Physical models of bacterial chromosomes

The interplay between bacterial chromosome organization and functions such as transcription and replication can be studied in increasing detail using novel experimental techniques. Interpreting the resulting quantitative data, however, can be theoretically challenging. In this minireview, we discuss how connecting experimental observations to biophysical theory and modeling can give rise to new insights on bacterial chromosome organization. We consider three flavors of models of increasing complexity: simple polymer models that explore how physical constraints, such as confinement or plectoneme branching, can affect bacterial chromosome organization; bottom-up mechanistic models that connect these constraints to their underlying causes, for instance chromosome compaction to macromolecular crowding, or supercoiling to transcription; and finally, data-driven methods for inferring interpretable and quantitative models directly from complex experimental data. Using recent examples, we discuss how biophysical models can both deepen our understanding of how bacterial chromosomes are structured, and give rise to novel predictions about bacterial chromosome organization.


Introduction
The genome of many bacterial species is contained in a single circular chromosome, which is compressed by orders of magnitude into the bacterial cell.Microscopy studies have revealed that various factors, such as transcription, Nucleoid Associated Proteins (NAPs), supercoiling, loop-extrusion by SMC complexes, and replication shape bacterial chromosome organization (Dame et al., 2019;Lioy et al., 2021;Yáñez-Cuna and Koszul, 2023;Gogou et al., 2021).However, since microscopy experiments cannot resolve the full 3D conformation of a bacterial chromosome, the effects of genetic and pharmocological perturbations on chromosome organization are often only explored indirectly, for instance by observing how they affect chromosome compaction or segregation.Alternatively, chromosome organization can be studied using sequencing-based methods, including Chromosome Conformation Capture experiments such as Hi-C (Le et al., 2013).Hi-C experiments measure how often pairs of loci are spatially proximate, or "in contact", averaged over a population of cells.Although interpreting Hi-C data remains challenging, these and other high-resolution quantitative data open new avenues to address old, yet unanswered questions: how are bacterial chromosomes organized across scales?how do various biological mechanisms control this organization?and, how does chromosome organization facilitate biological functions?
The recent surge in experimental techniques to quantitatively probe bacterial chromosome organization poses new and exciting challenges for biophysical modeling.Theoretical and computational biophysical models use concepts from polymer physics to explore how different mechanisms, such as macromolecular crowding (Rivas and Minton, 2016;Polson and Kerry, 2018) or bridging by NAPs (Dame et al., 2020;Amemiya et al., 2021), can affect chromosome organization and dynamics.This minireview focuses on recent models for bacterial chromosome organization, grouped according to their underlying modeling approach, and ordered by increasing complexity.First, we discuss polymer models that study how geometrical and topological constraints affect bacterial chromosomes, such as how cellular confinement or polymer branching influence chromosome conformation and dynamics.Second, we explore bottom-up models, which study how chromosome organization emerges from microscopic mechanisms, like how loop-extrusion by SMC complexes (∼ 50 nm) (Fudenberg et al., 2017;Banigan and Mirny, 2020) can organize chromosomes at the nucleoid scale (∼ 1 µm).Finally, we consider data-driven approaches, which seek to infer a model for chromosome organization, given experimental data, providing a physical interpretation for Hi-C maps.We discuss various benefits and limitations of these different modeling approaches, explore future modeling opportunities and challenges, and summarize key insights into bacterial chromosome organization gained via biophysical modeling.

Geometric and topological constraints determine chromosome organization
In its essence, a bacterial chromosome is a long polymer confined to a small volume.The simplest physical models for bacterial chromosome organization explore how different, often biologically motivated, forces and constraints affect the polymer's organization and dynamics (Fig. 1A).

Geometric constraints
The volume of confinement and constraining loci within the cell are examples of geometric constraints on a bacterial chromosome.Even these minimal constraints can explain well-known features of bacterial chromosome organization: tethering the origin of replication to a cell pole can give rise to linear chromosome organization (Buenemann andLenz, 2010, 2011), as seen in species such as C. crescentus, while tight confinement can drastically change the way that polymers interact, with implications for bacterial chromosome segregation.(Jun and Mulder, 2006).Constraining chromosomal regions corresponding to E. coli macrodomains to subvolumes of the nucleoid can give rise to order and chromosome segregation (Junier et al., 2014).Fixed loop topologies can orient bacterial chromosomes and enhance their segregation (Mitra et al., 2022b).Feather-boa models show that branching can affect chromosome compaction, segregation, and dynamics.Constraining origins of replication or other loci by tethering them to the cell membrane can give rise to linear ordering of the chromosome (Buenemann andLenz, 2010, 2011).B Examples of bottom-up mechanisms.Left to right: Macromolecular crowding due to RNA, ribosomes, and other large molecules can compact bacterial chromosomes (Rivas and Minton, 2016).Loop-extrusion by MukBEF can contribute to macrodomain formation in E. coli (Lioy et al., 2018), whereas condensin can tie together the chromosomal arms in C. crescentus and B. subtilis (Le et al., 2013;Wang et al., 2017).NAPs can impact chromosome organization by condensing macrodomains, by bridging, by stabilizing supercoils, or by associating with the cell membrane (Dame et al., 2020;Amemiya et al., 2021).Transcription gives rise to twin-supercoiled-domains, with positive supercoiling ahead and negative supercoiling behind RNA polymerases (Junier et al., 2023).The insertion of newly translated proteins into the cell membrane (transertion) causes certain genes to associate with the cell wall (Roggiani and Goulian, 2015).Jun and Mulder (2006) proposed that bacterial chromosome segregation could be explained by entropic forces acting on two confined polymers.Consider two polymers consisting of N monomers, confined to a long cylinder of diameter d.Since below length scales d, each polymer is unaffected by the confinement, we can split each polymer into subsections called "confinement blobs", inside which the polymer is unconstrained.Each blob is constrained to lie along the long axis of the cylinder, which introduces an entropic cost.The entropic cost of confinement can hence be shown to scale with the number of confinement blobs, N blobs ∼ N d −1/ν , where ν is the Flory exponent.Since overlap of any two blobs is entropically costly, the two polymers will entropically segregate.Such arguments can be extended for ring polymers and for shorter cylinders (Jun and Wright, 2010;Jung et al., 2012), suggesting that replicated bacterial chromosomes could entropically segregate in cellular confinement.

D
Despite theoretical arguments for entropic segregation, several simulation studies have shown that, without additional constraints, circular chromosomes do no segregate at intermediate replication stages.Constraints such as the concentric-shell model (Jun and Mulder, 2006), confining sections of the chromosome to sub-volumes of the nucleoid (modeling Macrodomains of the E. coli chromosome) (Junier et al., 2014), fixing the replisomes at mid-cell (El Najjar et al., 2020), or linking chromosomal arms with loopextruders (Harju et al., 2023) have been necessary to achieve segregation concurrent with replication.Why are such additional constraints needed?We recently argued that at intermediate replication stages, purely entropic forces can ac-tually inhibit bacterial chromosome segregation by pushing replication forks apart (Harju et al., 2023).Additionally, free energy calculations have shown that the time delay before entropic segregation begins can grow exponentially with the chain length (Minina andArnold, 2014, 2015;Polson and Kerry, 2018), and that two polymers of different lengths do not necessarily demix in confinement (Polson and Zhu, 2021).Both due to partially conflicting simulation results and the lack of experimental evidence, the role of entropy in bacterial chromosome organization remains a subject of debate.Furthermore, we still lack theoretical understanding for chromosome segregation in spherically shaped cocci (Pinho et al., 2013), and in species with multiple chromosomes of different topologies (Ren et al., 2022).

Topological constraints
The shape of a bacterial chromosome is characterized by topological constants: a linear, a circular, and a partially replicated chromosome all have different numbers of loops (0, 1, or 2) and are hence topologically distinct.We will now discuss how topological changes can affect the organization and dynamics of bacterial chromosomes.Mitra et al. (2022a,b) recently proposed that fixed loops at the boundaries of E. coli Macrodomains (Fig. 1A) could be sufficient to explain experimentally observed chromosome organization and segregation patterns.The authors showed that fixed loop architectures give rise to predictable and robust orientation of confined chromosomes, and that excluded volume interactions between loops can enhance chromosome segregation.Although it remains to be shown whether such stable loops at fixed genomic positions are common in bacte-D R A F T 3.1 Loop extrusion ria, these findings also suggest that more randomly placed loops could affect the direction of entropic forces at the single-cell level.Another example of a topological constraint on a bacterial chromosome is its supercoiling level.To illustrate, consider holding the ends of a piece of ribbon, and twisting them in opposite directions.This causes the ribbon to writhe around its central axis.If you now release tension by bringing the ends of the ribbon closer together, the ribbon will coil up into a plectoneme, but the number of turns (the linking number) will be conserved.Similarly, supercoiling of bacterial chromosomes by active mechanisms causes the DNA to branch into plectonemes (Dorman, 2019;Junier et al., 2023).This observation motivated the development of "feather-boa" models, where the chromosome is considered to consist of loops or branches emanating from a backbone (Reviewed in Ha and Jung (2015)).Branching can both compact chromosomes and enhance their segregation (Jun and Wright, 2010).Additionally, featherboa models have been shown to reproduce experimentally observed features of bacterial chromosome organization and dynamics, such as subdiffusive motion of chromosomal loci (Yu et al., 2021), and helical ordering of chromosomal arms (Swain et al., 2019).Feather-boa models hence remain an active area of research, and new computational advances are improving simulation resolutions and speeds (Goodsell et al., 2018;Ghobadpour et al., 2021).Despite these computational advances, an open-standing theoretical question is how branches and loops should be (dynamically) distributed in bacterial chromosome models, given what we know about their underlying causes.

Bottom-up modeling
In this section, we focus on bottom-up models, which model how chromosome organization emerges from proposed biological mechanisms (Fig. 1B).Such models can describe how transcription gives rise to plectoneme branches, or how macromolecular crowding confines the nucleoid to only 40-90% of the cell (Gray et al., 2019).A strength of bottom-up models is that they provide mechanistic insight and make novel predictions.A limitation is that more complex aspects of chromosome organization may be affected by several distinct mechanisms acting in unison.To illustrate, recent work by Joyeux (2021Joyeux ( , 2023) ) has shown that crowding and supercoiling can compact the chromosome in non-additive ways at high supercoiling densities, and that macromolecular crowding can enhance chromosome compaction by crosslinkers.These works illustrate that different bacterial chromosome organization mechanisms do not act in isolation.Despite such challenges, bottom-up models can provide conceptual insight into how various molecular mechanism control bacterial chromosome organization.

Loop extrusion
Hi-C experiments have revealed that bacterial chromosomes are more ordered than homogeneous, randomly oriented polymers in confinement.For instance, bacterial condensin mediates long-range contacts between the two chromosomal arms in species such as C. crescentus (Le et al., 2013), and B. subtilis (Wang et al., 2017), resulting in a prominent offdiagonal trace on Hi-C maps.Another SMC, MukBEF, on the other hand, enhances long-range contacts across large parts of the E. coli chromosome (Lioy et al., 2018).These findings have triggered the development of loop extrusion models for bacterial chromosomes.
One of the early questions addressed by biophysical modeling of SMCs in bacteria was whether these protein complexes move diffusively or via active loop extrusion.Whereas it was suggested that targeted loading of diffusive slip-links could be sufficient to model eukaryotic SMC behavior (Brackley et al., 2017) and that MukBEF clustering could arise due to Turing patterning (Murray and Sourjik, 2017), a simulation model for B. subtilis (Miermans and Broedersz, 2018) indicated that thousands of diffusive slip-links were needed to explain off-diagonal traces on bacterial Hi-C maps, while only tens of active loop-extruders were sufficient, more consistent with experimental reports of ∼ 30 condensin complexes per chromosome (Wilhelm et al., 2015).These and other simulations, as well as mounting experimental evidence, have led to loop extrusion becoming more broadly accepted (Fudenberg et al., 2017;Banigan and Mirny, 2020).
MukBEF loop extrusion in E. coli has been modeled in 1D (Mäkelä and Sherratt, 2020).The authors proposed that non-targeted loading of MukBEF gives rise to an array of loops that spans most of the chromosome.Non-targeted loading of MukBEF could hence result in a feather-boa structure on parts of the E. coli chromosome, reminiscent of microscopy observations of MukBEF distributions in widened E. coli cells (Japaridze et al., 2023).However, future work still needs to address how loop extrusion by MukBEF could affect the 3D organization of E. coli chromosomes.
The effects of loop extrusion on B. subtilis chromosome organization, by contrast, have been modeled by combining 1D loop-extruder dynamics with 3D polymer simulations (Brandão et al., 2019(Brandão et al., , 2021)).These models better recapitulate Hi-C data if loop-extruders slow down as they collide with RNA polymerases at highly transcribed regions.Patterns on Hi-C maps for strains with two loop-extruder loading sites, on the other hand, can be explained if loop-extruders can traverse each other upon collision, as seen in vitro (Kim et al., 2020).
More recently, we modeled how loop-extruders loaded at the origins of replication affect the segregation and organization of replicating bacterial chromosomes (Harju et al., 2023).This so-called topo-entropic segregation model explains how the geometry and effective topology of a replicating chromosome affect the direction of entropic forces.We found that at intermediate replication stages, purely entropic forces inhibit bacterial chromosome segregation.However, loop-extruders loaded at the origins of replication effectively linearize partially replicated chromosomes, and this change in effective topology redirects entropic forces to enable concurrent replication and segregation.

NAPs and phase separation
In vitro studies have shown that NAPs can locally twist, bend or bridge DNA (Song and Loparo, 2015;Dame et al., 2020;Amemiya et al., 2021).This shows that NAPs can locally affect DNA structure, but what is their impact on bacterial chromosome organization at larger scales?We will first discuss long-range bridging, which introduces transient crosslinks on bacterial chromosomes.We then turn to liquid-liquid phase separation, which may allow compartmentalization within bacterial cells (Cohan and Pappu, 2020;Azaldegui et al., 2021).Physically, NAPs can be modelled as particles that can diffuse, interact with each other, and bind to DNA.For bridging to occur, the number of DNA strands that the NAP can simultaneously bind to (its valency) should be at least two.Brackley et al. (2013) showed that, even in the absence of NAP-NAP interactions or cooperative binding, multivalent binding could be sufficient to give rise to NAP clustering.A bivalently binding NAP introduces a loop on the chromosome, which is entropically costly.Two bivalent NAPs can bind far apart, giving rise to two loops, or next to each other, effectively giving rise to just one loop.Hence, even in the absence of NAP-NAP interactions, bridging proteins can cluster for entropic reasons.Non-cooperative NAP binding can also affect chromosome dynamics; Subramanian and Murray (2023) showed that transient bridging can give rise to sub-diffusive motion of loci at timescales below the bridge lifetime.Consistent with this model, an H-NS mutant of E. coli showed weaker subdiffusive behavior of loci than the wild type.Although some NAPs might bind non-cooperatively, many are known to interact, which can be modelled by introducing NAP-NAP interactions in simulations.Joyeux (2021) studied how protein self-association impacts chromosome organization.Inspired by the E. coli NAP H-NS, two modes of NAP binding were modeled: filament-or cluster-forming.In simulations, filament-forming proteins stiffened chromosomal regions where they bound, but did not to compact DNA.Conversely, clustering proteins condensed the chromosome, but did not stiffen DNA.This work illustrates that even simple coarse-grained models can capture a variety of NAP behaviours.Certain NAP-NAP interactions can give rise to collective phenomena, such as biomolecular condensation.For instance, HU and Dps, two important NAPs in E. coli, have been observed to lead to phase separation of DNA segments in vitro (Gupta et al., 2023).Put simply, phase separation can occur when attractive interactions start to dominate over entropic effects; whereas entropy favors spreading NAPs across the accessible volume, attractive NAP-NAP interactions of suitable geometry and sufficient range can favor NAP condensation.Since phase separation can create long-range order, it can impact chromosome organization at large scales.Some of the earliest evidence for biomolecular condensation in bacterial cells came from observations of ParB clusters forming at parS sites on bacterial chromosomes and plasmids (Broedersz et al., 2014;Jalal and Le, 2020).Bottom-up models have been used to explore transport (Lim et al., 2014;Surovtsev et al., 2016;Hu et al., 2017;Walter et al., 2017;Köhler and Murray, 2023) and force generation (Hanauer et al., 2021) by the ParABS system, as well as ParB cluster formation (Broedersz et al., 2014;Sanchez et al., 2015;Walter et al., 2021).These works offer two key physical insights.First, as opposed to earlier works that assumed that ParB only spreads along the one-dimensional DNA strand (Murray et al., 2006;Breier and Grossman, 2007), the formation of ParB clusters is an inherently three-dimensional phenomenon; a well-known result from statistical physics states that phase separation cannot occur in one-dimensional systems with short range interactions.Hence a combination of 1D spreading, 3D bridging, and fluctuations of the chromosome are important for the formation of ParB clusters.Accordingly, in vitro experiments have confirmed that bridging is essential for ParB spreading (Graham et al., 2014), and that ParB-dimers can recruit each other in-trans and form dynamic clusters via bridging (Tišma et al., 2022(Tišma et al., , 2023)).Second, the maintenance of separate ParB clusters consumes energy; to minimize their surface area, phase-separated droplets are expected to merge either via Ostwald ripening (when constituents diffuse from smaller droplets to larger ones) or by collision.This implies that the maintenance of ParB condensates on separate plasmids and/or parS sites may require an active mechanism, such as ParA ATPase (Guilhas et al., 2020) and/or ParB CTPase activity (Osorio-Valeriano et al., 2021).

Effects of transcription
Transcription and translation can affect bacterial chromosome organization in multiple ways.Steric interactions with ribosomes and RNA can affect nucleoid compaction and localization (Xiang et al., 2021;Miangolarra et al., 2021).Transertion -the insertion of membrane proteins into the cell wall as they are translated and transcribed (Roggiani and Goulian, 2015;Spahn et al., 2023) -can cause loci to remain near the cell membrane.Highly transcribed genes have been proposed to colocalize, since RNA polymerases can cluster in fast growth conditions (Ladouceur et al., 2020).Finally, transcription introduces both positive and negative supercoils, and highly transcribed genes can act as topological barriers that inhibit plectoneme diffusion (Le et al., 2013;Le and Laub, 2016).Since single-molecule experiments are providing evidence that some NAPs (Guo et al., 2021) and potentially SMCs (Kim et al., 2022) are recruited to areas of high supercoiling, future models could explore the interplay of these different mechanisms of bacterial chromosome organization.By comparing Monte Carlo simulations of a confined polymer to experimental data, Xiang et al. (2021) showed that the mesh size of the E. coli nucleoid is compatible with the chromosome being embedded in an effective poor solvent.Since ribosome and DNA densities were found to be anticorrelated, the authors suggested that this effective poor solvent could be a result of excluded volume interactions between the chromosome and ribosomes and/or RNA.Miango-D R A F T 4.1 Consensus structure models larra et al. ( 2021) further explored steric interactions between bacterial chromosomes and the transcriptional-translational machinery.By modeling the coupled 1D dynamics of DNA, ribosomes, and mRNA, they showed how active transcription and translation can affect the shape, size, and position of the nucleoid.As reviewed by Junier et al. (2023), supercoiling due to transcription has not yet been modeled at scales of the bacterial chromosome.This is mainly due to computational limitations: bottom-up simulations for transcription-induced supercoiling in 3D have only been conducted for scales of tens of kilobases (Lepage and Junier, 2019).In light of these limitations, recent chromosome-scale models have considered branched polymers with plectoneme distributions that correlate with transcriptional activity (Hacker et al., 2017;Wasim et al., 2023a,b).To illustrate, Hacker et al. (2017) divided the E. coli chromosome into "plectoneme-rich" and "plectoneme-free" regions based on RNAP Chip-seq data, and then simulated branched polymers with sampled plectoneme configurations.Such use of complex, quantitative experimental data to constrain a model is a defining characteristic of modern data-driven modeling, which can offer new conceptual and mechanistic insights into bacterial chromosome organization.

Data-driven models
Over the last decade, Hi-C experiments have led to a breakthrough in studying chromosome organization quantitatively.A typical Hi-C map for a bacterial chromosome at a 5-10 kb resolution consists of ∼ 160000 data points, accurately probing features of chromosome organization over 3 orders of magnitude in genomic scales.Unlike microscopy methods, however, Hi-C experiments do not yield easily interpretable images, but rather a statistical metric for population-averaged pairwise contact counts.Using Hi-C data to faithfully extract information about the underlying distribution of threedimensional chromosome configurations is thus a daunting theoretical challenge.Data-driven theoretical approaches seek to exploit the quantitative potential of Hi-C maps by directly inferring a model for 3D chromosome organization from experimental data (Contessoto et al., 2022).Since Hi-C data represent an ensemble average of contact frequencies over the full distribution P ({r}) of 3D chromosome conformations {r}, datadriven models for bacterial chromosome organization usually seek to find either a single "average" chromosome consensus structure, {r} consensus , or an ensemble of chromosome configurations, P model ({r}) (McCord et al., 2020).Inference of both types of models is a technically challenging inverse problem, as we discuss below.

Consensus structure models
Most data-driven models don't use Hi-C data directly as input, and consensus structure algorithms are no exception (Fig. 2).To construct a consensus structure, Hi-C scores are first converted into average spatial distances between locus pairs.This can be done by assuming that the mean distance between loci has a power-law scaling with the contact frequency (Marbouty et al., 2015), or by using an experimentally determined calibration curve (Umbarger et al., 2011).Theoretically, however, the pairwise contact frequency between two foci is expected to depend not only on their mean distance, but also on the distance distribution's shape.Accordingly, experimental (Lioy et al., 2018) and simulation (Messelink et al., 2021) results show that mean distances between chromosomal loci can show large deviations from average scalings.Nevertheless, once an average distance map has been found, computational algorithms (reviewed by Liu et al. (2023)) can be used to find a single 3D structure where the pairwise distances between loci are as compatible with the estimated mean distances as possible.
How should we interpret a consensus structure?Unlike proteins that often fold into specific, robust shapes that are critical for their function, bacterial chromosomes are highly flexible and dynamic polymers; imaging experiments show that the positions of chromosomal loci can vary by as much as half a cell length (Viollier et al., 2004).This inherent conformational variability is neglected by consensus structure algorithms: they cannot predict population-level variations in chromosome organization.Nonetheless, consensus structures may offer intuition for global chromosome organization by providing a "convenient visualization tool" (Lioy et al., 2018) for estimated mean distances between chromosomal regions.Furthermore, comparison of consensus structures for mutant strains or for drug-treated cells might yield clues about how different perturbations affect global chromosome organization.Umbarger et al. (2011) applied an algorithm originally developed for macromolecular assemblies such as nuclear pore complexes (Russel et al., 2012) to predict consensus structures for a C. crescentus chromosome based on 5C data.A set of candidate structures were found by initializing the algorithm with different initial conditions, and the inferred structures were then grouped by similarity.The model suggested that the arms of the C. crescentus chromosome are wound in a loose helical structure.The authors also inferred structures for a mutant where the parS site was relocated, leading to a shift in the cross-diagonal line on the 5C map.The corresponding consensus structure showed that the end of the nucleoid shifted to the new location of the parS site, consistent with this site being tethered to a cell pole.
More recently, an error vector resultant algorithm was developed for faster and more accurate inference of consensus structures for prokaryotic chromosomes (Hua and Ma, 2019).The algorithm was applied to Hi-C data from C. crescentus, E. coli and B. subtilis.By comparing consensus structures for wild-type and a ∆fis mutant of E. coli, the authors concluded that the terminal region bends towards the rest of the chromosome in the mutant strain, reflecting increased Hi-C counts between the terminal region and the rest of the chromosome.Contrasting earlier consensus structures (Umbarger et al., 2011;Marbouty et al., 2015), helicity of the arms was only predicted for B. subtilis.Such contradictory results raise further questions about how consensus structures relate to the Consensus structure models ... the Hi-C map to a distance matrix, for instance by assuming a scaling between mean distances between loci d(i, j) and their contact counts Mi,j .Mean distances can be used to create a consensus structure, which depicts the estimated mean distances using a 3D curve.Subfigure shows consensus structure for E. coli, adapted from (Hua and Ma, 2019).Alternatively, mean distances can be used to constrain spring-based ensemble models, with effective harmonic potentials between loci.An ensemble model can also be inferred directly from Hi-C data: by maximizing the distribution entropy with constraints on contact probabilities ( S), one can choose the leastassuming chromosome configuration distribution P model ({r}) consistent with Hi-C data (Messelink et al., 2021).The MaxEnt procedure selects a model that features effective close-range interactions between monomers.For both spring-based and MaxEnt ensemble models, the effective interaction parameters need to be inferred using computational approaches.Once these parameters have been determined, the distribution can be sampled for single-cell chromosome configurations.Subfigure shows sampled configurations from the MaxEnt model for C. crescentus swarmer cells (Messelink et al., 2021).
underlying distribution of chromosome configurations in individual cells.

Ensemble models
Ensemble methods aim to capture population-level variability in bacterial chromosome organization by finding a distribution P model ({r}) of single-cell chromosome configurations, given Hi-C data (Fig. 2).Most approaches assume an underlying statistical model for this distribution, defined by a set of effective interaction parameters.Once the effective parameters have been inferred from data, the model ensemble can be sampled using statistical methods such as Monte Carlo simulations.Samples from the distribution can be interpreted as single-cell chromosome configurations.In this way, an ensemble model constructed using population-averaged data can be used to make predictions about chromosome organization both on the single-cell and population level.Like consensus structure models, most data-driven ensemble models for bacterial chromosome organization start by assuming a relation between Hi-C scores and average monomer distances.However, these distances are now typically used to define spring-like interactions between loci, which constrain the mean distances in the model to match input data.For example, Yildirim and Feig (2018) constructed an ensemble model for the C. crescentus chromosome by first converting Hi-C scores to expected distances between loci based on previous calibration data (Umbarger et al., 2011), and then constraining distances between a subset of monomer pairs in a plectonemic model using spring-like interactions.Equilibrium molecular dynamics simulations were used to produce an ensemble of chromosome configurations, and configura-tions were assigned statistical weights based on how closely their distance matrices matched input Hi-C data.The correlation between the model's contact map and the experimental Hi-C map (0.88) was comparable to that between Hi-C maps of C. crescentus and B. subtilis (0.878 for Hi-C maps from (Le et al., 2013;Wang et al., 2015)).This illustrates that models constrained with inferred distances do not necessarily reproduce the Hi-C map faithfully.Using a similar approach, Wasim et al. ( 2021) constructed a spring-based model for the E. coli chromosome.They have since studied the sub-diffusional behavior of loci in their model, compared models for wild-type cells and HUand MatP-mutants, and constructed a model with plectonemes (Bera et al., 2022;Wasim et al., 2023c,a).These works have suggested that locus (sub-)diffusion depends on genomic position, and that inclusion of plectonemes in the model slightly affected chromosome compaction, but not organization.These and other ensemble techniques have advanced datadriven modeling beyond consensus structure inference for both pro-and eukaryotic chromosomes (Marti-Renom et al., 2018;Oluwadare et al., 2019;McCord et al., 2020).However, several issues remain.Many ensemble approaches rely on strong assumptions, like thermal equilibrium or converting Hi-C counts to expected mean distances.Furthermore, the diversity of methods hints at a more fundamental concern: while all these approaches lead to a model based on a given Hi-C map, many distinct ensembles could be consistent with the same data.So, how do you select the right one?To address this challenge, our group developed a datadriven model for bacterial chromosome organization based D R A F T 4.2 Ensemble models on the Maximum Entropy (MaxEnt) principle.This principle selects the unique chromosome conformation distribution P MaxEnt ({r}) that reproduces a given Hi-C map but is otherwise as unstructured as possible.Notably, the model does not rely on converting Hi-C scores to mean pairwise distances between loci.We applied this approach to model chromosome organization in new-born C. crescentus swarmer cells (Messelink et al., 2021).As validation, we showed that the model accurately predicts the long-axis distributions of loci over the entire chromosome, as measured by independent experiments (Viollier et al., 2004).The MaxEnt model can also reveal novel features of chromosome organization.For example, our model predicted the presence of "super-domains", or clusters of high chromosomal density at the single-cell level.The presence of these super-domains was validated using super-resolution microscopy.Our MaxEnt model was limited to new-born cells with a single chromosome, constrained using Hi-C data from synchronized cells (Le et al., 2013).However, Hi-C experiments on E. coli, B. subtilis, and many other bacteria are conducted on asynchronous populations.The resulting Hi-C maps reflect an average over cells at different replication stages, which poses challenges for data-driven modeling.For instance, bacterial chromosome organization can vary over the cell cycle (Wang et al., 2014), and Hi-C experiments in replicating bacteria count both cis-and trans-contacts.Wasim et al. (2021) inferred models for E. coli at discrete replication stages by constraining each model with the same asynchronous Hi-C data, and by assuming that trans-contacts are negligible.Since the validity of these approximations has yet to be established, an open-standing question in the field is how to best infer a model using Hi-C data from an asynchronous population.This is clearly a challenging but worthwhile problem: Such a model could provide new insight into the dynamics of chromosome organization across the cell cycle.

Discussion and future challenges
We have discussed three biophysical approaches to modeling bacterial chromosomes: models based on imposed geometric or topological constraints, bottom-up models, and datadriven approaches.Simple models of (branched) polymers in confinement have revealed how different constraints affect bacterial chromosome organization.Bottom-up approaches link these constraints to their underlying biophysical causes, and can hence be used to gain mechanistic insights.Datadriven methods aim to capture detailed chromosome organization by inferring models from experimental Hi-C data.Importantly, we note that these modeling approaches can complement each other.For instance, data-driven approaches can help hypothesize simple physical principles of chromosome organization, which can then guide the construction of bottom-up or constraint-based models.Given that simplified constraint-based and bottom-up models are tailored to explain only certain aspects of chromosome organization, it can be difficult to test their predictions in vivo with controlled and targeted perturbations.To test whether effects seen in polymer simulations are relevant at biological length-and time-scales, these simplified models could be compared to artificial systems of chromosomes in confinement (Birnie and Dekker, 2021).Conversely, bottomup models could be constructed for "simpler" organisms: Stevens et al. (2023) recently presented a model for an entire, minimal bacterial cell, with a 543 kb long circular chromosome.For more complex systems, the increased availability of high-quality quantitative data creates opportunities for data-driven modeling.For instance, multiple types of data, such as Hi-C, imaging, and/or RNA-sequencing, can be combined to infer data-driven models that capture bacterial chromosome organization in its full complexity (Messelink et al., 2021;Wasim et al., 2023a).In conclusion, biophysical modeling has helped shape our understanding of how bacterial chromosomes are functionally organized.Since biophysical models can be easily adapted for different organisms, modeling can help search for divergent and unifying principles of prokaryotic genome organization.

Fig. 1 .
Fig. 1.Constraint-based and bottom-up models.A Examples of constraint-based models.Left to right: Polymer confinement can give rise to effects such as entropic segregation(Jun and Mulder, 2006).Constraining chromosomal regions corresponding to E. coli macrodomains to subvolumes of the nucleoid can give rise to order and chromosome segregation(Junier et al., 2014).Fixed loop topologies can orient bacterial chromosomes and enhance their segregation(Mitra et al., 2022b).Feather-boa models show that branching can affect chromosome compaction, segregation, and dynamics.Constraining origins of replication or other loci by tethering them to the cell membrane can give rise to linear ordering of the chromosome(Buenemann and Lenz, 2010, 2011).B Examples of bottom-up mechanisms.Left to right: Macromolecular crowding due to RNA, ribosomes, and other large molecules can compact bacterial chromosomes(Rivas and Minton, 2016).Loop-extrusion by MukBEF can contribute to macrodomain formation in E. coli(Lioy et al., 2018), whereas condensin can tie together the chromosomal arms in C. crescentus and B. subtilis(Le et al., 2013;Wang et al., 2017).NAPs can impact chromosome organization by condensing macrodomains, by bridging, by stabilizing supercoils, or by associating with the cell membrane(Dame et al., 2020; Amemiya et al., 2021).Transcription gives rise to twin-supercoiled-domains, with positive supercoiling ahead and negative supercoiling behind RNA polymerases(Junier et al., 2023).The insertion of newly translated proteins into the cell membrane (transertion) causes certain genes to associate with the cell wall(Roggiani and Goulian, 2015).

HiFig. 2 .
Fig. 2. Data-driven modeling of bacterial chromosomes.Data-driven models aim to infer a model for bacterial chromosomes from Hi-C data, which reflects the populationaveraged contact counts between chromosomal regions.Subfigure shows Hi-C data for C. crescentus swarmer cells (Le et al., 2013).Most models start by converting