Youssef Hmamouche
email: youssef.hmamouche@lis-lab.fr

Magalie Ochs
email: magalie.ochs@lis-lab.fr

Laurent Prévot
email: laurent.prevot@univ-amu.fr

Neuroscience to Investigate Social Mechanisms Involved in Human-Robot Interactions

Keywords: human-machine interactions, conversation, functional MRI, multimodal signals processing, feature selection, prediction

To what extent do human-robot interactions (HRI) rely on social processes similar to human-human interactions (HHI)? To address this question objectively, we use a unique corpus. Brain activity and behaviors were recorded synchronously while participants were discussing with a human (confederate of the experimenter) or a robotic device (controlled by the confederate). Here, we focus on two main regions of interest (ROIs), that form the core of "the social brain", the right temporoparietal junction [rTPJ] and right medial prefrontal cortex [rMPFC]. An new analysis approach derived from multivariate time-series forecasting is used. A prediction score describes the ability to predict brain activity for each ROI, and results identify which behavioral features, built from raw recordings of the conversations, are used for this prediction. Results identify some differences between HHI and HRI in the behavioral features predicting activity in these ROIs of the social brain, that could explain significant differences in the level of activity.

INTRODUCTION

Recently, "second-person neuroscience" was proposed as a new way for "the study of real-time social encounters in a truly interactive manner" [START_REF] Schilbach | Toward a second-person neuroscience[END_REF], and the approach described in this paper addresses this endeavor. It is part of a larger project that previously involved recording of the data and preparing the corpus for sharing with the community. This unique corpus combines synchronized behavioral and neurophysiological recordings during natural human-human and human-robot natural interactions (HHI and HRI). The participant's brain activity was continuously scanned with functional magnetic resonance imaging (fMRI) while having natural conversations with a human (confederate of the experimenter) or a robotic head (controlled by the confederate, unbeknownst of the participant who believed the robot to be autonomous).

The methodological challenge is to understand relationships between complex behaviors and activity in the social brain during unconstrained interactions. It poses one major difficulty: what to use as explanatory variable? The approach used here was described previously in [START_REF] Thierry | An experimental approach to study the physiology of natural social interactions[END_REF]. In a nutshell, behaviors are recorded and processed to build time-series, used in turn for the analysis of neurophysiological data. This approach allows the analysis to go further than the simple comparison between HHI and HRI already published for this corpus [START_REF] Rauchbauer | Brain activity during reciprocal social interaction investigated using conversational robots as control condition[END_REF].

Finding pertinent relationships between behavioral time-series and neurophysiological time-series is another challenge for which a new approach is presented here, based on a branch of machine learning, namely time-series forecasting [START_REF] Hmamouche | Exploring the Dependencies between Behavioral and Neuro-physiological Time-series Extracted from Conversations between Humans and Artificial Agents[END_REF]. The method is used to identify which behavioral time-series are required to predict brain activity in specific brain regions, and their respective weights. We present a preliminary result obtained using this methodology focusing on two core regions of the social brain, in the the Medial PreFrontal Cortex (MPFC) and TemporoParietal Junction (TPJ). Given the increase of activity between HHI and HRI reported previously in these areas, our hypothesis is that different behaviors will be associated to the activity in these areas during these two conditions. Alternatively, the same behaviors associated with different levels of activity will suggest a top-down "switch" akin to the "intentional stance" for these areas [START_REF] Dennett | The intentional stance[END_REF].

DESCRIPTION OF THE CORPUS 2.1 Justification of the paradigm

The theoretical grounds underlying the choice of experimental paradigm, the procedures to acquire and prepare the corpus both for behavioral and neurophysiological data, and the bases of the machine learning machinery developed for its analysis have already been published. We will summarize the main points for this paper to be self-contained, but suggest to read the references provided in each section for more details. In order to investigate natural social interactions, it is essential that participants are unaware of the real purpose of the experiment. For this purpose, a cover story was developed responding to a complex set of specifications. The experiment is described as a neuromarketing experiment. A company wants to know if discussing with a fellow or an Artificial Intelligence about the images of a forthcoming advertising campaign is enough to guess the objective of the campaign [START_REF] Thierry | An experimental approach to study the physiology of natural social interactions[END_REF]. The cover story is credible, there is a common, but loose, goal for the conversation, the conversations are truly bidirectional, and there is a legitimacy for talking with a robot.

Our objective isn't to test how an autonomous agent works, but how individuals' behavior changes depending on the nature of the agent. For this reason, participants believed the robot to be autonomous, but to avoid erratic responses during the recordings, a Wizard of Oz procedure was favored in which the confederate selects prerecorded responses expressed by the retroprojected conversational robotic head (Furhat robotics1 ) on a touchscreen.

FMRI experiment and data processing

Participants (n=25 in the full corpus) came to the MRI center and were presented with a fellow (a gender-matched confederate of the experimenter) and the robot. The cover story was presented and the participant installed in the scanner. The participant underwent four sessions of approximately 8 minutes of scanning comprising six experimental trials as follows: an image is presented, then a 1-minute discussion takes place, alternatively with the human and the robot. At the end of the recordings, we recorded 12 trials of 1-minute with the human and the robot for each participant. fMRI data processing followed standard procedures and has been described in details [START_REF] Rauchbauer | Brain activity during reciprocal social interaction investigated using conversational robots as control condition[END_REF]. fMRI data analysis first relied on the general linear model implemented in the toolbox Statistical Parametric Mapping [START_REF] Penny | Statistical Parametric Mapping: The Analysis of Functional Brain Images[END_REF], imported into the Conn toolbox [START_REF] Whitfield | Conn: A Functional Connectivity Toolbox for Correlated and Anticorrelated Brain Networks[END_REF] that automatizes the extraction of BOLD time series in regions of interest. A continuous time-series of 385 points covering the 8 minutes (repetition time: 1.205 seconds) were extracted for each Region of Interest (ROI), each session and each participant.

Regions of interest

For this paper, we investigated a question related to the theoretical framework of philosopher Daniel Dennett, namely the intentional stance [START_REF] Dennett | The intentional stance[END_REF]. It claims that when interacting with humans, we adopt a specific stance as we ascribe mental states to this rational agent. In contrast, we adopt a different stance when interacting with robots, despite the fact that they can appear and behave as humans. It is still debated in robotics today [START_REF] Wykowska | Embodied artificial agents for understanding human social cognition[END_REF], and neuroimaging offers an unique opportunity to address this question as we measure objective neurophysiological markers of cognitive processes.

The intentional stance entails the ascription of mental states such as beliefs, desires, knowledge to others. It parallels a theoretical framework at the core of social cognitive neuroscience, namely Theory of Mind (ToM), that describes one's ability to ascribe hidden mental states, such as intentions, desires or beliefs, to oneself and to others. The neural bases of this ability has been intensively investigated with neuroimaging techniques, identifying two key areas in the cortex that are differentially activated when ascribing mental states versus physical states. In the first significant finding, participants in a Positron Emission Tomography (PET) scanner believed they played stone-paper-scissors with a human or a computer [START_REF] Gallagher | Imaging the Intentional Stance in a Competitive Game[END_REF], and the only significant difference was an increase of activity in the medial prefrontal cortex (MPFC). The same region, as well as the right temporoparietal junction (TPJ), also had reduced activity when the partner was believed to be a robot endowed with with AI [START_REF] Thierry Chaminade | How do we think machines think? An fMRI study of alleged competition with an artificial intelligence[END_REF]. The very rude contrast HHI versus HRI we performed on this corpus [START_REF] Rauchbauer | Brain activity during reciprocal social interaction investigated using conversational robots as control condition[END_REF] identified increased response in the right TPJ. The statistical threshold for the contrast HHI versus HRI was lowered (p < 0.001 uncorrected, unpublished data) to identify a cluster in the MPFC. We selected the two "social brain" ROIs from the Brainnetome parcellation of the human brain [START_REF] Fan | The Human Brainnetome Atlas: A New Brain Atlas Based on Connectional Architecture[END_REF] that contained these rTPJ and rMPFC clusters.

There is a possible implication of the superior temporal gyrus (STS) in mentalizing, but this is controversial given the central role of this region in language (see for example [START_REF] Redcay | The superior temporal sulcus performs a common function for social and speech perception: Implications for the emergence of autism[END_REF]). We decided to include the left STS, given the left hemisphere dominance for language processing in right handed participants, to elucidate which aspects of behaviors are more pertinent for this region. Finally, we also included a control region in which the signal is not related to mental processing, a mask of brain white matter (WM).

METHODOLOGY

We implemented a machine learning process for the prediction of brain activity based on conversational behavioral data. A particularity of our approach compared to related work looking for features that cause the activation of a brain area ([4, 7, 15, 26]) is that it allows to quantify the predictive weights of the behavioral features. Therefore, it provides additional information about the association between behavioral features and the activity of a brain area.

The models of the proposed process are trained based on data of the corpus described in Section 2. The main steps are:

• Extracting behavioral features time-series from raw behaviors recorded in different modalities during conversations. • Resampling and synchronizing behavioral features with respect to the BOLD signal frequency. • Using feature selection method to select a subset of relevant features for each ROI. • Predicting the discretized BOLD signal of each ROI based on the selected features.

Features extraction

The multimodal recordings of behaviour contains 3 types of raw data: video, speech (audios and transcriptions) and eye-tracking. Note that we don't have the video of the participants, as they were inside the fMRI scanner during the experiment. Therefore, we have speech recorded for both the participant and the conversant (human or robot), the videos of the conversant only, and eyetracking recordings of the participant only. The aim of feature extraction is to derive high-level behavioral features from this multimodal raw data, features that describe the behavioral events happening during the conversation. These features are presented in Supplementary Table, and in more details in [START_REF] Rauchbauer | Multimodal corpus of bidirectional conversation of human-human and human-robot interaction during fMRI scanning[END_REF]. For speech, we built time-series IPU (Inter-Pausal Units 2 ) by IPU (e.g., for lexical richness) or word by word (e.g., for Feedback and Discourse Markers). For facial features, we used Openface software ( [START_REF] Baltrusaitis | OpenFace 2.0: Facial Behavior Analysis Toolkit[END_REF]) to detect facial landmarks, action units), head pose and gaze coordinates from the video of the conversant. Eyetracking features were extracted directly from the output of Eyelink1000 system. From this data we calculated other features characterizing where the participant was looking (Face, Eyes, Mouth), with the use of the detected landmarks points of the conversant combined with the gaze coordinate of the participant.

Features were re-sampled to have the same number of observations.

Feature selection

We use feature selection techniques before applying prediction models to improve the prediction by selecting the most relevant predictors, and at the same time to find the smallest set of features predicting the ROI. There are two categories of feature selection techniques. The first are Wrapper methods, i.e., methods that use the prediction model recursively to eliminate features with low importance scores. Here, ModelSelect uses the classifier by executing first the model with all features, then selecting the top-k features based on their importance scores. The second category are filter methods, that select variables based on the relationships between the variables and independently from the prediction model used.

We use two filter methods: Ranking based on mutual information (MI-rank) selects variables without the need of the classifier. It ranks variables based on their Shannon mutual information with the target variable. K-medoid based method (k-medoids) is an extension of the previous method. It first groups close predictive variables into clusters using the k-medoids algorithm (maximizing intra-class mutual information, and minimizing mutual information between classes), then selects one variable from each group based on the mutual information with the target variable. This method is designed to work with multimodal data, as grouping variables before ranking them partly solves the problem of dependencies between variables belonging to the same modality.

Prediction setup

Human-human and human-robot data are treated separately in order to compare them. For each agent, the obtained data consist of 13248 observations. The training data consist of 18 participants 2 An IPU is a speech block bounded by pauses and coming from a single speaker [START_REF] Levitan | Measuring acoustic-prosodic entrainment with respect to multiple levels and dimensions[END_REF].

from 24, the test data are the observations of the remaining 6 participants (i.e., 25% of all data). We applied the ADASYN algorithm on the training data [START_REF] He | ADASYN: Adaptive synthetic sampling approach for imbalanced learning[END_REF] to address the problem of imbalanced data. The problem of imbalanced data occurs in our case because some ROIs are activated rarely during conversations, depending on the participants and their behaviors. This algorithm generates new observations by considering the distribution of the data. Then, we performed a 10-fold-cross-validation on training data to find the appropriate parameters of the classifier based on the F-score measure. Finally, we train the model with the selected parameters and evaluations are made on test data.

Prediction formulation

we first binarize the BOLD signal measured by fMRI as we want to predict whether a brain area is active or not as the original signal is continuous. The discretization method uses the mean of observations as a threshold after normalizing data of each participant.

The BOLD signal recorded as fMRI response to a behavioral event follows the Hemodynamic Response Function (HRF) [START_REF] Gössl | Bayesian modeling of the hemodynamic response function in BOLD fMRI[END_REF]. This function peaks with a delay around five seconds that can vary somehow depending, for example, on the brain area. Other factors, such as saturation of the signal (see detailed examples in chapter 14 of [START_REF] Penny | Statistical Parametric Mapping: The Analysis of Functional Brain Images[END_REF]), make it difficult to consider a deconvolution of the BOLD signal. To handle this variability, our prediction model attempts to predict the discretized bold signal at time t based on sequence of observations of behavioral features that happened previously between 𝑡 -4𝑠 and 𝑡 -7𝑠. Therefore, lagged variables of each feature are considered. This is a temporal classification problem that can be expressed as follows:

𝑌 (𝑡) = 𝑓 (𝑋 𝑡 -𝜏 1 :𝑡 -𝜏 2 1 , . . . , 𝑋 𝑡 -𝜏 1 :𝑡 -𝜏 2 𝑘 ) + 𝑈 (𝑡),
where 𝑌 is the discretized BOLD signal, {𝑋 1 (𝑡), 𝑋 2 (𝑡), . . . , 𝑋 𝑘 (𝑡)} are 𝑘 behavioral variables. 𝑋 𝑡 -𝜏 1 :𝑡 -𝜏 2 𝑖 are the lagged variables of the 𝑖 𝑡ℎ behavioral feature 𝑋 𝑖 , 𝜏 1 = 7𝑠, 𝜏 2 = 4𝑠, and 𝑈 (𝑡) represents the error of the model.

We already used this methodology in previous works [START_REF] Hmamouche | Exploring the Dependencies between Behavioral and Neuro-physiological Time-series Extracted from Conversations between Humans and Artificial Agents[END_REF][START_REF] Hmamouche | BrainPredict: a Tool for Predicting and Visualising Local Brain Activity[END_REF], where we tested several classifiers (Logistic regression, SVM, Long Short Term Memory network) to predict brain areas involved in speech perception and production and face perception. We found that the Random Forest classifier outperforms the others. Here, we only use the Random Forest classifier, and we try to improve the predictions with feature selection.

ROI

RESULTS AND DISCUSSIONS

The results are presented in Table 1. It contains the best F-scores obtained, as well as the best feature selection method and the associated features and their importance. To reproduce the results, the input features and the code source are available online 3 . The results show that filter feature selection methods perform better than the wrapper method, which was selected once as best method. And as we expected, in one hand, the mi-rank method allows to have the best predictions for lSTS area in both HHI and HRI, as it involves one main modality. K-medoids works better for rTPJ and rMPFC, which involve features from multiple modalities. Results indicate that the system is able to predict brain activity in lSTS, rTPJ and rMPFC higher than chance, while it is not possible for the white matter (F-scores not different from 0.5), as expected given the absence of task-related activity in this tissue. A second remark relates to the contrast between lSTS on the one hand and rTPJ and rMPFC on the other hand. The former was selected as being an area strongly devoted to language, while some speculate about its involvement in social cognition. The MI Rank was sufficient for significant prediction in both HHI and HRI, and speech activity of the confederate strongly dominates the list of features used for predicting, even being the only feature in the case of HRI. In the case of HHI, only one other confederate linguistic features represents than 10% of importance. Perceiving voices is enough to activate the STS [START_REF] Pernet | The human voice areas: Spatial organization and inter-individual variability in temporal and extra-temporal cortices[END_REF]. Altogether, these results strongly comfort that the left posterior superior temporal sulcus region is devoted to speech perception. Concerning the "social cognition" areas, a number of preliminary remarks can be made. Importantly, increased activity for for the contrast HHI compared to HRI were found in the two ROIs [START_REF] Rauchbauer | Brain activity during reciprocal social interaction investigated using conversational robots as control condition[END_REF]. which has been interpreted as the neural consequence of the adoption of an intentional stance in HHI, but not HRI. The feature selection method and the F-scores are similar for the two areas and the two interlocutors. Such similarity is in line with the hypothesis that these two areas have both similar and complex (6 to 13 features used for prediction) functions. Considering the rTPJ, speech activity (feature 1 for participant and confederate) is responsible for more than 50% of the predictive power for both HHI and 3 https://github.com/Hmamouche/NeuroTSConvers HRI, while the other features (focusing on feature with importance > 0.1 indicated in bold in Table 1) are different between HHI and HRI. For HHI, we found one speech (Type-token ratio) and one facial (smiles) from the conversant, while the participant's saccades are used in HRI. Important features to predict the activity of the rMPFC differ between HHI and HRI. The conversant's speech and mouth movements represent 22% of the predictions for the former, while the participant's speed of gaze and the conversant's head rotations and action unit 2 together provide 55% of the prediction importance for the later. Altogether, there are differences between the features identified in HHI and HRI at this preliminary stage. For example, for both rTPJ and rMPFC, features used to predict HHI all relate to the conversant, emphasizing its importance in social cognition, while for HRI, the most important feature is from the participant's behavior (eye movements), which is not particularly pertinent for social cognition. Further work is needed to better compare the results for the two agents.

CONCLUSION

Here we have described an innovative methodology to investigate which behavioral features are required to explain the activity in brain areas devoted to social cognition, recorded while participants were having uncontrolled conversations with a human or a humanoid robot. We focused on four regions on the basis of simple assumptions: white matter should not be predictable, the left STS should be related to language, and regions of the social brain should be related to complex social signals. Preliminary results fit our hypotheses. Some differences were found between the behavioral features used to predict activity in social brain areas, not so much in the language perception area lSTS between HHI and HRI. We previously proposed that we adopt a different stance based on the level of activity of these areas. Further work is needed to interpret the differences found with the current analysis, still under development.

Figure 1 :

 1 Figure 1: The three regions of interest investigated in this paper, from left to right. In purple, the left posterior STS region (lSTS). In red, the right medial prefrontal cortex (rMPFC) and right temporoparietal junction (rTPJ).

Table 1 :

 1 Results of prediction evaluations for Human-Human and Human-Robot Interactions (HHI and HRI resp.). For each region of interest (ROI) indicating the codes of the selected features (see Supplementary Table), and their importance scores (Bold: Importance > 0.1). Features code ending with "-p" are for the scanned participant, with "-c" are for the confederate.

			Feature selection Codes of the selected features	Features importance	F-score
		lSTS	mi-rank	[1-c, 10-c, 4-c, 5-c]	[0.71, 0.17, 0.1, 0.02]	0.71
	HHI	rTPJ	k-medoids	[1-p, 1-c, 10-c, 34, 29, 7-p]	[0.33, 0.21, 0.17, 0.14, 0.09, 0.07]	0.62
		rMPFC	k-medoids	[1-c, 25, 21, 26, 16, 1-p, 28, 19, 27, 21, 38,	[0.19, 0.13, 0.09, 0.09, 0.08, 0.07, 0.07,	0.60
				9-c, 14]	0.05, 0.04, 0.03, 0.03, 0.02, 0.02]
		lSTS	mi-rank	[1-c]	[1.0]	0.68
	HRI	rTPJ	k-medoids	[1-c, 35, 1-p, 16, 12-p,11-p, 9-p, 29]	0.02] [0.62, 0.15, 0.06, 0.05, 0.03, 0.02, 0.02,	0.62
		rMPFC	modelSelect	[37, 19, 21, 26, 18, 25, 26]	[0.22, 0.21, 0.12, 0.06, 0.06, 0.06, 0.05]	0.62

https://www.furhatrobotics.com;[START_REF] Al Moubayed | Furhat: A Back-Projected Human-Like Robot Head for Multiparty Human-Machine Interaction[END_REF]

This research is supported by grants ANR-16-CONV-0002 (ILCB), ANR-11-LABX-0036 (BLRI) and AAP-ID-17-46-170301-11.1 by the Aix-Marseille Université Excellence Initiative (A*MIDEX).


Marc Joyeux
email: marc.joyeux@univ-grenoble-alpes.fr

Ivan Junier

Requirements for DNA-bridging proteins to act as topological barriers of the bacterial genome

INTRODUCTION

The genetic information of most bacteria is encoded in a circular DNA molecule comprising up to several millions of base pairs (bp). Such closed molecules are topologically constrained, because they lack free ends capable of rotating and releasing the torsional stress (1). In vivo alterations of the torsional state of circular DNA is mediated by the recruitment of enzymes called topoisomerases, which can either increase or decrease the twist of the double helix by opening transiently one or two strands (1). The net result of the action of all types of topoisomerases is that the genomic DNA of most bacteria is significantly undertwisted (negatively supercoiled) and winds about itself to transfer part of the torsional stress to the bending degrees of freedom, thereby forming plectonemes (1). If the DNA were not subject to any additional constraint beyond closure, then one single nick of one strand or one single break of the two strands would suffice to release the torsional stress of the full molecule. It has however long be known that at least several tens of nicks are required to achieve this goal in Escherichia coli (2,3). This indicates that there exist barriers which block the diffusion of the torsional stress along the DNA molecule and organize the chromosome of E. coli into many independent topological domains, whose torsional state is not affected by the relaxation of other domains. It is currently estimated that the genomic DNA of bacteria like E. coli is composed of several hundreds of different topological domains with variable size (average size ≈10 kbp) and position (4). More generally, the partitioning of the bacterial chromosome into ≈10 kbp independent topological domains is believed to hold in most bacteria (5) and to be related, in part, to the insulation of fundamental co-expression units that are neighbors along the genome (6).

The work reported here deals with the topological barriers which make this partitioning possible, that is, the barriers which block the diffusion of torsional stress along the DNA molecule and are responsible for its division into independent topological domains.

Although this question has now been addressed for nearly two decades (7)[START_REF] Thanbichler | The structure and function of the bacterial chromosome[END_REF][START_REF] Moulin | Topological insulators inhibit diffusion of transcription-induced positive supercoils in the chromosome of Escherichia coli[END_REF], the mechanisms underlying the formation of topological barriers remain mostly elusive. Most plausible models for the formation of topological barriers are (i) actively transcribing RNAP, which generate both positive and negative supercoils [START_REF] Liu | Supercoiling of the DNA template during transcription[END_REF] and may consequently block the displacement and dissipation of plectonemes [START_REF] Deng | Transcription-induced barriers to supercoil diffusion in the Salmonella typhimurium chromosome[END_REF][START_REF] Deng | Organization of supercoil domains and their reorganization by transcription[END_REF], and (ii) formation of DNA loops by certain nucleoid proteins, which may serve as topological barriers [START_REF] Saldanha | Recombination by resolvase is inhibited by lac repressor simultaneously binding operators between res sites[END_REF][START_REF] Wu | DNA looping alters local DNA conformation during transcription[END_REF][START_REF] Leng | Dividing a supercoiled DNA molecule into two independent topological domains[END_REF][START_REF] Yan | Protein-mediated loops in supercoiled DNA create large topological domains[END_REF]). In the present paper, we focus on this latter mechanism and establish under which conditions DNA-bridging proteins may serve as topological barriers.

Several DNA-bridging proteins have been studied in some detail, including transcription regulators like the LacI repressor [START_REF] Müller-Hill | Some repressors of bacterial transcription[END_REF], H-NS [START_REF] Dame | H-NS mediated compaction of DNA visualised by atomic force microscopy[END_REF] and H-NS-like proteins [START_REF] Bertin | The structural and functional organization of H-NS-like proteins is evolutionarily conserved in Gram-negative bacteria[END_REF], Lsr2 [START_REF] Chen | Lsr2 of Mycobacterium tuberculosis is a DNA-bridging protein[END_REF], Fis [START_REF] Skoko | Mechanism of chromosome compaction and looping by the Escherichia coli nucleoid protein Fis[END_REF] and Lrp [START_REF] Peterson | The role of high affinity nonspecific DNA binding by Lrp in transcriptional regulation and DNA organization[END_REF]. In their functional form, all of these proteins have at least two independent DNA-binding domains, so that they can interact with two DNA duplexes simultaneously and form a bridge between two sites that are widely separated from the genomic point of view. Most of these proteins also bind non-specifically but with high affinity to the DNA molecule. However, among all these DNA-bridging proteins, LacI is to date the only one that has been proved to work as a topological barrier [START_REF] Leng | Dividing a supercoiled DNA molecule into two independent topological domains[END_REF][START_REF] Fulcrand | DNA supercoiling, a critical signal regulating the basal expression of the lac operon in Escherichia coli[END_REF][START_REF] Fulcrand | Direct observation of a 91 bp LacI-mediated, negatively supercoiled DNA loop by atomic force microscope[END_REF].

The guiding line of the present paper is consequently the experimental demonstration in [START_REF] Leng | Dividing a supercoiled DNA molecule into two independent topological domains[END_REF] that the binding of a DNA-binding protein to its recognition sites in two different locations on a supercoiled DNA molecule can confine free supercoils to a defined region and divide the DNA molecule into two distinct topological domains [START_REF] Leng | Dividing a supercoiled DNA molecule into two independent topological domains[END_REF]. In order to gain more detailed information on this mechanism, we determined the minimal physical properties that must be added to a standard model of circular DNA to reproduce the results described in [START_REF] Leng | Dividing a supercoiled DNA molecule into two independent topological domains[END_REF].

The starting model for this study was similar to those we proposed recently to investigate facilitated diffusion [START_REF] Florescu | Description of non-specific DNA-protein interaction and facilitated diffusion with a dynamical model[END_REF][START_REF] Florescu | Dynamical model of DNA protein interaction: effect of protein charge distribution and mechanical properties[END_REF][START_REF] Florescu | Comparison of kinetic and dynamical models of DNA-protein interaction and facilitated diffusion[END_REF], the interactions of DNA and H-NS nucleoid proteins [START_REF] Joyeux | A model of H-NS mediated compaction of bacterial DNA[END_REF][START_REF] Joyeux | Equilibration of complexes of DNA and H-NS proteins on charged surfaces: A coarse-grained model point of view[END_REF][START_REF] Joyeux | Role of salt valency in the switch of H-NS proteins between DNAbridging and DNA-stiffening modes[END_REF], the formation of the bacterial nucleoid [START_REF] Joyeux | Compaction of bacterial genomic DNA: Clarifying the concepts[END_REF][START_REF] Joyeux | In vivo compaction dynamics of bacterial DNA: A fingerprint of DNA/RNA demixing ?[END_REF][START_REF] Joyeux | Coarse-grained model of the demixing of DNA and non-binding globular macromolecules[END_REF][START_REF] Joyeux | A segregative phase separation scenario of the formation of the bacterial nucleoid[END_REF][START_REF] Joyeux | Preferential localization of the bacterial nucleoid[END_REF], and the interplay of DNA demixing and supercoiling in compacting the nucleoid [START_REF] Joyeux | Bacterial nucleoid: Interplay of DNA demixing and supercoiling[END_REF].

The results presented in this article reveal that the formation of DNA loops is by no means sufficient to create topologically independent domains and that DNA-bridging proteins must exert rather strong constraints on their binding sites in order to act as topological barriers: They must block the diffusion of the excess of twist through both binding sites and must additionally block the rotation of one DNA segment relative to the other one.

METHODS

The coarse-grained bead-and-spring model developed for the present study is described in detail in Model and Simulations in the Supporting Material. In brief, bacterial DNA is modeled, as in [START_REF] Joyeux | Role of salt valency in the switch of H-NS proteins between DNAbridging and DNA-stiffening modes[END_REF][START_REF] Joyeux | Bacterial nucleoid: Interplay of DNA demixing and supercoiling[END_REF] ) was chosen so that the model reproduces the known persistence length of double-stranded DNA ( 50 = ξ nm).

As in [START_REF] Joyeux | Compaction of bacterial genomic DNA: Clarifying the concepts[END_REF][START_REF] Joyeux | Bacterial nucleoid: Interplay of DNA demixing and supercoiling[END_REF], the torsional energy term was borrowed from [START_REF] Chirico | Kinetics of DNA supercoiling studied by Brownian dynamics simulation[END_REF] and requires the introduction of a body-fixed frame ( , , )

k k k u f v ,
where k u denotes the unit vector pointing from bead k to bead 1 k + . The torsional rigidity opposing rotation of ( , , )

k k k u f v around k u ( B 25 k T τ =
) was adjusted so that at equilibrium the writhe contribution accounts for approximately 70% of the linking number difference [START_REF] Boles | Structure of plectonemically supercoiled DNA[END_REF], as is illustrated in Fig. S1. The values of the bending and torsional rigidities are close together, in agreement with experimental results [START_REF] Boles | Structure of plectonemically supercoiled DNA[END_REF]. It may be worth emphasizing that introduction of the body-fixed frame ( , , )

k k k
u f v is crucial for modeling correctly the torsion of double-stranded DNA with a "single-stranded" bead-and-spring model. This procedure is quite realistic and has already proved very useful for studying, for example, the interplay of DNA demixing and supercoiling in nucleoid compaction [START_REF] Joyeux | Bacterial nucleoid: Interplay of DNA demixing and supercoiling[END_REF], the buckling transition in double-stranded DNA and RNA [START_REF] Ott | Dynamics of the buckling transition in double-stranded DNA and RNA[END_REF], the influence of nucleoid-associated proteins on DNA supercoiling [START_REF] Dahlke | Influence of nucleoid-associated proteins on DNA supercoiling[END_REF], DNA supercoil relaxation [START_REF] Ivenso | Simulation of DNA supercoil relaxation[END_REF], the competition between B-Z and B-L transitions in a single DNA molecule [START_REF] Kwon | Competition between B-Z and B-L transitions in a single DNA molecule: Computational studies[END_REF], the relaxation of DNA supercoils by topoisomerase IB [START_REF] Lillian | A multiscale dynamic model of DNA supercoil relaxation by topoisomerase IB[END_REF], the sequencedependent duplex denaturation in superhelical DNA [START_REF] Mielke | Brownian dynamics simulations of sequence-dependent duplex denaturation in dynamically superhelical DNA[END_REF], transcription-driven twin supercoiling of DNA loops [START_REF] Mielke | Transcription-driven twin supercoiling of a DNA loop: A Brownian dynamics study[END_REF], site juxtaposition in supercoiled DNA [START_REF] Huang | Dynamics of site juxtaposition in supercoiled DNA[END_REF], and salt effects on the structure and internal dynamics of superhelical DNA [START_REF] Hammermann | Salt effects on the structure and internal dynamics of superhelical DNAs studied by light scattering and Brownian dynamics[END_REF].

Electrostatic repulsion between DNA beads is written as a sum of Debye-Hückel terms, which depend on effective electrostatic charges placed at the center of each bead. The values of these charges ( 3.52

q e ≈ -
, where e is the absolute charge of the electron) were derived from the net linear charge density along a DNA molecule immersed in a buffer with monovalent cations according to Manning's counterion condensation theory [START_REF] Manning | Limiting laws and counterion condensation in polyelectrolyte solutions. I. Colligative properties[END_REF][START_REF] Oosawa | Polyelectrolytes[END_REF]. The

RESULTS AND DISCUSSION

As a starting point, we recall the experimental results in [START_REF] Leng | Dividing a supercoiled DNA molecule into two independent topological domains[END_REF], where the torsional relaxation properties of 4100 bp plasmids with initial superhelical density 0.06 σ ≈ -( k 24 L ∆ ≈ -) were thoroughly investigated. Of special interest to us were the experiments performed with plasmids containing tandem copies of the binding site of the sequencespecific DNA-binding protein LacI at two different locations, such that the plasmid was divided into two stable loops of respective length 2900 bp and 1200 bp upon binding of LacI.

The 1200 bp region moreover contained the recognition site for one nicking enzyme (Nt.BbvC1) and the 2900 bp region the recognition site for a second nicking enzyme (Nb.BtsI). In the absence of LacI, addition of Nt.BbvC1 alone or Nb.BtsI alone was sufficient to release the full torsional stress of the plasmid. In contrast, in the presence of LacI, addition of Nt.BbvC1 alone removed only the 7 negative supercoils of the 1200 bp region, while addition of Nb.BtsI alone removed only the 17 negative supercoils of the 2900 bp region [START_REF] Leng | Dividing a supercoiled DNA molecule into two independent topological domains[END_REF]. Simultaneous addition of both enzymes was required to relax the full plasmid [START_REF] Leng | Dividing a supercoiled DNA molecule into two independent topological domains[END_REF]. This experiment demonstrates very clearly that pairs of LacI bridges block the diffusion of torsional stress and divide the plasmids into two independent topological domains.

We report below on our efforts to determine the minimal set of mandatory properties that allow molecular bridges to act as topological barriers and block the diffusion of torsional stress in out-of-equilibrium supercoiled chains.

Preparation of torsionally out-of-equilibrium circular chains.

Topological barriers are objects that block the diffusion of torsional stress and are able to maintain different torsional conditions on their left and right sides. Characterization of the underlying mechanism requires prior understanding of how torsional stress diffuses in out-ofequilibrium supercoiled chains.

β α = +
, which were chosen so that the distance between their centers is smaller than 10 nm. This step is illustrated in Fig. 1

T T T W W W W L L L W ∆ = ∆ + ∆ = + + ∆ ∆ + ∆ + (1)
where 12 r W quantifies the winding of one moiety of the chain around the other one, that is, loosely speaking, their intrication. The out-of-equilibrium chains prepared as described above

satisfy 12 r 0 W ≈ .
The main purpose of this work is to understand how DNA-bridging proteins like LacI [START_REF] Leng | Dividing a supercoiled DNA molecule into two independent topological domains[END_REF][START_REF] Fulcrand | DNA supercoiling, a critical signal regulating the basal expression of the lac operon in Escherichia coli[END_REF][START_REF] Fulcrand | Direct observation of a 91 bp LacI-mediated, negatively supercoiled DNA loop by atomic force microscope[END_REF] can maintain such an unbalance when the whole chain is relaxed. Since DNAbridging proteins form DNA loops by dynamically cross-linking widely separated DNA sites, the question amounts here to determine the constraints that must be imposed to beads α and β to divide the circular DNA chain into two topologically independent loops. Note that DNAbridging proteins are not introduced explicitly in the model: Rather, constraints on beads α and β are meant to model the mechanisms that allow the proteins to act as topological barriers.

Twist equilibrates much more rapidly than writhe.

In order to understand in more detail how the diffusion of torsional stress proceeds, the out-of-equilibrium circular chains prepared as described above were allowed to relax without constraint and all relevant quantities were monitored. From a practical point of view, four different initial out-of-equilibrium conformations were prepared for each value of 0 k L ∆ and 100 different relaxation trajectories were integrated for 1 ms for each initial conformation.

The evolution of k

j L ∆ , w j T ∆
, and r j W ( 1 j = , 2) was then averaged over these 100 trajectories. In the absence of any constraint on beads α and β, all conformations obtained at the end of the 1 ms integration time satisfy

1 2 k k L L ∆ ≈ ∆ and 1 2 w w T T ∆ ≈ ∆ , meaning that
torsional stress is distributed homogeneously in the whole chain. Information on the dynamics of the relaxation is provided by the examination of the time evolution of

T W t T T b t W W A B τ τ   ∆ -∆ ≈ -       - ≈ + -     (2)
where w T τ and r W τ are the characteristic time constants for the equilibration of the excess of twist and the writhe, respectively. Let us stress here that we are not aware of any reason why At equilibrium, the linking number difference decomposes into approximately 70% of writhe and 30% of excess of twist (see Fig. S1), but equilibration of the excess of twist is much faster than equilibration of the writhe. After a couple of microseconds, the linking number difference in the initially relaxed moiety of the chain is therefore composed of Several comments are in order here. First, the faster equilibration of twist compared to writhe for closed molecules or filaments with both torsional and bending energy (and comparable torsional and bending force constants) is well-known in the field of DNA biophysics. Yet, there are relatively few papers which discuss this point explicitly and provide estimates for the characteristic relaxation time scales. We note, however, that the theoretical study in [START_REF] Maggs | Twist and writhe dynamics of stiff polymers[END_REF] already pointed out that twist and writhe equilibrate on very different time scales. Second, it is emphasized that time scales deduced from coarse-grained models always underestimate real time scales by two to three orders of magnitudes, because coarse-grained models neglect many details which systematically slow down the dynamics. This point should of course be kept in mind when analyzing the results presented here and comparing them to experimental results. Third, we tested the importance of rotational noise (the last term in the right-hand side of Eq. (S9)) by launching simulations where this term was not taken into account. Characteristic equilibration times obtained as described above are shown in Fig. S3.

0 0 0.5 0.3 0.5 k 0.075 k L L × × × ∆ = ∆ excess

Comparison of Figs. 2 and S3

indicates that results obtained with and without rotational noise are identical within computational uncertainties. Rotational noise was consequently ignored in all subsequent simulations.

Closing a DNA loop does NOT suffice to create an independent topological domain.

Based on recent experimental results [START_REF] Deng | Organization of supercoil domains and their reorganization by transcription[END_REF][START_REF] Saldanha | Recombination by resolvase is inhibited by lac repressor simultaneously binding operators between res sites[END_REF][START_REF] Wu | DNA looping alters local DNA conformation during transcription[END_REF][START_REF] Leng | Dividing a supercoiled DNA molecule into two independent topological domains[END_REF][START_REF] Yan | Protein-mediated loops in supercoiled DNA create large topological domains[END_REF], one may wonder whether DNA loops are ipso facto independent topological domains, or whether some additional, more subtle properties of DNA-bridging proteins are required to achieve this goal. We emphasize that the word "loop" is used here according to its biological meaning and not the geometrical one.

More precisely, it does not refer to a closed curve, but rather to a conformation of the DNA chain, where two DNA sites that are widely separated from the genomic point of view are maintained close to each other by some other macromolecule, so that the segment of DNA located between these two sites looks like a loop when it is observed from a long distance or projected on a plane. However, the chain is not closed and torsional stress cannot diffuse directly from one extremity of the loop to the other one through the bridging macromolecule.

In order to ascertain this point, we launched a series of simulations with the crudest model for DNA-bridging proteins, namely a simple bond between beads α and β. The potential energy of the system was written as pot

BP E E +
, where pot E is the energy of the naked circular DNA chain described in Eqs. (S1)-(S5) and BP E an additional term, which models the action of DNA-bridging proteins. For the simplest model, we used

0 2 BP ( ) 2 h E d d αβ αβ = - , (3)
where d αβ is the distance between the centers of beads α and β and 0 4 d αβ = nm. For the sake of simplicity, we used the same value of the stretching rigidity as for the DNA chain, that is ).

Blocking diffusion of the twist is NOT sufficient to create independent topological domains

The reason why forming a loop is not sufficient to create independent topological domains is of course that the bond between beads α and β described in the previous subsection is not able to block the rapid diffusion of the excess of twist through beads α and β

and the subsequent equilibration of the writhe. In order to form independent topological domains, the diffusion of the excess of twist must absolutely be blocked. Stated in other words, rotation of the internal basis ( , , )

k k k u f v around k u must be forbidden at , k α β = .
From the biological point of view, diffusion of the excess of twist is probably blocked

tan (

) .

k k k α β α β δ - = - f r v r , (4)
( ,

k α β = ),
k k k k k k k k k k δ δ δ δ → + → - + f f v v f v (5)
for , k α β = . We note that the k δ remain small, and the corrections in Eq. ( 5 π . In this second model, the action of DNA-bridging proteins is consequently modeled by the potential energy terms in Eq. ( 6), plus the small, a posteriori corrections in Eq. ( 5).

The relaxation dynamics for this second model is much slower than for the first one, so that each trajectory had to be integrated for as long as 30 ms. Four different relaxation trajectories were integrated for each initial conformation, and four different initial conformations were again used for each value of (S17). In particular, r W does not display any sharp jump along single trajectories.

As can be checked in Fig. 4, the second model of DNA-bridging proteins succeeds in preventing equilibration of the twist and the writhe in the two moieties of the chain. Indeed, ) superimpose at short times, which indicates that the relaxation speed is proportional to the total amount of torsional stress. However, for longer times and/or larger values of the linking number difference, a slower regime sets in, which is probably due to the resistance opposed by plectonemes against the reorganization of the chain and the intrication of the two moieties. Finally, we note that after full relaxation the writhe contribution accounts for approximately 90% of the linking number difference, which is significantly larger than the 70% contribution in the absence of domains (see Fig. S1).

In conclusion, this second model of DNA-bridging proteins indicates that blocking the diffusion of the excess of twist at the two DNA loci bound by the proteins is sufficient to maintain a constant difference between the linking number difference, excess of twist, and writhe of the two moieties, but that it is not sufficient to insure true topological independence.

Indeed, the DNA chain evolves towards very compact and intricated final conformations, where the topology of each moiety is different from the initial one. The relaxation limit

Topological barriers must block diffusion of the twist AND the relative orientation of the DNA segments

The question that remains to be answered is consequently: What additional constraint must the DNA-bridging protein exert on its binding sites in order to preserve not only the difference in torsional stress between the two moieties of the DNA chain, but rather the precise initial torsional stress in each moiety ? The feeling that emerges upon examination of conformations like the one shown in Fig. 3(b), is that the winding of the initially relaxed moiety around the initially stressed one probably requires the relative rotation of the two DNA segments around the αβ bond, and that no winding would be possible if this rotation were blocked. In order to check this conjecture, a third bending term was added to BP E , relaxed moiety and one torsionally stressed moiety, as can be checked in Fig. 3(c). This third model of DNA-bridging proteins finally describes a topological barrier. In order to act as topological barriers, DNA-bridging proteins must consequently exert rather strong constraints on their binding sites: Indeed, they must block the diffusion of the excess of twist through both binding sites and must additionally block the rotation of one segment relative to the other one. It is worth emphasizing that this last condition is probably not satisfied by all DNAbridging proteins. For example, it is well known that H-NS dimers bind most often nonspecifically to the DNA, which means that the two C-terminal DNA-binding domains [START_REF] Gordon | Structural basis for recognition of AT-rich DNA by unrelated xenogeneic silencing proteins[END_REF] located at the end of flexible linkers [START_REF] Dorman | Domain organization and oligomerization among H-NS like nucleoid-associated proteins in bacteria[END_REF] [START_REF] Yan | Protein-mediated loops in supercoiled DNA create large topological domains[END_REF]. In this context, we surmise that the slow relaxation observed in [START_REF] Leng | Dividing a supercoiled DNA molecule into two independent topological domains[END_REF] for single DNA-bridges may correspond to the slow intrication obtained with our second model for DNA-bridging proteins, while the more stable topological separation observed in [START_REF] Leng | Dividing a supercoiled DNA molecule into two independent topological domains[END_REF] for bridges in tandem corresponds to true barriers obtained with the third model for DNA-bridging proteins. We finally note that the intrication properties discussed here are also expected to contribute to the formation of topological domains that are, in effect, larger than the domain delineated by the binding sites of LacI as recently observed in [START_REF] Yan | Protein-mediated loops in supercoiled DNA create large topological domains[END_REF].

namely 0 2 2 2 0 2 BP , , 5 5 5 (
) ( ) ( ) ( ) 2 2 2 2 2 2 h g g g E d d αβ αβ α β α β α β π π ξ ξ ψ ψ = - + - + - + - , (7) where

CONCLUSIONS

DNA-bridging proteins form DNA loops by dynamically cross-linking DNA sites that are widely separated from the genomic point of view [START_REF] Müller-Hill | Some repressors of bacterial transcription[END_REF][START_REF] Dame | H-NS mediated compaction of DNA visualised by atomic force microscopy[END_REF][START_REF] Bertin | The structural and functional organization of H-NS-like proteins is evolutionarily conserved in Gram-negative bacteria[END_REF][START_REF] Chen | Lsr2 of Mycobacterium tuberculosis is a DNA-bridging protein[END_REF][START_REF] Skoko | Mechanism of chromosome compaction and looping by the Escherichia coli nucleoid protein Fis[END_REF][START_REF] Peterson | The role of high affinity nonspecific DNA binding by Lrp in transcriptional regulation and DNA organization[END_REF]. They usually bind DNA nonspecifically but with high affinity. DNA-bridging proteins have now been investigated for more than two decades, but it is only very recently that it has been demonstrated that a DNAbridging protein like the LacI repressor is able to separate circular plasmids into two topologically independent loops [START_REF] Leng | Dividing a supercoiled DNA molecule into two independent topological domains[END_REF][START_REF] Fulcrand | DNA supercoiling, a critical signal regulating the basal expression of the lac operon in Escherichia coli[END_REF][START_REF] Fulcrand | Direct observation of a 91 bp LacI-mediated, negatively supercoiled DNA loop by atomic force microscope[END_REF]. To the best of our knowledge, this is at present the only demonstrated case of a DNA-bridging protein acting as a topological barrier. In the present work, we aimed at going one step further and determining the conditions that DNAbridging proteins must fulfill to act as topological barriers. We showed that these proteins must exert rather strong constraints on their binding sites: They must block the diffusion of the excess of twist through the two binding sites on the DNA molecule and must simultaneously prevent the rotation of one DNA segment relative to the other one. These criterions should be of great help when investigating the (existence, or lack of) topological barrier properties of other DNA-bridging proteins, like for example H-NS, which has been predicted to participate in the organization of the genome into topologically independent domains (6,[START_REF] Vora | Protein occupancy landscape of a bacterial genome[END_REF]. Indeed, it is probable that not all DNA-bridging proteins satisfy the second condition. If it is not satisfied, then intricate conformations are obtained, with a preserved difference in the torsional stress between the two domains, but different values of the stress in each domain. A rationale for the intriguing limit 12 r r

W W = (or equivalently 1 2 r r 0 W W + = ) is however missing.
This work also brings some light on another mechanism that has been proposed to explain the formation of independent topological domains, namely two proteins wrapping the DNA in two different locations (Ref. [START_REF] Leng | Dividing a supercoiled DNA molecule into two independent topological domains[END_REF], Fig. 6). There is experimental evidence that certain wrapping proteins, like GalR (15), the λ O protein [START_REF] Leng | Dividing a supercoiled DNA molecule into two independent topological domains[END_REF], and Fis [START_REF] Dages | Fis protein forms DNA topological barriers to confine transcription-coupled DNA supercoiling in Escherichia coli[END_REF], are able to form topological barriers by blocking the diffusion of twist. However, we checked that the torsional stress equilibrates rapidly in the two moieties of the DNA chain when simulations are performed with 0 10 d αβ = nm instead of 4 nm, because equilibration of twist and writhe is mediated in this case by the DNA chain passing repeatedly between beads α and β. The fact that GalR and λ O proteins are able to separate the plasmid into two stable topological domains [START_REF] Leng | Dividing a supercoiled DNA molecule into two independent topological domains[END_REF] therefore suggests that the two wrapping proteins are located very close to each other or are bound by some impenetrable molecular complex. Moreover, their relative orientation remains probably constant, because the intrication mechanism would otherwise set in. Our work therefore clearly supports the hypothesis of the authors, who state that they cannot fully exclude the role of protein-protein interactions of λ O-DNA complexes and GalR-DNA complexes in the formation of the two distinct topological domains [START_REF] Leng | Dividing a supercoiled DNA molecule into two independent topological domains[END_REF].

Still, our work suggests that the conditions for topological separation may be met by molecular geometries that do not require that the boundaries of the topological domains be close to each other. For example, one can imagine a topological barrier formed of two different proteins, which both anchor the DNA molecule to the cell membrane. According to our work, the DNA segment located between the two anchoring proteins may be topologically independent, provided that each protein is able to block the diffusion of twist along the DNA molecule and that the proteins cannot rotate with respect to the membrane. If this latter condition is not satisfied, then some complex winding of the DNA around itself is likely to occur, like for DNA-bridging proteins.

Another model which is frequently proposed for the formation of topological barriers, namely actively transcribing RNAP [START_REF] Liu | Supercoiling of the DNA template during transcription[END_REF][START_REF] Deng | Transcription-induced barriers to supercoil diffusion in the Salmonella typhimurium chromosome[END_REF][START_REF] Deng | Organization of supercoil domains and their reorganization by transcription[END_REF], is more difficult to discuss in terms of the results described in this paper, because several time scales need be considered. Indeed, it is likely that the two parts of the DNA molecule located respectively upstream and downstream from the transcribing RNAP will tend to increase their intricacy, in order to reduce the torsional stress induced by the blocking of the diffusion of twist at the RNAP. However, the RNAP is itself moving forward along the DNA, so that the ability of transcribing RNAP to act as topological barriers probably depends on the relative speeds of the two mechanisms. More work is needed to clarify this point. Similarly, the case of DNA-bending nucleoid proteins, like HU [START_REF] Berger | Genes on a wire: The nucleoid-associated protein HU insulates transcription units in Escherichia coli[END_REF], certainly deserves further investigations.

Finally, we note with interest that the intricate structure obtained with DNA-bridging proteins that block the diffusion of twist along the DNA but do not prevent the relative rotation of the DNA segments are highly compact. It turns out that the mechanism leading to the formation of the bacterial nucleoid is a longstanding but still lively debated question [START_REF] Joyeux | Compaction of bacterial genomic DNA: Clarifying the concepts[END_REF][START_REF] Joyeux | In vivo compaction dynamics of bacterial DNA: A fingerprint of DNA/RNA demixing ?[END_REF][START_REF] Zimmerman | Shape and compaction of Escherichia coli nucleoids[END_REF][START_REF] De Vries | DNA condensation in bacteria: Interplay between macromolecular crowding and nucleoid proteins[END_REF][START_REF] Benza | Physical descriptions of the bacterial nucleoid at large scales, and their biological implications[END_REF][START_REF] Lagomarsino | From structure to function of bacterial chromosomes: Evolutionary perspectives and ideas for new experiments[END_REF]. The point that has puzzled scientists for decades is that the volume of the unconstrained genomic DNA of bacteria in physiological solution, as estimated for example from the worm-like chain model [START_REF] Teraoka | Polymer solutions: An introduction to physical properties[END_REF], is approximately thousand times larger than the volume of the cell, while the nucleoid often occupies only a small fraction of the cell [START_REF] Woldringh | Molecular cytology of Escherichia coli[END_REF]. It would certainly be interesting to check whether the intrication resulting from the alternation of torsionally stressed and torsionally relaxed DNA domains could contribute significantly to the compaction of the nucleoid. (2) against the time evolution of (5) and ( 6)). Each curve was averaged over the 16 different trajectories with the same value of

E V V V V = + + + , (S1)
where

2 s 0 1 ( ) 2 n k k h V l l = = - ∑ (S2) 2 b 1 2 n k k g V θ = = ∑ (S3) 2 t 1 1 ( ) 2 n k k k V τ + = = Φ -Φ ∑ (S4) 2 2 e 1 2
( 2 )

n n k K k K k V q H a - = = + = - - ∑ ∑ r r (S5)
describe the stretching, bending, torsional, and electrostatic energy of the DNA chain, respectively. In Eqs. (S2)-(S5), k r denotes the position of bead k (with the convention that

n k k + = r r ), 1 k k k l + = - r
r the distance between two successive beads k and 1 k + , and The electrostatic energy of the DNA chain, e V , is expressed as a sum of repulsive Debye-Hückel terms with hard core. Function ( ) H r is defined according to where e is the absolute charge of the electron and 0.7 B = ℓ nm the Bjerrum length of water.

1 2 1 1 2 1 arccos(( )( ) / ( )) k k k k k k k k k θ + + + + + + = - - - - r
1 ( ) exp 4 D r H r r r πε   = -     , ( S6
In Eq. (S7), / B e ℓ is the net linear charge density along a DNA molecule immersed in a buffer with monovalent cations derived from Manning's counterion condensation theory (5,6). Note that electrostatic interactions between nearest neighbours are not included in Eq. (S5) because it is considered that they are already accounted for in the stretching and bending terms.

The dynamics of the system was investigated by integrating numerically overdamped Langevin equations. Practically, the updated positions and torsion angles at time step i+1 are computed from the positions and torsion angles at time step i according to The excess of twist w T ∆ , the writhe r W , and the linking number difference k L ∆ of the DNA chain were computed at regular time intervals according to (3) Finally, the global torsional state of each moiety can be described by the value of the partial linking number difference 5) and ( 7)). Each curve was averaged over 2 different trajectories starting from the same initial conformation.

( 1) ( ) ( ) ( ) 2 6 6 i i i i B k k k k k T t t x a a πη πη + ∆ ∆ = + + r r f (S8) ( 1) ( ) ( ) ( ) ( ) ( ) 1 1 2 2 0 0 2 ( 2 ) 4 4 i i i i i i B k k k k k k k T t t X a l a l τ πη πη + + - ∆ ∆ Φ = Φ + Φ -Φ + Φ + , ( S9
1 1 1 w= ( ) 2 n k k k T π + = ∆ Φ -Φ ∑ (S10) [ ] 1 1 3 1 1 ( ) ( ) .( ) 1 r 4 n n k k K K k K k K k K K k W π + + = = ≠ -× - - = - ∑ ∑
1 1 1 k = w r L T W ∆ ∆ + (S18)

  to a concentration of monovalent salt of 100 mM, which is the value that is generally assumed for the cytoplasm of bacterial cells. The torsional state of the DNA chain is characterized by the value of the of the number of base pairs of the DNA chain and the mean number of base pairs per turn of the torsionally relaxed double helix, and the linking number difference k= w+ r L T W ∆ ∆ is the sum of the excess of twist ( w T ∆ ) and the writhe ( r W ), which quantifies the winding of the DNA axis around itself. k L ∆ and σ are constants of the motion, while w T ∆ and r W fluctuate under the influence of thermal noise and external constraints. Bacterial DNA being negatively supercoiled (undertwisted), k L ∆ and σ are negative in this work. Investigated values of σ range from -0.033 to -0.121, that is approximately from the effective value for protein-bound DNA in living E. coli cells ( value generated by DNA gyrase and RNA polymerase (52-54) and corresponds to a range of superhelicity values where hyperplectonemes, a higher order DNA organization, have been observed (55,56). The properties of the model were determined by integrating numerically overdamped Langevin equations with time steps 10 t ∆ = ps. Temperature T was assumed to be 298 K throughout the study.

T τ and r W τ as a function of 0 σ-W.

 0 to an exponential law, that is, according to a process involving a single time scale, although such exponential decays might be rationalized by analyzing the relaxation dynamics of the DNA chain as a twist diffusion process[START_REF] Goldstein | Viscous nonlinear dynamics of twist and writhe[END_REF]. Eq. (2) should thus be considered here essentially as a convenient means to extract and compare characteristic equilibration time constants for the different models investigated in the present paper, that is the circular DNA chain without constraints on beads α and β discussed in the present subsection and the three different models of DNA-bridging proteins discussed in the following subsections. The evolution of w is shown in Fig.2. Each point in this figure was obtained from a fit of Eq. (2of the excess of twist between the two moieties of the circular chain is consequently rather fast, the reason being that it involves only some rotation of the chain around its own axis -more precisely, rotation of the internal basis ( around k u for all k. In contrast, Fig. 2 indicates that r W τ is typically one to two orders of magnitude larger than w T τ , which means that equilibration of the writhe is much slower than equilibration of the excess of twist. The reason for this longer time constant is that equilibration of the writhe requires global rearrangements of the whole chain, as can be checked in Fig. 1, instead of it mere rotation around its own axis. Moreover, the larger the difference in torsional stress between the two moieties, the larger the torsional force and torque exerted by each moiety on the other one, and the faster the motion of the beads involved in the rearrangement. This is probably the reason, why r Finally, the values of r W τ vary more broadly from one initial conformation to the other one for smaller values of 0 | | σ than for larger ones, as can be checked in Fig. 2.

  of twist and no writhe (0% writhe, 100% excess of twist), while in the initially stressed moiety it decomposes into twist (82% writhe, 18% excess of twist). As a consequence, during most part of the equilibration process the initially torsionally relaxed moiety of the chain has not enough writhe compared to the excess of twist, while the initially torsionally stressed moiety has too much writhe compared to the excess of twist. The need to equilibrate the writhe/twist ratio inside each moiety of the DNA chain may in turn accelerate the equilibration of writhe in the whole chain by favoring diffusion of the writhe from the initially stressed moiety to the initially relaxed one.

B 60 . 1 C is composed of beads α to 1 β

 6011 0 k T , for the second model discussed below, and obtained exactly the same results as with 0 4 d αβ = nm. This confirms that Eq. (2) describes an impenetrable αβ bond. The same initial conformations were used as in the preceding subsection and 100 different relaxation trajectories were again integrated for 1 ms for each initial conformation.Characteristic equilibration times obtained as described above are shown in Fig.S4.Comparison ofFig. S4 with Figs. 2 and S3 indicates that addition of a bond between beads α and β affects significantly neither w T τ nor r W τ . Stated in other words, not all DNA-bridgingproteins are topological barriers or, equivalently, not all DNA loops are independent topological domains. The ability of certain DNA-bridging proteins, like LacI, to act as topological barriers is consequently related to some property, which is subtler than the mere connection between two genetically widely separated DNA sites.Still, it is worth noting that the αβ bond has a significant effect on the time evolution of 12 r W , as can be checked in Fig.S5. The reason is the following. In order to compute the partial writhes r j W (see the Supporting Material), one has to introduce two "virtual" -, plus the "unphysical" segment between beads β and α, while 2 C is composed of beads β to 1 α -, plus the same "unphysical" segment between beads α and β. When beads α and β are bonded, they remain close to each other for 0 t > , as indicated by the positions of the two arrows in Fig.3(a), so that the unphysical segment between beads α and β remains small and 1 C is a good representation of the initially torsionally relaxed DNA loop (and 1 r W is a measure of its writhe),2 C is a good representation of the initially torsionally stressed DNA loop (and 2 r W is a measure of its writhe), and 12 r W is a measure of the intrication of the two loops. In this case, 12 r W remains close to zero at all times, as can be checked in Fig.S5(b). This indicates that, for bridged DNA, equilibration of the torsional stress is not accompanied by any significant intrication of the two circular moieties. In contrast, for the system without the BP E term, beads α and β do not remain close to each other for 0 t > , as indicated by the positions of the two arrows in Fig.1(c), so that the unphysical segment between beads α and β becomes arbitrarily large and 1 C , 2 C , the r j W and 12 r W lose their physical meaning. In this case 12 r W varies quite widely with changing DNA conformations, as is illustrated in Fig.S5(a) for an initial

  because the involved proteins bind to their recognition site by inserting loops in the major or minor grooves of the DNA duplex. For example, it has been shown that H-NS dimers insert one C-terminal loop inside the minor groove of double-stranded DNA[START_REF] Gordon | Structural basis for recognition of AT-rich DNA by unrelated xenogeneic silencing proteins[END_REF], thereby hindering rotation of the DNA around its own axis at this location. From the modeling point of view, diffusion of the twist can be blocked by imposing that α f and β f remain perpendicular to , is achieved by computing the angles , 1 , .

Fig. 4 .

 4 Fig. 4. At the level of single trajectories, the r j W , 12 r W and the k j L ∆ actually relax through a succession of plateaus followed by abrupt jumps, as is illustrated in Fig. S6 for 12 r W , while the time evolution is of course smoother when these quantities are averaged over several trajectories, as in Fig. 4(b). It is stressed that this behavior has no precise physical meaning but arises instead from the mathematical definition of the total writhe r W in Eq. (S11) and its somewhat arbitrary partitioning into the sum of the two partial quantities r j W and 12 r W in Eq.

  the integration interval. Nonetheless, this model still fails to describe topological barriersFig.4(b)). According to Eq. (1), this limit is equivalent to 12 intrication of the two moieties. Indeed, examination of the final conformations, like the one with Fig.3(b), reveals very compact structures, with the initially torsionally relaxed (green) moiety winding around the initially torsionally stressed (red) moiety. Clearly, the intrication of the two moieties is achieved by transforming part of the plectonemic supercoils of the red moiety into toroidal supercoils. Another general feature of the final conformations, is that the portion of the red moiety not involved in the winding with the green moiety has a compact geometry that probably results from the superposition of plectonemic and toroidal supercoils.More insight into the dynamics of the relaxation is gained by plotting the time linking number difference of the circular chain. The result is shown in Fig.5, where 12 r W has been averaged over the 16 relaxation trajectories that have been computed for each value of

  non trivial and intriguing to us. We suspect that the explanation involves subtle considerations that should deserve further work.

)

  figure, all quantities, linking numbers difference, excess of twist, and writhe, whether local or global, remain strictly constant along the trajectories. No relaxation actually ever occurs and conformations obtained after long integration times are still composed of one torsionally

Figure 1 :

 1 Figure 1 : (a) Representative conformation of an equilibrated circular chain with linking number difference

Figure 2 :

 2 Figure 2 : Evolution, as a function of

  starting from a single initial conformation (see Fig.S2). The four points shown for each value of 0 σ correspond to four different initial conformations. The dot-dashed lines are linear fits to the data and are provided as guidelines to the eyes and as a means of comparison with Figs. S3 and S4.

Figure 3 :

 3 Figure 3 : (a) Representative snapshot obtained upon relaxation of the out-of equilibrium conformation shown in Fig. 1(b) under the constraint that beads α and β are bonded (Eq. (3)). (b) Representative snapshot obtained upon relaxation of the out-of equilibrium conformation shown in Fig. 1(b) for the second model of DNA-bridging proteins (Eqs. (5) and (6)). (c)Representative snapshot obtained upon relaxation of the out-of equilibrium conformation shown in Fig.1(b) for the third model of DNA-bridging proteins (Eqs. (5) and (7)). The linking number difference of the three conformations

Figure 4 :

 4 Figure 4 : Time evolution of (a) the excess of twist, and (b) the writhe, during the relaxation of a torsionally out-of-equilibrium DNA conformation with

Figure 5 :

 5 Figure 5 : Time evolution of

FIGURE 5 Requirements

 5 FIGURE 5

  s V is a computational device without biological meaning, which is aimed at avoiding a rigid rod description. The stretching rigidity h was set to the variations of the distance between successive beads remain small enough (2).In contrast, the bending rigidity g in the expression of b V was obtained from the known persistence length of DNA, t V was borrowed from Ref. (3) and torsion forces and momenta were computed as described therein. writhe contribution r W accounts for approximately 70% of the linking number difference k L ∆ at equilibrium (4), see Fig. S1. The values of the bending and torsional rigidities of the model are close to each other, which agrees with experimental findings (4).

  buffer. This value of the Debye length corresponds to a concentration of monovalent salt of 100 mM, which is the value that is generally assumed for the cytoplasm of bacterial cells. q is the value of the electric charge, which is placed at the centre of each DNA bead

f

  are vectors of inter-particle forces arising from the potential energy pot E integration time step.

  of the number of base pairs of the DNA chain and the mean number of base pairs per turn of the torsionally relaxed double helix. Fig. S1 shows the plot of r / k W L ∆ , the mean contribution of the writhe to the linking number difference, as a function of σ-. This plot is in good agreement both with experimental results (4) and results obtained with another model(7).Partial values of the excess of twist and the writhe were also computed for each of the two moieties separated by beads α (to the total excess of twist

Figure S1 :

 S1 Figure S1 : Evolution of r / k W L ∆ , the mean contribution of the writhe to the linking number difference, as a function of σ -.

Figure S3 :

 S3 Figure S3 : Same as Fig. 2, except that rotational noise has been discarded from evolution equations Eq. (S9). Plots (a) and (b) show the evolution of w T τ and r W τ as a function of

Figure S4 :

 S4 Figure S4 : Same as Fig. S3, except that a bond has been added between beads α and β, as the crudest model for a DNA-bridging protein (see Eq. (3)). Plots (a) and (b) show the evolution of w T τ and r W τ as a function of

Figure, 2 r W , and 12 r

 212 Figure S5 : Time evolution of 1 r W ,

Figure S6 :

 S6 Figure S6 : Time evolution of 12 r W during the relaxation of a torsionally out-of-equilibrium DNA conformation with

Figure S7 :

 S7 Figure S7 : Time evolution of (a) the excess of twist, and (b) the writhe, during the relaxation of a torsionally out-of-equilibrium DNA conformation with 0 k 52 L ∆ = -( 0 0.121 σ ≈ -) constrained by the third model for DNA-bridging proteins (Eqs. (5) and (7)). Each curve was averaged over 2 different trajectories starting from the same initial conformation.

  To prepare such chains, we first let circular chains with

	n =	600	beads and different values of	∆	L	k	equilibrate for about 100 ms.	∆	L	k	was set to
	L ∆	k	0	14 = -, -20, -26, -34, -40, -46, and -52, corresponding to 0 σ ≈ -0.033	, -0.047,
	-0.061, -0.079, -0.093, -0.107, and -0.121, respectively. Equilibrated chains were then
	formally divided into two moieties of equal length on both sides of beads α (1	α ≤ ≤	n	/ 2	) and
						n	/ 2

  nm is small enough to ensure that no other DNA segment crosses the segment between α and β, which is formally occupied by the DNA bridging protein. The energetic barrier for any segment to cross this line is indeed about

	h =	100	k	B / T	l	2 0	(see Model and Simulation in the Supporting Material). The value 0 4 d αβ =

B 32.8 k T , which is virtually insuperable. For the peace of mind, we nevertheless ran several simulations with 0 3 d αβ = nm, corresponding to a barrier of about

  after each integration step and rotating k f and k v around k u by this angle k

		δ
	cos	sin
	sin	cos

  Note that the bending rigidity for these two angles was assumed to be five time larger than the bending rigidity of the DNA chain, in order for α

																											) consequently
	remain also small, only if , α β r is not perpendicular simultaneously to k f and k v , that is if
	vectors , α β r and k u (	, k α β =	) are not collinear. In order to prevent such collinearity, two
	bending terms were added to BP E , namely
	E	BP	=	2 h	(	d	αβ	-	d	0 2 ) αβ	+	5 2 g	( ξ α	-	2 π	)	2	+	5 2 g	( ξ	β	-	2 π	)	2	,	(6)
	where α ξ denotes the angle formed by beads β, α and	1 α + , and β ξ the angle formed by
	beads α, β and	1 β + . ξ and β ξ to deviate
	only moderately from / 2

  bind into the minor groove of the DNA (59) through generic electrostatic forces instead of a precise chemical bond. The consequence is that H-NS dimers can slide along the DNA molecule[START_REF] Joyeux | A model of H-NS mediated compaction of bacterial DNA[END_REF][START_REF] Joyeux | Equilibration of complexes of DNA and H-NS proteins on charged surfaces: A coarse-grained model point of view[END_REF], but they are also expected to rotate with respect to the DNA axis. Conversely, the two DNA segments can rotate with respect to the protein axis. Proteins binding specifically to the DNA are more likely to satisfy the norotation condition, provided that the bonds are strong enough to resist the torque exerted by the DNA. A second mechanism for satisfying the no-rotation condition is that the proteins systematically form pairs of neighboring bridges. In this respect, we note with interest that the authors of the experimental study about the topological barrier formed by LacI tetramers reported that the barrier formed by a single LacI tetramer is much less stable comparing with those containing multiple LacI-lac O1 nucleoprotein complexes[START_REF] Leng | Dividing a supercoiled DNA molecule into two independent topological domains[END_REF]. The most straightforward explanation for this result is that two bridges in tandem are required to sustain the torque arising from the difference in torsional stress in the two DNA loops. Yet, single molecule experiments have demonstrated that a single LacI bridge can sustain the torque

	associated with physiological values of supercoiling, that is	0.06 σ ≈ -

  W , respectively) according to equations similar to Eq. (S11). Note that 1 C and 2 Care "virtual" chains, because torsional stress cannot flow directly between beads α and β. W quantify the winding of each closed chain around itself. In contrast, 12 r W quantifies the winding of 1 C around 2 C , that is, loosely speaking, the intrication of the two loops when beads α and β remain close to each other.

	T ∆	w	T = ∆	w	1	T + ∆	w	2	.	(S16)
			The case of the writhe is more complex. Partial values were obtained by defining two
	closed chains of length	/ 2 n , namely	1 C ( , α α = r r r 1 , α +	+	2	,...,	2 r r r 1 , , ) β β α --	and
	2 C ( , β β = r r r 1 , β +	+	2	1 ,..., , ,..., n r r	2 r r r , and computing the writhe for 1 1 , , ) α α β --C and 2 C (labeled
	1 r W and 2 r
	Most importantly, r W is not necessarily close to the sum of 1 r W and	2 r W , because of the
	double sums in the expressions for r W , 1 r W , and 2 r W . Instead, one has
	1 r W W W W 2 r r r 12 = + +	.	(S17)
	r W , 1 r W , and 2 r

ACKNOWLEDGMENTS M. J. thanks Prof. C. Hegedus (Université Lumière Lyon 2) for her hospitality during the Covid-19 epidemic confinement. I. J. thanks B. Houchmandzadeh (CNRS and Université

Grenoble Alpes) for valuable discussions on the preliminary simulations.

SUPPORTING MATERIAL

Model and Simulations section. Figures S1 to S7.

SUPPORTING CITATIONS

References (70,71) appear in the Supporting Material.

AUTHOR CONTRIBUTIONS

I.J. imagined the research project and performed preliminary simulations. M.J. developed the models, ran the simulations and analyzed the results. M.J. and I.J. wrote the manuscript.