Linking topology of tethered polymer rings with applications to chromosome segregation and estimation of the knotting length

Heterogeneous flexibility can contribute to chromatin segregation in the cell nucleus

The highly and slightly condensed forms of chromatin, heterochromatin and euchromatin, respectively, segregate in the cell nucleus. Heterochromatin is more abundant in the nucleus periphery. Here we study the mechanism of heterochromatin segregation by modeling interphase chromosomes as diblock ring copolymers confined in a rigid spherical shell using molecular dynamics simulations. In our model, heterochromatin and euchromatin are distinguished by their bending stiffnesses only, while an interaction potential between the spherical shell and chromatin is used to model lamin-associated proteins. Our simulations indicate that in the absence of attractive interactions between the nuclear shell and the chromatin, most heterochromatin segregates towards the nuclear interior due to the depletion of less flexible heterochromatin segments from the nuclear periphery. This inverted chromatin distribution,which is opposite to the conventional case with heterochromatin dominating at the periphery, is in accord with experimental observations in rod cells. This "inversion" is also found to be independent of the heterochromatin concentration and chromosome number. The chromatin distribution at the periphery found in vivo can be recovered by further increasing the bending stiffness of heterochromatin segments or by turning on attractive interactions between the nuclear shell and heterochromatin. Our results indicate that the bending stiffness of chromatin could be a contributor to chromosome organization along with differential effects of HP1α-driven phase segregation and of loop extruders and interactions with the nuclear envelope and topological constraints.

Dynamics of chromosome organization in a minimal bacterial cell

Computational models of cells cannot be considered complete unless they include the most fundamental process of life, the replication and inheritance of genetic material. By creating a computational framework to model systems of replicating bacterial chromosomes as polymers at 10 bp resolution with Brownian dynamics, we investigate changes in chromosome organization during replication and extend the applicability of an existing whole-cell model (WCM) for a genetically minimal bacterium, JCVI-syn3A, to the entire cell-cycle. To achieve cell-scale chromosome structures that are realistic, we model the chromosome as a self-avoiding homopolymer with bending and torsional stiffnesses that capture the essential mechanical properties of dsDNA in Syn3A. In addition, the conformations of the circular DNA must avoid overlapping with ribosomes identitied in cryo-electron tomograms. While Syn3A lacks the complex regulatory systems known to orchestrate chromosome segregation in other bacteria, its minimized genome retains essential loop-extruding structural maintenance of chromosomes (SMC) protein complexes (SMC-scpAB) and topoisomerases. Through implementing the effects of these proteins in our simulations of replicating chromosomes, we find that they alone are sufficient for simultaneous chromosome segregation across all generations within nested theta structures. This supports previous studies suggesting loop-extrusion serves as a near-universal mechanism for chromosome organization within bacterial and eukaryotic cells. Furthermore, we analyze ribosome diffusion under the influence of the chromosome and calculate in silico chromosome contact maps that capture inter-daughter interactions. Finally, we present a methodology to map the polymer model of the chromosome to a Martini coarse-grained representation to prepare molecular dynamics models of entire Syn3A cells, which serves as an ultimate means of validation for cell states predicted by the WCM.

Spatial Organization of Chromatin: Emergence of Chromatin Structure During Development

Nuclei are central hubs for information processing in eukaryotic cells. The need to fit large genomes into small nuclei imposes severe restrictions on genome organization and the mechanisms that drive genome-wide regulatory processes. How a disordered polymer such as chromatin, which has vast heterogeneity in its DNA and histone modification profiles, folds into discernibly consistent patterns is a fundamental question in biology. Outstanding questions include how genomes are spatially and temporally organized to regulate cellular processes with high precision and whether genome organization is causally linked to transcription regulation. The advent of next-generation sequencing, super-resolution imaging, multiplexed fluorescent in situ hybridization, and single-molecule imaging in individual living cells has caused a resurgence in efforts to understand the spatiotemporal organization of the genome. In this review, we discuss structural and mechanistic properties of genome organization at different length scales and examine changes in higher-order chromatin organization during important developmental transitions.

Linker histone H1.8 inhibits chromatin binding of condensins and DNA topoisomerase II to tune chromosome length and individualization

DNA loop extrusion by condensins and decatenation by DNA topoisomerase II (topo II) are thought to drive mitotic chromosome compaction and individualization. Here, we reveal that the linker histone H1.8 antagonizes condensins and topo II to shape mitotic chromosome organization. In vitro chromatin reconstitution experiments demonstrate that H1.8 inhibits binding of condensins and topo II to nucleosome arrays. Accordingly, H1.8 depletion in Xenopus egg extracts increased condensins and topo II levels on mitotic chromatin. Chromosome morphology and Hi-C analyses suggest that H1.8 depletion makes chromosomes thinner and longer through shortening the average loop size and reducing the DNA amount in each layer of mitotic loops. Furthermore, excess loading of condensins and topo II to chromosomes by H1.8 depletion causes hyper-chromosome individualization and dispersion. We propose that condensins and topo II are essential for chromosome individualization, but their functions are tuned by the linker histone to keep chromosomes together until anaphase.

3D genomics across the tree of life reveals condensin II as a determinant of architecture type

We investigated genome folding across the eukaryotic tree of life. We find two types of three-dimensional (3D) genome architectures at the chromosome scale. Each type appears and disappears repeatedly during eukaryotic evolution. The type of genome architecture that an organism exhibits correlates with the absence of condensin II subunits. Moreover, condensin II depletion converts the architecture of the human genome to a state resembling that seen in organisms such as fungi or mosquitoes. In this state, centromeres cluster together at nucleoli, and heterochromatin domains merge. We propose a physical model in which lengthwise compaction of chromosomes by condensin II during mitosis determines chromosome-scale genome architecture, with effects that are retained during the subsequent interphase. This mechanism likely has been conserved since the last common ancestor of all eukaryotes.

XerD unloads bacterial SMC complexes at the replication terminus

Bacillus subtilis structural maintenance of chromosomes (SMC) complexes are topologically loaded at centromeric sites adjacent to the replication origin by the partitioning protein ParB. These ring-shaped ATPases then translocate down the left and right chromosome arms while tethering them together. Here, we show that the site-specific recombinase XerD, which resolves chromosome dimers, is required to unload SMC tethers when they reach the terminus. We identify XerD-specific binding sites in the terminus region and show that they dictate the site of unloading in a manner that depends on XerD but not its catalytic residue, its partner protein XerC, or the recombination site dif. Finally, we provide evidence that ParB and XerD homologs perform similar functions in Staphylococcus aureus. Thus, two broadly conserved factors that act at the origin and terminus have second functions in loading and unloading SMC complexes that travel between them.

Loop extrusion: theory meets single-molecule experiments

Chromosomes are organized as chromatin loops that promote segregation, enhancer-promoter interactions, and other genomic functions. Loops were hypothesized to form by 'loop extrusion,' by which structural maintenance of chromosomes (SMC) complexes, such as condensin and cohesin, bind to chromatin, reel it in, and extrude it as a loop. However, such exotic motor activity had never been observed. Following an explosion of indirect evidence, recent single-molecule experiments directly imaged DNA loop extrusion by condensin and cohesin in vitro. These experiments observe rapid (kb/s) extrusion that requires ATP hydrolysis and stalls under pN forces. Surprisingly, condensin extrudes loops asymmetrically, challenging previous models. Extrusion by cohesin is symmetric but requires the protein Nipbl. We discuss how SMC complexes may perform their functions on chromatin in vivo.

Chromosome organization by one-sided and two-sided loop extrusion

SMC complexes, such as condensin or cohesin, organize chromatin throughout the cell cycle by a process known as loop extrusion. SMC complexes reel in DNA, extruding and progressively growing DNA loops. Modeling assuming two-sided loop extrusion reproduces key features of chromatin organization across different organisms. In vitro single-molecule experiments confirmed that yeast condensins extrude loops, however, they remain anchored to their loading sites and extrude loops in a 'one-sided' manner. We therefore simulate one-sided loop extrusion to investigate whether 'one-sided' complexes can compact mitotic chromosomes, organize interphase domains, and juxtapose bacterial chromosomal arms, as can be done by 'two-sided' loop extruders. While one-sided loop extrusion cannot reproduce these phenomena, variants can recapitulate in vivo observations. We predict that SMC complexes in vivo constitute effectively two-sided motors or exhibit biased loading and propose relevant experiments. Our work suggests that loop extrusion is a viable general mechanism of chromatin organization.

Organization of Chromosomal DNA by SMC Complexes

Structural maintenance of chromosomes (SMC) complexes are key organizers of chromosome architecture in all kingdoms of life. Despite seemingly divergent functions, such as chromosome segregation, chromosome maintenance, sister chromatid cohesion, and mitotic chromosome compaction, it appears that these complexes function via highly conserved mechanisms and that they represent a novel class of DNA translocases.

Chromosome disentanglement driven via optimal compaction of loop-extruded brush structures

Eukaryote cell division features a chromosome compaction-decompaction cycle that is synchronized with their physical and topological segregation. It has been proposed that lengthwise compaction of chromatin into mitotic chromosomes via loop extrusion underlies the compaction-segregation/resolution process. We analyze this disentanglement scheme via considering the chromosome to be a succession of DNA/chromatin loops-a polymer "brush"-where active extrusion of loops controls the brush structure. Given type-II DNA topoisomerase (Topo II)-catalyzed topology fluctuations, we find that interchromosome entanglements are minimized for a certain "optimal" loop that scales with the chromosome size. The optimal loop organization is in accord with experimental data across species, suggesting an important structural role of genomic loops in maintaining a less entangled genome. Application of the model to the interphase genome indicates that active loop extrusion can maintain a level of chromosome compaction with suppressed entanglements; the transition to the metaphase state requires higher lengthwise compaction and drives complete topological segregation. Optimized genomic loops may provide a means for evolutionary propagation of gene-expression patterns while simultaneously maintaining a disentangled genome. We also find that compact metaphase chromosomes have a densely packed core along their cylindrical axes that explains their observed mechanical stiffness. Our model connects chromosome structural reorganization to topological resolution through the cell cycle and highlights a mechanism of directing Topo II-mediated strand passage via loop extrusion-driven lengthwise compaction.

DNA-segment-capture model for loop extrusion by structural maintenance of chromosome (SMC) protein complexes

Cells possess remarkable control of the folding and entanglement topology of long and flexible chromosomal DNA molecules. It is thought that structural maintenance of chromosome (SMC) protein complexes play a crucial role in this, by organizing long DNAs into series of loops. Experimental data suggest that SMC complexes are able to translocate on DNA, as well as pull out lengths of DNA via a 'loop extrusion' process. We describe a Brownian loop-capture-ratchet model for translocation and loop extrusion based on known structural, catalytic, and DNA-binding properties of the Bacillus subtilis SMC complex. Our model provides an example of a new class of molecular motor where large conformational fluctuations of the motor 'track'-in this case DNA-are involved in the basic translocation process. Quantitative analysis of our model leads to a series of predictions for the motor properties of SMC complexes, most strikingly a strong dependence of SMC translocation velocity and step size on tension in the DNA track that it is moving along, with 'stalling' occuring at subpiconewton tensions. We discuss how the same mechanism might be used by structurally related SMC complexes (Escherichia coli MukBEF and eukaryote condensin, cohesin and SMC5/6) to organize genomic DNA.

DNA Mechanics and Topology

We review the current understanding of the mechanics of DNA and DNA-protein complexes, from scales of base pairs up to whole chromosomes. Mechanics of the double helix as revealed by single-molecule experiments will be described, with an emphasis on the role of polymer statistical mechanics. We will then discuss how topological constraints- entanglement and supercoiling-impact physical and mechanical responses. Models for protein-DNA interactions, including effects on polymer properties of DNA of DNA-bending proteins will be described, relevant to behavior of protein-DNA complexes in vivo. We also discuss control of DNA entanglement topology by DNA-lengthwise-compaction machinery acting in concert with topoisomerases. Finally, the chapter will conclude with a discussion of relevance of several aspects of physical properties of DNA and chromatin to oncology.

Topologically Linked Chains in Confinement

We examine how channel confinement affects the equilibrium properties of topologically linked ring polymers and, by contrast, of equivalent unlinked rings, too. By performing extensive simulations of semiflexible rings of different chain length, N, and channel diameter, D, we discover three notable properties purely due to linking. First, upon entering the weak confinement regime, the length of the physically linked portion, lLKThe, becomes independent of chain length. Next, even when confinement is strong enough to pull apart and segregate unlinked rings, lLK stays much larger than in the highly stretched limit. Finally, at fixed N, lLK varies approximately as D^0.5, and we provide a simple scaling argument for this power-law behavior. These properties, which may hold for different link topologies, can be tested by current experimental setups on DNA rings confined in microchannels. Moreover, they could be relevant for the efficient in vivo unlinking of newly replicated bacterial chromosomes.

Subcellular Organization: A Critical Feature of Bacterial Cell Replication

Spatial organization is a hallmark of all living systems. Even bacteria, the smallest forms of cellular life, display defined shapes and complex internal organization, showcasing a highly structured genome, cytoskeletal filaments, localized scaffolding structures, dynamic spatial patterns, active transport, and occasionally, intracellular organelles. Spatial order is required for faithful and efficient cellular replication and offers a powerful means for the development of unique biological properties. Here, we discuss organizational features of bacterial cells and highlight how bacteria have evolved diverse spatial mechanisms to overcome challenges cells face as self-replicating entities.

An integrative view of cell cycle control in Escherichia coli

Bacterial proliferation depends on the cells' capability to proceed through consecutive rounds of the cell cycle. The cell cycle consists of a series of events during which cells grow, copy their genome, partition the duplicated DNA into different cell halves and, ultimately, divide to produce two newly formed daughter cells. Cell cycle control is of the utmost importance to maintain the correct order of events and safeguard the integrity of the cell and its genomic information. This review covers insights into the regulation of individual key cell cycle events in Escherichia coli. The control of initiation of DNA replication, chromosome segregation and cell division is discussed. Furthermore, we highlight connections between these processes. Although detailed mechanistic insight into these connections is largely still emerging, it is clear that the different processes of the bacterial cell cycle are coordinated to one another. This careful coordination of events ensures that every daughter cell ends up with one complete and intact copy of the genome, which is vital for bacterial survival.

Condensin controls mitotic chromosome stiffness and stability without forming a structurally contiguous scaffold

During cell division, chromosomes must be folded into their compact mitotic form to ensure their segregation. This process is thought to be largely controlled by the action of condensin SMC protein complexes on chromatin fibers. However, how condensins organize metaphase chromosomes is not understood. We have combined micromanipulation of single human mitotic chromosomes, sub-nanonewton force measurement, siRNA interference of condensin subunit expression, and fluorescence microscopy, to analyze the role of condensin in large-scale chromosome organization. Condensin depletion leads to a dramatic (~ 10-fold) reduction in chromosome elastic stiffness relative to the native, non-depleted case. We also find that prolonged metaphase stalling of cells leads to overloading of chromosomes with condensin, with abnormally high chromosome stiffness. These results demonstrate that condensin is a main element controlling the stiffness of mitotic chromosomes. Isolated, slightly stretched chromosomes display a discontinuous condensing staining pattern, suggesting that condensins organize mitotic chromosomes by forming isolated compaction centers that do not form a continuous scaffold.

Compaction and segregation of sister chromatids via active loop extrusion

The mechanism by which chromatids and chromosomes are segregated during mitosis and meiosis is a major puzzle of biology and biophysics. Using polymer simulations of chromosome dynamics, we show that a single mechanism of loop extrusion by condensins can robustly compact, segregate and disentangle chromosomes, arriving at individualized chromatids with morphology observed in vivo. Our model resolves the paradox of topological simplification concomitant with chromosome 'condensation', and explains how enzymes a few nanometers in size are able to control chromosome geometry and topology at micron length scales. We suggest that loop extrusion is a universal mechanism of genome folding that mediates functional interactions during interphase and compacts chromosomes during mitosis.

Annealed random copolymer model of the B-Z transition in DNA: torsional responses

Both in vivo and in vitro, specific sequences in double-stranded DNA can adopt the left-handed Z-form when underwound. Recently, the B-Z transition of DNA has been studied in detail in magnetic tweezers experiments by several groups. We present a theoretical description of this transition, based on an annealed random copolymer model. The transition of a switchable sequence is discussed as a function of energetic and geometric parameters of the B- and Z-forms, of the applied boundary conditions, and of the characteristics of the B-Z interface. We address a possible torsional softening upon the B-Z transition. The model can be also applied to other biofilaments with annealed torsional/flexural degrees of freedom.

Condensin promotes the juxtaposition of DNA flanking its loading site in Bacillus subtilis

SMC condensin complexes play a central role in compacting and resolving replicated chromosomes in virtually all organisms, yet how they accomplish this remains elusive. In Bacillus subtilis, condensin is loaded at centromeric parS sites, where it encircles DNA and individualizes newly replicated origins. Using chromosome conformation capture and cytological assays, we show that condensin recruitment to origin-proximal parS sites is required for the juxtaposition of the two chromosome arms. Recruitment to ectopic parS sites promotes alignment of large tracks of DNA flanking these sites. Importantly, insertion of parS sites on opposing arms indicates that these "zip-up" interactions only occur between adjacent DNA segments. Collectively, our data suggest that condensin resolves replicated origins by promoting the juxtaposition of DNA flanking parS sites, drawing sister origins in on themselves and away from each other. These results are consistent with a model in which condensin encircles the DNA flanking its loading site and then slides down, tethering the two arms together. Lengthwise condensation via loop extrusion could provide a generalizable mechanism by which condensin complexes act dynamically to individualize origins in B. subtilis and, when loaded along eukaryotic chromosomes, resolve them during mitosis.

The bacterial nucleoid: nature, dynamics and sister segregation

Recent studies reveal that the bacterial nucleoid has a defined, self-adherent shape and an underlying longitudinal organization and comprises a viscoelastic matrix. Within this shape, mobility is enhanced by ATP-dependent processes and individual loci can undergo ballistic off-equilibrium movements. In Escherichia coli, two global dynamic nucleoid behaviors emerge pointing to nucleoid-wide accumulation and relief of internal stress. Sister segregation begins with local splitting of individual loci, which is delayed at origin, terminus and specialized interstitial snap regions. Globally, as studied in several systems, segregation is a multi-step process in which internal nucleoid state plays critical roles that involve both compaction and expansion. The origin and terminus regions undergo specialized programs partially driven by complex ATP burning mechanisms such as a ParAB Brownian ratchet and a septum-associated FtsK motor. These recent findings reveal strong, direct parallels among events in different systems and between bacterial nucleoids and mammalian chromosomes with respect to physical properties, internal organization and dynamic behaviors.

Chromosome organization and replisome dynamics in Mycobacterium smegmatis

Subcellular organization of the bacterial nucleoid and spatiotemporal dynamics of DNA replication and segregation have been studied intensively, but the functional link between these processes remains poorly understood. Here we use quantitative time-lapse fluorescence microscopy for single-cell analysis of chromosome organization and DNA replisome dynamics in Mycobacterium smegmatis. We report that DNA replication takes place near midcell, where, following assembly of the replisome on the replication origin, the left and right replication forks colocalize throughout the replication cycle. From its initial position near the cell pole, a fluorescently tagged chromosomal locus (attB, 245° from the origin) moves rapidly to the replisome complex just before it is replicated. The newly duplicated attB loci then segregate to mirror-symmetric positions relative to midcell. Genetic ablation of ParB, a component of the ParABS chromosome segregation system, causes marked defects in chromosome organization, condensation, and segregation. ParB deficiency also results in mislocalization of the DNA replication machinery and SMC (structural maintenance of chromosome) protein. These observations suggest that ParB and SMC play important and overlapping roles in chromosome organization and replisome dynamics in mycobacteria.

IMPORTANCE

We studied the spatiotemporal organization of the chromosome and DNA replication machinery in Mycobacterium smegmatis, a fast-growing relative of the human pathogen Mycobacterium tuberculosis. We show that genetic ablation of the DNA-binding proteins ParB and SMC disturbs the organization of the chromosome and causes a severe defect in subcellular localization and movement of the DNA replication complexes. These observations suggest that ParB and SMC provide a functional link between chromosome organization and DNA replication dynamics. This work also reveals important differences in the biological roles of the ParABS and SMC systems in mycobacteria versus better-characterized model organisms, such as Escherichia coli and Bacillus subtilis.

Topological energy storage of work generated by nanomotors

Most macroscopic machines rely on wheels and gears. Yet, rigid gears are entirely impractical on the nano-scale. Here we propose a more useful method to couple any rotary engine to any other mechanical elements on the nano- and micro-scale. We argue that a rotary molecular motor attached to an entangled polymer energy storage unit, which together form what we call the "tanglotron" device, is a viable concept that can be experimentally implemented. We derive the torque-entanglement relationship for a tanglotron (its "equation of state") and show that it can be understood by simple statistical mechanics arguments. We find that a typical entanglement at low packing density costs around 6kT. In the high entanglement regime, the free energy diverges logarithmically close to a maximal geometric packing density. We outline several promising applications of the tanglotron idea and conclude that the transmission, storage and back-conversion of topological entanglement energy are not only physically feasible but also practical for a number of reasons.

Biophysics of protein-DNA interactions and chromosome organization

The function of DNA in cells depends on its interactions with protein molecules, which recognize and act on base sequence patterns along the double helix. These notes aim to introduce basic polymer physics of DNA molecules, biophysics of protein-DNA interactions and their study in single-DNA experiments, and some aspects of large-scale chromosome structure. Mechanisms for control of chromosome topology will also be discussed.

Chromosome organization: original condensins

Two new studies reveal the main actors involved in the resolution and segregation of newly replicated origins in bacteria. These results have important implications for our understanding of the mechanisms involved in precisely coordinating chromosome organization, segregation and replication.

Bacillus subtilis chromosome organization oscillates between two distinct patterns

Bacterial chromosomes have been found to possess one of two distinct patterns of spatial organization. In the first, called "ori-ter" and exemplified by Caulobacter crescentus, the chromosome arms lie side-by-side, with the replication origin and terminus at opposite cell poles. In the second, observed in slow-growing Escherichia coli ("left-ori-right"), the two chromosome arms reside in separate cell halves, on either side of a centrally located origin. These two patterns, rotated 90° relative to each other, appear to result from different segregation mechanisms. Here, we show that the Bacillus subtilis chromosome alternates between them. For most of the cell cycle, newly replicated origins are maintained at opposite poles with chromosome arms adjacent to each other, in an ori-ter configuration. Shortly after replication initiation, the duplicated origins move as a unit to midcell and the two unreplicated arms resolve into opposite cell halves, generating a left-ori-right pattern. The origins are then actively segregated toward opposite poles, resetting the cycle. Our data suggest that the condensin complex and the parABS partitioning system are the principal driving forces underlying this oscillatory cycle. We propose that the distinct organization patterns observed for bacterial chromosomes reflect a common organization-segregation mechanism, and that simple modifications to it underlie the unique patterns observed in different species.

The SMC condensin complex is required for origin segregation in Bacillus subtilis

SMC condensin complexes play a central role in organizing and compacting chromosomes in all domains of life [1, 2]. In the bacterium Bacillus subtilis, cells lacking SMC are viable only during slow growth and display decondensed chromosomes, suggesting that SMC complexes function throughout the genome [3, 4]. Here, we show that rapid inactivation of SMC or its partner protein ScpB during fast growth leads to a failure to resolve newly replicated origins and a complete block to chromosome segregation. Importantly, the loss of origin segregation is not due to an inability to unlink precatenated sister chromosomes by Topoisomerase IV. In support of the idea that ParB-mediated recruitment of SMC complexes to the origin is important for their segregation, cells with reduced levels of SMC that lack ParB are severely impaired in origin resolution. Finally, we demonstrate that origin segregation is a task shared by the condensin complex and the parABS partitioning system. We propose that origin-localized SMC constrains adjacent DNA segments along their lengths, drawing replicated origins in on themselves and away from each other. This SMC-mediated lengthwise condensation, bolstered by the parABS system, drives origin segregation.

Variation of the folding and dynamics of the Escherichia coli chromosome with growth conditions

We examine whether the Escherichia coli chromosome is folded into a self-adherent nucleoprotein complex, or alternately is a confined but otherwise unconstrained self-avoiding polymer. We address this through in vivo visualization, using an inducible GFP fusion to the nucleoid-associated protein Fis to non-specifically decorate the entire chromosome. For a range of different growth conditions, the chromosome is a compact structure that does not fill the volume of the cell, and which moves from the new pole to the cell centre. During rapid growth, chromosome segregation occurs well before cell division, with daughter chromosomes coupled by a thin inter-daughter filament before complete segregation, whereas during slow growth chromosomes stay adjacent until cell division occurs. Image correlation analysis indicates that sub-nucleoid structure is stable on a 1 min timescale, comparable to the timescale for redistribution time measured for GFP-Fis after photobleaching. Optical deconvolution and writhe calculation analysis indicate that the nucleoid has a large-scale coiled organization rather than being an amorphous mass. Our observations are consistent with the chromosome having a self-adherent filament organization.

Organization and segregation of bacterial chromosomes

The bacterial chromosome must be compacted more than 1,000-fold to fit into the compartment in which it resides. How it is condensed, organized and ultimately segregated has been a puzzle for over half a century. Recent advances in live-cell imaging and genome-scale analyses have led to new insights into these problems. We argue that the key feature of compaction is the orderly folding of DNA along adjacent segments and that this organization provides easy and efficient access for protein-DNA transactions and has a central role in driving segregation. Similar principles and common proteins are used in eukaryotes to condense and to resolve sister chromatids at metaphase.

Physical descriptions of the bacterial nucleoid at large scales, and their biological implications

Recent experimental and theoretical approaches have attempted to quantify the physical organization (compaction and geometry) of the bacterial chromosome with its complement of proteins (the nucleoid). The genomic DNA exists in a complex and dynamic protein-rich state, which is highly organized at various length scales. This has implications for modulating (when not directly enabling) the core biological processes of replication, transcription and segregation. We overview the progress in this area, driven in the last few years by new scientific ideas and new interdisciplinary experimental techniques, ranging from high space- and time-resolution microscopy to high-throughput genomics employing sequencing to map different aspects of the nucleoid-related interactome. The aim of this review is to present the wide spectrum of experimental and theoretical findings coherently, from a physics viewpoint. In particular, we highlight the role that statistical and soft condensed matter physics play in describing this system of fundamental biological importance, specifically reviewing classic and more modern tools from the theory of polymers. We also discuss some attempts toward unifying interpretations of the current results, pointing to possible directions for future investigation.

Loops determine the mechanical properties of mitotic chromosomes

We introduce a new polymer model for mitotic chromosomes. The key assumption of the model is the ability of the chromatin fibre to cross-link to itself due to binding proteins. These protein-chromatin interactions are included by a probabilistic and dynamic mechanism. The hypothesis is motivated by the observation of high repulsive forces between ring polymers. We performed computer simulations to validate our model. Our results show that the presence of loops leads to a tight compaction and contributes significantly to the bending rigidity of chromosomes. Moreover, our qualitative prediction of the force elongation behaviour is close to experimental findings. The Dynamic Loop Model presented here indicates that the internal structure of mitotic chromosomes is based on self-organization of the chromatin fibre rather than attachment of chromatin to a protein scaffold. It also shows that the number and size of loops have a strong influence on the mechanical properties. We suggest that changes in the mechanical characteristics of chromosomes in different stages of the cell cycle, for example, can be explained by an altered internal loop structure.

Centromeres: unique chromatin structures that drive chromosome segregation

Fidelity during chromosome segregation is essential to prevent aneuploidy. The proteins and chromatin at the centromere form a unique site for kinetochore attachment and allow the cell to sense and correct errors during chromosome segregation. Centromeric chromatin is characterized by distinct chromatin organization, epigenetics, centromere-associated proteins and histone variants. These include the histone H3 variant centromeric protein A (CENPA), the composition and deposition of which have been widely investigated. Studies have examined the structural and biophysical properties of the centromere and have suggested that the centromere is not simply a 'landing pad' for kinetochore formation, but has an essential role in mitosis by assembling and directing the organization of the kinetochore.

Scaling of Linking and Writhing Numbers for Spherically Confined and Topologically Equilibrated Flexible Polymers

Scaling laws for Gauss linking number Ca and writhing number Wr for spherically confined flexible polymers with thermally fluctuating topology are analyzed. For ideal (phantom) polymers each of N segments of length unity confined to a spherical pore of radius R there are two scaling regimes: for sufficiently weak confinement (R ⪢ N(1/3)) each chain has |Wr| ≈ N(1/2), and each pair of chains has average |Ca| ≈ N/R(3/2); alternately for sufficiently tight confinement (N(1/3) ⪢ R), |Wr| ≈ |CA| ≈ N/R(3/2). Adding segment-segment avoidance modifies this result: for n chains with excluded volume interactions |Ca| ≈ (N/n)(1/2)f(ϕ) where f is a scaling function that depends approximately linearly on the segment concentration ϕ = nN/R(3). Scaling results for writhe are used to estimate the maximum writhe of a polymer; this is demonstrated to be realizable through a writhing instability that occurs for a polymer which is able to change knotting topology and which is subject to an applied torque. Finally, scaling results for linking are used to estimate bounds on the entanglement complexity of long chromosomal DNA molecules inside cells, and to show how "lengthwise" chromosome condensation can suppress DNA entanglement.

Linking topology of large DNA molecules

Double helix DNA molecules, the carriers of genetic instructions in cells, are strongly affected by their topological properties. Two distinct and biologically important types of linking are associated with double helix DNAs: 'internal' linking of the two strands of individual double helices, and 'external' linking of separate double helix DNAs. Constraint of internal linking gives rise to internal torsional stress and supercoiling of circular DNAs. External linking is a likely outcome of DNA replication, and must be eliminated by the cell in order to separate duplicated DNAs. I outline some of the physics and biology connected with both internal and external linking.

Strong intranucleoid interactions organize the Escherichia coli chromosome into a nucleoid filament

The stochasticity of chromosome organization was investigated by fluorescently labeling genetic loci in live Escherichia coli cells. In spite of the common assumption that the chromosome is well modeled by an unstructured polymer, measurements of the locus distributions reveal that the E. coli chromosome is precisely organized into a nucleoid filament with a linear order. Loci in the body of the nucleoid show a precision of positioning within the cell of better than 10% of the cell length. The precision of interlocus distance of genomically-proximate loci was better than 4% of the cell length. The measured dependence of the precision of interlocus distance on genomic distance singles out intranucleoid interactions as the mechanism responsible for chromosome organization. From the magnitude of the variance, we infer the existence of an as-yet uncharacterized higher-order DNA organization in bacteria. We demonstrate that both the stochastic and average structure of the nucleoid is captured by a fluctuating elastic filament model.

Simulation of DNA catenanes

DNA catenanes are important objects in biology, foremost as they appear during replication of circular DNA molecules. In this review we analyze how conformational properties of DNA catenanes can be studied by computer simulation. We consider classification of catenanes, their topological invariants and the methods of calculation of these invariants. We briefly analyze the DNA model and the simulation procedure used to sample the equilibrium conformational ensemble of catenanes with a particular topology. We consider how to avoid direct simulation of many DNA molecules when we need to account for the linking-unlinking process. The simulation methods and their comparisons with experiments are illustrated by some examples. We also describe an approach that allows simulating the steady state fraction of DNA catenanes created by type II topoisomerases.