Abstract
The spatiotemporal organization of chromatin influences many nuclear processes: from chromosome segregation to transcriptional regulation. To get a deeper understanding of these processes, it is essential to go beyond static viewpoints of chromosome structures, to accurately characterize chromatin’s diffusion properties. We present GPFBM: a computational framework based on Gaussian processes and fractional Brownian motion to extract diffusion properties from stochastic trajectories of labeled chromatin loci. GPFBM uses higherorder temporal correlations present in the data, therefore, outperforming existing methods. Furthermore, GPFBM allows to interpolate incomplete trajectories and account for substrate movement when two or more particles are present. Using our method, we show that average chromatin diffusion properties are surprisingly similar in interphase and mitosis in mouse embryonic stem cells. We observe surprising heterogeneity in local chromatin dynamics, correlating with potential regulatory activity. We also present GPTool, a userfriendly graphical interface to facilitate usage of GPFBM by the research community.
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Introduction
The spatiotemporal organization of chromatin plays a crucial role in several nuclear processes: from cell division, where chromatin is compacted to facilitate chromosome segregation during mitosis, to gene regulation, where precise control of transcription correlates with specific longrange chromatin contacts^{1}. Chromosome conformation capture techniques and imaging approaches have revealed fundamental structural features of chromatin at different resolutions. Of special interest are topological associated domains (TADs) which are characterized by an increased frequency of interactions between genomic loci within the same domain with reduced interactions across domains^{2,3}. Remarkably, it has been shown that TAD organization can influence regulation of transcription^{4} and that TADs are dismantled during mitosis where chromatin density dramatically increases^{5,6}. However, much less is known about the diffusion properties of chromatin and how they depend on the genomic context. For instance, it is not clear whether or how transcriptional activation affects chromatin mobility, with previous studies giving seemingly conflicting results^{7,8}. Insights into the dynamic properties of chromatin motion are required to understand how gene regulatory elements communicate within the nuclear space^{9}.
The simplest model to describe the diffusion of microscopic systems is Brownian motion, whereby movements are caused by random collisions of small particles within the system^{10,11}. However, as a bulky polymer interacting with itself and the nuclear environment^{12,13,14}, chromatin displays subdiffusive behavior, more constrained than classical Brownian motion for short periods of time^{15,16,17}. Therefore, the mean squared displacement (MSD) of chromatin is expected to follow this relationship with time: MSD ∝ D_{α}t^{α}. Two parameters thus describe the diffusion properties of chromatin: the apparent diffusion coefficient D_{α}, indicating the “speed” of motion, and the anomalous coefficient α, which for subdiffusive behavior is <1 indicating greater constraint of movement. The traditional method used to estimate the diffusion parameters is based on calculating the MSD over time from measured trajectories and fitting the above theoretical expression to the data. More sophisticated methods based on particle displacement use higherorder moments^{18} or probability density functions^{19,20,21} (henceforth referred to as displacement distributionbased methods or DDB) to obtain more accurate estimations of the diffusion parameters. However, these methods do not use all the information contained in the trajectories as higherorder temporal correlations are discarded. Furthermore, errors due to measurement noise cannot easily be included into the analysis, and it is not possible to recover missing data points due to misdetection or occlusions.
We propose GPFBM, a computational method based on Gaussian Processes (GP)^{22,23} and fractional Brownian motion (FBM)^{24,25,26}, which improves and extends the concepts presented in^{27,28}. Importantly, GP provides a consistent probabilistic framework that considers entire trajectories and thus utilizes all the available information. Trials on simulated data demonstrate a greater precision of GPFBM in measuring diffusion parameters over MSD and DDB. Furthermore, as it is applied directly on trajectories, GPFBM naturally takes into account localization errors and occlusions without the need to establish a fitted MSD curve or displacement distributions. We further extend this model to account for external sources of movement (e.g. displacement of whole nuclei or chromosomes) using underlying correlations between multiple trajectories, without the need to further develop substrate motion models and experiments for calibration^{29}. Finally, we applied GPFBM to two experimental systems to study chromatin diffusion properties in different contexts. First, we characterized chromatin dynamics in interphase and mitosis using tagged arrays inserted at random genomic locations in mouse embryonic stem (ES) cells. Although chromatin density increases by a factor of three during mitosis^{30}, our results surprisingly indicate that there are no significant differences on average in the apparent diffusion or anomalous coefficients. Second, to compare the diffusion properties of different specific genomic regions, we performed doublelabeling and live tracking experiments around the HoxA locus in mouse ES cells before and after induction of the genes with retinoic acid. We discover that, instead of having homogeneous diffusion properties across euchromatin, genomic loci significantly differ in both their apparent diffusion and anomalous coefficients. In some cases, altered chromatin diffusion properties correlate with underlying functions such as gene regulation or CTCF binding. The methods we have developed are integrated into a userfriendly package, GPTool, for use in the scientific community. Chromatin mobility has been overlooked in previous studies of genome functions, and we anticipate that GPFBM will facilitate research in that area.
Results
Modeling diffusion dynamics with GPFBM
Traditional methods to analyze particle diffusion dynamics rely on particle displacements calculated between two frames at different time intervals, hence information on how precisely the particle moves between the two points is not considered. This has important drawbacks: higherorder temporal correlations within the trajectories are discarded; errors due to framedependent measurement noise cannot be easily included into the analysis; and missing data points due to misdetection or occlusions are ignored and cannot be recovered by inference. To address these problems we built a consistent probabilistic framework based on Gaussian Processes (GP)^{22,23}. Briefly, a GP is defined as a collection of random variables such that every finite subset of them follows a multivariate normal distribution which is fully determined by its mean and kernel functions μ(t) and \({{\Sigma }}(t,t^{\prime} )\). We assume that a stochastic diffusion trajectory x(t) of a given chromatin locus can be modeled as a Gaussian process with the following fractional Brownian kernel^{24,25,26}:
where D_{α} is the apparent diffusion coefficient and α is the anomalous coefficient defined in the range 0 < α < 2. This kernel produces a generalized Brownian motion with a mean squared displacement 〈r^{2}〉 = 2nD_{α}t^{α}, where n corresponds to the number of degrees of freedom. Notice that the traditional Brownian dynamics is recovered with α = 1. Then the probability of observing a discrete trajectory r = {r_{i}} measured at a set of times t = {t_{i}} is given by the multivariate Gaussian distribution,
where the covariance matrix is defined as \({{{\Sigma }}}_{ij}\equiv {{{\Sigma }}}_{{D}_{\alpha },\alpha }({t}_{i},{t}_{j})\) and we take a constant μ without loss of generality. Furthermore, we can easily incorporate localization errors by adding the diagonal term \({\sigma }_{i}^{2}{\delta }_{ij}\) to the covariance matrix Σ, which assumes that errors are decorrelated and normally distributed with standard deviations σ = {σ_{i}} (see Methods). Ultimately, providing a trajectory r and the localization errors σ, the likelihood (2) can be used to calculate estimates of the diffusion parameters D_{α} and α either via optimization or sampling using the MetropolisHastings algorithm^{23,31}.
We first test the performance of the GPFBM method on synthetic trajectories simulated from a FBM model with a given time step (dt). To mimic the measurement noise observed in real trajectories, we introduce the localization error σ and the occlusion rate o. An example of simulated trajectory and the effect of the measurement noise is shown in Fig. 1a. We then obtain a posterior distribution over the parameters D_{α} and α given the trajectory and the localization errors by combining the likelihood (2) with flat priors over all parameters, which can be sampled using Markov Chain Monte Carlo (MCMC) (see Fig. 1b and Methods). Interestingly, once the diffusion parameters are estimated, the power of the GP framework can be used to infer the most probable trajectory of the particle by removing measurement noise and predicting the particle position where occlusions or misdetections occurred (Fig. 1a). To systematically evaluate the performance of our method to infer diffusion parameters, we generated 50000 synthetic trajectories using uniformly distributed random values of D_{α} and α in the range 0 < D_{α} < 1.5 and 0 < α < 2. For generality, we also sample the simulation time step (dt), localization error (σ) and occlusion ratio (o) from a uniform random distribution in the respective ranges: 0.1 < dt < 1.0, 0.001 < σ < 0.25 and 0 < o < 0.8. We compared the results obtained using our approach on the simulated data with the traditional MSD and DDB methods. GPFBM clearly outperforms both methods, producing smaller relative errors of parameter estimation over all (Fig. 1c) and across different parameter ranges (Supplementary Figs. 1 and 2). Unlike GPFBM, both MSD and DDB methods require trajectories to be split into individual displacements, thus neglecting higherorder temporal correlations that the trajectories may contain. Therefore GPFBM method can optimally infer diffusion parameters from single trajectories, using all the information contained in the data and thus achieving greater precision.
Accounting for substrate movement with GPFBM
Often a particle may be subject to secondary movement that is entangled with its diffusion dynamics. In chromatin dynamics, this movement is frequently associated with the substrate in which the particle is diffusing, such as cell displacement, membrane fluctuations or chromatin reallocation, as well as technical considerations such as thermal drift and undesired media flow. If overlooked, this may result in overestimation of the diffusive properties. However, when two or more particles are measured in the same context, this substrate movement can be accounted for by analyzing the crosscorrelation introduced between the particle trajectories. To that end, we developed a covariance model that takes advantage of the GPFBM framework to quantify substrate movement and handle the crosscorrelation that it may introduce into the movement of all particles (see Methods). In the case of two particles, we obtain the probability distribution,
where Σ_{1}, Σ_{2} and Σ_{R} are FBM covariance matrices for the two particles and substrate respectively with diffusion parameters D_{α} = {D_{α,1}, D_{α,2}, D_{α,R}} and α = {α_{1}, α_{2}, α_{R}}. This method can easily be extended for higher number of particles, limited only by required computational power in practice, even though most of the correction is already achieved with two particles (Supplementary Fig. 3). In this study, we restrict our analysis to five particles per cell.
To demonstrate the utility of this approach, we generated 2000 synthetic trajectories as before, but now including substrate movement, generating vectors r_{i} as a combination of substrate displacement R and the actual particle displacement a_{i} (Fig. 2a). Simulations are generated with 10% occlusion rate and localization error, which are values commonly found in our experiments. As expected, D_{α} and α tend to be overestimated if substrate movement is unconsidered; however, the parameters are more precisely determined when the substrate correction is incorporated into the model (Fig. 2b,c). The method is also able to estimate the dynamic properties of the substrate and, albeit with less precision, the substrate movement itself (see Fig. 2d, Fig. S4 and Methods). Furthermore, we tested the performance of the method depending on the number of tracked particles subjected to the same substrate movement. Precision is increased with use of more particles, but the bulk of the error is already removed with only two particles (see Supplementary Fig. 3). Finally, we showed that GPFBM outperforms the DDB method even when the substrate movement is taken into account (see Methods and Supplementary Fig. 5). In conclusion, GPFBM has the ability to remove the substrate movement from the analysis which is demonstrably important for precise measurement of diffusion parameters. Other approaches can be applied to estimate cell movements and correct trajectories^{32,33}, but GPFBM has the advantage of being able to derive this information directly from the trajectories themselves, provided that two or more particles are tracked per cell. Consequently, we are able to automatically characterize the substrate movement and remove it from the analysis without the need of extra image processing steps thanks to the crosscorrelation that this external movement imprints in the particle diffusion dynamics.
Analyzing chromatin dynamics in interphase and mitosis
Due to chromosome compaction and condensation, chromatin density increases by a factor of three during mitosis^{30}. Although the structure of mitotic chromatin has been intensively studied^{5,34,35}, it is unknown if or how the higher density and rearrangement of chromatin fibers affects chromatin diffusion properties. To measure chromatin dynamics in interphase and mitosis, we used a mouse ES cell line carrying approximately 20 TetO arrays of 7 kb length inserted at random genomic locations^{36}. GFP::TetR is stably expressed in these cells, where it binds to the TetO arrays for the simultaneous visualization of several chromatin loci in each cell. We performed confocal liveimaging and distinguished interphase and mitotic cells by DNA staining using Hoechst 33342, recording images at 4 frames per second for 75 s. To increase the number of mitotic cells, we also performed liveimaging experiments on cells arrested in prometaphase with nocodazole (see Fig. 3a and Methods). We tracked spots using ICY^{37} and enhanced particle localization precision by fitting a 2D Gaussian function to the signal of the tracked spots (see Methods and Supplementary Fig. 6). Before applying the GPFBM probabilistic framework, we first determined whether the measured stochastic trajectories present, to a certain approximation, selfsimilar Gaussian distributed displacements and a FBM velocity autocorrelation function (see Methods). Interestingly, that seems to be the case for chromatin movements at the time scale of this study, hence GPFBM is an appropriate approach for the analysis (Supplementary Figs. 7 and 8).
Comparing the performance of GPFBM with and without substrate movement correction, it was apparent that actively dividing mitotic cells had greater substrate movement (presumably due to coordinated alignment and movement of chromosomes by the mitotic spindle), but that appreciable correction was required for precise chromatin dynamics measurements in all conditions (Fig. 3b). Surprisingly, we observed no significant differences in the mean apparent diffusion or the mean anomalous coefficients between interphase and mitotic chromosomes (p >= 0.05), suggesting that condensation may not necessarily affect the average local diffusion dynamics of chromatin (Fig. 3c and d). We observed a small but significant increase in the anomalous coefficient of mitoticarrested cells compared to interphase, which might be related to the effect that nocodazole has on microtubule formation and thus mitotic chromosome stability^{38}.
Interestingly, we obtained a wide range of estimated D_{α} and α coefficients indicating a remarkable spottospot variability in their diffusion dynamics, even when correcting for substrate movement. This variability could partially be caused by differences in the state of the analyzed cells (intercell variability) leading to different overall chromatin dynamics. Alternatively, differences in the chromatin context of the genomic loci could lead to specific diffusion dynamics (intracell variability). Applying the law of total variance (see Methods), we quantified the contribution of intercell vs intracell variability (Fig. 3e and f). Strikingly, as much as 75% of the variability in D_{α} and 65% in α could be explained by differences within the same cells. This estimate is even higher when substrate movement is taken into account, especially in the case of mitotic cells, when mouse ES cells tend to detach from their colonies, becoming more prone to movement. In contrast, the nocodazole arrested cells are allowed to sediment onto the glass surface, thus are less mobile during imaging. Together, this suggests that different genomic loci may have characteristic local diffusion properties due to their specific chromatin or nuclear context.
Distinguishing locus and cellspecific diffusion properties
Except for a tendency for chromatin mobility to be reduced at centromeric or telomeric locations in yeast^{39}, little is known about how different genomic contexts may affect dynamics of the underlying chromatin. Further, previous studies give conflicting views on whether transcriptional activation can increase local confinement of a gene (as observed in the same cell before and after estrogen stimulation^{7}) and/or increase gene mobility (as observed comparing cells before and after differentiation^{8}). To compare the diffusion properties of different specific genomic regions, we performed doublelabeling and live tracking experiments with the ANCHOR system^{40} around the HoxA locus in mouse ES cells before and after induction of Hox genes with retinoic acid. We engineered the ANCH1 and ANCH3^{7} labels into different locations within the same allele to generate two ES lines with equidistant probes assessing interTAD (T1T2) or intraTAD (T2T3) associations (Fig. 4a, b) and imaged at 2 frames per second for 2 min. As may be expected, the average interprobe distance was higher for the interTAD than intraTAD combination, but with large heterogeneity in the distance distributions (Fig. 4c;^{41}). Interestingly, Hox gene induction had no effect on intraTAD distances within the neighboring domain, but increased interTAD distances, supporting the idea of general TAD reinforcement as cell differentiation is induced^{42}. As tests of Gaussianity and velocity autocorrelation again verified approximation of chromatin dynamics to FBM (see Methods and Supplementary Figs. 7 and 8), we performed GPFBM for the three loci and found that, in undifferentiated ES cells, although they have equivalent apparent diffusion rates, region T1 is significantly more constrained than T2 or T3 (Fig. 4d, e). Closer inspection of the ES (and differentiated neuronal precursor cell) epigenomic profiles around these regions showed that T1 is close (<15 kb) to a putative active enhancer of Halr1 (Supplementary Fig. 9). This gene encodes the long noncoding RNA Haunt, whose specific expression in ES cells is linked to suppression of the HoxA genes^{43}. Active histone modifications around T1, compared to the silent T2 and T3 regions, correlates with a greater constraint of the chromatin, in line with a previous study of an estrogeninduced gene^{7} and predictive polymer models^{14}. Hox gene induction by retinoic acid had no significant effect on the diffusive rate of T1 but did reduce locus constraint (Fig. 4d, e). In contrast, the region T2, which lacks any known epigenomic or regulatory features, had increases in D_{α} and α, perhaps indicative of general chromatin remodeling caused on onset of differentiation. Curiously, T3 became more constrained on retinoic acid treatment, with a concomitant increase in mobility. This region contains sites bound by the architectural protein CTCF, whose binding is either lost or reduced on differentiation to neuronal precursors (Supplementary Fig. 9). CTCF is proposed to form a roadblock for cohesinmediated loop extrusion processes^{44,45}, and this may be expected to play out in alterations to local chromatin dynamics, although this has been largely unexplored. Overall, these results show previously unappreciated locusspecific variation in chromatin diffusive properties, which in some cases correlate with underlying histone modifications or CTCF binding.
GPTool allows userfriendly application of GPFBM
To facilitate use of GPFBM by the community, we developed a freely available graphical user interface called GPTool (Fig. 5; github.com/guilmont). GPTool contains 4 plugins: movie, alignment, trajectories and gprocess. The movie plugin allows the user to open TIFF files, display basic ImageJ and OME metadata, define colormaps for each channel and manually correct for contrast. The alignment plugin runs the algorithm described in Methods to digitally correct chromatin aberration and possible camera alignment issues. Alternatively, the user can manually modify each of the parameters. Finally, the gprocess plugin allows to infer optimal values for the apparent diffusion and anomalous coefficients for several cells in the same movie whilst correcting for substrate movement if two or more particles are selected. It is also possible to use a MetropolisHastings sampler to obtain the posterior probability distribution associated with each of these parameters. Once the analysis is complete, the tool provides the possibility to save the results into JSON files. It also provides export functions to save tables in CSV format. All these formats are easily parsed in all major computing languages, such as C/C++, Python and R. Finally, GPTool provides shared libraries and C/C++ examples for batching multiple movies. A complete documentation of the software can be found in the aforementioned Github account and in Supplementary Materials.
Discussion
We developed GPFBM, a Bayesian framework that combines the inference power of Gaussian processes with fractional Brownian motion, a flexible model to describe Gaussianlike diffusion dynamics. Importantly, chromatin loci show Gaussian and selfsimilar displacement distributions, indicating that FBM is an adequate approximation to assessing chromatin movement, at least for the time scales over which experiments are commonly performed (from a few up to hundreds of seconds). Notice that for longer time scales a crossover has been observed between different diffusion regimes and our model would have to be modified to incorporate this behavior^{16}. Note also that a myriad of other biological systems have nonGaussian dynamics, so would not be suitably analyzed by GPFBM^{20,21,46,47}.
GPFBM treats stochastic trajectories as a whole without preprocessing or extracting limited statistics from them. Therefore, this approach utilizes optimally all the information contained in the data by incorporating higherorder temporal correlations into the analysis. In addition, the Gaussian process framework allows easy integration of spotdependent localization errors which translate into a consistent weighting of time points, depending on the precision at which the spot position is determined. Furthermore, missing data due to spot misdetection or occlusion does not hinder the analysis and, on the contrary, GPFBM can be used to probabilistically assign spot positions for any given time point. Finally, when two or more particles are tracked in similar context, GPFBM uses possible crosscorrelations between trajectories to characterize substrate movement and, therefore, remove it from the analysis. A number of other methods have been developed over recent years to better characterize diffusion of particles, employing variations of MSD^{18,48}, probability density functions for particle displacements^{19,20,21}, Bayesian inference^{27,28} or even machine learning approaches^{49,50}. However, to our knowledge, these methods are either not readily applicable to experimental data, require extra experiments to precisely measure complementary parameters to determine background movement, require large amounts of varying training sets to account for different shapes/types of input data, and/or are not robust to mislocalization and occlusion events that are commonplace in imaging experiments. Benchmarking against MSD and DDB, GPFBM show improved results over all combinations and ranges of tested parameters. GPFBM is thus a precise and robust tool. Providing that the model assumptions are fulfilled, this increase in accuracy can be crucial to study changes in diffusion dynamic properties in different conditions.
We applied GPFBM to two ES cell systems and observed a large variability in chromatin dynamics when comparing individual cells and comparing different loci. A fraction of this variability can be explained by differences across cells, especially in interphase cells, indicating that cell state (cell cycle or metabolism) may globally influence chromatin dynamics. However, the majority of the observed variability is related to differences across loci. Unexpectedly, chromatin exhibits similar average diffusion dynamics in interphase and mitosis despite a large difference in chromatin density. This result may be related with recent findings showing that mitotic chromatin is not as inaccessible and inert as previously thought. Indeed, several studies have shown that mitotic chromatin is bound by transcription factors^{51,52} and some genes are even transcribed during mitosis^{53}. In contrast, different genomic loci can have striking differences from average chromatin dynamic properties, which in some cases correlate with underlying functional chromatin marks. It has been previously proposed that chromatin mobility is affected directly by transcription, although results were seemingly conflicting^{7,8}. More widespread application of GPFBM to labeled transcribed loci and other specific regulatory elements, such as enhancers or TAD borders, will likely uncover more interesting functional links between genome function and the dynamics of its component chromatin.
Finally, we present GPTool a graphical user interface that helps to perform GPFBM analysis on microscopy movies with only a few mouse clicks. Importantly, this tool and the GPFBM framework can be applied to study not only chromatin dynamics but potentially any labeled particle that can be tracked over time providing that Gaussianity and other assumptions of FBM are met. Alternatively, the FBM kernel used in this study can potentially be replaced by alternative kernels that may better describe dynamics of other systems. We thus anticipate that GPFBM and GPTool will greatly facilitate the analysis of diffusion dynamics in biology.
Methods
Cell lines, culture, and treatments
Transgenic TetO ES line
The mouse ES cell line was kindly provided by Dr. Luca Giorgetti. It is derived from an X0 clone of the PGKT2 subclone of the feederindependent PGK12.1 mouse ESC line which was engineered by cotransfection with pBROAD3TetRICP22NLSeGFP and pcDNA3.1Hygro to stably express the TetReGFP recombinant protein after random integration and hygromycin selection (250 μg/ml) as described in ref. ^{36,54}. The piggyBac transposon system was then used to generate cells with 20–25 stable random integrations of a 150 TetO binding site array as described in^{36}. Cells were cultured on 0.1% gelatincoated culture plates in DMEM (4.5 g/l glucose) supplemented with GLUTAMAXI, 15% fetal calf serum (ES cell culture tested), 0.1 mM betamercaptoethanol, 1,500 U/ml leukemia inhibitory factor (LIF; produced in house), and 0.1 mM nonessential amino acids in 5% CO_{2} at 37 °C. Mitotic arrest was performed by treating the cells for 5 h with 100 ng/ml Nocodazole (Sigma, M14042MG). This cell line is available upon reasonable request.
Transgenic ANCHOR ES lines
J1 mouse ES cells were grown on gammairradiated mouse embryonic fibroblast cells under standard conditions (4.5 g/L glucoseDMEN, 15% FCS, 0.1 mM nonessential amino acids, 0.1 mM betamercaptoethanol, 1 mM glutamine, 500 U/mL LIF, gentamicin), then passaged onto feederfree 0.2% gelatincoated plates for at least two passages to remove feeder cells before subsequent transfections. The two ("interTAD” and “intraTAD”) ANCHOR transgenic lines were generated by sequential CRISPR/Cas9mediated knockin experiments in the following manner. First, flanking homology arms (mm9 chr6: 52,320,06152,321,144, and chr6: 52,321,14552,322,244) were introduced by PCR amplification and Gibson assembly into a vector containing ANCH1 sequence^{40}. This vector (1 μg) was cotransfected with 3 μg of a vector containing Cas9GFP, a puromycin resistance marker, and the scaffold to transcribe the sgRNA specific to the T2 insertion site (CGGCGCGCACTTAACACCAA; vector generated by the IGBMC Molecular Biology platform) in 1 million cells with Lipofectamine2000. Two days after transfection, the cells were cultured for 24 h with 3 μg/ml puromycin, then 48 h with 1 μg/ml puromycin to enrich for transfected cells, before sorting individual GFPpositive cells on to feeders to amplify individual clones. Clones with the correct sequence were screened by PCR and sequencing, then the CRISPR knockin process was repeated to insert the ANCH3 sequence^{7} into either the T1 site ("interTAD” line; homology arms at chr6: 52,013,47152,014,370 and chr6: 52,014,37152,015,270; gRNA sequence AATCGAGCTCACGCCATTAG) or the T3 site ("intraTAD” line; homology arms at chr6: 52,622,95552,623,855 and chr6: 52,623,85652,624,755; gRNA sequence TATGCTGAGGCGTGTCGCAA). Final clones were verified for maintained pluripotency by qRTPCR to assess Oct4, Nanog (e.g. Supplementary Fig. 9), and Sox2 expression. Subsequent microscopy experiments (see below) confirmed heterozygous incorporation of the ANCH sequences (detection of one specific spot per ANCH sequence per cell) within the same allele (two spots were always in close proximity). This cell line is available upon reasonable request.
OR transfection
150,000 cells are plated two days prior to imaging off feeder cells onto laminin511coated 35 mm glass bottom petri dishes, and transfected with 3 μg OR1EGFP and 3 μg OR3IRFP plasmids (vectors available from NeoVirTech (contact@neovirtech.com); were modified from original source by changing the Cterminal fluorescent protein sequence, introducing Kozak sequence before the translation start site and replacing the CMV promoter with EF1α) using Lipofectamine2000. After two days, the medium is changed to remove dead cells, before passing directly to microscopy.
Hox induction
ES cells were passaged without feeders and cultured on laminin511 for two days without LIF, then for a subsequent three days without LIF and with the addition of 5 μM retinoic acid. One day after the addition of retinoic acid, the cells are transfected with the OR proteins as previously.
Microscopy
Live cell imaging of TetO ES cells
35 mm glassbottom dishes (Ibidi 81158) were coated with 10 μg/ml fibronectin human plasma (Sigma, F20061MG) in PBS for 45 min at room temperature. A total of 3–5 × 10^{5} cells were seeded one day before imaging, then the medium was replaced by phenolredfree medium containing 500 ng/ml Hoechst 33342 (Invitrogen, H3570). Cells arrested in mitosis were collected on the day of imaging by “shakeoff”, incubated with 0.25% Trypsin1 mM EDTA (Invitrogen, 25200072) for 1 min at 37 °C and washed, and placed on fibronectincoated glassbottom dishes in phenolredfree medium containing 100 ng/ml Nocodazole and 500 ng/ml Hoechst 33342. Confocal livecell imaging was performed on a Nikon Eclipse TiE inverted widefield microscope (Perfect Focus System) equipped with a CSUX1 confocal scanner unit and an Evolve backilluminated EMCCD camera (Photometrics). Images were recorded using 100 × HC Plan APO oil immersion objective (Leica, NA 1.4). Intensities were set to 10% for the 405 nm and 30% or 50% for the 491 nm lasers, with exposure times of 100 ms and 50 ms or 25 ms, respectively. 5 zstacks with 0.5 μm distances were recorded for each channel. 301 timelapse images were recorded only in the 491 channel.
Live cell imaging of ANCHOR ES cells
Imaging experiments were performed on an inverted Nikon Eclipse Ti microscope equipped with a PFS (perfect focus system), a Yokogawa CSUX1 confocal spinning disk unit, two sCMOS Photometrics Prime 95B cameras for simultaneous dual acquisition to provide 95% quantum efficiency at 11 μm × 11 μm pixels and a Leica 100× oil objective (HC PL APO 1,4 oil immersion). We excited EGFP and IRFP with a 491 nm (100 mw) and a 635nm laser (> 28 mW), respectively. We detected green and far red fluorescence with an emission filter using a 525/50 nm and a 708/75 nm detection window, respectively. A thermostated heater (Tokai Hit Stage Top Incubator) allowed for heating at 37 °C, humidity, and CO_{2} control (5%). Timelapse analysis of GFP and IRFP foci was performed in 2D acquiring 241 time points at a 0.5 s time interval. The system was controlled using Metamorph 7.10 software. Timelapse was concatenated into single TIFF file.
RTqPCR
RNA was extracted from cells using the Nucleospin RNA extraction kit (MacheryNagel), then cDNA was prepared with SuperScript IV (Invitrogen), following the manufacturer’s instructions and using random hexanucleotides as primers. The cDNA was quantified by qPCR on a LC480 LightCycler (Roche), using QuantitTect SYBR Green PCR kit (Qiagen). Amplification was normalized to GAPDH. Primer sequences are given in Supplementary Table 1.
Image preprocessing
Spot detection and tracking
Spot detection and tracking for all movies was performed with ICY, an image analysis software^{37}. Localization precision was then enhanced by assuming that the spots have the shape of a 2D Gaussian function as follows,
with μ_{i} representing the center of mass of the spot, L_{i} its size in directions x and y, − 1 < θ < 1 a possible rotation, while B_{G} and I_{o} are background and spot signal, respectively. We optimize its localization using the NMSimplex method^{55} and estimate localization error using the MetropolisHastings algorithm^{23,31}. This method is implemented and automatically runs when trajectories are loaded in GPTool. For more information see Supplementary Figs. 6 and 10.
Multichannel alignment correction
For the ANCHOR ES cell line experiments, we used a spinning disk microscope setup with 2 cameras, i.e., one per channel. Even though these cameras were aligned manually using fluorescent beads, we could still observe nonnegligible differences between images captured in both cameras. Furthermore, even in rare situations when both cameras were properly aligned, we could observe effects of chromatic aberrations towards the edges of the image due the different wavelengths used. To correct for such problems, we performed digital postalignment using a generic set of affine transformations including translation, rotation and scaling as defined in
where, s_{i} accounts for scaling in directions x and y, d_{i} accounts for translation in both directions and θ is the angle of rotation between both channels in relation to point c_{i}.
To infer optimal parameters for correction, we used 5 frames from all the movies recorded in the session and maximize the following likelihood using the NelderMead simplex method^{55}
where W and H correspond to width and height of images and I_{r}(k, l∣A) is the value of pixel (k,l) in channel r given transformation A. Here, \({\mathbb{1}}\) represents the identity matrix. Supplementary Fig. 11 shows examples of misaligned images and how the alignment improves greatly after applying our algorithm.
Derivation of GPFBM models
Fractional Brownian motion
The covariance function of FBM can easily be derived from the assumption of two basic properties^{56}: stationary increments B(t) − B(s) ∝ B(t − s) and a powerlaw variance, \(\left\langle B{(t)}^{2}\right\rangle \propto  t{ }^{\alpha }\). Then, the offdiagonal terms of the covariance function can be determined as follows:
Finally, the apparent diffusion coefficient D_{α} is introduced as a proportionality factor to rescale mobility, leading to the final kernel as presented in the main text:
We can also calculate the velocity autocorrelation function for the FBM model^{26}, which can be easily calculated from experimental trajectories using
where velocity is defined as \(\nu (\tau )={\epsilon }^{1}\left[x(\tau +\epsilon )x(\tau )\right]\). Using that, the theoretical curve for velocity autocorrelation function for FBM is calculated to be
To show that FBM is a viable approximation for the dynamics displayed by chromatin in the time range of our experimental measurements, we verified that displacements are selfsimilar Gaussian distributed with aforementioned covariance matrix and that its velocity autocorrelation agrees with theoretical predictions (Supplementary Figs. 7 and 8).
Bayesian inference of diffusion parameters
The GP provides the probability of observing a trajectory r given D_{α} and α. Then, we applied Bayes theorem^{23} to obtain the posterior distribution over the diffusion parameters given the measured trajectory:
where P(D_{α}, α, μ) represents the prior distribution of the model parameters. Assuming a flat prior on μ, D_{α} and α, the logposterior can be expressed as
where N represents the number of points measured and ∣ ⋅ ∣ is the determinant function. To obtain maximum posterior estimates, we optimized (12) using the NelderMead Simplex method^{55}. In addition, we used the MetropolisHastings method^{23,31} to sample the posterior probability distribution in order to calculate confidence intervals for our estimations. Note that, thanks to this Bayesian approach, available prior knowledge of the diffusion parameters can easily be incorporated into the analysis. For more information regarding MH sampler, view Supplementary Fig. 12.
Incorporating of substrate movement in the GPFBM framework
In the main text, we introduced an extended GPFBM model to deal with external sources of movement. Here, we present the derivation for two particles subject to a common substrate movement, however, it can be extended for an arbitrary number of particles using the marginalization rule of multivariate Gaussian distributions^{23}. The key idea is to assume that the movement of the particles with respect to the substrate as well as the movement of the substrate itself can be described by independent fractional Brownian motions. Therefore, the probability of observing the particle trajectories a_{1} and a_{2} with respect to a given frame of reference R that moves with the substrate is:
where Σ_{1}, Σ_{2} and Σ_{R} are FBM covariance matrices that are fully characterized given the diffusion parameters D_{α} = {D_{α,1}, D_{α,2}, D_{α,R}} and α = {α_{1}, α_{2}, α_{R}}.
Next, to obtain the probability distribution over the trajectories r_{1} and r_{2}, we applied the change of coordinates r_{i} = a_{i} + R (see scheme in Fig. 2a) leading to the matrix expression,
Then, to marginalize the unobserved trajectory of the moving frame R_{i}, we need to calculate the inverse of the block matrix in equation (14). To do so, we use results from^{57} on inverting a 2x2 block matrix such as
according to the following result
Taking \(A=\left[({{{\Sigma }}}_{1}^{1},\,0);\,(0,\,{{{\Sigma }}}_{2}^{1})\right]\), \(B=\left[{{{\Sigma }}}_{1}^{1};\,{{{\Sigma }}}_{2}^{1}\right]\), C = B^{T} and \(D={{{\Sigma }}}_{1}^{1}+{{{\Sigma }}}_{2}^{1}+{{{\Sigma }}}_{R}^{1}\), it is easily shown that the topleft corner of Λ^{−1} is given by \(\left[({{{\Sigma }}}_{1}+{{{\Sigma }}}_{R},\,{{{\Sigma }}}_{R});\,({{{\Sigma }}}_{R},\,{{{\Sigma }}}_{2}+{{{\Sigma }}}_{R})\right]\). Using this result, we can marginalize R in equation (14) giving the expression,
This result clearly shows that the substrate movement induces a correlation between the particle trajectories as the offdiagonal elements of the block matrix in equation (17) are non zero. In addition, the covariance matrix of the substrate movement appears also in the diagonal terms, increasing the overall variance of the particles and their total movement. Consequently, if this correction is ignored the diffusion parameters are overestimated.
Inference of substrate movement
Similarly as before, the diffusion parameters of the particles as well as the substrate can be estimated using equation (17) and Bayesian inference. We can also estimate how the substrate moves. For that, we can calculate the conditional distribution of R given the particle trajectories as
with covariance matrix given by \({{{\Sigma }}}_{\left\langle {{{{{{{\boldsymbol{R}}}}}}}}\right\rangle }={\left({{{\Sigma }}}_{1}^{1}+{{{\Sigma }}}_{2}^{1}+{{{\Sigma }}}_{R}^{1}\right)}^{1}\).
Unfortunately, we could not find an analytical solution for this problem. Nonetheless, we can solve it numerically. In Supplementary Fig. 4.a we show an example of the estimated movement for the substrate with one standard deviation compared to the real simulated trajectory for a system with two particles. In Supplementary Fig. 4.bc we display the overall accuracy when working with two or more particles.
Benchmarking GPFBM
Simulated trajectories
We simulated single trajectories with 250 time points using the aforementioned Gaussian process with FBM kernel. To keep the benchmark as general and unbiased as possible, we uniformly sampled values for our parameters in the ranges 0.01 < D_{α} < 1.5, 0.01 < α < 1.9, 0.1 < dt < 1.0 and 0.001 < σ < 0.25. To benchmark a system of N particles affected by substrate movement, we generated N+1 trajectories and add the latter to all the others. Finally, a uniform distribution is used to remove 0% to 80% of points from each trajectory to simulate experimental occlusions.
Mean squared displacement (MSD) implementation
To calculate D_{α} and α for single trajectories, we estimate a MSD using a sliding window method. This method is mathematically defined as follows
for a trajectory with N points and step interval n. Due to implicit correlations present in single trajectories, we use only initial 10% step intervals. To improve accuracy, we also estimate an average localization error σ. Finally, this experimental curve is approximated by the theoretical mean squared displacement equation
from which diffusion parameters are inferred using linear regression. For more information^{58}.
Displacement distribution based (DDB) implementation
The theoretical expressions for the displacement distribution is obtained as a solution of the FokkerPlanck equation with localization error σ. In polar coordinates it takes the form
In order to calculate experimental distributions for single cells, we resort to a sliding window method similar to the one present for MSD. Differently, we calculate normalized histograms with all the absolute displacement values. As before, we calculate an average localization error σ to improve localization and use only histograms calculated for initial 10 step intervals. With these measurements, we optimize the equation above for D_{α} and α using Bayes approach with noninformative priors for both parameters.
Law of total variance
The law of total variance is used to determine how much of the measured variance comes from within or across samples. Starting off from the law of total expectation:
we can calculate
Subtracting \({\left\langle \left\langle x y\right\rangle \right\rangle }^{2}\) from both sides
Upon algebraic manipulation, we obtain the final result
which states that the total measured variance in x is composed by the \(\,{{\mbox{var}}}\,\left(x\right)\) given sample y and \(\left\langle x\right\rangle\) calculated for each y.
Statistical analysis and reproducibility
Data for interphase and mitotic cells are from two independent experiments, while Nocodazoletreated cells are from one experiment. Anchor data is accumulated from 11 independent experiments.
To compare diffusive properties or interprobe distances across different loci or conditions (Fig. 3c, d, and 4b–d), we performed Wilcoxon rank sum tests. For interprobe distances, the distributions of the median distances for each movie were used. For Fig. 4c, d, where fifteen pairwise comparisons are possible, the pvalues were corrected for multiple testing with the BenjaminiHochberg method and differences are considered statistically significant for pvalues inferior to 0.05.
Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
Data availability
The data that support this study are available from the corresponding authors upon reasonable request. Microscopy data with analysis files that support the findings of this study have been deposited in Zenodo and can be accessed with https://doi.org/10.5281/zenodo.5359893^{59}, https://doi.org/10.5281/zenodo.5360028^{60} and https://doi.org/10.5281/zenodo.5361054^{61}. The source data are provided with this paper.
ES HiC sequence data from^{42} were taken from Gene Expression Omnibus (GSE96107 [https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE96107]), and mapped to mm9 and normalized with FANC^{62}. The normalized submatrix (chr6:5150000053000000) was then extracted for visualization. ES and neuronal precursor cell H3K27ac and CTCF ChIPseq data were taken as bigWig files from Gene Expression Omnibus; GSE96107 [https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE96107] for all except ES H3K27ac, taken from GSE49847 [https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE49847]) and visualized in R using the package rtracklayer. Source data are provided with this paper.
Code availability
C++ libraries, batch templates and graphical user interface are available at https://github.com/guilmont^{63}.
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Acknowledgements
We thank Luca Giorgetti for providing the TetO ES line and for critical reading of the manuscript. This work was possible thanks to funding from grants by LabEx INRT (ANR10LABX0030INRT, a French State fund managed by the Agence Nationale de la Recherche under the frame program Investissements d’Avenir ANR10IDEX000202), CNRS << Osez l’interdisciplinarité ! >> , ERC (Starting Grant 678624  CHROMTOPOLOGY) and ATIPAvenir. The microscopy was performed at the Imaging Center of the IGBMC.
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G.M.O. developed mathematical models, algorithms, and GPTool, also performed overall data analysis. D.K. generated anchor cell lines. A.O., D.K., and M.M. did experiments and validations. K.B. provided ANCHOR constructs and consulted on ANCHOR experiments. T.S. and N.M. conceived, designed, and supervised the study. G.M.O., T.S., and N.M. wrote this manuscript with input from all other authors.
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Oliveira, G.M., Oravecz, A., Kobi, D. et al. Precise measurements of chromatin diffusion dynamics by modeling using Gaussian processes. Nat Commun 12, 6184 (2021). https://doi.org/10.1038/s41467021264667
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DOI: https://doi.org/10.1038/s41467021264667
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