O Shukron

A Seeber

A Amitai

D Holcman
email: david.holcman@ens.fr

Advances using single particle trajectories to reconstruct chromatin organization and dynamics

Keywords: chromatin, loci, single particle trajectories, analysis, statistical methods, polymer models

Chromatin organization remains complex and far from understood. In this article, we review recent statistical methods of extracting biophysical parameters from in vivo single-particle trajectories of loci to reconstruct chromatin reorganization in response to cellular stress such as DNA damage. We look at methods for analyzing both single locus and multiple loci tracked simultaneously and explain how to quantify and describe chromatin motion using a combination of extractable parameters. These parameters can be converted into information about chromatin dynamics and function. Furthermore, we discuss how the time scale of recurrent encounter between loci can be extracted and interpreted. We also discuss the effect of sampling rate on the estimated parameters. Finally, we review a polymer method to reconstruct chromatin structure using cross-linkers between chromatin sites. We list and refer to some software packages that are now publicly available to simulate polymer motion. To conclude, chromatin organization and dynamics can be reconstructed from locus trajectories and predicted based on polymer models.

Exploring chromatin mobility

Fundamental physical properties of DNA, such as its internal elasticity, bending properties, and rotational energy, have been estimated ex-vivo [START_REF] Strick | Twisting and stretching single dna molecules[END_REF][START_REF] Strick | Stretching of macromolecules and proteins[END_REF]. But what about the properties of chromatin in vivo? Not only has it been difficult to visualize specific chromatin loci, but it remains unclear what physical properties should be measured. Shall we simply use classical physical observables or shall we focus on extracting other latent parameters? The choice of what to measure is open to bias. In the case of chromatin, we still do not have a complete and definitive physical model that would recapitulate and explain its dynamics and structure. However, recent works have advanced chromatin models to the point that they can predict changes in organization and dynamics [START_REF] Hauer | Histone degradation in response to dna damage enhances chromatin dynamics and recombination rates[END_REF][START_REF] Amitai | Visualization of chromatin decompaction and break site extrusion as predicted by statistical polymer modeling of single-locus trajectories[END_REF]. Here, we review work combining singleparticle trajectories (SPTs) of chromatin loci with polymer modeling. The aim of this review is to clarify how experimental observation of chromatin dynamics can be related to its underlying structure using those models. As an example, we focus our attention on the chromatin response to DNA damage. While this review focuses also on connecting chromatin dynamics and with polymer models, others have reviewed different aspects of chromatin mobility (see [START_REF] Tiana | Integrating experiment, theory and simulation to determine the structure and dynamics of mammalian chromosomes[END_REF][START_REF] Parmar | How the genome folds: The biophysics of four-dimensional chromatin organization[END_REF][START_REF] Zimmer | Chromatin mobility upon dna damage: state of the art and remaining questions[END_REF][START_REF] Fabre | From dynamic chromatin architecture to dna damage repair and back[END_REF]).

Early on, it was recognized that DNA is compacted into chromatin. Because this structure coils back on itself and has a repeating unit (see early studies showing chromatin as beads on a string), it was natural to start describing it as a polymer. Taking a coarse-grained approach, one could use a polymer model where the beads are connected by springs, repulse one another (Lennard-Jones forces) and have bending elasticity as a starting point for modeling chromatin (reviewed in [START_REF] Amitai | Polymer physics of nuclear organization and function[END_REF]) (see Box 1). Building on such a model and accounting for new experimental results coming from HiC, long-range interactions between distant chromatin sites have been addressed in simulations [START_REF] Bohn | Random loop model for long polymers[END_REF][START_REF] Heermann | Physical nuclear organization: loops and entropy[END_REF][START_REF] Barbieri | Complexity of chromatin folding is captured by the strings and binders switch model[END_REF] and theory [START_REF] Amitai | Polymer model with long-range interactions: Analysis and applications to the chromatin structure[END_REF][START_REF] Shukron | Statistics of randomly cross-linked polymer models to interpret chromatin conformation capture data[END_REF]. Although these models are used to simulate chromatin in some defined conditions, how they relate to the biological analog remains unclear.

To determine polymer models that infer the actual behavior of chromatin inside the nucleus, one must be able to compare extracted physical param-eters from real experimental chromatin data to those predicted from the model. To this end, the adoption of fluorophores, such as green fluorescent protein (GFP), combined with bacterial operator array systems to visualize single chromosomal loci live and in-vivo created the sub-field of chromatin dynamics [START_REF] Chen | Imaging specific genomic dna in living cells[END_REF]. More recent developments using dCas9 and modified guide RNAs as well as new bacterial arrays have expanded the live cell imaging chromatin toolbox -reviewed in [START_REF] Seeber | Chromosome dynamics in response to dna damage[END_REF][START_REF] Smith | Poetry in motion: Increased chromosomal mobility after dna damage[END_REF]. These tools have advanced our knowledge of what regulates chromatin movement. In particular, research in budding yeast, performed by many laboratories over two decades, has elucidated a partial framework of the regulators of chromatin movement. While the important fine details are better reviewed elsewhere [START_REF] Seeber | Chromosome dynamics in response to dna damage[END_REF] we highlight a few key regulators.

To start, chromatin motion depends on intracellular ATP levels. Glucose starvation or treatment of cells with either sodium azide or the mitochondrial uncoupler CCCP (carbonyl cyanide m-chlorophenylhydrazone), all of which deplete ATP, reduce chromatin motion [START_REF] Heun | Chromosome dynamics in the yeast interphase nucleus[END_REF][START_REF] Gartenberg | Sir-mediated repression can occur independently of chromosomal and subnuclear contexts[END_REF][START_REF] Liu | Nanoscale histone localization in live cells reveals reduced chromatin mobility in response to dna damage[END_REF][START_REF] Joyner | A glucose-starvation response regulates the diffusion of macromolecules[END_REF][START_REF] Levi | Chromatin dynamics during interphase explored by single-particle tracking[END_REF][START_REF] Weber | Nonthermal atpdependent fluctuations contribute to the in vivo motion of chromosomal loci[END_REF]. In addition to ATP, the cell cycle stage has a dramatic effect on chromatin movement where chromatin moves less in S phase and is much more dynamic in G1 phase. This effect is conserved from yeast to human cells [START_REF] Heun | Chromosome dynamics in the yeast interphase nucleus[END_REF][START_REF] Nozaki | Dynamic organization of chromatin domains revealed by super-resolution livecell imaging[END_REF][START_REF] Krawczyk | Chromatin mobility is increased at sites of dna double-strand breaks[END_REF][START_REF] Pliss | Chromatin dynamics is correlated with replication timing[END_REF]. The effect depends on DNA replication and artificial cleavage of cohesin in S phase restores mobility to G1 levels [START_REF] Dion | Cohesin and the nucleolus constrain the mobility of spontaneous repair foci[END_REF]. Chromosome tethers also play a role in restricting chromatin movement. A clear example is the Rabl configuration of chromosomes in budding yeast whereby the telomeres transiently associate with the nuclear periphery and the centromeres are attached to the spindle pole body [START_REF] Taddei | Structure and function in the budding yeast nucleus[END_REF][START_REF] Jin | Centromere clustering is a major determinant of yeast interphase nuclear organization[END_REF]. These attachments restrict motion and when artificially removed increase chromatin dynamics. Although yeast do not have a nuclear lamina, the loss chromosome tethering by ablation of lamin A in mammalian cells leads to increased chromosome dynamics [START_REF] Gartenberg | Sir-mediated repression can occur independently of chromosomal and subnuclear contexts[END_REF][START_REF] Hediger | Live imaging of telomeres: yku and sir proteins define redundant telomere-anchoring pathways in yeast[END_REF][START_REF] Spichal | Evidence for a dual role of actin in regulating chromosome organization and dynamics in yeast[END_REF][START_REF] Strecker | Dna damage signalling targets the kinetochore to promote chromatin mobility[END_REF][START_REF] Verdaasdonk | Centromere tethering confines chromosome domains[END_REF][START_REF] Bronshtein | Loss of lamin a function increases chromatin dynamics in the nuclear interior[END_REF]. Another master regulator of chromatin movement is nucleosome occupancy. Work on the chromatin remodeling enzyme, INO80, showed that direct targeting of INO80 to a chromosomal locus was sufficient to increase the dynamics of that locus [START_REF] Neumann | Targeted ino80 enhances subnuclear chromatin movement and ectopic homologous recombination[END_REF]. Loss of nucleosomes on DNA through other means, either by the depletion of Nhp6, the budding yeast version of high mobility group protein B1 (HMGB1), or the transcriptional downregulation of histone genes, also increases global chromatin dynamics [START_REF] Hauer | Histone degradation in response to dna damage enhances chromatin dynamics and recombination rates[END_REF]. Thus, many regulators of chromatin dynamics have been described. What is still little understood is the function of chromatin movement.

While chromatin mobility can be regulated experimentally, proving that it has a role in cellular processes has been difficult. An early postulated role for mobility was transcription. This was supported by evidence that artificial activation of transcription by targeting the viral transactivator protein VP16 to a transgene array in mammalian cells causes the locus to undergo directional movement away from the nuclear periphery [START_REF] Chuang | Long-range directional movement of an interphase chromosome site[END_REF]. Consistently, targeting of VP16 to a yeast telomeric locus also drives this locus away from the periphery and VP16 targeting to a locus on the middle of a chromosomal arm increases its dynamics [START_REF] Neumann | Targeted ino80 enhances subnuclear chromatin movement and ectopic homologous recombination[END_REF]. However, targeting of another transcriptional activator, Gal4, did not increase locus dynamics nor did the inhibition of transcription in yeast cells [START_REF] Neumann | Targeted ino80 enhances subnuclear chromatin movement and ectopic homologous recombination[END_REF]. The effect of transcription in higher eukaryotes, in general, is less clear and has been correlated with decreased motion [START_REF] Chen | Dynamic interplay between enhancer-promoter topology and gene activity[END_REF]. Others have shown that inhibition of RNA polymerase II increases chromatin motion and that transcription may serve to stabilize networks of chromatin domains [START_REF] Ochiai | Simultaneous live imaging of the transcription and nuclear position of specific genes[END_REF][START_REF] Germier | Real-time imaging of a single gene reveals transcription-initiated local confinement[END_REF][START_REF] Nagashima | Single nucleosome imaging reveals loose genome chromatin networks via active rna polymerase ii[END_REF]. Measuring chromatin movement during enhancer and promoter contact still remains relatively unstudied. Another role in which chromatin movement may serve is the efficiency of repair of DNA double-strand breaks (DSBs) by a process called homologous recombination (HR), discussed in [START_REF] Hauer | Histone degradation in response to dna damage enhances chromatin dynamics and recombination rates[END_REF]. Here, a broken DNA strand must physically scan the nucleus to find its homologous partner if the replicated sister is not present or is otherwise damaged. While definitive proof that chromatin motion facilitates repair by HR remains elusive, there are a number of correlations. In budding yeast, induction of a DSB and activation of the DNA damage checkpoint, a control mechanism in the cell-division cycle that ensures that mitosis does not occur until damaged DNA has been repaired, increases chromatin motion both locally and globally [START_REF] Dion | Increased mobility of double-strand breaks requires mec1, rad9 and the homologous recombination machinery[END_REF][START_REF] Seeber | Checkpoint kinases and the ino80 nucleosome remodeling complex enhance global chromatin mobility in response to dna damage[END_REF]. Ablation of proteins that result in reduced movement decreases repair efficiency by HR [START_REF] Dion | Increased mobility of double-strand breaks requires mec1, rad9 and the homologous recombination machinery[END_REF], while mutations that increase movement improve HR efficiency [START_REF] Hauer | Histone degradation in response to dna damage enhances chromatin dynamics and recombination rates[END_REF][START_REF] Neumann | Targeted ino80 enhances subnuclear chromatin movement and ectopic homologous recombination[END_REF]. Ultimately, these still remain correlations and an experiment directly showing that enhanced movement is still lacking.

In general, chromatin movement has been studied by analyzing single locus trajectories and plotting the mean square displacement (MSD). From this, a radius of constraint or volume of space explored by the locus can be extracted. A number of recent studies have published methods on extracting hidden directed motion within trajectories using methods such as directional change distribution (DCD) analysis or computational methods to identify long-lasting directed motions (LDMs) [START_REF] Caridi | Nuclear f-actin and myosins drive relocalization of heterochromatic breaks[END_REF][START_REF] Burov | Distribution of directional change as a signature of complex dynamics[END_REF]. In addition to these useful tools, we and others have asked the question, "what other information can be obtained from trajectories?". The goal of this review is to present advances single particle trajectory analysis and to illustrate how extracted parameters can be used in polymer models to make biological predictions.

This review is organized as follows: we first describe diffusion models that have been used to analyze chromatin trajectories. We introduce four key physical parameters that can be extracted from chromatin loci trajectories and can be used in combination to characterize motion. We then focus on the selection of polymer models and describe how the sampling time step influences the extracted statistical observables from the trajectories. Next we will present how to perform a correlation analysis for two loci recorded simultaneously and we will discuss physical constraints on chromatin and describe an approach to estimate the number of cross-linkers between chromatin loci. Finally, we summarize publicly available software packages for polymer simulations (see supplementary material). Although many examples described here concern yeast, the statistical methods, and data analysis can be applied to many other organisms, from insects to mammalian cells [START_REF] Chen | Dynamic interplay between enhancer-promoter topology and gene activity[END_REF][START_REF] Tiana | Structural fluctuations of the chromatin fiber within topologically associating domains[END_REF][START_REF] Maass | Spatiotemporal allele organization by allele-specific crispr live-cell imaging (snp-cling)[END_REF].

Chromatin locus dynamics revealed by SPTs

Choice of a diffusion model: free, confined or anomalous SPTs consist of an ensemble of successive points acquired at a sampling rate ∆t. What physical observables can be extracted from these trajectories? To answer this question a physical diffusion model (not to be confused with a polymer model) should be selected. The model chosen to fit a MSD curve must provide plausible physical mechanisms and accurate predictions. Examples of physical models include Brownian motion or Random walk (which is a discretized version of a Brownian motion at a time step ∆t) and anomalous diffusion. Selection of the correct model remains challenging because not all loci behave similarly. Failure to select the correct model will prevent accurate extraction of parameters. A recent study used a Bayesian inference approach to automate MSD model selection across trajectories throughout the nucleus [START_REF] Shaban | Nanoscale mapping of dna dynamics in live human cells[END_REF]. Others have published approaches to test for the goodness-of-fit of model assumptions and estimated parameters [START_REF] Calderon | Quantifying transient 3d dynamical phenomena of single mrna particles in live yeast cell measurements[END_REF][START_REF] Calderon | Inferring latent states and refining force estimates via hierarchical dirichlet process modeling in single particle tracking experiments[END_REF]. In the following section, we will briefly describe some physical models and what physical observable should be measured.

The most well-known example of passive molecular dynamics is classical free diffusion (see Glossary) described by Brown in 1827 and quantified statistically by Einstein in 1905. A particle driven by free diffusion at scale ∆t is characterized by random jumps that follow a Gaussian distribution with zero mean and variance 2D∆t, where D is the diffusion coefficient. In this case, the physical observable to estimate is simply the diffusion coefficient D. However, the locus motion can be restricted due to obstacles or to some resulting tethering force, which is not accounted in classical Brownian motion. In that case, the extent of the restricted region and the nature of the restriction (confinement vs tethering force) must be determined. Another model used to describe the motion of a locus is a deviation of classical Brownian motion, called anomalous diffusion. Here the forces underlying the locus dynamics are correlated in time [START_REF] Metzler | The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics[END_REF]. In this case, the correlation modifies the nature of the dynamics (see section "Anomalous exponent α"). Another possibility is that the particle motion observed by SPTs result from a combination of deterministic (generated by a force) and Brownian motion. The motion could also result from alternating between a deterministic force and diffusion. The nature of the force should be identified as well as the switching rate, often approximated as Poissonian (characterized by a single exponential parameter).

To summarize, we need to select a physical model, making an informed decision based on trajectory data. When successful, the underlying model can be used to extract specific parameters that characterize chromatin motion.

Parameters that describe chromatin motion

In this section, we describe four parameters computed from SPTs that are used to characterize chromatin motion. They provide independent, complementary information on first and second moment statistics.

Length of Constraint L c to characterize confined motion

How much space does a locus explore? The majority of studies up to this point have used the plateau of an MSD curve to estimate the radius of constraint. This number coupled with the size of the nucleus can be used to estimate the relative volume of space a locus will move through. However, not all MSD graphs plateau and thus a plateau-independent parameter is required. We and others have proposed using another parameter, such as the standard deviation of the locus position with respect to its mean average over time [START_REF] Amitai | Visualization of chromatin decompaction and break site extrusion as predicted by statistical polymer modeling of single-locus trajectories[END_REF][START_REF] Verdaasdonk | Centromere tethering confines chromosome domains[END_REF], that we term the length of constraint L c (see Box 2). This parameter, like the radius of constraint, can be used to estimate the degree of confinement of a locus. A small L c compared to the radius of the nucleus is thus the signature of a high confined motion, while a large L c means that the locus motion is not restricted by any nuclear bodies or chromatin. In summary, the length of constraint L c provides a measure of confinement, but it does not reveal the underlying mechanism of motion.

Anomalous exponent α

Anomalous diffusion is a type of random motion characterized by an anomalous exponent α, which is classically extracted from the slope of the (MSD) |X(t + ∆t) -X(t)| 2 that behaves like At α for ∆t small over the time (Box 2). When α = 1 it reflects Brownian motion, α > 1 is called super-diffusion, which may represent a dynamics containing an element of deterministic (ballistic) directed motion. Finally, α < 1 is a sub-diffusive motion, which could result from forces between monomers/chromatin sites (see below) and/or the visco-elastic properties of the nuclear plasm [START_REF] Amitai | Visualization of chromatin decompaction and break site extrusion as predicted by statistical polymer modeling of single-locus trajectories[END_REF][START_REF] Amitai | Polymer model with long-range interactions: Analysis and applications to the chromatin structure[END_REF][START_REF] Kepten | Improved estimation of anomalous diffusion exponents in single-particle tracking experiments[END_REF][START_REF] Kepten | Guidelines for the fitting of anomalous diffusion mean square displacement graphs from single particle tracking experiments[END_REF]. When ∆t is in the range of 30 to 300ms, we estimate α from the first six time points of an MSD. However, we point out that the initial slope of an MSD curve is calculated using all time points and thus longer trajectories will provide a more accurate approximation of α. One way to compute the MSD is by taking the ensemble-average -averaging the displacements of multiple particles. The alternative is a time-averaged MSD, which averages all the displacements made by a single trajectory over a given period. The majority of single particle data on chromosomal loci is done on time-averaged MSDs.

As mentioned above, directed motion is often hidden in the MSD analysis. In that case, extracting the anomalous exponent is not enough to account for the entire dynamics. In parallel, a drift analysis has to be performed in order to extract the spatial distribution of vector fields, [START_REF] Hoze | Heterogeneity of ampa receptor trafficking and molecular interactions revealed by superresolution analysis of live cell imaging[END_REF][START_REF] Hozé | Statistical methods for large ensembles of super-resolution stochastic single particle trajectories in cell biology[END_REF][START_REF] Holcman | Single particle trajectories reveal active endoplasmic reticulum luminal flow[END_REF] and directional changes [START_REF] Burov | Distribution of directional change as a signature of complex dynamics[END_REF][START_REF] Oshidari | Nuclear microtubule filaments mediate non-linear directional motion of chromatin and promote dna repair[END_REF]. The exponent α computed for a single locus remains difficult to interpret and in particular to connect with local chromatin organization. α reveals much of the nature of a locus that belongs to a polymer but clearly remain insufficient, alone, to recover all the properties of chromatin. In general, the exponent α for chromatin loci varies in the range 0.3-0.5, when no external forces are applied to the polymer model [START_REF] Amitai | Visualization of chromatin decompaction and break site extrusion as predicted by statistical polymer modeling of single-locus trajectories[END_REF]. But in the presence of deterministic forces, generated for example by nuclear oscillation, actin or microtubule networks, the range of α can increase to (0.5 -2). To give a few examples, for a ∆t larger than 5s, α was estimated around 0.4 for the GAL loci in Yeast [START_REF] Cabal | Saga interacting factors confine sub-diffusion of transcribed genes to the nuclear envelope[END_REF] under glucose or galactose. Another study [START_REF] Backlund | Correlations of three-dimensional motion of chromosomal loci in yeast revealed by the double-helix point spread function microscope[END_REF] found α = 0.75 for the same locus when the yeast were grown under glucose and 0.64 for galactose. Possible reasons for the discrepancy between these values could be differences between yeast strain backgrounds, localization error or nuclear motion. For telomeres of mammalian cells, α ≈ 0.32, when the time series analysis is performed in the interval [10 -2 ; 1]s [START_REF] Bronstein | Transient anomalous diffusion of telomeres in the nucleus of mammalian cells[END_REF], which is consistent with other studies [START_REF] Weber | Nonthermal atpdependent fluctuations contribute to the in vivo motion of chromosomal loci[END_REF][START_REF] Weber | Bacterial chromosomal loci move subdiffusively through a viscoelastic cytoplasm[END_REF], reporting 0.39 for chromosomal loci in bacteria. In general, phenomenological models of anomalous dynamics such as Fractional Brownian motion [START_REF] Weber | Bacterial chromosomal loci move subdiffusively through a viscoelastic cytoplasm[END_REF] have been used to interpret these observations. However, as we will see below much could be learned by using a polymer models which would recapitulate these behaviors by considering independent physical sources of motion such as local interactions, drift and more.

The effective spring coefficient K c

What restricts the motion of chromatin loci? As described in the introduction, a number of chromatin tethers exist in the nucleus, such as the centromere and telomeres in budding yeast. The force of these and other tethers cannot easily be measured in vivo but they can be inferred from SPTs by estimating an effective spring coefficient K c [START_REF] Amitai | Visualization of chromatin decompaction and break site extrusion as predicted by statistical polymer modeling of single-locus trajectories[END_REF][START_REF] Verdaasdonk | Centromere tethering confines chromosome domains[END_REF][START_REF] Amitai | Analysis of single locus trajectories for extracting in vivo chromatin tethering interactions[END_REF]. A trajectory could be confined either by obstacles or by a field of force. How can we differentiate between these two possibilities? This question can be addressed by using a polymer model, where a tethering force is generated by a potential well

U ext (R n ) = 1 2 K c (µ -R n ) 2 centered
at a point µ and applied at a monomer R n . The spring constant K c has to be estimated from data (Fig. 1A, Key Figure and Box 2). The field of force U ext should influence the motion of any locus and thus could be extracted from trajectories. This tethering not only affects the motion of the whole polymer (chromosome) but can arise from interactions of the locus with other chromosomes or nuclear substructures such as the nuclear envelope. A procedure has been developed for freely moving particles [START_REF] Hozé | Statistical methods for large ensembles of super-resolution stochastic single particle trajectories in cell biology[END_REF] requiring many redundant trajectories exploring the same micro-environment. However, for a single-long trajectory that does not come back too many times on itself, there may not be necessarily enough data points. To overcome this difficulty, a recent method [START_REF] Amitai | Analysis of single locus trajectories for extracting in vivo chromatin tethering interactions[END_REF] considers consecutive displacements R c (t + ∆t) -R c (t) as independent when a trajectory is recurrent returning back on itself many times. In this case, it is possible to estimate the spring constant K c , as described in Box 2 and [START_REF] Amitai | Analysis of single locus trajectories for extracting in vivo chromatin tethering interactions[END_REF].

To conclude, when a trajectory folds back on itself many times, it is possible to extract a field of force [START_REF] Zidovska | Micron-scale coherence in interphase chromatin dynamics[END_REF].

As described in the introduction, artificial removal of cohesin in S phase cells increases chromatin motion. Interestingly, removal of cohesin in G1 phase cells, where the sister chromosome is absent, also marginally increases chromatin motion. This suggests a role for cohesin as a possible connector or tether in budding yeast [START_REF] Dion | Cohesin and the nucleolus constrain the mobility of spontaneous repair foci[END_REF][START_REF] Verdaasdonk | Centromere tethering confines chromosome domains[END_REF]. Removal of such a tether (cohesin) decreases K c [START_REF] Verdaasdonk | Centromere tethering confines chromosome domains[END_REF].

For motion confined due to crowding only, applying the above procedure to a Brownian particle would not lead to the estimation of any forces (which means K c = 0 [START_REF] Amitai | Analysis of single locus trajectories for extracting in vivo chromatin tethering interactions[END_REF]), because there is no field of force, except for the points in very close vicinity of the large obstacles [START_REF] Hoze | Heterogeneity of ampa receptor trafficking and molecular interactions revealed by superresolution analysis of live cell imaging[END_REF] that restrict motion (Fig. 1B, Key Figure). For a monomer located on a polymer, we expect K c = 0 to reflect the external forces acting on the chain. Hence, estimating K c for a locus near obstacles such as the nucleolus or the boundary of the nucleus would result in a non-zero force estimate (K c > 0). When obstacles are completely intermingled, these estimations are less robust.

Effective diffusion coefficient D c

Accurately estimating the diffusion coefficient from an SPT is not an easy task and much work has been done to develop estimators to do so [START_REF] Vestergaard | Optimal estimation of diffusion coefficients from single-particle trajectories[END_REF]. Much like the anomalous exponent, D c relies on a suitable number of points. A study measuring the diffusion of fluorescent beads in a glycerol solution generated extremely long trajectories [START_REF] Ernst | Measuring a diffusion coefficient by singleparticle tracking: statistical analysis of experimental mean squared displacement curves[END_REF] which could be decomposed into short trajectories of variable length and enabled measurement of the relative error of the diffusion coefficient. Segments consisting of 100-time points had a relative error of 25% and increasing the number of time points to 1000 decreased the error to 10%. Most chromatin loci tracking experiments record 200 time points. Therefore, it is unsurprising that the same loci in different cells can have very different diffusion coefficients, demonstrating the heterogeneity of this parameter. Others have tried to make sense of the diffusion coefficient by segmenting trajectories at points where D c changes [START_REF] Yin | Detection of velocity and diffusion coefficient change points in single-particle trajectories[END_REF]. Since the diffusion coefficient D c is highly variable between different cells, estimating it from multiple trajectories will give far greater statistical power than from a single one. Indeed, computing the diffusion coefficient of a massive number (10 4 -10 5 ) of super-resolution trajectories allows for the reconstruction of a diffusion map inside the cell. This can be used for a variety of cellular of fluorescently tagged proteins in different cellular compartments including histones and receptor proteins [START_REF] Shaban | Nanoscale mapping of dna dynamics in live human cells[END_REF][START_REF] Hoze | Heterogeneity of ampa receptor trafficking and molecular interactions revealed by superresolution analysis of live cell imaging[END_REF][START_REF] Hozé | Statistical methods for large ensembles of super-resolution stochastic single particle trajectories in cell biology[END_REF][START_REF] Manley | High-density mapping of singlemolecule trajectories with photoactivated localization microscopy[END_REF][START_REF] Frost | Singlemolecule discrimination of discrete perisynaptic and distributed sites of actin filament assembly within dendritic spines[END_REF][START_REF] English | Single-molecule investigations of the stringent response machinery in living bacterial cells[END_REF][START_REF] Gebhardt | Single-molecule imaging of transcription factor binding to dna in live mammalian cells[END_REF][START_REF] Izeddin | Single-molecule tracking in live cells reveals distinct targetsearch strategies of transcription factors in the nucleus[END_REF][START_REF] Liu | Imaging live-cell dynamics and structure at the single-molecule level[END_REF][START_REF] Hoze | Residence times of receptors in dendritic spines analyzed by stochastic simulations in empirical domains[END_REF][START_REF] Holcman | Analysis and interpretation of superresolution single-particle trajectories[END_REF][START_REF] Holcman | Single particle trajectories reveal active endoplasmic reticulum luminal flow[END_REF].

To conclude, the four parameters described above (L c , α, K c , D c , ) can be estimated from trajectories and they provide complementary information about the statistical properties of chromatin loci [START_REF] Hauer | Histone degradation in response to dna damage enhances chromatin dynamics and recombination rates[END_REF][START_REF] Amitai | Visualization of chromatin decompaction and break site extrusion as predicted by statistical polymer modeling of single-locus trajectories[END_REF]. In addition, their estimation does not assume or require any underlying physical structure such as a polymer model, which will come in a second stage for the interpretation and the reconstruction of the underlying chromatin organization at a given spatial and temporal scale (see Box 1).

Constructing polymer models that represent chromatin dynamics and structure

How to select a polymer model from the statistics of SPTs? For the Rouse polymer model, consisting of monomers connected by springs, the anomalous exponent of a monomer whose position in time is

X(t) is α = 0.5 ( |X(t + ∆t) -X(t)| 2 ∼ t 0.5
). This is true at a time shorter than the slowest relaxation time of the polymer. This statistical property deviates from Brownian motion where α = 1. This difference in the exponent α exemplifies the polymer nature of single locus dynamics. Recently, new classes of polymer models were introduced where mid and long-range forces are added between all monomers. These long-range forces result in the additional coupling between monomers reviewed in [START_REF] Amitai | Polymer physics of nuclear organization and function[END_REF]. The strength of the long range forces coupling can be tuned to modify the anomalous exponent of a monomer in the range of α ∈ (0 -0.5] such that the anomalous exponent can be prescribed [START_REF] Amitai | Polymer model with long-range interactions: Analysis and applications to the chromatin structure[END_REF]. Stronger coupling leads to lower anomalous exponent, while the limit of weak coupling between distant monomers, is the limit of the linear chain -the Rouse model (α = 0.5) [START_REF] Amitai | Polymer model with long-range interactions: Analysis and applications to the chromatin structure[END_REF].

Following DSB induction, the overall dynamics of a broken chromatin locus increased. The anomalous exponent of a locus on the DNA increase from 0.38 (before break) to 0.53 (after break induction) [START_REF] Amitai | Visualization of chromatin decompaction and break site extrusion as predicted by statistical polymer modeling of single-locus trajectories[END_REF]. A similar effect was observed in that study for the MAT locus. Reconstructing chromatin when the motion of loci is characterized by α > 0.5 is more complicated, as different types of forces are possible: combining a β-polymer model with a deterministic drift [START_REF] Amitai | Visualization of chromatin decompaction and break site extrusion as predicted by statistical polymer modeling of single-locus trajectories[END_REF] increases the anomalous exponents. Oscillatory motion of the whole nucleus can also lead to such an increase in α [START_REF] Amitai | Visualization of chromatin decompaction and break site extrusion as predicted by statistical polymer modeling of single-locus trajectories[END_REF]. At this stage, polymer models and numerical simulations have revealed an expansion of chromatin at the site of a DSB, a prediction that was confirmed by superresolution microscopy [START_REF] Amitai | Visualization of chromatin decompaction and break site extrusion as predicted by statistical polymer modeling of single-locus trajectories[END_REF]. An increase in α and a reduction of K c (Box 2), was associated with an increased motion of the adjacent loci [START_REF] Hauer | Histone degradation in response to dna damage enhances chromatin dynamics and recombination rates[END_REF][START_REF] Amitai | Visualization of chromatin decompaction and break site extrusion as predicted by statistical polymer modeling of single-locus trajectories[END_REF]. This reduction could be due to reduced tethering. Supporting this hypothesis, chromatin decompacts at the site of a DSB likely due to the loss of histone proteins. Loss of histones may reduce the ability of the locus to interact with other chromatin loci or nuclear components. To conclude polymer model can drive testable hypothesis and can lead to the design experiments to confirm the theoretical predictions.

We would like to mention an alternative approach to use polymer model to reconstruct chromatin dynamics. It consists in using a kernel representation of anomalous diffusion, suggesting that a change in α is associated with the properties visco-elastic nucleoplasm, which would result in additional correlations acting on chromatin [START_REF] Weber | Bacterial chromosomal loci move subdiffusively through a viscoelastic cytoplasm[END_REF][START_REF] Weber | Subdiffusive motion of a polymer composed of subdiffusive monomers[END_REF][START_REF] Lampo | Physical modeling of dynamic coupling between chromosomal loci[END_REF][START_REF] Lampo | Cytoplasmic rna-protein particles exhibit non-gaussian subdiffusive behavior[END_REF].

Polymer models with connectors reflects in vivo chromatin dynamics

There are different ways to study the effect of dynamics coming from interactions between distant chromatin sites. One way is to use a model where the monomers of a polymer are connected randomly with transient connectors (binder model or random loop model [START_REF] Bohn | Random loop model for long polymers[END_REF][START_REF] Barbieri | Complexity of chromatin folding is captured by the strings and binders switch model[END_REF]) or fixed connectors (randomly cross-linked model -RCL [START_REF] Shukron | Statistics of randomly cross-linked polymer models to interpret chromatin conformation capture data[END_REF][START_REF] Shukron | Transient chromatin properties revealed by polymer models and stochastic simulations constructed from chromosomal capture data[END_REF]) (Fig. 1C, Key Figure). The advantage of using connectors to describe chromatin dynamics/interactions is that connectors exist in biology, specifically in the form of CTCF and cohesin that mediate chromatin looping [START_REF] Ghirlando | Ctcf: making the right connections[END_REF]. Adding connectors leads to an anomalous exponent < 0.5. An exact relationship between the number of connectors and the anomalous exponent has not been derived. However, the underlying mechanism is similar to the β-polymer model: the connection between monomers couples their dynamics and reduces α.

Connectors offer an alternative confinement (restriction) mechanism to crowding through cross-linking (Fig. 1B, Key Figure ). Changing the number and distribution of connectors affect the four parameters described in section 1(Fig. 1C, Key Figure ). For example, a reduction of the length of constraint L c in parallel with a reduction of the anomalous exponent α and an increase in the local tethering force spring K c is associated with an increase in the number of connectors and chromatin condensation. However, no changes are expected to be seen in the diffusion coefficient [START_REF] Amitai | Visualization of chromatin decompaction and break site extrusion as predicted by statistical polymer modeling of single-locus trajectories[END_REF]. Conversely, a decrease of the diffusion coefficient is directly connected to an increase of crowding [START_REF] Holcman | Narrow escape through a funnel and effective diffusion on a crowded membrane[END_REF] obtained by adding sufficiently large obstacles compared to the size of the observed loci [START_REF] Shukron | Transient chromatin properties revealed by polymer models and stochastic simulations constructed from chromosomal capture data[END_REF].

To conclude, changing the number of connectors between chromatin loci has several consequences: First, only a few connectors are required to see a deviation from Rouse polymer dynamics. Second, connectors can be positioned at random and not at specific locations, leading to a spectrum of anomalous exponents at each locus. The number of connectors can be estimated using the encounter probability of loci coming from existing HiC data [START_REF] Shukron | Transient chromatin properties revealed by polymer models and stochastic simulations constructed from chromosomal capture data[END_REF]. The number of connectors seems to play a key role in defining topologically associating domains (TADs), gene regulation and organization across cell differentiation [START_REF] Shukron | Transient chromatin properties revealed by polymer models and stochastic simulations constructed from chromosomal capture data[END_REF][START_REF] Shukron | Heterogeneous cross-linked polymers to reconstruct chromatin reorganization during cell differentiation[END_REF].

Classifying forces that constrain locus dynamics

What restricts the motion of a chromatin locus? restriction can results from confinement and/or tethering. In experiments, we usually observe one or two loci simultaneously. Hence, it is not yet possible to reconstruct all independent forces acting on each monomer in the polymer. Attempt to reconstruct tethering forces acting on a locus assumes that the restriction could result from a parabolic potential energy acting on an unknown site on the chromosome, as discussed in subsection 1 and Box 2. This classical model is used in the theory of chemical reactions and is characterized by a finite parabolic well, truncated at a specific height [START_REF] Holcman | Analysis and interpretation of superresolution single-particle trajectories[END_REF]. Another possibility as discussed above is to account for motion restriction is to add random or fixed connectors to a Rouse polymer, that leads to slower motion (smaller α) of its monomers (Fig. 1B, Key Figure). This modified polymer would lead to an apparent restricted motion. Tethering, associated with external forces acting on a locus, limits the span of space sampled by the locus.

Once an overall force is computed from SPTs, due to the nature of the polymer, an additional step is needed to differentiate the internal (due to chromatin internal properties) from the external forces. This step involves a deconvolution procedure [START_REF] Amitai | Analysis of single locus trajectories for extracting in vivo chromatin tethering interactions[END_REF]. When the chromatin is approximated as a Rouse polymer model, it is then possible to recover the spring constant K c of the effective external force from the total force, when we know the genomic distance between the tagged locus and the one where the force is applied to (Box 2). The case of multiple forces was resolved in the Supplementary Info of [START_REF] Amitai | Analysis of single locus trajectories for extracting in vivo chromatin tethering interactions[END_REF]. For more elaborate models, numerical simulations are used to recover external forces.

To conclude, the origin of deterministic forces acting on chromatin is still poorly understood, but polymer models suggest that connectors that restrict chromatin loci motion, lead to apparent forces. Measuring the correlated motion of multiple loci may be useful to better estimate those forces.

How does the sampling time step ∆t influence the four parameters?

How does the sampling time ∆t influence the biophysical parameters extracted from SPTs? The four parameters mentioned above depend on ∆t (Box 2). For example, in yeast, for the MAT locus, when ∆t = 80msec, α = 0.49 (see Fig. 2B table)), while for the same locus for ∆t = 300msec, α increases to 0.66. A similar effect is observed for the other parameters [START_REF] Amitai | Polymer physics of nuclear organization and function[END_REF]. In principle, if we were to estimate α in a simple polymer (using Brownian motion or the Rouse model for example), it would have been independent of ∆t [START_REF] Doi | The Theory of Polymer Dynamics Clarendon[END_REF]. Indeed, at intermediate times, the dynamics of a monomer obeys a power-law, and thus is independent of the time scale. So why do we see a difference when observing chromatin at different time scales?

Chromatin, like other polymers, has multiple relaxation times. The shorter relaxation times are associated with smaller genomic lengths, and larger times, are associated with larger lengths (Fig. 2A). By choosing a specific time step, we establish a cutoff below which locus dynamics are no longer observed. For example, if we take ∆t = 80msec, all faster processes occurring on chromatin are averaged when estimating the parameters. Imagine a fly in a room. If we observe it every two hours, it will have had enough time to explore the whole room (sample all positions), meaning that we will observe the fly in its favorite positions (equilibrium distribution) but cannot learn anything about its dynamics. However, if we observe the fly every second, we capture its trajectory from one place to the next, allowing us to extract dynamical parameters such as its velocity. Applying the same concept to measuring chromatin dynamics it is important to choose a time scale that is relevant to the dynamics we wish to measure.

The choice of ∆t is thus critical: it should not be too small (that would only capture the motion of a single monomer) or not too large (when the entire polymer can move by diffusion in a confined environment), but be chosen in an intermediate time regime, as predicted from the anomalous behavior of a monomer [START_REF] Amitai | Visualization of chromatin decompaction and break site extrusion as predicted by statistical polymer modeling of single-locus trajectories[END_REF][START_REF] Amitai | Polymer model with long-range interactions: Analysis and applications to the chromatin structure[END_REF]. We note that a time-scale analysis, changing ∆t from 10ms, 100ms to seconds has recently revealed much less impact of chromatin on single locus dynamics with an anomalous exponent around 0.5 [START_REF] Miné-Hattab | Multi-scale tracking reveals scale-dependent chromatin dynamics after dna damage[END_REF]. This is in contrast with an early study [START_REF] Bronstein | Transient anomalous diffusion of telomeres in the nucleus of mammalian cells[END_REF] of telomere dynamics in mammalian cells, where α changed from 0.32, when estimated in the range of [10 -2 ; 1]s, to 0.51 in [1; 100s]. When looking at chromatin at ∆t = 300msec, the larger value of α suggests that this scale is compromised by additional factors such as nuclear rotation. Another effect of ∆t concerns the interpretation for the tracking localization noise (Fig. 2C), where a Gaussian error of amplitude σ is added to the real position X, so that the measured position is X = X + σξ. Such localization error affects the value of the measured (effective) diffusion coefficient compared to the real diffusion coefficient D real , leading to a shift

D measured = D real + σ 2 2∆t + Aσ 2 divf
, where A is a constant and f is the force applied to the tagged loci [START_REF] Hoze | Recovering a stochastic process from superresolution noisy ensembles of single-particle trajectories[END_REF]. Increasing the localization error changes the estimation of all parameters. The effect of Deltat, through the localization error, influences α, D c , and k c (Fig. 2C). The diffusion coefficient can also depend on ∆t (Fig. 2B). In a region with well-spread obstacles, small ∆t captures motion in the space between obstacles. However, increasing ∆t means looking at chromatin at larger steps each frame, which would result in it bumping into obstacles and a reduced diffusion coefficient. An analysis of the effect of obstacle density on the diffusion coefficient of a Brownian particle can be found in [START_REF] Holcman | Narrow escape through a funnel and effective diffusion on a crowded membrane[END_REF].

In summary, the choice of ∆t affects both the localization error and the biological processes we can follow. Ideally, localization error of the microscope will be known for the ∆t chosen. As a general guideline, if the process we wish to follow occurs on chromatin, ∆t must be small enough such that nuclear drift or other motion artifacts do not affect the measurements. It is worthwhile noting that if one is looking for a biological effect on chromatin mobility it may be beneficial to look for it at multiple timescales. Future investigations are needed to clarify what exactly can be revealed from the chromatin at small and large ∆t and how to reconcile the different values of parameters computed for different ∆t.

Statistic of two simultaneous tracked loci reveals chromatin dynamics at the 250nm scale

Are there new features or biophysical information contained in trajectories of two loci tracked simultaneously that are not contained in loci tracked individually? A number of studies have tracked two loci to extract additional information about chromatin dynamics and interactions [START_REF] Hauer | Histone degradation in response to dna damage enhances chromatin dynamics and recombination rates[END_REF][START_REF] Lampo | Physical modeling of dynamic coupling between chromosomal loci[END_REF][START_REF] Shukron | Two loci single particle trajectories analysis: constructing a first passage time statistics of local chromatin exploration[END_REF][START_REF] Dickerson | High resolution imaging reveals heterogeneity in chromatin states between cells that is not inherited through cell division[END_REF]. Using the four biophysical parameters mentioned above, each locus could be analyzed separately. However, there are a number of biological processes that could be better understood by studying the interaction of two genomic loci simultaneously (Fig. 3A). For instance, chromatin looping in the context of enhancer/promoter contacts and repair of DNA double-strand breaks by homologous recombination. Such processes have an associated encounter (looping) time τ E . Once the two loci meet, they will stay to together until drifting apart and this process has a dissociation time τ D which is related to the binding energy between them (Fig. 3BC). The statistics of these two times seem to obey a Poissonian distribution (Fig. 3C). We recall that the Poissonian decay of the encounter time distribution τ E was predicted by the polymer looping theory [START_REF] Amitai | Polymer physics of nuclear organization and function[END_REF][START_REF] Shukron | Two loci single particle trajectories analysis: constructing a first passage time statistics of local chromatin exploration[END_REF][START_REF] Pastor | Diffusion limited first contact of the ends of a polymer: Comparison of theory with simulation[END_REF][START_REF] Amitai | Computation of the mean firstencounter time between the ends of a polymer chain[END_REF] because it is a rare event.

The method of analysis is divided into several steps: first, two-locus trajectories X(t) and Y (t) are collected (Fig. 3A). Then an arbitrary threshold length T = ε is chosen that defines an encounter distance, which we can vary between tens to hundreds of nanometers. In the second step, the distance d(t) = |X(t) -Y (t)| is computed from trajectories. The third consists in a segmentation, where the time intervals before the two-locus encounter for the first time, defined by d(t) > ε, are collected. It leads to the time series [τ E1 , .., τ En ] (Fig. 3B, yellow band). Then a second time series for the distribution of times before the two loci dissociate [τ D1 , .., τ Dm ] (Fig. 3B blue band) is also collected . The last step of analysis consists of plotting the distribution of time. Interestingly, the two time distributions τ E and τ D can be well fitted by exponentials (continuous black lines in Fig. 3C). This procedure confirms the Poissonian distribution (non-correlated events) of the distribution. This Poissonian property is valid for several genomic distances ∆ between the two loci [START_REF] Shukron | Two loci single particle trajectories analysis: constructing a first passage time statistics of local chromatin exploration[END_REF]. We recall that the Poissonian decay of the encounter time distribution τ E was predicted by the polymer looping theory [START_REF] Amitai | Polymer physics of nuclear organization and function[END_REF][START_REF] Pastor | Diffusion limited first contact of the ends of a polymer: Comparison of theory with simulation[END_REF][START_REF] Amitai | Computation of the mean firstencounter time between the ends of a polymer chain[END_REF] because it is a rare event. However, the mechanism of two loci dissociation is not that clear: if a physical mechanism prevents the two loci from separating, then the dis-sociation time distribution can also be approximated by a single exponential distribution, similar to two molecules that dissociate. Two-locus dynamics were collected for 5 genomic distances ∆ in the range [23.5, 100.8] kbps [START_REF] Dickerson | High resolution imaging reveals heterogeneity in chromatin states between cells that is not inherited through cell division[END_REF], by tracking trajectories at a time interval of 300ms over 60s. The passage time analysis [START_REF] Shukron | Two loci single particle trajectories analysis: constructing a first passage time statistics of local chromatin exploration[END_REF] revealed that the mean encounter time τ E decreases in the range ε ∈ [0, 0.25]µm, but was independent of the encounter distance for ε > 0.25µm. In comparison, the mean dissociation time τ D increases with ε in the range [0, 0.5]µm. The passage time analysis was also applied before and after the use of the drug Zeocin 500µg/ml, which induces DNA damages at random positions of the chromatin (Fig. 3D, top, red crosses): two loci located at a distance ∆ = 50kbp apart, tracked over 60 -120s at 300ms rate [START_REF] Hauer | Histone degradation in response to dna damage enhances chromatin dynamics and recombination rates[END_REF], have a mean duration τ E that decreases with respect to the distance threshold ε, both in the untreated and Zeocin treated cases (Fig. 3D): a plateau is reached in 500ms for ε = 0.25µm, showing that the encounter time does not depend on ε above 0.25µm. Interestingly, the mean encounter time τ E (Fig. 3E) is slower following induction of a DSB, while the mean dissociation time τ D is faster (Fig. 3F). Thus panels E-F of Fig. 3 reveal that changes in chromatin organization are possible below 250nm following DSB. In addition, the changes in < τ E > and < τ D > can be due to a local decompensation of chromatin below 250nm, so that it takes less time for two loci to dissociate after DSB, while it takes more time to meet. This result suggested that chromatin is decondensed and probably less crowded at the spatial scale of the order of the distance ∆ between the two loci, while it is more crowded far away. This crowding may be due to a local depletion of material locally after DSB, which is relocated further away from the break site. To conclude, distributed DSBs impair recurrent chromatin motions only at a scale below 0.25µm, suggesting that this spatial scale characterizes the local chromatin organization in which undamaged loci can freely move, but become restricted above it (Fig. 3G). Possibly future recordings of multiple loci at the same time could reveal higher-order chromatin organization and could be used to determine the minimum number of cross-linkers compatible with the statistics of the recurrent loci behavior.

Local chromatin environment revealed by coarsegrained polymer models

Understanding how chromatin moves in a confined environment remains a challenge. What could be the mechanism that confines chromatin? a modeling approach can be used to address this question by transforming the information contained in the four parameters mentioned in section 1 into cross-linkers to be added on polymer models.

Cross-linkers constrain chromatin motion

We recall that short and long-range interactions of cross-linkers can constrain chromatin motion. Indeed, there are around ∼100,000 CTCF binding sites identified in the human genome [START_REF] Kim | Analysis of the vertebrate insulator protein ctcf-binding sites in the human genome[END_REF][START_REF] Rao | A 3d map of the human genome at kilobase resolution reveals principles of chromatin looping[END_REF], but not all binding sites are occupied at any moment of time. Genomic loops mediated by structural proteins vary in the range of 15-123 kbp in HeLa cells, with a mean of 86 kbp [START_REF] Jackson | The size of chromatin loops in hela cells[END_REF]. A significantly larger cohesin loop of ∼ 370 kbp was measured in [START_REF] Haarhuis | The cohesin release factor wapl restricts chromatin loop extension[END_REF]. Bound cohesin is not distributed uniformly along the chromosome, having higher concentration near centromeres and overall average spacing of 10kb apart [START_REF] Stigler | Singlemolecule imaging reveals a collapsed conformational state for dnabound cohesin[END_REF] see also [START_REF] Blat | Cohesins bind to preferential sites along yeast chromosome iii, with differential regulation along arms versus the centric region[END_REF][START_REF] Weber | The kinetochore is an enhancer of pericentric cohesin binding[END_REF][START_REF] Yeh | Pericentric chromatin is organized into an intramolecular loop in mitosis[END_REF]. In addition, cohesin binding sites correlate with the position of DNA replication origins [START_REF] Guillou | Cohesin organizes chromatin loops at dna replication factories[END_REF]. Although CTCFs, cohesin, and condensin molecules are bound to chromatin, it is still unclear how exactly they are participating to loop formation. But CTCF residence time is ∼ 1min in mouse embryonic stem cells [START_REF] Hansen | Ctcf and cohesin regulate chromatin loop stability with distinct dynamics[END_REF]. Approximately ∼80% of CTCF molecules bind transiently and non-specifically to the chromatin with a residence time of ∼0.2-0.6s [START_REF] Agarwal | Direct observation of cell-cycle-dependent interactions between ctcf and chromatin[END_REF], whereas the residence time for a residual fraction of CTCF is in a wide range between ∼4s to > 15min. Residence times for cohesin are longer than those of CTCF and are cell cycle dependent. In G1, about 30% of cohesin is bound to chromatin with a residence of ∼6 hours, whereas in G2 about 45% of cohesin is bound with a residence time of ∼24 min, as measured in [START_REF] Ghirlando | Ctcf: making the right connections[END_REF]. Similar times of ∼20min were also measured in [START_REF] Hansen | Ctcf and cohesin regulate chromatin loop stability with distinct dynamics[END_REF].

In that context, two types of polymer models have been used to account for these binding molecules: those with transient [START_REF] Bohn | Random loop model for long polymers[END_REF][START_REF] Barbieri | Complexity of chromatin folding is captured by the strings and binders switch model[END_REF] and the others with permanent but randomly positioned connectors [START_REF] Shukron | Statistics of randomly cross-linked polymer models to interpret chromatin conformation capture data[END_REF][START_REF] Shukron | Transient chromatin properties revealed by polymer models and stochastic simulations constructed from chromosomal capture data[END_REF]. The local chromatin environment including constraints can be simulated using polymer models such as the β-polymer or RCL model [START_REF] Shukron | Statistics of randomly cross-linked polymer models to interpret chromatin conformation capture data[END_REF][START_REF] Shukron | Transient chromatin properties revealed by polymer models and stochastic simulations constructed from chromosomal capture data[END_REF]. The later model describes chromatin containing cross-linkers, positioned at random places. We recall that the RCL polymer is composed of a linear Rouse backbone of N monomers, connected sequentially by harmonic springs. In each realization of the polymer, N c spring connectors are added (Fig. 4A) between randomly chosen non-nearest monomers pairs. This parsimonious addition of connectors accounts for binding molecules such as CTCF and cohesin or condensin, and their semi-random position can generate the heterogeneity in chromatin architecture observed across a cell population.

To conclude, the key parameter that reflects the constraint of the chromatin encounter probability is the number of connectors in a defined subchromatin region, but not necessarily their position, as long as they are uniformly distributed. Transient and static connectors lead to similar statistical properties over a time scale longer than the transient detachment and re-attachment of each connector.

How many cross-linkers are needed to agree with a small length of constraint? CTCF, cohesin, and condensin contribute to chromatin organization and crosslinker proteins play an important role in maintaining the structure of the nucleolus [START_REF] Hult | Enrichment of dynamic chromosomal crosslinks drive phase separation of the nucleolus[END_REF]. Condensin and High Mobility Group protein 1 (Hmo1) could form dynamic crosslinks that phase separates the nucleolus from the rest of chromatin ensuring that rDNA loci stay within a single nuclear subdomain. As a consequence of these heterogeneous phases, chromatin motion is restricted and could be described by adding connectors to a polymer model. In that case, how many connectors should be added and where? In a recent method [START_REF] Shukron | Transient chromatin properties revealed by polymer models and stochastic simulations constructed from chromosomal capture data[END_REF] using HiC data, the encounter probability was used to find short-range (intra-TAD organization) and long-range connectors. The simulated encounter probability map, which appears to be very similar to the empirical HiC one, suggested that the reconstructed polymer model is an adequate representation of chromatin at the scale of the HiC data. Once such a polymer is constructed, it is then possible to explore transient properties such as the time and the probability for two loci to meet before meeting the third one. Thus, this analysis reveals the local chromatin properties.

Another possible calibration of cross-linkers is obtained by using the length of constraint L c described above: Before and after the induction of a DSB, the length L c increases from 0.13 (before) to 0.23 µm (after) DSB [START_REF] Amitai | Visualization of chromatin decompaction and break site extrusion as predicted by statistical polymer modeling of single-locus trajectories[END_REF].

In that case, the number of needed connectors for a polymer with a total length of N = 100 monomers varied from N c = 130 to 125, respectively (Fig. 4A) [START_REF] Shukron | Two loci single particle trajectories analysis: constructing a first passage time statistics of local chromatin exploration[END_REF]. Interestingly, if a large number of cross-linkers is required to condense a polymer to a small blob, surprisingly, only ∼ 4% are removed to create the decondensed DSB phase. In addition, the mean radius of gyration R g ≈150nm is mostly unchanged between the unbroken locus and DSB. By removing the number of connectors, the mean maximal distance between the two monomers increases from 0.33µm in the unbroken case to 0.75µm after DSB. Separation of the two DSB ends cannot be too large due to remaining connectors that maintain the two extremities. To conclude, randomly positioned cross-linkers can be used to model chromatin condensation and dynamics. Their minimum number can be adjusted to match experimental data of SPTs and HiC, offering a simple tool to represent chromatin organization at a given spatial scale, that can be used to explore the local chromatin environment and dynamical properties.

Concluding Remarks

As the world of chromatin molecular biology and biophysics become more entangled it is essential to build a set of reliable tools to easily analyze biological data. Here we have described techniques and tools to study SPT data as well as how to use polymer models to interpret these results in the context of chromatin sub-organization. We further presented how to extract information from correlated loci analysis and how to interpret the changes in parameter values when the sampling rate ∆t is changing.

Models are a step towards a unified tool of SPT data with data generated by chromatin conformation capture techniques such as HiC (Fig. 4B). Such a tool would be immensely useful as it would take into account the physical dynamics of chromatin at a local level while also giving an overview of interactions throughout globally. This type of unified model would allow us to better understand a number of complicated biological processes such as enhancer/promoter organization during transcriptional reprogramming, homology search during DSB repair and CRISPR/Cas9 editing. In particular, simulating the effects of integrating viral DNA on the chromatin environment would give new insight into how viruses cause mutations and ultimately disease.

Figures

Figure 1: From SPT of chromatin locus to polymer models. A. Extraction of four parameters from a trajectory of a single locus (green). The four parameters extracted are: the diffusion coefficient D c , the length of constraint Lc, the anomalous exponent α, and the spring constant Kc of a parabolic potential well (red box). B. Two possible confinement mechanisms: crowding can be generated from the exclusion of a polymer from chromosome territories, thus leading to the restriction of a locus (left), or confinement can be induced by added cross-linkers (such as CTCF, cohesin, condensin) to the chromatin fiber. C. Construction of polymer models constrained by SPT statistics: values of parameters such as Lc or α can be used to constrain polymer such as the RCL polymer (top) or the beta polymer (bottom). For example, Lc and α small can be recovered by adding the right number of connectors (green) or long-range interactions (arrows) to RCL and beta polymer model, respectively.

Glossary

• Observables: physical quantities that can be measured.

• Single particle trajectory (SPT): causal collection of points represented by a time series of a single tagged molecule moving through a medium. The sampling time is ∆t. In the context of the genome, SPTs are traces of the motion trajectory of a chromatin locus acquired in 2 or 3 dimension.

• Polymer modeling: mathematical idealization of a continuum polymer chain, approximated by point (bead) connected by springs. More forces can be added such as bending elasticity, torsion forces and many more. A polymer model is an Avatar of chromatin inside a computer.

• Lennard-Jones force: a physical force that model short-range repulsion and long-range attraction between two monomers using a potential well. It is used to prevent a polymer chain from collapsing to a point at equilibrium.

• HiC data: a certain type of chromosomal capture data, providing the histogram of encounters between any two genomic loci resulting from physical proximity or DNA-protein cross-linking.

• Cohesin: A protein complex that holds replicated sister chromatids together after DNA replication. It is involved in DSB repair. This complex has a role in chromatin organization in G1 phase of the cell cycle. Cohesin removal from DNA increases chromatin dynamics.

• INO80 chromatin remodeling complex: This protein complex is involved in nucleosome remodeling at the site of DNA DSBs. It consists of many proteins including the catalytic Ino80, and the actin related proteins, Arp5 and Arp8. Artificial targeting of this protein to a chromatin locus increases its dynamics in yeast.

• DNA double-strand break (DSB): This is a type of DNA damage where both strands of DNA are cleaved.

• Homologous recombination (HR): also called homology directed repair (HDR), this DNA double-strand break repair pathway relies on using a donor template to repair the damage site.

• Mean-Square-Displacement (MSD): second moment of a stochastic process X(t) computed as |X(t + ∆t) -X(t)|2 ≈ A∆t α (for ∆t small). It is computed by averaging over trajectory realization. When X(t) = x, the MSD is computed at point x. When there are not enough trajectories, the MSD is often approximated by its value along a single trajectory, that however could create a bias in the statistics especially when the medium is not homogeneous.

• Diffusion: collective motion of Brownian particles. The probability density function or the concentration (number of particle per unit of volume) satisfies the diffusion equation.

• Brownian motion: mathematical limit of a random walk when the time steps ∆t tends to zero (construction of Paul Levy).

• Random walk: sequences of points, where the increment between two consecutive points is obtained from a Gaussian distribution with a fixed time step ∆t.

• Anomalous diffusion: stochastic processes, the statistics of which deviates from that of a classical Brownian motion. It is characterized by an MSD with a non-linear relationship with time. When α > 1, the locus motion is described as super-diffusive (the motion contains a drift component) while for α < 1 the motion is called subdiffusive (constrained by obstacles or tethering forces).

• Confined motion: result of a motion restricted by impenetrable obstacles or by a field of force.

• Length of constraint (Lc):The standard deviation of locus position with respect to its mean averaged over time. Estimates the space explored by a locus trajectory.

• Effective spring coefficient (Kc): A spring force applied to a monomer of a polymer results in a tether or anchor. Used to infer the resulting tethering interactions around a chromatin locus.

• Rouse model: the most elementary polymer model whereby beads are connected by springs. A Rouse polymer has an anomalous exponent α = 0.5. In numerical simulations, the chain is allow to cross itself.

• β-polymer model: A polymer model with long-range interactions between monomers, that can be modulated so that the anomalous exponent α of any internal monomer is prescribed to a given value α in the range ]0-0.5].

• Randomly Cross-Linked (RCL) model: Polymer model where monomer pairs are connected randomly. The RCL model is used to infer the minimum number of permanent cross-links in chromatin region from chromosome conformation capture data.

• CTCF: CCCTC-binding factor containing 11 zinc finger binding domains. It is involved in three-dimensional chromatin organization by loop formation, as anchor points and in boundary formation.

• Passage time analysis: analysis of the statistics associated with continuous random processes crossing a threshold. Examples are the escape from potential well (activation escape), escape though a small hole (narrow escape), or polymer looping (time for the two extremities to enter into a ball of small radius ).

• Zeocin: A radiomimetic drug that induces random DNA damages including DSBs.

Outstanding questions

While more and more data have been collected on chromatin dynamics in various organisms and for many different genomic loci, it has yet to be directly shown how changes in chromatin dynamics benefits a biological process. Specifically, how chromatin movement influences DNA repair?

What is the relationship between chromatin structure and its dynamics? While we know that in yeast removal of nucleosomes from DNA is correlated with increased loci mobility, little is known in higher eukaryotes.

Which universal polymer model should be used to explain the dynamics of chromatin and its structure in living cells? Probably additional data on both the movement of chromatin and its structure changes during that movement will be required to refine such model. This will likely come from developments in live-cell microscopy of multiple chromatin loci.

Single particle trajectories of a chromatin locus reveals the local chromatin dynamics. What else can be revealed by the correlated motion of many loci tracked simultaneously?

In the multiscale chromatin organization, the spatial scale of 200 -300nm seems to correspond to the control of short and long-range loops. What is the relation between this spatial scale, chromatin regulation, and gene expression?

How can we combine statistics of population and single cell HiC? How the distribution of the encounter probability extracted from HiC compare with SPTs (which provide statistics of local time dynamics)?

Polymer models have been used to represent DNA reorganization following DSB. How to converge to a unified polymer model? Shall we keep models as simple as possible, adjusting them for each situation?

Box 1

How to model chromatin? Polymer modeling is a powerful tool, but finding the appropriate polymer model remains challenging and often depends on the scale and the question to be answered. The simplest model, that has already provided many insight into chromatin dynamics in free and confined environment, is the Rouse model that consists of beads connected by harmonic springs. The potential energy is

φ Rouse = (R 1 , . . . , R N ) = κ 2 N n=1 (R n -R n-1 ) 2 ,
where the spring constant κ = 3k B T /b 2 is related to the standard-deviation b of the distance between adjacent monomers [START_REF] Doi | The Theory of Polymer Dynamics Clarendon[END_REF], with k B the Boltzmann coefficient and T the temperature. The dynamics of monomer R n is

dR n dt = -D∇ Rn φ Rouse + √ 2D dω dt ,
for n = 1, . . . , N and ω n are independent three-dimensional white noise with mean zero and variance 1. It is possible to add the Lennard Jones (LJ) potential. To account for the LJ forces, we use the potential energy defined by

U (R 1 , . . . , R N ) = U spring (R 1 , . . . , R N ) + U LJ (R 1 , . . . , R N ),
where the spring potential is

U spring (R 1 , . . . , R N ) = κ 2 N -1 i=1 (|r i,j+1 -l 0 ) 2 ,
where l 0 is the equilibrium length of the bond. When we write κ = 3

s 2 l 0
, then s l 0 is the standard-deviation of the bond length (for example, s l 0 = 0.2l 0 ). Note that U i,j LJ (r i,j ), with r i,j = R i -R j and

U i,j LJ (r i,j ) =    4 σ r i,j 1
2 -2 σ r i,j 6 + 1 4 , for|r i,j | ≥ 2 1/6 σ 0, for|r i,j | < 2 1/6 σ, where σ is the distance at which the potential is negligible. This distance remains to be found. With the choice l 0 = 2σ, the springs which materialize bonds, cannot cross each other in stochastic simulations. We briefly summarize the construction of a Randomly Cross-Linked (RCL) polymer [START_REF] Shukron | Statistics of randomly cross-linked polymer models to interpret chromatin conformation capture data[END_REF]: it consists of a linear backbone of monomers connected by springs, with an additional N c added connectors between randomly chosen non nearest neighbor monomer pairs. The energy of the RCL is the sum of two contributions, one from the backbone (Rouse) plus a contribution from the random connectors:

U (R 1 , . . . , R N ) = κ 2 U Rouse + U G random ,
where

U G random = m,n (R m -R n ) 2
, is the potential energy from N c added spring connectors (realization G), and the sum extends to pairs (m, n) such that |m -n| > 1, i.e. R m and R n are not immediate neighbors along the linear backbone. A realization G means that we chose randomly N c monomer pairs to connect (see [START_REF] Shukron | Statistics of randomly cross-linked polymer models to interpret chromatin conformation capture data[END_REF] for a description of physical properties).

The encounter probability between monomers, which depends on the number of added connectors N c is usually a free parameter to be fitted from empirical data. Using the RCL model it is possible to fit the empirical loci encounter probability from HiC data and derive the minimal number of crosslinks in a genomic section at a coarse-grained resolution. The RCL model could therefore be used to obtain structural parameters not necessarily contained in the HiC data, such as the radius of gyration, the MSD of any loci, and the mean first encounter time between any two loci.

Finally, we summarize the procedure of construction of a polymer model:

1. First, select the resolution to coarse-grain chromatin, so that one bead represents a given Xkbps. For example, X = 3kbps with 100 beads, leads to a representation of 3MBp.

2. Add connectors if necessary (see RCL description above) to represent chromatin compaction having a similar encounter probability matrix as the one obtained from HiC data. This procedure is non trivial, and described in [START_REF] Shukron | Statistics of randomly cross-linked polymer models to interpret chromatin conformation capture data[END_REF].

3. Generate numerical simulations of the polymer. Make sure to simulate enough time to reach steady-state.

4. Collect enough realizations (see Box 2) and apply the statistical estimators for evaluating quantities of interest.

5. Special care should be given to the case of transient simulations, such as the consequences of a DSB. Set time t = 0 exactly when connectors are removed post DSB. In that case, the polymer is locally out of equilibrium. Removing local forces or tethering forces at the initial time can lead to chromatin reorganization at short and long scale [START_REF] Shukron | Heterogeneous cross-linked polymers to reconstruct chromatin reorganization during cell differentiation[END_REF] time scale.

Supplementary material: available chromatin polymer packages

There are different types of available packages to compute and analyze the properties of loci mentioned in the past section: 1) Extract parameters from SPTs data 2) Reconstruct chromatin from HiC 3) Simulate polymer dynamics; single loci, compute MSD, 4) Reconstruct polymer model with specific properties such as β-polymer, accounting for viscosity or RCL.

Type 1

Matlab codes to compute the four parameters from SPTs [START_REF] Hauer | Histone degradation in response to dna damage enhances chromatin dynamics and recombination rates[END_REF][START_REF] Amitai | Visualization of chromatin decompaction and break site extrusion as predicted by statistical polymer modeling of single-locus trajectories[END_REF] are available at http : //bionewmetrics.org/analysis -of -spts/. Fast algorithms of RCLpolymers [START_REF] Shukron | Statistics of randomly cross-linked polymer models to interpret chromatin conformation capture data[END_REF] are available at http : //bionewmetrics.org/randomlycrosslinkedpolymer//.

Trajectories and polymer can be visualized in 3D and quantities such as mean-square-radius of gyration (MSRG), encounter probability (EP), and variance of the distance between loci, the MSD and mean first encounter time (MFET) between loci are computed. Extension of this framework to include a single and multiple DSB, is provided in a Julia framework at http : //bionewmetrics.org/randomlycross-linked-polymer -f ollowing -dsb/.

Types 2-3

The numerical code to generate a generalized RCL polymer framework, consisting of two interacting Topologically Associating Domains (TADs) [START_REF] Shukron | Transient chromatin properties revealed by polymer models and stochastic simulations constructed from chromosomal capture data[END_REF] is available at http : //bionewmetrics.org/reconstruction-of -a-polymermodel -f rom -the -chromatin -capture -data/. Short random and fixed long-range connectors are added between TADs. Chromatin parameters can be calibrated directly from chromatin conformation capture data (3C, 5C, HiC) given TAD boundaries positions. Peaks within and between TADs located in the encounter probability matrix are detected and converted in a number N c of connectors. This framework enables to study the influence of short-range random connectors within TADs and long-range fixed connectors within and between TADs on the dynamic properties of loci in cross-linked chromatin.

Figure 2 :

 2 Figure2: Influence of the sampling rate ∆t and localization noise on SPT statistics. A. The region of influence on a locus (green) depends on the sampling rate ∆t: as ∆t increases the fraction of the polymer influencing the motion of a single locus increases, at 80ms (brown) the region is smaller than the one obtained at 300ms (green). B. Example of changes in the four parameters with respect to the sampling rate ∆t, obtained from yeast SPTs[START_REF] Amitai | Visualization of chromatin decompaction and break site extrusion as predicted by statistical polymer modeling of single-locus trajectories[END_REF]. C. Influence of the localization noise on SPTs. The localization noise generates a Gaussian error on the position of loci at each time step ∆t. Note, this noise can affect the four parameters: α, D c , Lc, Kc.

Figure 3 :

 3 Figure 3: Multiple loci SPT statistics. A. Two loci (green, orange), separated by a genomic distance ∆, generate two trajectories X(t), Y (t), showing regions where they are located inside balls of radius (blue) separated by regions (yellow) where the trajectories are far apart. B. Distance D(X)t, Y (t)) versus time. When D(X(t), Y (t)) is lower than a threshold , trajectories are classified as associated (blue, Panel A) and characterized by the time (dissociation time τ D ) it takes for the two trajectories to separate. Conversely, when D(X(t), Y (t)) > , trajectories are dissociated (yellow, Panel A), and characterized by the time (association time τ A ) it takes for the two trajectories to enter for the first time into a region of radius ≤ . C. Distribution of association and dissociation time, characterized by a single exponential. D. Schematic representation of two loci association (left) and dissociation (right) dynamics with multiple DNA damages (red crosses). E. Mean first association time τ A versus the encounter distance for zeocin (red, squares) and untreated (blue, circles) chromatin. F. Mean first dissociation time τ D versus the encounter distance for zeocin (red, squares) and untreated (blue, circles) chromatin. G. consequences of DNA damages of local chromatin reorganization. Material is extruded following DSB.

Figure 4 :

 4 Figure 4: Chromatin modification represented by polymer models before and after DSB. A. An increase of Lc following DSB can be obtained by removing connectors (green) in RCL polymer model (top) or by removing long-range forces in the beta polymer model (bottom). B. Summary of the general procedure of converting information contained in the four parameters extracted from SPT of chromatin loci into a polymer representation. This polymer can be used to generate statistics not contained in the original trajectories.

Type 3-4

Fast numerical polymer simulations, incorporating internal and external forces, such as those from nuclear boundaries and diffusing molecules and structural proteins (CTCF) are implemented in an open source platform OpenMM [START_REF] Eastman | Openmm 7: Rapid development of high performance algorithms for molecular dynamics[END_REF] (http : //openmm.org/). OpenMM framework is used in [START_REF] Doyle | Chromatin loops as allosteric modulators of enhancer-promoter interactions[END_REF] to simulate polymers as a model of chromatin in confined nano-domains. Coarse-grained polymer simulations consisting of transient genomic loops with OpenMM are available at https : //mimakaev.bitbucket.io/index.html.

A general molecular dynamics computational framework is LAMMPS [START_REF] Plimpton | Fast parallel algorithms for short-range molecular dynamics[END_REF] (https://lammps.sandia.gov). LAMMPS provides a flexible interface and packages to simulate polymer in confined domains. LAMMPS could also be used to simulate realizations of many types of molecular dynamics scenarios, such as blood flow in capillaries. LAMMPS was used in [START_REF] Zhang | Topology, structures, and energy landscapes of human chromosomes[END_REF][START_REF] Shi | Interphase human chromosome exhibits out of equilibrium glassy dynamics[END_REF][START_REF] Di Pierro | De novo prediction of human chromosome structures: Epigenetic marking patterns encode genome architecture[END_REF] to study the folding landscape of chromatin calibrated from CC contact data.

Reconstruction of a three-dimensional (3D) chromatin structures from ensemble CC contact maps remains a challenging task [START_REF]Chapter nine -computational models of large-scale genome architecture[END_REF]. Models that suggest a solution to this problem take as an input CC contact maps and output an averaged (consensus) 3D structures by employing optimization algorithms to convert loci-loci encounter probability to a spatial distance, commonly based on the expected decay of the CC encounter probability with genomic distance [START_REF] Wang | Inferential modeling of 3d chromatin structure[END_REF]. In the model developed in [START_REF] Olarte-Plata | The folding landscape of the epigenome[END_REF], the average loci distances are computed from CC maps and additional constraints on chromatin folding from the expression of epigenomic markers see also [START_REF] Zhang | Topology, structures, and energy landscapes of human chromosomes[END_REF].

3D structure reconstruction from a single cell HiC data was developed in [START_REF] Stevens | 3d structures of individual mammalian genomes studied by single-cell hi-c[END_REF] based on NMR principle, codes are available at (https : //github.com/T heLaueLab/nuc d ynam Although average distances between genomic loci can be computed from the 3D structure, the dynamics of the reconstructed chromatin cannot be retrieved directly from the resulting structure because the final structure does not include information about cross-linkers, their distribution, and dynamics. Bio-informatic tools [START_REF] Ay | Analysis methods for studying the 3d architecture of the genome[END_REF] provide algorithms to extract parameters directly from CC contact maps, such as TAD boundaries and peak (see [START_REF] Rao | A 3d map of the human genome at kilobase resolution reveals principles of chromatin looping[END_REF]), which enables to extract the position of fixed long-range connectors. The parameters extracted can be used for polymer simulations.