## ABSTRACT

Principles of genome folding and their relationship to function depend on understanding conformational changes of the chromatin fiber. Analysis of bulk chromatin motion at high resolution is still lacking. We developed Hi-D, a method to quantitatively map DNA dynamics for every pixel simultaneously over the entire nucleus from real-time fluorescence images. Hi-D combines reconstruction of chromatin motion using computer vision and classification of local diffusion processes by Bayesian inference. We found that DNA dynamics in the nuclear interior are spatially organized into 0.3 – 3 µm domains of distinct types of diffusion was uncoupled from chromatin compaction. Reorganization of the network of dynamic domains between quiescent and active cells suggest that the microenvironment plays a predominant role in stochastic chromatin motion. Hi-D opens new perspectives towards understanding of chromatin organization placing global motion of nuclear molecules in the context of nuclear architecture.

## INTRODUCTION

Spatial organization and dynamics of chromatin correlate with cell function and fate (Serizay and Ahringer 2018). In mammalian cells, chromosomes occupy territories whose localization is stable over the cell cycle although their positioning relative to each other and to other nuclear bodies, and degree of intermingling varies between cells(Cremer et al. 2015). Each chromosome contains dense, heterochromatin regions and open, euchromatin areas whose extent is indicative of cellular activity (Banerjee et al. 2006). Transitions within and between eu- and heterchromatin involve multiple levels of chromatin organization from nucleosome density, long-range looping and domain folding that adapt to and enable DNA processing (Fraser and Bickmore 2007). Structural models derived from contact and crosslinking frequencies(Lieberman-Aiden et al. 2009; Dixon et al. 2012; Beagrie et al. 2017) are consistent with the view that the genome is organized in compartments(Nora et al. 2012) but the true physical nature and localization of these compartments is not known. Understanding formation of nuclear substructures and even the folding of the nucleosome fiber itself requires imaging the behavior of chromatin in living cells. Tracking of labelled single DNA loci (Chubb et al. 2002; Levi et al. 2005; Chuang et al. 2006; Chen et al. 2013; Germier et al. 2017) or chromatin domains (Bornfleth et al. 1999; Nozaki et al. 2017) demonstrated that chromatin motion is highly heterogeneous at short time intervals. A few studies showed that chromatin motion was sub-diffusive, however, sparse loci are difficult to place in the context of global chromatin organization which is subject to highly complex dynamics (Di Pierro et al. 2018). Locally restricted genomic processes can also not be inferred from averaging motion over the entire nucleus (Zidovska et al. 2013; Shinkai et al. 2016).

To decipher the relationship between genome dynamics, organization and activity, we developed a new method, called Hi-D, to overcome the limitations of sparse and ensemble approaches for imaging dense structures such as chromatin and nuclear proteins. Hi-D, a combination of dense optical flow reconstruction to track the local motion of chromatin at sub-pixel accuracy in live cell imaging(Shaban et al. 2018), and a Bayesian inference approach to precisely classify local types of diffusion. Biophysical properties such as diffusion coefficients and anomalous diffusion exponents were determined for each pixel and assigned to nuclear compartments. We present for the first time a two dimensional map of fluorescently labelled chromatin dynamics at sub pixel resolution down to 65 nm over the entire nucleus in living single cells. We examine the capacity to study dynamic changes of labelled chromatin in quiescent and stimulated cells. We find that in general, contrary to common belief, DNA compaction and dynamics do not necessarily correlate and strengthen the hypothesis that instead extrinsic mechanisms such as macromolecular crowding and association with the nuclear lamina dictate chromatin dynamics at local scales.

## RESULTS

### Hi-D determines and classifies DNA dynamics at nanoscale

We reconstructed trajectories of virtual particles from a series of conventional confocal fluorescence microscopy images analyzed by a dense Optical Flow method (Shaban et al. 2018) (supplementary note 1). First, we validated that MSD curves and derived diffusion constants were significantly greater in living than in formaldehyde fixed cells (Supplementary Figure 2). The type of diffusion characterizing each pixel’s chromatin motion was determined in an unbiased manner using Bayesian inference to simultaneously test a set of common models to fit each MSD curve (Monnier et al. 2012). The five principal models are shown as a coloured map directly on the nucleus (Figure 1b right panel) (Methods section). This classification of local diffusion processes revealed that diffusion of DNA is highly heterogeneous throughout the nucleus presumably due to numerous mechanisms influencing chromatin fiber physical properties including molecular crowding and energetic processes such as active transport or chromatin remodeling during transcription.

We found that only a small fraction of trajectories displayed directed diffusion (Figure 1b), while the bulk of chromatin exhibited sub-diffusive behaviour. By examining a wide range of parameters governing these types of diffusion, our results suggest that chromatin diffusion can be adequately described as anomalous (Supplementary note 2 and Supplementary Figure 4). Biophysical parameters calculated for each pixel (diffusion constant *D*, anomalous exponent αand drift velocity *V*) and presented in color-coded 2D heatmaps (Figure 1c) (Methods section) show islands of irregular shape and dimensions that contain fast diffusing DNA with large variations in the anomaly of DNA motion (Figure 1c). We determine that chromatin mobility groups spatially partition the nucleus into a network of domains.

### Hi-D mapping resolves heterogeneous motion of chromatin independently of compaction

We investigated whether chromatin motion and compaction are correlated, because it is widely assumed that heterochromatin is less dynamic (Nozaki et al. 2017). Deconvolution of the distribution of diffusion constants revealed three subpopulations (Methods section) (Figure 2b). A distinct 100 - 200 *nm* peripheral rim of slow freely-diffusing DNA (D values < 1.1 x10^{-3} μm^{2}/s) and reminiscent of lamin-associated domains (LADs)(Kind et al. 2013) colocalized with dense chromatin (Figure 2c,h). Inside the nucleus, a network of fast sub-diffusive areas of irregular dimensions spanning 0.3-3 μ*m* in diameter was imbedded in the bulk of moderately dynamic chromatin. Surprisingly, intra-nuclear heterochromatin domains and areas of these moderately sub-diffusive areas only overlapped partially in U2OS cells (Figure 2g, h). The analysis was confirmed on n=13 U2OS cells and similar results were obtained in MCF-7 cells (Supplementary Figure 6). These results reveal that chromatin dynamics are largely uncoupled from compaction.

### Transcription status alters nuclear diffusion processes at the chromatin level

We assessed the effect of stimulating transcriptional activity on chromatin dynamics. Compared to serum-starved U2OS nuclei, average D decreased by nearly one order of magnitude for all three mobility groups upon addition of serum (Figure 3a). In serum-stimulated cells, sub-diffusive and super-diffusive regimes were strengthened (Figure 3b). The presence of a population with αclose to 0.5 in both conditions was indicative of a Rouse-like chromatin fiber architecture (Hajjoul et al. 2013; Ghosh and Jost 2018). We hence hypothesize that the global decrease in D upon transcription activation stems from factor binding to the chromatin fiber or viscoelastic properties of the microenvironment (Lucas et al. 2014) including non-chromatin nuclear bodies such as speckles(Kim et al. 2018). A subset of these activities could be involved in active processes including long-range interactions or transport resulting in apparent super-diffusion.

In serum starved cells, large variations of D are suggestive of a range of diffusion modes in the absence of DNA related activities (Figure 3c,d) and increased α(˜0.75) in a large fraction of DNA further corroborates particle-like behavior of the fiber in a crowded solution(Banks and Fradin 2005). In contrast, upon serum stimulation anomalous diffusion (α˜0.33) was predominant and indicative of entangled polymers(Doi and Edwards 1988) Independence of D and α confirmed that compaction, chromatin mobility and forces hindering normal diffusion are not or only very weakly correlated (Supplementary Figure 7). Upon serum stimulation, reduced D again suggests that DNA-related processes hamper local chromatin diffusion and that αis altered due to functional reorganization of the chromatin fiber.

### Single-cell biophysical property maps of genome conformation and behaviour

To concomitantly monitor position and distribution of the three mobility populations in quiescent and activated cells, we determined Hi-D maps for a single cell (Figure 4a, b). The low D population occupying ˜6% of the nuclear area was invariant to transcriptional changes (Figure 4c). In contrast, upon serum stimulation, a large fraction of the fast moving population in quiescent cells was re-classified as intermediate mobility population with moderate changes in the mode of diffusion (Figure 4d). We conclude that decreased mobility results from a change in the microenvironment rather than from fiber folding.

Anomalous diffusion dominated across the entire nucleus (0.3 ≤ α≤ 0.73) forming a connected network which was reinforced in transcriptionally active nuclei (Figure 4f). Within this network, patches of highly anomalous (blue: < 0.3) and super-diffusive (red: α> 1) motion segregated into distinct islands which became more fragmented upon serum stimulation (Figure 4g, h). Hi-D thus reveals spatio-temporal changes in hindered diffusion of chromatin domains at high resolution in single cells. Further investigation may tell us if all or a subset of these physical domains correspond to the ones determined using Hi-C or GAM.

## DISCUSSION

Hi-D is a single cell approach that enables tracking of dense structures such as chromatin and condensates directly and further classification of the highly heterogeneous underlying type of motion without losing active fluorophore density and with no need for prior experience in sophisticated labelling preparations or advanced microscopy (Manley et al. 2008). The information gain through image analysis afforded by Hi-D alleviates the incompatibility of conventional microscopy for nanoscale mapping of chromatin dynamic properties in living cells.

Hi-D analysis revealed three distinct populations of DNA diffusing in a comparable manner. The first, a slow mobility fraction at the nuclear rim, is reminiscent of lamina associated domains (LADs) (Kind et al. 2013) Decrease of mobility in the active state supports the hypothesis that LADs play an important role for attenuating transcription activity and of integrity of gene expression(Akhtar et al. 2013). Our analysis points out that although the periphery of nuclear chromatin in most cases is less mobile and largely connected to the existence of heterochromatin, it is independent of chromatin compaction levels. The intermediate and highly diffusive regimes were distributed in a mosaic-like pattern within the nucleus, with a small extent being nevertheless present at the nuclear periphery. Heterochromatin therefore does not exhibit low mobility in general, but may be divided into rigid elastic LAD chromatin with reduced mobility and a more viscous component. The extent of the third, highly mobile fraction which dominated chromatin behaviour in the quiescent state, decreased dramatically when cells were serum stimulated. This switch in chromatin mobility in distinct nuclear domains indicates changes in local molecular crowding which may result from accumulation of actors regulating gene expression (Ghamari et al. 2013; Cisse et al. 2013; Norred et al. 2018; Shaban et al. 2018). Heterogeneous chromatin motion arises due to irregular protein binding along the chromosomes and can lead to thermodynamic or electrostatic self-organisation of nuclear compartments (Haddad et al. 2017; Sewitz et al. 2017). Local patches of large anomalous exponents indicate super-diffusive behaviour of chromatin which may result from loop extrusion by structural maintenance of chromosome (SMC) complexes (Vian et al. 2018; Ganji et al. 2018). Strikingly, quiescent cells show an increase of chromatin condensation by accumulation of the SMC complex condensin during quiescence entry (Swygert et al. 2018), which is indicative of dynamically active chromatin in quiescence. Furthermore, chromatin patches with α< 0.3 and α> 1 respectively correspond in size to one or a few TADs (Giorgetti et al. 2014). These two types of patches are present as a network within the general chromatin fraction governed by an anomalous exponent 0.3 < α< 1. This organization is in good agreement with the chromosome territory – interchromatin compartment model (Cremer et al. 2015). In conclusion, the combination of diffusion and anomalous exponent maps are complementary and provide new insights into chromatin organization and long-range interactions (active transport) during genomic processes.

We focus here on DNA, but Hi-D can be applied to real-time imaging of any fluorescent molecule to obtain comprehensive maps of their dynamic behavior in response to stimuli, inhibitors or disruptors of nuclear functions and integrity. Hi-D could be combined with tracking specific loci to zoom in on their environment. It will also be exciting to probe phase separating condensates to gain a better understanding of their physical nature(Erdel and Rippe 2018).

## METHODS

### Cell Culture

A Human U2OS osterosarcoma and MCF-7 cells (ATCC) were maintained in Dulbecco’s modified Eagle’s medium (DMEM) containing phenol red-free and DMED-12 (Sigma-Aldrich), respectively. Medium was supplemented with Glutamax containing 50 μg/ml gentamicin (Sigma-Aldrich), 10% Fetal bovine serum (FBS), 1 mM sodium pyruvate (Sigma-Aldrich) and G418 0.5 mg/ml (Sigma-Aldrich) at 37°C with 5% CO2. Cells were plated for 24 h on 35 mm petri dishes with a #1.5 coverslip like bottom (μ-Dish, Ibidi, Biovalley) with a density of about 10^{5} cells/dish.

### Cell starvation and stimulation

For starvation mode, cells were incubated for 24 h at 37°C before imaging with serum-free medium (DMEM, Glutamax containing 50 μg/ml gentamicin, 1 mM sodium pyruvate, and G418 0.5 mg/ml). Just before imaging, cells were mounted in L-15 medium. For stimulation, 10% FBS was added to the L-15 medium for 10 minutes.

### Chromatin staining

U2OS and MCF-7 cell lines were labelled by using SiR-DNA (SiR-Hoechst) kit (Spirochrome AG). For DNA was labelled as described in (Lukinavicius et al. 2015). Briefly, we diluted 1 mM stock solution in cell culture medium to concentration of 2 μM and vortexed briefly. On the day of the imaging, the culture medium was changed to medium containing SiR-fluorophores and incubated at 37°C for 30-60 minutes. Before imaging, the medium was changed to L-15 medium (Liebovitz’s, Gibco) for live imaging.

### Cell fixation

U2OS cells were washed with a pre-warmed (37 °C) phosphate buffered saline (PBS) and followed by fixation with 4% (vol/vol) Paraformaldehyde in PBS for 10-20 min at room temperature. Imaging movies were recorded at room temperature in PBS, after washing the cells with PBS (three times, 5 min per each).

### Imaging

Cells were placed in a 37?°C humid incubator by controlling the temperature and CO2 flow using H201- couple with temperature and CO2 units. Live chromatin imaging was acquired using a DMI8 inverted automated microscope (Leica Microsystems) featuring a confocal spinning disk unit (CSU-X1-M1N, Yokogawa). Integrated laser engine (ILE 400, Andor) was used for excitation with a selected wavelength of 647 nm and 140mW as excitation power. A 100x oil immersion objective (Leica HCX-PL-APO) with a 1.4 NA was chosen for a high resolution imaging. Fluorescence emission of the SiR–Hoechst was filtered by a single-band bandpass filter (FF01-650/13-25, Semrock, Inc.). Image series of 150 frames (5 fps) were acquired using Metamorph software (Molecular Devices), and detected using sCMOS cameras (ORCA-Flash4.0 V2) and (1×1 binning), with sample pixel size of 65 nm. All series were recorded at 37°

### Image processing

#### Denoising

Raw images were denoised using non-iterative bilateral filtering (Tomasi and Manduchi 1998). While Gaussian blurring only accounts for the spatial distance of a pixel and its neighbourhood, bilateral filtering additionally takes the difference in intensity values into account and is therefore an edge-preserving method. Abrupt transitions from high- to low-intensity regions (e.g. heterochromatin to euchromatin) are not over-smoothed.

#### MSD analysis and model selection by using Bayesian inference

In order to carry out a MSD analysis locally, the spatial dependency of the Mean Squared Displacement (MSD) can be written explicitly:
where is the position at time *t* of a virtual particle with initial position are time lags where Δ*t* is the time difference between subsequent images and the average < >_{t} is taken over time. The resulting MSD is a function of the particle’s initial position and the time lag τ.

#### MSD models

The MSD can be expressed analytically for anomalous diffusion (DA), confined diffusion (DR) and directed motion (V) in two dimensions as
where *D* is the particles diffusion coefficient, αis its anomalous exponent, 𝒱 its velocity and *R*_{C} is the radius of a sphere within the particle is confined(Saxton and Jacobson 1997). The case α= 1 is known as free diffusion, 0 < α< 1 corresponds to anomalous diffusion and 1 < α≤ 2 corresponds to superdiffusion. Additionally to eq. (1)-(3), different types of motion can appear overlaying, resulting in a linear combination of the equations above. For example, confined motion can be superimposed on an underlying drift and the resulting *MSD* reads *MSD*_{DRV} (τ) = *MSD*_{DR}(τ) + *MSD*_{V}(τ). The abbreviations used in this study correspond are summarized in Table 1. As experimental data is usually subject to noise, a constant offset *o* is added to every model.

#### MSD model selection

The MSD is calculated for every pixel independently, resulting in a space- and time lag-dependent MSD. It is known that living cells can behave largely heterogeneous(Banerjee et al. 2006; Dickerson et al. 2016). Ad-hoc, it is not known which particle undergoes which kind of diffusion. Fitting a MSD curve with a wrong model might result in poor fits and highly inaccurate determination of the mentioned parameters. For this reason, we use a Bayesian inference approach to test different models for any given MSD curve as proposed by Monnier *et al.*(Monnier et al. 2012). Given the data *Y* = {*Y*_{1},…, *Y*_{n}} and *K* model candidates *M* = {*M*_{1},…, *M*_{K}}, each with its own (multidimensional) parameter set *θ* = {*θ*_{1},…, *θ*_{K}}, we want to find the model *M*_{k}(*Y*,*θ*_{k}) such that the probability that *M*_{k}(*Y*,*θ*_{k}) describes the data, given the set of models to test, is maximal. By Bayes’ theorem, the probability for each model is given by

If there is no reason to prefer one model over the other, the prior probability of each model *P*(*M*_{k}) is equal. The parameter set which is used to describe the data, given a fixed model, strongly influences the probability. Therefore, it is crucial to estimate the optimal parameters for every model in order to calculate the model probabilities. The probability that the data *Y* is observed, given the model *M*_{k} described by the model function *M*_{k}(*x; θ*_{k}) and any parameter set *θ*_{k} is approximated by a general multivariate Gaussian function (Seber and Wild 2003)
where *C* is the empirical covariance matrix of the data and the prefactor is a normalizing factor. This equation has an intuitive meaning. Assume we test a model *M*_{k} parametrized by *θ*_{k} to find out if it describes the data *Y*. The exponential function consists of the term [*Y* -*M*_{k}(*x; θ*_{k})], i.e. the residuals of the data and the describing model. If the residuals are small, i.e. the model describes the data well, the exponent is small and the probability *P*(*Y*|*θ*_{k},*M*_{k}) seeks 1. On the other hand, the worse the fit, the greater the resulting residuals and the probability seeks asymptotically to 0. The factor *C*^{-1} accounts for the covariance in the data. The covariance matrix for a set of MSD curves normally shows large values for large time lags as the uncertainty increases and MSD curves diverge. The covariance matrix implicitly introduces a weight to the data, which is small for large variances and large where the data spreads little. This fact avoids cutting of the MSD curve after a specific number of time lags, but instead includes all available time lags weighted by the covariance matrix. The approach is illustrated in (Supplementary Figure 3b) with the covariance matrix exemplary shown in the inset. In case of uncorrelated errors, non-diagonal elements are zero, but the approach keeps its validity(He et al. 2012) and follows an ordinary least-squares regression.

Given the best estimate of the parameter set for a model, the model and its corresponding parameters are chosen so that their probability to describe the data is maximal: . It has to be stressed that values of the anomalous exponent scatter around 1, but do not assume the value 1. This is due to the model selection procedure, selecting the simplest model consisting with the data. In case that the underlying motion is well described by free diffusion, αis inherently set to 1 and classified as free diffusion rather than anomalous diffusion. The descriptions of free diffusion or anomalous diffusion with α= 1 are equivalent, but the free diffusion model contains one parameter less and is therefore preferred leading to abundance of αvalues close to 1 in the parameter maps and histograms. To carry out the MSD analysis locally, we choose to take the 3×3 neighborhood of a pixel, detect possible outliers therein by the interquartile range criterion (Rousseeuw and Croux 1993) and calculate the error covariance matrix of the data within the pixel’s neighborhood. The restriction to a single pixel and its neighborhood allows us to carry out the MSD analysis of trajectories locally, in contrast to an ensemble MSD in previous studies (Zidovska et al. 2013), revealing only average information over many trajectories. The choice of a 3×3 window is reasonable with regard to the equivalently chosen filter size in the Optical Flow estimation. The flow field in this region is therefore assumed to be sufficiently smooth. All calculations, except for the General Mixture Model analysis, were carried out using MATLAB (MATLAB Release 2017a, The MathWorks, Inc., Natick, Massachusetts, United States) on a 64-bit Intel Xeon CPUE5-2609 1.90 GHz workstation with 64 GB RAM and running Microsoft Windows 10 Professional.

### Deconvolution of sub-populations

Regarding especially the distribution of the diffusion coefficient, an analytical expression can be found assuming that the diffusion coefficient was calculated from a freely diffusing particle (α= 1)(Vrljic et al. 2002). However, we find anomalous diffusion to a large extent in our data (e.g. Figure 2e, f and Figure 4d) and, to our knowledge, an analytical expression cannot be found for distributions of anomalous exponent, radius of confinement and drift velocity. We therefore aim to deconvolve the parameter sets in a rather general manner, for which we use a General Mixture model (GMM), a probabilistic model composed of multiple distributions and corresponding weights. We describe each data point as a portion of a normal or log-normal distribution described by
respectively. The logarithmic mean *m* and standard deviation *s* are related to the mean and standard deviation of the normal distribution via (Mood et al. 1974)

We consider up to three subpopulations to be found in our data and model the total density estimate as a superposition of one, two or three subpopulations, i.e. the Mixture Model reads
for both normal and log-normal distributions, where to sum goes to 1, 2 or 3. The variable *w*_{k} describes the weights for each population (or component), which satisfy 0 ≤ *w*_{k} ≤ 1 and sum up to unity. The weights of each component are directly proportional to the area of the histogram covered by this component and therefore its existence in the data set.

### General Mixture Model analysis

Let *Y* = {*Y*_{1},…, *Y*_{n}} denote *n* data points. For the scope of this description, assume *Y* to be a one-dimensional variable. Further assume that the data cannot be described by a single distribution, but by a mixture of distributions. A deconvolution of the data into sub-populations faces the following problem: Given a label for each data point, denoting the affiliation to a population, one could group corresponding data points and find the parameters of each population separately using a maximum likelihood estimation or other methods. On the other hand, if we had given the model parameters for each population, labels could in principle be inferred from the likelihood of a data point being described by a population or another. The problem can be formulated by Bayes’ rule (*M* indicates model, *D* indicates data)

Here, *P*(*M*|*D*) is the posterior probability of the model given the data, which is the aim to calculate. We assign a data point to the component, which maximizes *P*(*M*|*D*). The probability to observe the data given a model is described by *P*(*D*|*M*), i.e. the likelihood function. *P*(*M*) is the prior for the models to be chosen from. In our case, we have no prior beliefs on the models (all models are equally likely) such that *P*(*M*) is uniform. Lastly, the probability *P*(*D*) does not depend on the models and can therefore be dropped.

Unfortunately, neither labels, that is *P*(*M*|*D*), nor model parameters and weights are known a priori. The problem can be approached by an Expectation-Maximization (EM) scheme: Without any prior beliefs about the data distribution, one starts with a simple estimate of model parameters, e.g. a k- means clustering estimate and iterates subsequently between the two following steps until convergence:

### Expectation step

Calculation of the probability that the component with the current parameter estimate generated the sample, i.e. *P*(*D*|*M*).

### Maximization step

Update the current parameter estimate for each component by means of a weighted maximum likelihood estimate, where the weight is the probability that the component generated the sample.

We illustrate the results of the EM algorithm exemplary in Supplementary Figure 5. From the input data (Supplementary Figure 5a), represented as histogram, both the likelihood *P*(*D*|*M*) (Supplementary Figure 5b) and the posterior (Supplementary Figure 5c) is obtained. The sum of sub-populations corresponds to the overall probability distribution (shown in black) with different model parameters and weights found by maximizing the likelihood function. The posterior describes the probability of data points to fall under each population, i.e. ∑_{k} *P*(*M*_{k}|*D*) = 1. The data points are assigned to those population, for which *P*(*M*_{k}|*D*) is maximum, resulting in labeled data. The labels are subsequently mapped in two dimensions, visualizing spatial correspondence of slow, intermediate and fast sub-populations (Supplementary Figure 5d). The GMM analysis is carried out using the pomegranate machine learning package for probabilistic modeling in Python (Schreiber 2017).

### Selection of subpopulations by the Bayesian Information Criterion (BIC)

A priori, it is not unambiguously clear from how many populations the data is sampled and which form the subpopulations take. We therefore assess the suitability of each model by means of the Bayesian Information Criterion (BIC), which is calculated by(Schwarz 1978)
where is the maximum likelihood of the maximum likelihood estimation (MLE) estimate, *p* denotes the number of parameters in the model and*n* is the number of data points used for the fit. Among a family of models, the one with the lowest BIC is considered to describe the data best, taking into account competing complexity of models. A large likelihood of a model favors it to describe the data well, while on the other hand the model is penalized if many parameters are involved in the model by the second term in (3). Therefore, the BIC prevents overfitting. In order to judge which model is appropriate for our data, we tested all considered models for each histogram and assessed the optimal model by means of the BIC. The fraction of all histograms which described best by one of the six models considered is given in Supplementary Table 2. Based on the objective judgement of the fit using the BIC, we chose for each parameter the model which best describes the largest fraction of histograms (**Error! Reference source not found.**, bold cells).

## AUTHOR CONTRIBUTIONS

H.A.S. conceived the method, performed the experimental work, developed the data processing algorithm and analyzed the data; R.B. wrote the code, developed the data processing algorithm and analyzed the data; K. B. funded the research; H.A.S. and K. B. interpreted the results; H.A.S., R. B. and K. B. wrote the manuscript.

## CONFLICT OF INTEREST

The authors declare no competing financial interests.

## ACKNOWLEDGMENTS

We thank Raphael Mourad (CBI Toulouse) and Genevieve Fourel (ENS Lyon) for fruitful discussions, and Alain Kamgoué for assistance with computation. We acknowledge support from the LITC imaging platform, CBI Toulouse. This work was supported by the ANR ANDY, IDEX ATs NudGENE, the Foundation ARC and INSERM Cancer and Epigenetics.