

The analysis presented in this theoretical work is aimed at further
development of this minimalistic model with the aim of describing a larger
body of potentially accessible experimental data. More specifically, we
describe the joint statistics of the two contacts lying on the line
perpendicular to the main diagonal of the contact map. The opportunity to
extract this kind of data for chromatin in a living cell appeared thanks to
the development of the single-cell Hi-C method [15, 16, 17, 18, 19, 20, 21,
22, 23]. Unlike the population averaged contact maps, generated by Hi-C method
which detect only a small fraction of all contacts in single cells, this
technique provides high coverage of contacts at the single-cell level (see
Fig. 1). Besides, the joint statistics of two contacts is potentially
available due to modern tools for high-throughput super-resolution imaging
enabling direct visualization of the spatial positions of many genomic loci in
individual cells [15, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33].

##  II Model formulation

The interphase chromatin can be modelled as a polymer chain with dynamic array
of cohesin-mediated loops, see Fig. 1. The growth of a loop starts when a
cohesin complex binds to chromatin, and stops either at the moment of
dissociation, when free cohesin returns to the nucleoplasm, or as a result of
blocking of motor activity by another cohesin complex or by CTCF protein [34].
The mean processivity $\lambda$ of extrusion is defined as the average contour
length of the loop generated by a cohesin in the absence of interaction with
any unit that can block the loop growth. Another important characteristic of
extrusion process is the average contour distance $d$ between the binding
sites of neighboring cohesin complexes. For interphase conditions, the
inequality $\lambda/d\lesssim 1$ is valid [11, 35], so that for a typical
chain conformation most of the cohesin-mediated loops are separated by fairly
large gaps, as shown in Fig. 1, whereas the fraction of nested and spliced
loops is relatively small. Due to the stochasticity of the binding and
unbinding events associated with exchange by cohesin between chromatin and
nucleoplasm the resulting contour lengths of loops should be considered a
random variables. This argument together with irregular pattern of the binding
sites of SMC-proteins mean that the contour lengths of the chain segments
enclosed between adjacent loops should also be treated statistically. In our
analytical calculations we will consider the asymptotic limit $\lambda/d\ll
1$. In this case, the contour lengths of loops, $L$, and gaps, $h$, can be
considered as independent random variables with exponential probability
density functions

$p_{\text{loop }}(L)=\frac{1}{\lambda}\exp\left(-\frac{L}{\lambda}\right),$
(1)

and

$p_{\text{gap }}(h)=\frac{1}{d}\exp\left(-\frac{h}{d}\right).$ (2)

Next, we will assume that chromatin is an ideal chain with the Kuhn segment
length $l_{\text{eff}}$ [36]. The conformation of the chain changes randomly
over time due to thermal fluctuations and the motor activity of loop-extruding
complexes. Focusing on the slow extrusion limit, we will assume that the loop
disorder is frozen, so that random conformations of the polymer chain belong
to equilibrium ensemble. More specifically, this means that chain conformation
can be thought of as alternating free Brownian paths and Brownian bridges
[13].

Event | Probability
---|---
$\sigma_{i_{1}j_{1}}=1$, $\sigma_{i_{2}j_{2}}=1$ | $C(s_{1},s_{2})$
$\sigma_{i_{1}j_{1}}=1$, $\sigma_{i_{2}j_{2}}=0$ | $P(s_{1})-C(s_{1},s_{2})$
$\sigma_{i_{1}j_{1}}=0$, $\sigma_{i_{2}j_{2}}=1$ | $P(s_{2})-C(s_{1},s_{2})$
$\sigma_{i_{1}j_{1}}=0$, $\sigma_{i_{2}j_{2}}=0$ | $1-P(s_{1})-P(s_{2})+C(s_{1},s_{2})$
Table 1: The probabilities of different values of a pair of random variables
$\sigma_{i_{1}j_{1}}\sigma_{i_{2}j_{2}}$ can be expressed via the pairwise
contact probability $P$ and the pair correlation function $C$.

The probability density function of a random vector $\vec{R}$ connecting two
points of an equilibrium ideal chain belonging to the gap region between the
cohesin-mediated loop is determined by the expression (see, e.g., [36])

$P_{\text{free }}(\vec{R}\mid s)=\frac{1}{\left(2\pi\sigma_{\text{free
}}^{2}(s)\right)^{3/2}}\exp\left(-\frac{R^{2}}{2\sigma_{\text{free
}}^{2}(s)}\right),$ (3)

where $\sigma_{\text{free}}^{2}(s)=\frac{l_{\text{eff}}^{2}s}{3}$ and $s\gg
l_{\text{eff}}$ denotes the contour distance between points of interest.

For the probability distribution of the random vector $\vec{R}$ connecting two
points belonging to a cohesin-mediated loop one has (see, e.g., [36])

$P_{\mathrm{coil}}(\vec{R}\mid s,L)=\frac{1}{\left(2\pi\sigma_{\text{coil
}}^{2}(s,L)\right)^{3/2}}\exp\left(-\frac{R^{2}}{2\sigma_{\text{coil
}}^{2}(s,L)}\right).$ (4)

Here $\sigma_{\text{coil }}^{2}(s,L)=\frac{l_{\text{eff}}^{2}s(L-s)}{3L}$, $L$
is the contour length of the loop, and $s$ is the contour distance between
points of interest (assuming $l_{\text{eff}}s<L$).

Formulas (1), (2), (3) and (4) are sufficient to assign the probability
measure to various conformations of ideal chain with array of random loops.

##  III Metrics of interest and sketch of calculations

Let us introduce the indicator of contact, i.e. a binary random variable
$\sigma_{ij}$, which is equal to unity if there is contact between points of
the polymer chain with contour coordinates $i$ and $j$, and equal to zero
otherwise. This can be formally expressed as

$\sigma_{ij}=I(|\vec{r}_{i}-\vec{r}_{j}|\leq a),$ (5)

where the vectors $\vec{r}_{i}$ and $\vec{r}_{j}$ determine the positions of
the loci $i$ and $j$ in space, $I(\dots)$ is an indicator function, variable,
and $a$ denotes the contact radius whose value depends on the details of the
experimental technique.

It follows from definition (5) that probability $P(s)$ to find two points
separated by genomic distance $s=|i-j|$ in contact is given by the expected
value of the contact indicator, i.e.

$\displaystyle P(s)=\langle\sigma_{ij}\rangle.$ (6)

Next, the pair correlation function of contact indicators

$C(s_{1},s_{2})=\langle\sigma_{i_{1}j_{1}}\sigma_{i_{2}j_{2}}\rangle,$ (7)

gives the probability to find a contact between both pairs of points:
$(i_{1},j_{1})$ and $(i_{2},j_{2})$. As shown in Table 1, two functions $P(s)$
and $C(s_{1},s_{2})$ completely characterise joint statistics of any two
pixels in single-cell contact map. In this paper we calculate these objects
analytically using the ideal chain model with quenched disorder of loops
described in the previous section. For the sake of simplicity of the
calculations we focus on the scenarios $i_{1}\leq i_{2}\leq j_{2}\leq j_{1}$
and additionally assume the condition $i_{2}-i_{1}=j_{1}-j_{2}$. In other
words, our analysis considers pairs of contacts belonging to the line
perpendicular to the main diagonal of the contact map, see Fig. 1.

Let us briefly describe the key steps of calculations required to arrive at
$P(s)$ and $C(s_{1},s_{2})$.

Figure 2: Three types of one-loop diagrams contributing to contact probability
$P(s)$ between two points of ideal chain with disorder of rare random loops.
The bases of the cohesin-mediated loops are depicted as black dots. Figure 3:
Fourteen classes of one-loop diagrams contributing to pair correlation
function $C(s_{1},s_{2})$ of the ideal chain with disorder of rare random
loops. The bases of the cohesin-mediated loops are depicted as black dots.

Strictly speaking, there is an infinitely large set of scenarios for the
mutual arrangement of points $i$, $j$ and the bases of loops formed through
the motor activity of SMC complexes. The route to the integral representation
for contact probability accounting all types of loop patterns at
$\lambda/d\lesssim 1$ was described in Ref. [11]. The resulting expression
however contains multiple integrals which should be evaluated numerically. In
the rare loop limit adopted here the relatively simple analytical answer for
$P(s)$ can be obtained. Namely, if $\lambda/d\ll 1$, then in the region $s\ll
1$ the leading contribution to the expectation $\langle\sigma_{ij}\rangle$ is
determined by 3 one-loop diagrams, see Fig. 2, i.e.

$\displaystyle P(s)=\sum_{\alpha=a,b,c}\langle{\cal
Q}_{\alpha}(s|\\{A\\}_{\alpha})\rangle_{\text{loops}},$ (8)

where $\alpha$ enumerates the diagrams, ${\cal
Q}_{\alpha}(s|\\{A\\}_{\alpha})$ \- is the conditional expectation obtained by
averaging of $\sigma_{ij}$ over thermal noise at fixed pattern of random
loops, $\\{A\\}_{\alpha}$ represents the set of random variables parametrising
the corresponding diagram, and $\langle\cdots\rangle_{\text{loops}}$ denotes
the averaging over the variables $\\{A\\}_{\alpha}$. The conditional
expectations associated with different diagrams are given by

$\displaystyle{\cal Q}_{\alpha}(s|\\{A\\}_{\alpha})\approx\frac{4}{3}\pi
a^{3}\int{\cal P}_{\alpha}(\vec{R}|\\{A\\}_{\alpha};s)\delta(\vec{R})d^{3}R=$
(9) $\displaystyle=\frac{4}{3}\pi a^{3}{\cal
P}_{\alpha}(0|\\{A\\}_{\alpha};s).$ (10)

Here ${\cal P}_{\alpha}(\vec{R}|\\{A\\}_{\alpha};s)$ \- is the probability
density function of the vector $\vec{R}=\vec{r}_{i}-\vec{r}_{j}$ conditioned
to the particular parameters of cohesin-mediated loop. The variable $s$ enters
this function as a parameter. While deriving Eq. (10) we assumed that
$a\ll\sqrt{\langle R^{2}\rangle}$.

Next, assuming again the rare loop limit $\lambda/d\ll 1$, we find that at
scales $s_{1},s_{2}\ll d$ the leading contribution to the pair correlator is
determined by 14 one-loop diagrams, see Fig. 3, i.e.

$\displaystyle C(s_{1},s_{2})=\sum_{\beta=1}^{14}\langle{\cal
C}_{\beta}(s_{1},s_{2}|\\{B\\}_{\beta})\rangle_{\text{loops}},$ (11)

where $\beta$ enumerates the diagrams, ${\cal
C}_{\beta}(s_{1},s_{2}|\\{B\\}_{\beta})$ \- is the conditional pair
correlation function obtained by averaging of the product
$\sigma_{i_{1}j_{1}}\sigma_{i_{2}j_{2}}$ over thermal noise at the fixed
pattern of random loops, $\\{B\\}_{\beta}$ represents the set of random
variables parametrising the corresponding diagram, and
$\langle\cdots\rangle_{\text{loops}}$ denotes the averaging over the variables
$\\{B\\}_{\beta}$. The contributions coming from different diagrams can be
expressed as

$\displaystyle{\cal
C}_{\beta}(s_{1},s_{2}|\\{A\\}_{\beta})\approx(\frac{4}{3}\pi
a^{3})^{2}\int{\cal
P}_{\beta}(\vec{R}_{1},\vec{R}_{2}|\vec{R}_{3}=0,\\{B\\}_{\beta};s_{1},s_{2})\delta(\vec{R}_{1})\delta(\vec{R}_{2})d^{3}R_{1}d^{3}R_{2}=$
(12) $\displaystyle=\frac{16}{9}\pi^{2}a^{6}{\cal
P}_{\beta}(\vec{R}_{1}=0,\vec{R}_{2}=0|\vec{R}_{3}=0,\\{B\\}_{\beta};s_{1},s_{2}).$
(13)

Here ${\cal
P}_{\beta}(\vec{R}_{1},\vec{R}_{2}|\vec{R}_{3}=0,\\{B\\}_{\beta};s_{1},s_{2})$
denotes the joint probability density of vectors
$\vec{R}_{1}\equiv\vec{r}_{i_{1}}-\vec{r}_{j_{1}}$ and
$\vec{R}_{2}\equiv\vec{r}_{i_{2}}-\vec{r}_{j_{2}}$ conditioned to particular
values of random parameters $\\{B\\}_{\beta}$ and to the contact
$\vec{R}_{3}\equiv\vec{j}_{3}-\vec{i}_{3}=0$ between points $i_{3}$ and
$j_{3}$ (see Fig. 3).

Thus, our task reduces to finding conditional probabilities according to Eqs.
(10) and (13) and further averaging of the obtained expressions over the
statistics of random loops. The averaging procedure implies integration of
${\cal Q}_{\alpha}$ (or ${\cal C}_{\beta}$) over set of variables
$\\{A\\}_{\alpha}$ (or $\\{B\\}_{\beta}$) with some weights. We implement this
calculation program in Appendix. The next section is devoted to presenting the
final results.

Note that conditional averages ${\cal C}_{\beta}(s_{1},s_{2}|\\{B\\}_{\beta})$
were the subject of calculations in the paper [37]. Here we calculate them by
using a methodologically more simple approach. Comparison of our answers with
those presented in [37] demonstrates the difference for diagrams 6A, 6B, 9A
and 9B. In contrast to [37], our expressions for these diagrams exhibit
correct behaviour in limiting cases, which speaks in support of their
correctness.

##  IV Results and discussion

The conditional probability density ${\cal
P}_{\alpha}(\vec{R}|\\{A\\}_{\alpha};s)$ is given by Eqs. (30)-(33). For the
corresponding weights see Eqs. (34)-(36). Collecting all contributions
according to decomposition (8) we find that pairwise contact probability is
given by

$P(s)=\frac{b}{s^{3/2}}\left(1+\frac{\lambda}{d}f\left(\frac{s}{\lambda}\right)\right),$
(14)

where

$\displaystyle
f(s)=\left(3s+1\right)\left(2\sqrt{s}D\left(\sqrt{s}\right)-1\right)+$ (15)
$\displaystyle+\sqrt{\pi}s^{2}e^{-s}\left(2U(\frac{1}{2};2;s)+U(\frac{1}{2};3;s)\right),$
(16)

and
$U(a;b;s)=\frac{1}{\Gamma(a)}\int_{0}^{+\infty}y^{a-1}(1+y)^{b-a-1}e^{-sy}dy$
\- confluent hypergeometric function, $D(s)=e^{-s^{2}}\int_{0}^{s}e^{y^{2}}dy$
\- Dawson integral. We also introduced notation
$b=\sqrt{\frac{6}{\pi}}\frac{a^{3}}{l_{\text{eff}}^{3/2}}$. The intuition
behind the structure of this result is quite transparent: at $\lambda/d\ll 1$,
the leading contribution to contact probability $P(s)$ is determined by the
loops-free answer $\propto s^{-3/2}$, while the diagrams containing a single
cohesin-mediated loop produce the linear-order correction in small parameter
$\lambda/d$.

It is straightforward to show that $f(s)\to 0$ at $s\to 0$ so that the loop-
free result $P_{0}(s)=bs^{-3/2}$ recovers at scales $s\ll\lambda$. This is
because sufficiently short segments of ideal chain do not feel the loop
constraints. In the opposite limit $s\to\infty$ one obtains $f(s)\to 3/2$: the
random loops reduce the effective contour distance between distant loci, thus,
increasing their contact frequency. It should be immediately noted, however,
that the one-loop approximation underlying the above calculations is justified
if $s\ll d$. For this reason, one may expect that the large-scale behaviour
dictated by Eq. (14) is inaccurate. However, the direct calculations for the
region $s\gg d$ give exactly the same asymptotic $\lim_{s\to 0}f(s)=3/2$ as
the one-loop approximation [13].

Next the conditional probability densities ${\cal
P}_{\beta}(\vec{R}_{1},\vec{R}_{2}|\vec{R}_{3}=0,\\{B\\}_{\beta};s_{1},s_{2})$
entering Eq. (13) are determined by Eqs. (47), (52), (54), (57), (59), (61),
(64), (68), (73) and (78) in Appendix, whereas the statistical weights
required to perform averaging over disorder of loops are specified by Eqs.
(81)-(89). Summing up contributions coming from all diagrams depicted in Fig.
3 we obtain

$C(s_{1},s_{2})=\frac{b^{2}}{(s_{1}-s_{2})^{3/2}s_{2}^{3/2}}\left(1+\frac{\lambda}{d}F\left(\frac{s_{1}}{\lambda},\frac{s_{2}}{\lambda}\right)\right),$
(17)

where $F(s_{1},s_{2})$ is given by Eq. (99) in Appendix. We see again that
one-loop diagrams generate the linear correction in small parameter
$\lambda/d$ to loops-free answer. As is clear from Eq. (17) and the left panel
of Fig. 5, for a fixed value of $s_{2}$ correlations decay as a power law with
an exponent $-3/2$ at large $s_{1}$. Being considered as a function of
$s_{2}$, correlator $C(s_{1},s_{2})$ is non-monotonic and attains its minimum
at $s_{2}\approx s_{1}/2$, see left panel of Fig. 6. This is explained by the
structure of the loops-free correlation function
$C_{0}(s_{1},s_{2})=\frac{b^{2}}{(s_{1}-s_{2})^{3/2}s_{2}^{3/2}}$ whose
minimum $s_{2}=s_{1}/2$ is weakly affected by presence of rare random loops.

Figure 4: Log-log derivative $\frac{d\log_{10}C(s_{1},\alpha
s_{1})}{d\log_{10}\frac{s_{1}}{\lambda}}$ (see Eq. (23)) in dependence on
$s_{1}$ for different values of $\alpha$ at $\lambda/d=0.1$. Note that for
$\alpha\ll 1$ the first argument of the pair correlation function is much
larger than the second so that $C(s_{1},\alpha s_{1})\approx P(s_{1})P(\alpha
s_{1})$. This explains the presence of the second maximum at
$s_{1}\approx\lambda/\alpha$ in the left most panel.

Equations (14) and (17) suggest that an appropriate graphical representation
of the data on contact statistics may help to estimate the mean loop length.
Namely, Fig. 5 demonstrates the logarithmic derivative of the contact
probability

$\displaystyle\frac{d\log_{10}P\left(s\right)}{d\log_{10}\frac{s}{\lambda}}\approx-\frac{3}{2}+\frac{\lambda}{d}s\frac{d}{ds}f(\frac{s}{\lambda}),$
(18)

and the logarithmic derivative of the pair correlator for sufficiently small
$s_{2}/\lambda$

$\displaystyle\frac{d\log_{10}C\left(s_{1},s_{2}\right)}{d\log_{10}\frac{s_{1}}{\lambda}}\approx-\frac{3s_{1}}{2(s_{1}-s_{2})}+$
(19)
$\displaystyle+\frac{\lambda}{d}\frac{s_{1}}{\lambda}F_{z_{1}}^{\prime}\left(\frac{s_{1}}{\lambda},\frac{s_{2}}{\lambda}\right),$
(20)

attains maximum value at $s\sim\lambda$. Analogous observation has been
previously made in Ref. [11] for log-log derivative of $P(s)$ based on non-
perturbative semi-analytical calculations accounting all many-loop diagrams in
our model. Note that for $s_{2}\ll s_{1}$ the function
$\frac{d\log_{10}C\left(s_{1},s_{2}\right)}{d\log_{10}\frac{s_{1}}{\lambda}}$
approaches
$\frac{d\log_{10}P\left(s_{1}\right)}{d\log_{10}\frac{s_{1}}{\lambda}}$ since
in this case contacts become statistically independent from each other so that
$C(s_{1},s_{2})\approx P(s_{1})P(s_{2})$ (see also Fig. (7) below). Similar
observations takes place for the logarithmic derivative of the pair
correlation function

$\displaystyle\frac{d\log_{10}C\left(s_{1},s_{2}\right)}{d\log_{10}\frac{s_{2}}{\lambda}}\approx-\frac{3}{2}+\frac{3s_{2}}{2(s_{1}-s_{2})}+$
(21)
$\displaystyle+\frac{\lambda}{d}\frac{s_{2}}{\lambda}F_{z_{2}}^{\prime}\left(\frac{s_{1}}{\lambda},\frac{s_{2}}{\lambda}\right),$
(22)

in its dependence on $s_{2}$, see Fig. 6. When $s_{1}\gg s_{2}$, it approaches
$\frac{d\log_{10}P\left(s_{2}\right)}{d\log_{10}\frac{s_{2}}{\lambda}}$.

Figure 5: Left: Pair correlation function $C(s_{1},s_{2})$ in dependence on
$s_{1}/s_{2}$ at $\lambda/d=0.1$ for different values of $s_{2}/\lambda$.
Right: Log-log derivative of pair-correlation function $C(s_{1},s_{2})$ (see
Eq. (17)) in dependence on $s_{1}/\lambda$ at $\lambda/d=0.1$ for different
values of $s_{2}/\lambda$. The dotted line corresponds to the log-log
derivative of the contact probability $P(s_{1})$ (see Eq. (14)).

Next let us pass to effectively one-dimensional representation of the pair
correlator $C(s_{1},s_{2})$ by introducing the following function

$\displaystyle\frac{d\log_{10}C(s_{1},\alpha
s_{1})}{d\log_{10}\frac{s_{1}}{\lambda}}=-3+\frac{\lambda}{d}s_{1}\frac{dF\left(\frac{s_{1}}{\lambda},\frac{\alpha
s_{1}}{\lambda}\right)}{ds_{1}},$ (23)

where $0<\alpha<1$ is some constant. As shown in Fig. 4 this object in its
dependence on $s_{1}$ exhibits multiple extrema whose number depends on the
value of parameter $\alpha$. Namely, there are four extrema at $\alpha<1/4$.
Next, in the case $1/4<\alpha<3/4$ we observe two extrema. Finally, one has
three extrema at $\alpha>3/4$. Importantly, for any value of $\alpha$ there
exists a maximum at $s\sim\lambda$. Such rich behavior, if confirmed by
analysis of the corresponding experimental data, would serve as another good
confirmation of the effectiveness of our model. Note, however, that for
sufficiently small $\alpha$ the right-most extrema may not be observed
experimentally since their positions correspond to large genomic scales (see
first panel of Fig. 4) where contact statistics is affected by
compartmentalization [38] \- an effect which is not incorporated into the
simple model considered here.

Figure 6: Left: Pair correlation function $C(s_{1},s_{2})$ in dependence on
$s_{2}/s_{1}$ at $\lambda/d=0.1$ for different values of $s_{1}/\lambda$.
Right: Log-log derivative of pair-correlation function $C(s_{1},s_{2})$ (see
Eq. (17)) in dependence on $s_{2}/\lambda$ at $\lambda/d=0.1$ for different
values of $s_{1}/\lambda$. The dotted line corresponds to the log-log
derivative of the contact probability $P(s_{2})$ (see Eq. (14)). Figure 7:
Left: The Pearson correlation coefficient $\rho(s_{1},s_{2})$ (see Eq. (26))
of the random variables $\sigma_{i_{1}j_{1}}$ and $\sigma_{i_{2}j_{2}}$ in
dependence on $s_{1}$ for different values of $s_{2}$ at $\lambda/d=0.1$.
Right: The logarithmic derivative of the Pearson coefficient
$\frac{d\log_{10}\rho(s_{1},s_{2})}{d\log_{10}\frac{s_{1}}{\lambda}}$ (see Eq.
(27)).

Finally, let us consider the Pearson correlation coefficient defined as

$\displaystyle\rho=\frac{\langle(\sigma_{i_{1}j_{1}}-\langle\sigma_{i_{1}j_{1}}\rangle)(\sigma_{i_{2}j_{2}}-\langle\sigma_{i_{2}j_{2}}\rangle)\rangle}{\sqrt{\langle(\sigma_{i_{1}j_{1}}-\langle\sigma_{i_{1}j_{1}}\rangle)^{2}\rangle}\sqrt{\langle(\sigma_{i_{2}j_{2}}-\langle\sigma_{i_{2}j_{2}}\rangle)^{2}\rangle}},$
(24)

to quantify linear correlation between random variables $\sigma_{i_{1}j_{1}}$
and $\sigma_{i_{2}j_{2}}$. It is easy to check that $-1\leq\rho\leq+1$.

From Eqs. (14) and (17) we immediately find that

$\displaystyle\rho=\frac{C(s_{1},s_{2})-P(s_{1})P(s_{2})}{\sqrt{P(s_{1})P(s_{2})(1-P(s_{1}))(1-P(s_{2}))}}.$
(25)

Next, taking into account that $P(s_{1}),P(s_{2})\ll 1$ at genomic scales
$s_{1},s_{2}\gg a^{2}/l_{\text{eff}}$, in the leading order approximation in
small parameters $s_{1}/d$, $s_{2}/d$, $b/s_{1}^{3/2}$, $b/s_{2}^{3/2}$ and
$\lambda/d$ one obtains

$\displaystyle\rho(s_{1},s_{2})\approx\frac{C(s_{1},s_{2})-P(s_{1})P(s_{2})}{\sqrt{P(s_{1})P(s_{2})}}\approx
b\frac{s_{1}^{3/2}-(s_{1}-s_{2})^{3/2}}{s_{1}^{3/4}s_{2}^{3/4}(s_{1}-s_{2})^{3/2}}\left(1+\frac{b}{2s_{1}^{3/2}}+\frac{b}{2s_{2}^{3/2}}+\frac{\lambda}{d}\Phi(\frac{s_{1}}{\lambda},\frac{s_{2}}{\lambda})\right),$
(26)

and

$\displaystyle\frac{d\log_{10}\rho(s_{1},s_{2})}{d\log_{10}\frac{s_{1}}{\lambda}}\approx-\frac{3}{4}+\frac{3}{2}\frac{s_{1}^{3/2}-s_{1}\cdot(s_{1}-s_{2})^{1/2}}{s_{1}^{3/2}-(s_{1}-s_{2})^{3/2}}-\frac{3s_{1}}{2(s_{1}-s_{2})}-\frac{3b}{4s_{1}^{3/2}}+\frac{s_{1}}{\lambda}\frac{\lambda}{d}\Phi_{z_{1}}^{\prime}\left(\frac{s_{1}}{\lambda},\frac{s_{2}}{\lambda}\right),$
(27)

where

$\displaystyle\Phi(s_{1},s_{2})=-\frac{1}{2}(f(s_{1})+f(s_{2}))+$ (28)
$\displaystyle+$
$\displaystyle\frac{s_{1}^{3/2}F(s_{1},s_{2})-(s_{1}-s_{2})^{3/2}(f(s_{1})+f(s_{2}))}{s_{1}^{3/2}-(s_{1}-s_{2})^{3/2}}.$
(29)

Figure (7) illustrates Eqs. (26) and (27). For sufficiently small $s_{2}$ the
logarithmic derivative
$\frac{d\log_{10}\rho(s_{1},s_{2})}{d\log_{10}\frac{s_{1}}{\lambda}}$ exhibits
non-monotonic behaviour with maximum at $s_{1}\sim\lambda$ (see red curve in
right panel of Fig. (7)). Similar representation from $s_{2}$-perspective for
fixed $s_{1}$ is not shown here as the corresponding dependence is monotonic
and thus it cannot be used to infer information regarding characteristic loop
size.

##  V Conclusion

The study of contact statistics of chromatin may shed light onto relationship
between the spatial structure of the genome and its functioning [39]. The main
regulatory mechanism that initiates gene expression in a living cell is the
formation of chromatin loops, bringing enhancers and promoters into spatial
proximity. Importantly, some enhancers and promoters are located at genomic
distances of several hundred thousand base pairs from each other. To
understand to what extent loop extrusion may take a part in the regulation of
enhancer-promoter communication in a cell of particular type one should be
able to estimate the typical genomic size of cohesin-mediated loops. At the
moment, several predictors of the mean loop size have been proposed it the
literature: maximum of the log-log derivative of the contact probability [11],
maximum of the log-log derivative of the conditional contact probability [12],
minimum of the log-log derivative of mean squared physical distance between
genomic loci [13], maximum of kurtosis of the separation vector [13]. In this
work we proposed several new predictors based on the behaviour of pair
correlation function $C(s_{1},s_{2})$ and Pearson coefficient of the contact
indicators associated with single-cell contact maps.

###### Acknowledgements.

The work was supported by the Russian Science Foundation, project no.
22-72-10052.

##  Appendix A Derivation of Eq. (14)

Let us consider three one-loop diagrams contributing to the pairwise contact
probability $P_{c}(s)$ in the rare loop limit, as shown in the figure above.
Following results of [11] we start with the expression

$\displaystyle{\cal P}_{\alpha}(0|\\{A\\}_{\alpha};s)=\frac{1}{(\frac{2}{3}\pi
l_{\text{eff}}\tilde{s}_{\alpha}[s,\\{A\\}_{\alpha}])^{3/2}},$ (30)

where

$\displaystyle\tilde{s}_{a}[s,x_{s}]=(1-x_{s})s,$ (31)
$\displaystyle\tilde{s}_{b}[s,l_{1},l_{2}]=s-l_{2}+\frac{l_{1}l_{2}}{l_{1}+l_{2}},$
(32)
$\displaystyle\tilde{s}_{c}[s,l_{1},l_{2}]=\left(1-\frac{s}{l_{1}+l_{2}}\right)s.$
(33)

Here $x_{s}=L/s$ and the contour lengths $l_{1}$ and $l_{2}$ are shown in the
figure above.

The linearized statistical wights of diagrams are given by

$\displaystyle{\cal
W}_{a}(x_{s}|s)\approx\delta(x_{s})+\frac{\lambda}{d}\left(-(1+\frac{s}{\lambda})\delta(x_{s})+\frac{(1-x_{s})s^{2}}{\lambda}p_{\text{loop}}(x_{s}s)\right),\
\ \text{for}\ \ 0\leq x_{s}<1,$ (34) $\displaystyle{\cal W}_{b}(L,q|s)\approx
2\frac{\lambda}{d}\frac{L}{\lambda}p_{\text{loop}}(L),\ \ \text{for}\ \ 0\leq
q\leq min[1,\frac{s}{L}],L\geq 0,$ (35) $\displaystyle{\cal
W}_{c}(L,q|s)\approx\frac{\lambda}{d}\frac{L}{\lambda}p_{\text{loop}}(L),\ \
\text{for}\ \ \frac{s}{L}\leq q\leq 1,L\geq s,$ (36)

where $q=l_{2}/L$.

Substituting Eqs. (10) with ${\cal P}_{\alpha}(0|\\{A\\}_{\alpha};s)$ given by
Eqs. (30) into Eq. (8) and using Eqs. (31)-(36) we find

$\displaystyle P(s)=\frac{4}{3}\pi a^{3}\sum_{\alpha=a,b,c}\langle{\cal
P}_{\alpha}(0|\\{A\\}_{\alpha};s)\rangle_{\text{loops}}=\sqrt{\frac{6}{\pi}}\frac{a^{3}}{l^{3/2}_{\text{eff}}}\int\limits_{0}^{1}\frac{{\cal
W}_{a}(x_{s}|s)}{\tilde{s}_{a}[s,x_{s}]^{3/2}}dx_{s}+$ (37)
$\displaystyle+\sqrt{\frac{6}{\pi}}\frac{a^{3}}{l^{3/2}_{\text{eff}}}\int_{0}^{\infty}dL\int_{0}^{min[1,s/L]}dq\frac{{\cal
W}_{b}(L,q|s)}{\tilde{s}_{b}[s,l_{1},l_{2}]^{3/2}}+$ (38)
$\displaystyle+\sqrt{\frac{6}{\pi}}\frac{a^{3}}{l^{3/2}_{\text{eff}}}\int_{s}^{\infty}dL\int_{s/L}^{1}dq\frac{{\cal
W}_{c}(L,q|s)}{\tilde{s}_{c}[s,l_{1},l_{2}]^{3/2}}=\sqrt{\frac{6}{\pi}}\frac{a^{3}}{l^{3/2}_{\text{eff}}s^{3/2}}\left[1+\frac{\lambda}{d}\left(-1-\frac{s}{\lambda}+\frac{s^{2}}{\lambda}\int\limits_{0}^{1}\frac{p_{\text{loop}}(ys)dy}{(1-y)^{1/2}}\right)\right]+$
(39)
$\displaystyle+\sqrt{\frac{6}{\pi}}\frac{a^{3}}{l^{3/2}_{\text{eff}}s^{3/2}}\frac{s^{2}}{d}\left[\int\limits_{0}^{1}\frac{p_{\text{loop}}(ys)ydy}{(1-y)^{1/2}}+\int\limits_{0}^{+\infty}\frac{(y+1)^{1/2}p_{\text{loop}}((y+1)s)dy}{y^{1/2}}\right]+$
(40)
$\displaystyle+\sqrt{\frac{6}{\pi}}\frac{a^{3}}{l^{3/2}_{\text{eff}}s^{3/2}}\frac{s^{2}}{d}\int\limits_{0}^{+\infty}\frac{(y+1)^{3/2}p_{\text{loop}}((y+1)s)dy}{y^{1/2}}=$
(41)
$\displaystyle=\frac{b}{s^{3/2}}\left(1+\frac{\lambda}{d}\left[\left(3\frac{s}{\lambda}+1\right)\left(2\sqrt{\frac{s}{\lambda}}D\left(\sqrt{\frac{s}{\lambda}}\right)-1\right)+\sqrt{\pi}(\frac{s}{\lambda})^{2}e^{-\frac{s}{\lambda}}\left(2U(\frac{1}{2};2;\frac{s}{\lambda})+U(\frac{1}{2};3;\frac{s}{\lambda})\right)\right]\right).$
(42)

This result coincides with Eq. (14) in the main text.

##  Appendix B Derivation of Eq. (17)

We pass to calculation of the conditional probability densities ${\cal
P}_{\beta}(\vec{R}_{i_{1}j_{1}}=0,\vec{R}_{i_{2}j_{2}}=0|\vec{R}_{i_{3}j_{3}}=0,\\{B\\}_{\beta};s_{1},s_{2})$
and of the statistical weights $\omega_{\beta}(\\{B\\}_{\beta},s_{1},s_{2})$
required to derive analytical expression for the pair correlator
$C(s_{1},s_{2})$. Since $i_{2}-i_{1}=j_{1}-j_{2}$ by assumption (see the main
text), 5 diagrams out of 14 depicted in Fig. 3 are mirror images of others, so
it is enough to consider 9 scenarios. For the sake of brevity of notation, we
will denote the spatial position of the points having contour coordinates
$i_{k}$ and $j_{k}$ as $\vec{i}_{k}$ and $\vec{j}_{k}$, respectively. Also,
let us denote the contact event $\vec{R}_{i_{k}j_{k}}=0$ as $c_{k}$ and write
shortly $p_{\beta}(c_{1},c_{2}|c_{3})$ instead of ${\cal
P}_{\beta}(\vec{R}_{i_{1}j_{1}}=0,\vec{R}_{i_{2}j_{2}}=0|\vec{R}_{i_{3}j_{3}}=0,\\{B\\}_{\beta};s_{1},s_{2})$.

###  B.1 Diagram 1

First we consider diagram 1 in Fig. 3. Obviously

$p_{1}(c_{1},c_{2}|c_{3})=\frac{\text{Pr}[c_{1},c_{2},c_{3}]}{\text{Pr}[c_{3}]}=\frac{\text{Pr}[c_{1}|c_{2},c_{3}]\text{Pr}[c_{2},c_{3}]}{\text{Pr}[c_{3}]}=\text{Pr}[c_{1}|c_{2},c_{3}]\text{Pr}[c_{2}|c_{3}].$
(43)

The conditional probability of contact $\text{Pr}[c_{2}|c_{3}]$ can be
represented as

$\text{Pr}[c_{2}|c_{3}]=\int_{R\leq
a}P(\vec{R})d^{3}\vec{R}\approx\frac{4}{3}\pi a^{3}P(\vec{R}=0),$ (44)

where $P(\vec{R})$ \- is the probability density function of random vector
$\vec{R}=(\vec{i_{3}}-\vec{i_{2}})+(\vec{j_{2}}-\vec{j_{3}})$ which is equal
to the sum of two independent normally distributed random vector. Recalling
that the sum of independent Gaussian variables has normal statistics with the
variance given by the sum of variances of the underlying terms and taking into
account (3) and (44), we obtain

$\text{Pr}[c_{2}|c_{3}]=\frac{b}{(s_{2}-L)^{3/2}}.$ (45)

Similarly it can be shown that

$\text{Pr}[c_{1}|c_{2},c_{3}]=\frac{b}{(s_{1}-s_{2})^{3/2}}.$ (46)

Thus, from Eq. (43) one obtains

$p_{1}(c_{1},c_{2}|c_{3})=\frac{b^{2}}{(s_{2}-L)^{3/2}(s_{1}-s_{2})^{3/2}}.$
(47)

For the statistical weight corresponding to the diagram 1 we find

$\displaystyle\omega_{1}(h,L|s_{1},s_{2},l)=\pi_{gap}p_{gap}(l+h)p_{loop}(L)\text{Pr}[t\geq
s_{1}-l-h-L]=$ (48)
$\displaystyle=\frac{d}{d+\lambda}\frac{1}{d}\exp(-\frac{l+h}{d})\frac{1}{\lambda}\exp(-\frac{L}{\lambda})\int_{s_{1}-l-h-L}^{\infty}\frac{1}{d}\exp(-\frac{t}{d})dt=$
$\displaystyle=\frac{1}{(d+\lambda)\lambda}\exp(-\frac{l+h}{d}-\frac{L}{\lambda})\cdot\exp(-\frac{s_{1}-l-h-L}{d}),$

where $l=i_{2}-i_{1}=j_{1}-j_{2}$, $h=i_{3}-i_{2}$,
$\pi_{gap}=\frac{d}{d+\lambda}$ \- is the probability that the point $i_{1}$
belongs to the gap region of the chain, $p_{gap}(l+h)$ \- is the probability
density to observe a gap of contour length $l+h$, $p_{loop}(L)$ \- probability
density of loop length $L$, $t$ \- is the random variable with probability
density $p_{gap}(l)$, and $\text{Pr}[t\geq s_{1}-l-h-L]$ represents the
probability that there are no additional loops between points $j_{3}$ and
$j_{1}$. When writing Eq. (48), we took into account the memoryless property
of the exponentially distributed random variables. Note that the random
parameters $L$ and $h$ entering Eq. (48) belong to the intervals
$L\in[0,s_{2}]$ and $h\in[0,s_{2}-L]$, respectively.

###  B.2 Diagram 2

The conditional probability associated with diagram 2 can be represented as

$p_{2}(c_{1},c_{2}|c_{3})=\text{Pr}[c_{1}|c_{2},c_{3}]\text{Pr}[c_{2}|c_{3}].$
(49)

Following line of arguments similar to that presented in the previous
subsection, we obtain

$\text{Pr}[c_{1}|c_{2},c_{3}]=\frac{b}{(s_{1}-L)^{3/2}}.$ (50)

Next, using Eq. (4) from the main text we obtain

$\text{Pr}[c_{2}|c_{3}]=\frac{bL^{3/2}}{s_{2}^{3/2}\cdot(L-s_{2})^{3/2}}.$
(51)

Substituting (50) and (51) into (49) we find

$p_{2}(c_{1},c_{2}|c_{3})=\text{Pr}[c_{1}|c_{2},c_{3}]\text{Pr}[c_{2}|c_{3}]=\frac{b^{2}L^{3/2}}{s_{2}^{3/2}\cdot(L-s_{2})^{3/2}\cdot(s_{1}-L)^{3/2}}.$
(52)

The statistical weight of the diagram 2 is given by

$\displaystyle\omega_{2}(h,L|s_{1},s_{2},l)=\pi_{gap}p_{gap}(h)p_{loop}(L)\text{Pr}[t\geq
s_{1}-h-L]=$ (53)
$\displaystyle=\frac{d}{d+\lambda}\frac{1}{d}\exp(-\frac{h}{d})\frac{1}{\lambda}\exp(-\frac{L}{\lambda})\int_{s_{1}-h-L}^{\infty}\frac{1}{d}\exp(-\frac{t}{d})=\frac{1}{(d+\lambda)\lambda}\exp(-\frac{s_{1}-L}{d}-\frac{L}{\lambda}).$

Now $L\in[s_{2},s_{1}]$ and
$h\in[Max[0,\frac{s_{1}+s_{2}}{2}-L],Min[\frac{s_{1}-s_{2}}{2},s_{1}-L]]$.
Such a choice guarantees that mutual positions of points
$i_{1},i_{2},i_{3},j_{1},j_{2}$ and $j_{3}$ corresponds to situation depicted
in Fig. 3(2).

###  B.3 Diagram 3

As previously, let us use the following representation

$p_{3}(c_{1},c_{2}|c_{3})=\text{Pr}[c_{2}|c_{1},c_{3}]\text{Pr}[c_{1}|c_{3}].$
(54)

The second term in the right hand side of the above expression can be easily
calculated using Eq. (4). As for the first term, let us note that
$\text{Pr}(c_{2}|c_{1},c_{3})=\text{Pr}(c_{2}|c_{1})$ due to the Markovian
property of the ideal chain. Given this fact, the corresponding term can also
be calculated based on Eq. (4). This finally yields

$p_{3}(c_{1},c_{2}|c_{3})=\frac{b^{2}L^{3/2}}{s_{2}^{3/2}(s_{1}-s_{2})^{3/2}(L-s_{1})^{3/2}}.$
(55)

To arrive at the statistical weight of the diagram 3 let us introduce a
variable $q=h/L$ defined as the ratio in which the point $i_{1}$ divides the
cohesin-mediated loop. In terms of variables $L$ and $q$ the statistical
weight is given by

$\omega_{3}(q,L|s_{1},s_{2},l)=\pi_{loop}\frac{L}{\lambda}p_{loop}(L)\Theta(q)\Theta(1-q)=\frac{L}{d+\lambda}\frac{1}{\lambda}e^{-\frac{L}{\lambda}}\Theta(q)\Theta(1-q),$
(56)

where $\pi_{loop}=\frac{\lambda}{d+\lambda}$ represents the probability that
point $i_{1}$ lies at the loop, $\frac{L}{\lambda}p_{loop}(L)$ is the
probability density of the length of this loop, and
$\rho(q)=\Theta(q)\Theta(1-q)$ is the probability density of $q$. Clearly
$L\in[0,\infty]$, while $h\in[0,L-s_{1}]$ and, thus, $q=h/L\in[0,1-s_{1}/L]$.

###  B.4 Diagram 4

For the conditional probability corresponding to the diagram 4 we find

$p_{4}(c_{1},c_{2}|c_{3})=\text{Pr}[c_{1}|c_{2},c_{3}]\text{Pr}[c_{2}|c_{3}]=\text{Pr}[c_{1}|c_{2}]\text{Pr}[c_{2}]=\frac{b^{2}}{(s_{1}-s_{2})^{3/2}s_{2}^{3/2}},$
(57)

where we exploited the Markovian property of the ideal chain. The probability
of observing chain conformation describing by this diagram is given by

$\omega_{4}(h,L|s_{1},s_{2})=\pi_{gap}\text{Pr}[t\geq
s_{1}]=\frac{d}{d+\lambda}\exp(-\frac{s_{1}}{d}).$ (58)

###  B.5 Diagrams 5A and 5B

Repeating arguments similar to those presented in (B.1), we find

$p_{5}(c_{1},c_{2}|c_{3})=\text{Pr}[c_{1}|c_{2},c_{3}]\text{Pr}[c_{2}|c_{3}]=\text{Pr}[c_{1}|c_{2},c_{3}]\text{Pr}[c_{2}]=\frac{b^{2}}{(s_{1}-s_{2}-L)^{3/2}s_{2}^{3/2}}.$
(59)

The statistical weight of diagram 5 is as follows

$\displaystyle\omega_{5}(h,L|s_{1},s_{2},l)=\pi_{gap}p_{gap}(h)p_{loop}(L)\text{Pr}[t\geq
s_{1}-h-L]=$ (60)
$\displaystyle=\frac{d}{d+\lambda}\frac{1}{d}\exp(-\frac{h}{d})\frac{1}{\lambda}\exp(-\frac{L}{\lambda})\int_{s_{1}-h-L}^{\infty}\frac{1}{d}\exp(-\frac{t}{d})=\frac{1}{(d+\lambda)\lambda}\exp(-\frac{s_{1}-L}{d}-\frac{L}{\lambda}).$

The random variables $L$ and $h$ parametrising this diagram obey the
conditions $L\in[0,l]$ and $h\in[0,l-L]$.

Similarly for diagram 5B we find

$p_{5B}(c_{1},c_{2}|c_{3})=\text{Pr}[c_{1}|c_{2},c_{3}]\text{Pr}[c_{2}|c_{3}]=\frac{b^{2}}{(s_{1}-s_{2}-L)^{3/2}s_{2}^{3/2}},$
(61)

and

$\omega_{5B}(h,L|s_{1},s_{2},l)=\pi_{gap}p_{gap}[s_{1}-h-L]p_{loop}(L)\text{Pr}[t\geq
h]=\frac{1}{(d+\lambda)\lambda}\exp(-\frac{s_{1}-L}{d}-\frac{L}{\lambda}),$
(62)

where $L\in[0,l]$ and $h\in[0,l-L]$.

###  B.6 Diagram 6A and 6B

We start again with representation

$p(c_{1},c_{2}|c_{3})=\text{Pr}[c_{2}|c_{1},c_{3}]\text{Pr}[c_{1}|c_{3}].$
(63)

The second term in the right hand side is similar to (45), so let us analyse
the first one. It is convenient to introduce the notations
$\vec{R}_{1}=\vec{i}_{2}-\vec{j}_{3}$ and
$\vec{R}_{2}=\vec{j}_{2}-\vec{j}_{3}$. The probability densities of the random
vectors $\vec{R_{1}}$ and $\vec{R_{2}}$ are given by normal distributions
$P_{coil}(\vec{R}_{1}|j_{3}-i_{2},s_{3})$ and
$P_{coil}(\vec{R}_{2}|j_{2}-j_{3},s_{1}-s_{3})$, respectively (see Eq. (4)).
Then the contact probability $\text{Pr}[c_{2}|c_{1},c_{3}]$ can be found from
the probability density of the vector $\vec{R}=\vec{R_{2}}-\vec{R_{1}}$.
Taking into account that variances of $\vec{R}_{1}$ and $\vec{R}_{2}$ behave
as $\sigma_{1}^{2}\sim\frac{(s_{2}-h)(L-s_{2}+h)}{L}$ and
$\sigma_{2}^{2}\sim\frac{h(s_{1}-L-h)}{s_{1}-L}$, we obtain

$\displaystyle
p_{6}(c_{1},c_{2}|c_{3})=\frac{b^{2}}{(s_{1}-L)^{3/2}\cdot\left(\frac{(s_{2}-h)(L-s_{2}+h)}{L}+\frac{h(s_{1}-L-h)}{s_{1}-L}\right)^{3/2}}=$
(64)
$\displaystyle=\frac{b^{2}L^{3/2}}{\left[(s_{1}-L)(s_{2}-h)(L-s_{2}+h)+L\cdot
h\cdot(s_{1}-L-h)\right]^{3/2}}=\frac{b^{2}L^{3/2}}{\left[s_{2}\cdot(s_{1}-L)\cdot(L+2h-s_{2})-h^{2}s_{1}\right]^{3/2}}.$

For the statistical weight of the diagram 6 one obtains

$\displaystyle\omega_{6}(h,L|s_{1},s_{2},l)=\pi_{gap}p_{gap}(s_{1}-l-h-L)p_{loop}(L)\text{Pr}[t\geq(h+l)]=$
(65)
$\displaystyle=\frac{d}{\lambda+d}\frac{1}{d}\exp\left(-\frac{1}{d}(s_{1}-L)\right)\frac{1}{\lambda}\exp(-\frac{L}{\lambda}),$

where $L\in[0,s_{1}-l]$ and
$h\in[max(0,s_{2}-L),min(s_{2},\frac{s_{1}+s_{2}}{2}-L)]$.

The non-trivial range for $h$ is explained by the need to simultaneously
satisfy three conditions: (i) the point $i_{2}$ should belong to the cohesin-
mediated loop so that $h\leq s_{2}$ and $h\geq s_{2}-L$, (ii) the total
contour length of the internal segments should not exceed the contour distance
between the external contacts, i.e. $h\leq s_{1}-l-L$, (iii) the right point
of internal contact lies to the right of the loop, i.e. $h\geq 0$.

###  B.7 Diagrams 7A and 7B

$p_{7}(c_{1},c_{2}|c_{3})=\text{Pr}[c_{1}|c_{2},c_{3}]\text{Pr}[c_{2}|c_{3}]=\text{Pr}[c_{1}|c_{2},c_{3}]\text{Pr}[c_{2}].$
(66)

From Eq. (3) we immediately find

$\displaystyle\text{Pr}[c_{2}]=\frac{b^{2}}{s_{2}^{3/2}}.$ (67)

To calculate $\text{Pr}[c_{1}|c_{2},c_{3}]$ let us introduce notations
$\vec{R}=\vec{R_{1}}+\vec{R_{2}}=\vec{j_{1}}-\vec{i_{1}}$,
$\vec{R_{1}}=\vec{j_{1}}-\vec{j_{3}}$ and
$\vec{R_{2}}=\vec{j_{3}}-\vec{i_{1}}$.

The conditional contact probability $\text{Pr}[c_{1}|c_{2},c_{3}]$ can be
found from probability density of the random vector
$\vec{R}=\vec{R_{1}}+\vec{R_{2}}$. Since independent vectors $\vec{R_{1}}$ and
$\vec{R_{2}}$ have normal statistics with variances
$\sigma_{1}^{2}\sim(j_{1}-j_{3})-s_{2}=s_{1}-h-s_{2}$ and
$\sigma_{2}^{2}\sim\frac{h(L-h)}{L}$, one obtains

$p_{7}(c_{1},c_{2}|c_{3})=\frac{b^{2}}{s_{2}^{3/2}\cdot\left(s_{1}-h-s_{2}+\frac{h(L-h)}{L}\right)^{3/2}}=\frac{b^{2}L^{3/2}}{s_{2}^{3/2}\cdot\left(L(s_{1}-s_{2})-h^{2}\right)^{3/2}}.$
(68)

The statistical weight of the diagram 7 can be expressed as

$\omega_{7}(q,L|s_{1},s_{2},l)=\pi_{loop}\frac{L}{\lambda}p_{loop}(L)\Theta(q)\Theta(1-q)\text{Pr}[t\geq
s_{1}-h],$ (69)

where $q=h/L$ and $h=j_{3}-i_{1}$. As follows from graphical representation,
see Fig. 3, the random variables $L$ and $h$ lie in the ranges
$L\in[0,\infty)$ and $h\in[0,min(L,s_{1}-s_{2}-l)]$. Therefore,
$h\in[0,min(1,\frac{s_{1}-s_{2}}{2L})]$.

###  B.8 Diagrams 8A and 8B

Let us use the representation

$p_{8}(c_{1},c_{2}|c_{3})=\text{Pr}[c_{1}|c_{2},c_{3}]\text{Pr}[c_{2}|c_{3}].$
(70)

For the second term one readily obtains from Eq. (4)

$\displaystyle\text{Pr}[c_{2}|c_{3}]\sim\frac{L^{3/2}}{s_{2}^{3/2}\cdot(L-s_{2})^{3/2}}.$
(71)

To arrive at analytical expression for $\text{Pr}[c_{1}|c_{2},c_{3}]$ let us
introduce the vectors $\vec{R_{1}}=\vec{j_{3}}-\vec{i_{1}}$ and
$\vec{R_{2}}=\vec{j_{1}}-\vec{j_{3}}$. Then conditional probability
$\text{Pr}[c_{1}|c_{2},c_{3}]$ can be derived based on the probability density
of random vector $\vec{R}=\vec{R_{1}}+\vec{R_{2}}=\vec{j_{1}}-\vec{i_{1}}$.
Since $\vec{R}_{1}$ and $\vec{R}_{2}$ have normal statistics with variances
$\sigma_{1}^{2}\sim(j_{3}-i_{1})-s_{3}=i_{3}-i_{1}=s_{1}-h$ and
$\sigma_{2}^{2}\sim\frac{(j_{3}-j_{1})(s_{3}-s_{2}-j_{3}+j_{1})}{s_{3}-s_{2}}=\frac{(s_{3}-h)(s_{3}-s_{2}-s_{3}+h)}{s_{3}-s_{2}}=\frac{(s_{3}-h)(h-s_{2})}{s_{3}-s_{2}}$,
we find

$\text{Pr}[c_{1}|c_{2},c_{3}]\sim\frac{1}{\left(s_{1}-h+\frac{(L-h)(h-s_{2})}{L-s_{2}}\right)^{3/2}}.$
(72)

Inserting expressions (71) and (72) into Eq. (70), we obtain

$\displaystyle
p^{(8)}(c_{1},c_{2}|c_{3})=A^{2}\frac{L^{3/2}}{s_{2}^{3/2}\cdot(L-s_{2})^{3/2}}\cdot\frac{1}{\left(s_{1}-h+\frac{(L-h)(h-s_{2})}{L-s_{2}}\right)^{3/2}}=$
(73)
$\displaystyle=A^{2}\frac{L^{3/2}}{s_{2}^{3/2}\cdot\left((s_{1}-h)(L-s_{2})+(L-h)(h-s_{2}))^{3/2}\right)}.$

The corresponding statistical weight equals to

$\omega^{(8)}(h,L|s_{1},s_{2})=\pi_{gap}p_{gap}(s_{1}-h)p_{loop}(L),$ (74)

where $L\in[s_{2}+l,\infty)$and $h\in[s_{2}+l,min(L,s_{1})]$.

###  B.9 Diagrams 9A and 9B

Le us introduce the notations
$\vec{r_{1}}=\vec{R_{3}}-\vec{R_{2}}-\vec{R_{1}}$ and
$\vec{r_{2}}=-\vec{R_{2}}+\vec{R_{3}}-\vec{R_{4}}$. The conditional
probability $p(c_{1},c_{2}|c_{3})=P(\vec{r_{1}}=0,\vec{r_{2}}=0|c_{3})$ can be
calculated via averaging over realizations of vectors
$\vec{R_{1}},\vec{R_{2}},\vec{R_{3}}$ and $\vec{R_{4}}$. Namely

$\displaystyle
P(\vec{r_{1}},\vec{r_{2}}|c_{3})=\langle\delta(\vec{r_{1}}-\vec{R_{3}}+\vec{R_{1}}+\vec{R_{2}})\delta(\vec{r_{2}}+\vec{R_{2}}-\vec{R_{3}}+\vec{R_{4}})\rangle_{\vec{R_{1}},\vec{R_{2}},\vec{R_{3}},\vec{R_{4}}}=$
(75) $\displaystyle=\int
d^{3}R_{1}d^{3}R_{2}d^{3}R_{3}d^{3}R_{4}\delta(\vec{r_{1}}-\vec{R_{3}}+\vec{R_{1}}+\vec{R_{2}})\delta(\vec{r_{2}}+\vec{R_{2}}-\vec{R_{3}}+\vec{R_{4}})P(\vec{R_{1}},\vec{R_{2}},\vec{R_{3}},\vec{R_{4}})=$
$\displaystyle=\int
d^{3}R_{1}d^{3}R_{2}d^{3}R_{3}d^{3}R_{4}\delta(\vec{r_{1}}-\vec{R_{3}}+\vec{R_{1}}+\vec{R_{2}})\delta(\vec{r_{2}}+\vec{R_{2}}-\vec{R_{3}}+\vec{R_{4}})P_{f}(\vec{R_{1}}|i_{2}-i_{1})P_{f}(\vec{R_{2}}|i_{3}-i_{2})P(\vec{R_{3}},\vec{R_{4}}),$

where we exploited the fact that random vectors $\vec{R_{1}}$ and
$\vec{R_{2}}$ are statistical independent from $\vec{R}_{3}$ and
$\vec{R}_{4}$. Note also that probability densities of $\vec{R_{1}}$ and
$\vec{R_{2}}$ are determined by Eq. 3).

The joint probability density $P(\vec{R_{3}},\vec{R_{4}})$ cannot be
factorized into the product of marginal probability densities
$P_{coil}(\vec{R_{3}}|s_{3},L)\cdot P_{coil}(\vec{R_{4}}|s_{3},L)$ since two
vectors connecting pairs of point belonging the same loop are statistically
correlated. Exploiting results of the work [12] we can write

$P\left(\vec{R_{3}},\vec{R_{4}}\mid
l,h,L\right)=\left(\frac{L}{4\pi^{2}hl\left(L-l-h\right)}\right)^{3/2}\exp\left(-\frac{\left(\vec{R_{3}}+\vec{R_{4}}\right)^{2}}{2h}-\frac{\vec{R_{3}}^{2}}{2\left(L-l-h\right)}-\frac{\vec{R_{4}}^{2}}{2l}\right),$
(76)

where $l=j_{1}-j_{2}$ and $h=j_{2}-i_{3}$. There we can rewrite Eq. (75) as

$\displaystyle p(c_{1},c_{2}|c_{3})=P(\vec{r_{1}}=0,\vec{r_{2}}=0|c_{3})=$
(77) $\displaystyle\int
d^{3}R_{2}d^{3}R_{3}P_{f}(\vec{R_{3}}-\vec{R_{2}}|s_{1}-s_{2}-l)P_{f}(\vec{R_{2}}|s_{2}-h)P(\vec{R_{3}},\vec{R_{3}}-\vec{R_{2}}|l,h,L)\sim$
$\displaystyle\sim\frac{1}{(s_{1}-s_{2}-l)^{3/2}}\cdot\frac{1}{(s_{2}-h)^{3/2}}\cdot\left(\frac{L}{hl\left(L-l-h\right)}\right)^{3/2}\cdot$
$\displaystyle\cdot\int
d^{3}R_{2}d^{3}R_{3}\exp\left(-\frac{(\vec{R_{3}}-\vec{R_{2}})^{2}}{2(s_{1}-s_{2}-l)}-\frac{\vec{R_{2}}^{2}}{2(s_{2}-h)}-\frac{\vec{R_{2}}^{2}}{2h}-\frac{(\vec{R_{3}}-\vec{R_{2}})^{2}}{2l}-\frac{\vec{R_{3}}^{2}}{2(L-l-h)}-\right)\sim$
$\displaystyle\sim\frac{1}{(s_{1}-s_{2}-l)^{3/2}}\cdot\frac{1}{(s_{2}-h)^{3/2}}\cdot\left(\frac{L}{hl\left(L-l-h\right)}\right)^{3/2}\cdot\frac{(L-l-h)^{3/2}\cdot(s_{1}-s_{2}-l)^{3/2}\cdot(lh)^{3/2}\cdot(s_{2}-h)^{3/2}}{\left((s_{1}-s_{2})(Ls_{2}-h^{2})-l^{2}s_{2}\right)^{3/2}}.$

Restoring the prefactor, we finally get

$p_{9}(c_{1},c_{2}|c_{3})=A^{2}\frac{L^{3/2}}{\left((s_{1}-s_{2})(Ls_{2}-h^{2})-l^{2}s_{2}\right)^{3/2}}.$
(78)

Finally, the probability of observing the diagram 9 is given by the following
expression

$\omega^{(9)}(h,L|s_{1},s_{2})=\pi_{gap}p_{gap}(s_{1}-l-h)p_{loop}(L),$ (79)

where $L\in[l,\infty)$ and $h\in[0,min(L-l,s_{2})]$.

Due to symmetry reasons, the diagram 9B, representing the mirror image of 9A,
can be accounted by multiplying the contribution coming from 9A by factor $2$.

##  Appendix C Linearized statistical weights

As explained in the main text we restrict ourselves to one-loop diagrams. Such
an approximation is justified under assumption of smallness of dimensionless
parameters

$\frac{\lambda}{d}\sim\frac{s_{1}}{d}\sim\frac{s_{2}}{d}\sim\varepsilon\ll 1.$
(80)

Since our calculations are correct only up to first order in small parameter
$\varepsilon$, before calculating the pair correlator $C(s_{1},s_{2})$, the
previously obtained expressions $\omega^{\alpha}(h,L|s_{1},s_{2})$ should be
linearized

$\displaystyle\omega^{(1)}(h,L|s_{1},s_{2})=\frac{1}{(d+\lambda)\lambda}\exp(-\frac{s_{1}-L}{d}-\frac{L}{\lambda})=\frac{e^{-\frac{L}{\lambda}}}{\lambda
d}\left(1-\frac{s_{1}}{d}\right)\cdot\left(1+\frac{L}{d}\right)\cdot\left(1-\frac{\lambda}{d}\right)+O(\varepsilon^{2})=$
(81) $\displaystyle=\frac{e^{-\frac{L}{\lambda}}}{\lambda
d}\left(1-\frac{s_{1}+\lambda-L}{d}\right)+O(\varepsilon^{2}),$
$\displaystyle\omega^{(2)}(h,L|s_{1},s_{2})=\frac{1}{(d+\lambda)\lambda}\exp(-\frac{s_{1}-L}{d}-\frac{L}{\lambda})=\frac{e^{-\frac{L}{\lambda}}}{\lambda
d}\left(1-\frac{s_{1}}{d}\right)\cdot\left(1+\frac{L}{d}\right)\cdot\left(1-\frac{\lambda}{d}\right)+O(\varepsilon^{2})=$
(82) $\displaystyle=\frac{e^{-\frac{L}{\lambda}}}{\lambda
d}\left(1-\frac{s_{1}+\lambda-L}{d}\right)+O(\varepsilon^{2}),$
$\displaystyle\omega^{(3)}(q,L|s_{1},s_{2},l)=\pi_{loop}\frac{L}{\lambda}p_{loop}(L)\Theta(q)\Theta(1-q)=\frac{L}{d+\lambda}\frac{1}{\lambda}e^{-\frac{L}{\lambda}}\Theta(q)\Theta(1-q)=$
(83) $\displaystyle=\frac{Le^{-\frac{L}{\lambda}}}{\lambda
d}\left(1-\frac{\lambda}{d}\right)\Theta(q)\Theta(1-q)+O(\varepsilon^{2}),$
$\omega^{(4)}(h,L|s_{1},s_{2},l)=\frac{d}{d+\lambda}\exp(-\frac{s_{1}}{d})=\left(1-\frac{s_{1}}{d}\right)\cdot\left(1-\frac{\lambda}{d}\right)+O(\varepsilon^{2})=1-\frac{s_{1}+\lambda}{d}+O(\varepsilon^{2})$
(84)
$\displaystyle\omega^{(5)}(h,L|s_{1},s_{2})=\frac{1}{(d+\lambda)\lambda}\exp(-\frac{s_{1}-L}{d}-\frac{L}{\lambda})=\frac{e^{-\frac{L}{\lambda}}}{\lambda
d}\left(1-\frac{s_{1}}{d}\right)\cdot\left(1+\frac{L}{d}\right)\cdot\left(1-\frac{\lambda}{d}\right)+O(\varepsilon^{2})=$
(85) $\displaystyle=\frac{e^{-\frac{L}{\lambda}}}{\lambda
d}\left(1-\frac{s_{1}+\lambda-L}{d}\right)+O(\varepsilon^{2}),$
$\displaystyle\omega^{(6)}(h,L|s_{1},s_{2})=\frac{1}{(d+\lambda)\lambda}\exp(-\frac{s_{1}-L}{d}-\frac{L}{\lambda})=\frac{e^{-\frac{L}{\lambda}}}{\lambda
d}\left(1-\frac{s_{1}}{d}\right)\cdot\left(1+\frac{L}{d}\right)\cdot\left(1-\frac{\lambda}{d}\right)+O(\varepsilon^{2})=$
(86) $\displaystyle=\frac{e^{-\frac{L}{\lambda}}}{\lambda
d}\left(1-\frac{s_{1}+\lambda-L}{d}\right)+O(\varepsilon^{2}),$
$\displaystyle\omega^{(7)}(h,L|s_{1},s_{2})=\frac{L}{d+\lambda}\frac{1}{\lambda}\exp(-\frac{L}{\lambda}-\frac{s_{1}-qL}{d})\Theta(q)\Theta(1-q)=\frac{Le^{-\frac{L}{\lambda}}}{\lambda
d}\left(1-\frac{s_{1}+\lambda-
qL}{d}\right)\Theta(q)\Theta(1-q)+O(\varepsilon^{2})$ (87)
$\displaystyle\omega^{(8)}(h,L|s_{1},s_{2})=\frac{1}{\lambda+d}\frac{1}{\lambda}\exp(-\frac{s_{1}-h}{d}-\frac{L}{\lambda})=\frac{e^{-\frac{L}{\lambda}}}{\lambda
d}\left(1-\frac{s_{1}}{d}\right)\cdot\left(1+\frac{h}{d}\right)\cdot\left(1-\frac{\lambda}{d}\right)+O(\varepsilon^{2})=$
(88) $\displaystyle=\frac{e^{-\frac{L}{\lambda}}}{\lambda
d}\left(1-\frac{s_{1}+\lambda-h}{d}\right)+O(\varepsilon^{2}).$
$\displaystyle\omega^{(9)}(h,L|s_{1},s_{2})\frac{1}{\lambda+d}\frac{1}{\lambda}\exp(-\frac{\frac{s_{1}+s_{2}}{2}-h}{d}-\frac{L}{\lambda})=\frac{e^{-\frac{L}{\lambda}}}{\lambda
d}\left(1-\frac{s_{1}+s_{2}}{2d}\right)\cdot\left(1+\frac{h}{d}\right)\cdot\left(1-\frac{\lambda}{d}\right)+O(\varepsilon^{2})=$
(89) $\displaystyle=\frac{e^{-\frac{L}{\lambda}}}{\lambda
d}\left(1-\frac{s_{1}+s_{2}+2\lambda-2h}{2d}\right)+O(\varepsilon^{2}).$

##  Appendix D Pair correlation function

Multiplying conditional probabilities of contacts by linearized statistical
weights and integrating over random variables associated with loops disorder
according to Eq. (11) we arrive at Eq. (17) where

$\displaystyle
F(z_{1},z_{2})=-1-z_{1}+z_{2}^{2}\int_{0}^{1}d\tilde{L}\frac{e^{-\tilde{L}z_{2}}}{(1-\tilde{L})^{1/2}}+$
(90)
$\displaystyle+z_{2}^{2}\int_{1}^{\frac{1}{2}(1+z_{1}/z_{2})}d\tilde{L}\frac{\tilde{L}^{3/2}}{(\tilde{L}-1)^{1/2}}\cdot\frac{1}{\left(1-\frac{z_{2}(\tilde{L}-1)}{z_{1}-z_{2}}\right)^{3/2}}\cdot
e^{-\tilde{L}z_{2}}+$ (91)
$\displaystyle+z_{1}^{2}\int_{\frac{1}{2}(1+z_{2}/z_{1})}^{1}d\tilde{L}\frac{\tilde{L}^{3/2}}{(1-\tilde{L})^{1/2}}\cdot\frac{1}{\left(1-\frac{z_{1}(1-\tilde{L})}{z_{1}-z_{2}}\right)^{3/2}}\cdot
e^{-\tilde{L}z_{1}}+$ (92)
$\displaystyle+z_{1}^{2}\int_{1}^{\infty}d\tilde{L}\frac{\tilde{L}^{3/2}}{(\tilde{L}-1)^{1/2}}\cdot
e^{-\tilde{L}z_{1}}+$ (93)
$\displaystyle+\frac{(z_{1}-z_{2})^{2}}{2}\int_{0}^{1}d\tilde{L}\frac{(1-\tilde{L})}{(1-\tilde{L}/2)^{3/2}}\cdot
e^{-\frac{\tilde{L}}{2}(z_{1}-z_{2})}+$ (94)
$\displaystyle+2z_{2}^{2}\int_{0}^{1}d\tilde{h}\int_{1}^{\frac{z_{1}+z_{2}}{2z_{2}}}d\tilde{L}\frac{(L-h)^{3/2}\cdot\left(1-\frac{z_{2}}{z_{1}}\right)^{3/2}\cdot
e^{-z_{2}(L-h)}}{\left[(L+h-1)(1-\frac{z_{2}}{z_{1}}(L-h))-h^{2}\right]^{3/2}}+$
(95)
$\displaystyle+2(z_{1}-z_{2})^{2}\cdot\int_{0}^{1/2}d\tilde{L}\int_{0}^{1}dq\frac{\tilde{L}}{(1-q^{2}\tilde{L})^{3/2}}\cdot
e^{-\tilde{L}(z_{1}-z_{2})}+$ (96)
$\displaystyle+2(z_{1}-z_{2})^{2}\cdot\int_{1/2}^{\infty}d\tilde{L}\int_{0}^{\frac{1}{2\tilde{L}}}dq\frac{\tilde{L}}{(1-q^{2}\tilde{L})^{3/2}}\cdot
e^{-\tilde{L}(z_{1}-z_{2})}+$ (97)
$\displaystyle+\frac{(z_{1}+z_{2})^{2}}{2}\int_{1}^{\infty}d\tilde{L}\int_{1}^{min(\tilde{L},\frac{2z_{1}}{z_{1}+z_{2}})}d\tilde{h}\frac{\tilde{L}^{3/2}\cdot
e^{-\frac{\tilde{L}}{2}(z_{1}+z_{2})}}{\left(\tilde{L}+\tilde{h}\frac{2z_{2}}{z_{1}-z_{2}}-\tilde{h}^{2}\frac{z_{1}+z_{2}}{2(z_{1}-z_{2})}-\frac{2z_{1}z_{2}}{z_{1}^{2}-z_{2}^{2}}\right)^{3/2}}+$
(98)
$\displaystyle+\frac{(z_{1}-z_{2})^{2}}{2}\int_{1}^{\infty}d\tilde{L}\int_{0}^{min(\tilde{L}-1,\frac{2z_{2}}{z_{1}-z_{2}})}d\tilde{h}\frac{\tilde{L}^{3/2}\cdot
e^{-\tilde{L}\frac{z_{1}-z_{2}}{2}}}{(\tilde{L}-\tilde{h}^{2}\cdot\frac{z_{1}-z_{2}}{2z_{2}}-\frac{1}{2})^{3/2}}+O(\varepsilon),$
(99)

and we passed to dimensionless variables of integration.

## References

  * Ganji  _et al._ [2018] M. Ganji, I. A. Shaltiel, S. Bisht, E. Kim, A. Kalichava, C. H. Haering, and C. Dekker, Real-time imaging of dna loop extrusion by condensin, Science 360, 102 (2018).
  * Golfier  _et al._ [2020] S. Golfier, T. Quail, H. Kimura, and J. Brugués, Cohesin and condensin extrude dna loops in a cell cycle-dependent manner, Elife 9, e53885 (2020).
  * Kong  _et al._ [2020] M. Kong, E. E. Cutts, D. Pan, F. Beuron, T. Kaliyappan, C. Xue, E. P. Morris, A. Musacchio, A. Vannini, and E. C. Greene, Human condensin i and ii drive extensive atp-dependent compaction of nucleosome-bound dna, Molecular cell 79, 99 (2020).
  * Davidson  _et al._ [2019] I. F. Davidson, B. Bauer, D. Goetz, W. Tang, G. Wutz, and J.-M. Peters, Dna loop extrusion by human cohesin, Science 366, 1338 (2019).
  * Kim  _et al._ [2019] Y. Kim, Z. Shi, H. Zhang, I. J. Finkelstein, and H. Yu, Human cohesin compacts dna by loop extrusion, Science 366, 1345 (2019).
  * Ryu  _et al._ [2020] J.-K. Ryu, A. J. Katan, E. O. van der Sluis, T. Wisse, R. de Groot, C. H. Haering, and C. Dekker, The condensin holocomplex cycles dynamically between open and collapsed states, Nature Structural & Molecular Biology 27, 1134 (2020).
  * Banigan and Mirny [2020] E. J. Banigan and L. A. Mirny, Loop extrusion: theory meets single-molecule experiments, Current opinion in cell biology 64, 124 (2020).
  * Kimura  _et al._ [1999] K. Kimura, V. V. Rybenkov, N. J. Crisona, T. Hirano, and N. R. Cozzarelli, 13s condensin actively reconfigures dna by introducing global positive writhe: implications for chromosome condensation, Cell 98, 239 (1999).
  * Nasmyth [2001] K. Nasmyth, Disseminating the genome: joining, resolving, and separating sister chromatids during mitosis and meiosis, Annual review of genetics 35, 673 (2001).
  * Riggs [1990] A. Riggs, Dna methylation and late replication probably aid cell memory, and type i dna reeling could aid chromosome folding and enhancer function, Philosophical Transactions of the Royal Society of London. B, Biological Sciences 326, 285 (1990).
  * Polovnikov  _et al._ [2023] K. E. Polovnikov, H. B. Brandão, S. Belan, B. Slavov, M. Imakaev, and L. A. Mirny, Crumpled polymer with loops recapitulates key features of chromosome organization, Physical Review X 13, 041029 (2023).
  * Belan and Starkov [2022] S. Belan and D. Starkov, Influence of active loop extrusion on the statistics of triple contacts in the model of interphase chromosomes, JETP Letters 115, 763 (2022).
  * Belan and Parfenyev [2023] S. Belan and V. Parfenyev, Footprints of loop extrusion in statistics of intra-chromosomal distances: an analytically solvable model, arXiv preprint arXiv:2301.03856 (2023).
  * Slavov and Polovnikov [2023] B. Slavov and K. Polovnikov, Intrachain distances in a crumpled polymer with random loops, JETP Letters 118, 208 (2023).
  * Cattoni  _et al._ [2017] D. I. Cattoni, A. M. Cardozo Gizzi, M. Georgieva, M. Di Stefano, A. Valeri, D. Chamousset, C. Houbron, S. Déjardin, J.-B. Fiche, I. González, _et al._ , Single-cell absolute contact probability detection reveals chromosomes are organized by multiple low-frequency yet specific interactions, Nature communications 8, 1 (2017).
  * Gassler  _et al._ [2017] J. Gassler, H. B. Brandão, M. Imakaev, I. M. Flyamer, S. Ladstätter, W. A. Bickmore, J.-M. Peters, L. A. Mirny, and K. Tachibana, A mechanism of cohesin-dependent loop extrusion organizes zygotic genome architecture, The EMBO journal 36, 3600 (2017).
  * Nagano  _et al._ [2013] T. Nagano, Y. Lubling, T. J. Stevens, S. Schoenfelder, E. Yaffe, W. Dean, E. D. Laue, A. Tanay, and P. Fraser, Single-cell hi-c reveals cell-to-cell variability in chromosome structure, Nature 502, 59 (2013).
  * Stevens  _et al._ [2017] T. J. Stevens, D. Lando, S. Basu, L. P. Atkinson, Y. Cao, S. F. Lee, M. Leeb, K. J. Wohlfahrt, W. Boucher, A. O’Shaughnessy-Kirwan, _et al._ , 3d structures of individual mammalian genomes studied by single-cell hi-c, Nature 544, 59 (2017).
  * Nagano  _et al._ [2015] T. Nagano, Y. Lubling, E. Yaffe, S. W. Wingett, W. Dean, A. Tanay, and P. Fraser, Single-cell hi-c for genome-wide detection of chromatin interactions that occur simultaneously in a single cell, Nature protocols 10, 1986 (2015).
  * Ramani  _et al._ [2020] V. Ramani, X. Deng, R. Qiu, C. Lee, C. M. Disteche, W. S. Noble, J. Shendure, and Z. Duan, Sci-hi-c: a single-cell hi-c method for mapping 3d genome organization in large number of single cells, Methods 170, 61 (2020).
  * Galitsyna and Gelfand [2021] A. A. Galitsyna and M. S. Gelfand, Single-cell hi-c data analysis: safety in numbers, Briefings in bioinformatics 22, bbab316 (2021).
  * Kos  _et al._ [2021] P. I. Kos, A. A. Galitsyna, S. V. Ulianov, M. S. Gelfand, S. V. Razin, and A. V. Chertovich, Perspectives for the reconstruction of 3d chromatin conformation using single cell hi-c data, PLoS Computational Biology 17, e1009546 (2021).
  * Zhang  _et al._ [2022] R. Zhang, T. Zhou, and J. Ma, Multiscale and integrative single-cell hi-c analysis with higashi, Nature biotechnology 40, 254 (2022).
  * Ou  _et al._ [2017] H. D. Ou, S. Phan, T. J. Deerinck, A. Thor, M. H. Ellisman, and C. C. O’shea, Chromemt: Visualizing 3d chromatin structure and compaction in interphase and mitotic cells, Science 357, eaag0025 (2017).
  * Bintu  _et al._ [2018] B. Bintu, L. J. Mateo, J.-H. Su, N. A. Sinnott-Armstrong, M. Parker, S. Kinrot, K. Yamaya, A. N. Boettiger, and X. Zhuang, Super-resolution chromatin tracing reveals domains and cooperative interactions in single cells, Science 362, eaau1783 (2018).
  * Nir  _et al._ [2018] G. Nir, I. Farabella, C. Pérez Estrada, C. G. Ebeling, B. J. Beliveau, H. M. Sasaki, S. D. Lee, S. C. Nguyen, R. B. McCole, S. Chattoraj, _et al._ , Walking along chromosomes with super-resolution imaging, contact maps, and integrative modeling, PLoS genetics 14, e1007872 (2018).
  * Boettiger and Murphy [2020] A. Boettiger and S. Murphy, Advances in chromatin imaging at kilobase-scale resolution, Trends in Genetics 36, 273 (2020).
  * Kempfer and Pombo [2020] R. Kempfer and A. Pombo, Methods for mapping 3d chromosome architecture, Nature Reviews Genetics 21, 207 (2020).
  * Su  _et al._ [2020] J.-H. Su, P. Zheng, S. S. Kinrot, B. Bintu, and X. Zhuang, Genome-scale imaging of the 3d organization and transcriptional activity of chromatin, Cell 182, 1641 (2020).
  * Liu  _et al._ [2020] M. Liu, Y. Lu, B. Yang, Y. Chen, J. S. Radda, M. Hu, S. G. Katz, and S. Wang, Multiplexed imaging of nucleome architectures in single cells of mammalian tissue, Nature communications 11, 1 (2020).
  * Xie and Liu [2021] L. Xie and Z. Liu, Single-cell imaging of genome organization and dynamics, Molecular Systems Biology 17, e9653 (2021).
  * Li  _et al._ [2021] Y. Li, A. Eshein, R. K. Virk, A. Eid, W. Wu, J. Frederick, D. VanDerway, S. Gladstein, K. Huang, A. R. Shim, _et al._ , Nanoscale chromatin imaging and analysis platform bridges 4d chromatin organization with molecular function, Science advances 7, eabe4310 (2021).
  * Gabriele  _et al._ [2022] M. Gabriele, H. B. Brandão, S. Grosse-Holz, A. Jha, G. M. Dailey, C. Cattoglio, T.-H. S. Hsieh, L. Mirny, C. Zechner, and A. S. Hansen, Dynamics of ctcf-and cohesin-mediated chromatin looping revealed by live-cell imaging, Science 376, 496 (2022).
  * Fudenberg  _et al._ [2017] G. Fudenberg, N. Abdennur, M. Imakaev, A. Goloborodko, and L. A. Mirny, Emerging evidence of chromosome folding by loop extrusion, in  _Cold Spring Harbor symposia on quantitative biology_ , Vol. 82 (Cold Spring Harbor Laboratory Press, 2017) pp. 45–55.
  * Goloborodko  _et al._ [2016] A. Goloborodko, J. F. Marko, and L. A. Mirny, Chromosome compaction by active loop extrusion, Biophysical Journal 110, 2162 (2016).
  * Grosberg and Khokhlov [1994] A. Y. Grosberg and A. Khokhlov,  _Statistical Physics of Macromolecules_ (Woodbury, NY: AIP Press, 1994).
  * Zhang and Xia [2001] L.-X. Zhang and A. Xia, Probability of triple contacts in polymer chains, European polymer journal 37, 1263 (2001).
  * Nuebler  _et al._ [2018] J. Nuebler, G. Fudenberg, M. Imakaev, N. Abdennur, and L. Mirny, Chromatin organization by an interplay of loop extrusion and compartmental segregation, Biophysical Journal 114, 30a (2018).
  * Hafner and Boettiger [2022] A. Hafner and A. Boettiger, The spatial organization of transcriptional control, Nature Reviews Genetics , 1 (2022).