Induction of entropic segregation: the first step is the hardest

Loop-extruders alter bacterial chromosome topology to direct entropic forces for segregation

Entropic forces have been argued to drive bacterial chromosome segregation during replication. In many bacterial species, however, specifically evolved mechanisms, such as loop-extruding SMC complexes and the ParABS origin segregation system, contribute to or are even required for chromosome segregation, suggesting that entropic forces alone may be insufficient. The interplay between and the relative contributions of these segregation mechanisms remain unclear. Here, we develop a biophysical model showing that purely entropic forces actually inhibit bacterial chromosome segregation until late replication stages. By contrast, our model reveals that loop-extruders loaded at the origins of replication, as observed in many bacterial species, alter the effective topology of the chromosome, thereby redirecting and enhancing entropic forces to enable accurate chromosome segregation during replication. We confirm our model predictions with polymer simulations: purely entropic forces do not allow for concurrent replication and segregation, whereas entropic forces steered by specifically loaded loop-extruders lead to robust, global chromosome segregation during replication. Finally, we show how loop-extruders can complement locally acting origin separation mechanisms, such as the ParABS system. Together, our results illustrate how changes in the geometry and topology of the polymer, induced by DNA-replication and loop-extrusion, impact the organization and segregation of bacterial chromosomes.

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.

Equilibrium behaviour of two cavity-confined polymers: effects of polymer width and system asymmetries

Experiments using nanofluidic devices have proven effective in characterizing the physical properties of polymers confined to small cavities. Two recent studies using such methods examined the organization and dynamics of two DNA molecules in box-like cavities with strong confinement in one direction and with square and elliptical cross sections in the lateral plane. Motivated by these experiments, we employ Monte Carlo and Brownian dynamics simulations to study the physical behaviour of two polymers confined to small cavities with shapes comparable to those used in the experiments. We quantify the effects of varying the following polymer properties and confinement dimensions on the organization and dynamics of the polymers: the polymer width, the polymer contour length ratio, the cavity cross-sectional area, and the degree of cavity elongation for cavities with rectangular and elliptical cross sections. We find that the tendency for polymers to segregate is enhanced by increasing polymer width. For sufficiently small cavities, increasing cavity elongation promotes segregation and localization of identical polymers to opposite sides of the cavity along its long axis. A free-energy barrier controls the rate of polymers swapping positions, and the observed dynamics are roughly in accord with predictions of a simple theoretical model. Increasing the contour length difference between polymers significantly affects their organization in the cavity. In the case of a large linear polymer co-trapped with a small ring polymer in an elliptical cavity, the small polymer tends to lie near the lateral confining walls, and especially at the cavity poles for highly elongated ellipses.

Topology-driven spatial organization of ring polymers under confinement

Entropic repulsion between DNA ring polymers under confinement is a key mechanism governing the spatial segregation of bacterial DNA before cell division. Here we establish that "internal" loops within a modified-ring polymer architecture enhance entropic repulsion between two overlapping polymers confined in a cylinder. Interestingly, they also induce entropy-driven spatial organization of polymer segments as seen in vivo. Here we design polymers of different architectures in our simulations by introducing a minimal number of cross-links between specific monomers along the ring-polymer contour. The cross-links are likely induced by various bridging proteins inside living cells. We investigate the segregation of two polymers with modified topologies confined in a cylinder, which initially had spatially overlapping configurations. This helps us to identify the architectures that lead to higher success rates of segregation. We also establish the mechanism that leads to localization of specific polymer segments. We use the blob model to provide a theoretical understanding of why certain architectures lead to enhanced entropic repulsive forces between the polymers. Lastly, we establish a correspondence between the organizational patterns of the chromosome of the C.crescentus bacterium and our results for a specifically designed polymer architecture. However, the principles outlined here pertaining to the organization of polymeric segments are applicable to both synthetic and biological polymers.

Machine learning classification of trajectories from molecular dynamics simulations of chromosome segregation

In contrast to the well characterized mitotic machinery in eukaryotes it seems as if there is no universal mechanism organizing chromosome segregation in all bacteria. Apparently, some bacteria even use combinations of different segregation mechanisms such as protein machines or rely on physical forces. The identification of the relevant mechanisms is a difficult task. Here, we introduce a new machine learning approach to this problem. It is based on the analysis of trajectories of individual loci in the course of chromosomal segregation obtained by fluorescence microscopy. While machine learning approaches have already been applied successfully to trajectory classification in other areas, so far it has not been possible to use them to discriminate segregation mechanisms in bacteria. A main obstacle for this is the large number of trajectories required to train machine learning algorithms that we overcome here by using trajectories obtained from molecular dynamics simulations. We used these trajectories to train four different machine learning algorithms, two linear models and two tree-based classifiers, to discriminate segregation mechanisms and possible combinations of them. The classification was performed once using the complete trajectories as high-dimensional input vectors as well as on a set of features which were used to transform the trajectories into low-dimensional input vectors for the classifiers. Finally, we tested our classifiers on shorter trajectories with duration times comparable (or even shorter) than typical experimental trajectories and on trajectories measured with varying temporal resolutions. Our results demonstrate that machine learning algorithms are indeed capable of discriminating different segregation mechanisms in bacteria and to even resolve combinations of the mechanisms on rather short time scales.

Equilibrium organization, conformation, and dynamics of two polymers under box-like confinement

Motivated by recent nanofluidics experiments, we use Brownian dynamics and Monte Carlo simulations to study the conformation, organization and dynamics of two polymer chains confined to a single box-like cavity. The polymers are modeled as flexible bead-spring chains, and the box has a square cross-section of side length L and a height that is small enough to compress the polymers in that dimension. For sufficiently large L, the system behaviour approaches that of an isolated polymer in a slit. However, the combined effects of crowding and confinement on the polymer organization, conformation and equilibrium dynamics become significant when where is the transverse radius of gyration for a slit geometry. In this regime, the centre-of-mass probability distribution in the transverse plane exhibits a depletion zone near the centre of the cavity (except at very small L) and a 4-fold symmetry with quasi-discrete positions. Reduction in polymer size with decreasing L arises principally from confinement rather than inter-polymer crowding. By contrast, polymer diffusion and internal motion are strongly affected by inter-polymer crowding. The two polymers tend to occupy opposite positions relative to the box centre, about which they diffuse relatively freely. Qualitatively, this static and dynamical behaviour differs significantly from that previously observed for confinement of two polymers to a narrow channel. The simulation results for a suitably chosen box width are qualitatively consistent with results from a recent experimental study of two λ-DNA chains confined to a nanofluidic cavity.

Free energy and segregation dynamics of two channel-confined polymers of different lengths

Polymers confined to a narrow channel are subject to strong entropic forces that tend to drive the molecules apart. In this study, we use Monte Carlo computer simulations to study the segregation behavior of two flexible hard-sphere polymers under confinement in a cylindrical channel. We focus on the effects of using polymers of different lengths. We measure the variation of the conformational free energy, F, with the center-of-mass separation distance, λ. The simulations reveal four different separation regimes, characterized by different scaling properties of the free energy with respect to the polymer lengths and the channel diameter, D. We propose a regime map in which the state of the system is determined by the values of the quantities N_{2}/N_{1} and λ/(N_{1}+N_{2})D^{-β}, where N_{1} and N_{2} are the polymer lengths, and where β≈0.64. The observed scaling behavior of F(λ) in each regime is in reasonable agreement with predictions using a simple theoretical model. In addition, we use MC dynamics simulations to study the segregation dynamics of initially overlapping polymers by measurement of the incremental mean first-passage time with respect to λ. For systems characterized by a wide range of λ in which a short polymer is nested within a longer one, the segregation dynamics are close to that expected for two noninteracting one-dimensional random walkers undergoing unbiased diffusion. When the free-energy gradient is large, segregation is rapid and characterized by out-of-equilibrium effects.

Geometric principles underlying the proliferation of a model cell system

Many bacteria can form wall-deficient variants, or L-forms, that divide by a simple mechanism that does not require the FtsZ-based cell division machinery. Here, we use microfluidic systems to probe the growth, chromosome cycle and division mechanism of Bacillus subtilis L-forms. We find that forcing cells into a narrow linear configuration greatly improves the efficiency of cell growth and chromosome segregation. This reinforces the view that L-form division is driven by an excess accumulation of surface area over volume. Cell geometry also plays a dominant role in controlling the relative positions and movement of segregating chromosomes. Furthermore, the presence of the nucleoid appears to influence division both via a cell volume effect and by nucleoid occlusion, even in the absence of FtsZ. Our results emphasise the importance of geometric effects for a range of crucial cell functions, and are of relevance for efforts to develop artificial or minimal cell systems.

Chromosome Segregation in Bacillus subtilis Follows an Overall Pattern of Linear Movement and Is Highly Robust against Cell Cycle Perturbations

We have followed the segregation of origin regions on the Bacillus subtilis chromosome in the fastest practically achievable temporal manner, for a large fraction of the cell cycle. We show that segregation occurred in highly variable patterns but overall in an almost linear manner throughout the cell cycle. Segregation was slowed down, but not arrested, by treatment of cells that led to transient blocks in DNA replication, showing that segregation is highly robust against cell cycle perturbation. Computer simulations based on entropy-driven separation of newly synthesized DNA polymers can recapitulate sudden bursts of movement and segregation patterns compatible with the observed in vivo patterns, indicating that for Bacillus, segregation patterns may include entropic forces helping to separate chromosomes during the cell cycle.

Although several proteins have been identified that facilitate chromosome segregation in bacteria, no clear analogue of the mitotic machinery in eukaryotic cells has been identified. In order to investigate if recognizable patterns of segregation exist during the cell cycle, we tracked the segregation of duplicated origin regions in Bacillus subtilis for 60 min in the fastest practically achievable resolution, achieving 10-s intervals. We found that while separation occurred in random patterns, often including backwards movement, overall, segregation of loci near the origins of replication was linear for the entire cell cycle. Thus, the process of partitioning can be best described as directed motion. Simulations with entropy-driven separation of polymers synthesized by two polymerases show sudden bursts of movement and segregation patterns compatible with the observed in vivo patterns, showing that for Bacillus, segregation patterns can be modeled based on entropic forces. To test if obstacles for replication forks lead to an alteration of the partitioning pattern, we challenged cells with chemicals inducing DNA damage or blocking of topoisomerase activity. Both treatments led to a moderate slowing down of separation, but linear segregation was retained, showing that chromosome segregation is highly robust against cell cycle perturbation.

IMPORTANCE We have followed the segregation of origin regions on the Bacillus subtilis chromosome in the fastest practically achievable temporal manner, for a large fraction of the cell cycle. We show that segregation occurred in highly variable patterns but overall in an almost linear manner throughout the cell cycle. Segregation was slowed down, but not arrested, by treatment of cells that led to transient blocks in DNA replication, showing that segregation is highly robust against cell cycle perturbation. Computer simulations based on entropy-driven separation of newly synthesized DNA polymers can recapitulate sudden bursts of movement and segregation patterns compatible with the observed in vivo patterns, indicating that for Bacillus, segregation patterns may include entropic forces helping to separate chromosomes during the cell cycle.

Self-organised segregation of bacterial chromosomal origins

The chromosomal replication origin region (ori) of characterised bacteria is dynamically positioned throughout the cell cycle. In slowly growing Escherichia coli, ori is maintained at mid-cell from birth until its replication, after which newly replicated sister oris move to opposite quarter positions. Here, we provide an explanation for ori positioning based on the self-organisation of the Structural Maintenance of Chromosomes complex, MukBEF, which forms dynamically positioned clusters on the chromosome. We propose that a non-trivial feedback between the self-organising gradient of MukBEF complexes and the oris leads to accurate ori positioning. We find excellent agreement with quantitative experimental measurements and confirm key predictions. Specifically, we show that oris exhibit biased motion towards MukBEF clusters, rather than mid-cell. Our findings suggest that MukBEF and oris act together as a self-organising system in chromosome organisation-segregation and introduces protein self-organisation as an important consideration for future studies of chromosome dynamics.

Segregation of polymers under cylindrical confinement: effects of polymer topology and crowding

Monte Carlo computer simulations are used to study the segregation behaviour of two polymers under cylindrical confinement. Using a multiple-histogram method, the conformational free energy, F, of the polymers was measured as a function of the centre-of-mass separation distance, λ. We examined the scaling of the free energy functions with the polymer length, the length and diameter of the confining cylinder, the polymer topology (i.e. linear vs. ring polymers), and the packing fraction and size of mobile crowding agents. In the absence of crowders, the observed scaling of F(λ) is similar to that predicted using a simple model employing the de Gennes blob model and the approximation that the free energy of overlapping chains in a tube is equal to that of two isolated chains each in a tube of half the cross-sectional area. Simulations were used to test the latter approximation and reveal that it yields poor quantitative predictions. This, along with generic finite-size effects, likely gives rise to the discrepancies between the predicted and measured values of scaling exponents for F(λ). For segregation in the presence of crowding agents, the free energy barrier generally decreases with increasing crowder packing fraction, thus reducing the entropic forces driving segregation. However, for fixed packing fraction, the barrier increases as the crowder/monomer size ratio decreases.

Surface Segregation of Cyclic Chains in Binary Melts of Thin Polymer Films: The Influence of Constituent Concentration

We carry out extensive molecular dynamics simulations of thin films of bead-spring models of binary mixtures composed of cyclic and linear polymer chains. We study the equilibrium behavior of the polymer chains for two very different chain lengths, which resemble short (10-mers) and long (100-mers) chains, at different concentrations of the binary mixture. We clearly show how the concentration variable affects the enrichment of either of the two polymer species at the interface, and also how the chain length influences this process.

Escherichia coli Chromosomal Loci Segregate from Midcell with Universal Dynamics

The structure of the Escherichia coli chromosome is inherently dynamic over the duration of the cell cycle. Genetic loci undergo both stochastic motion around their initial positions and directed motion to opposite poles of the rod-shaped cell during segregation. We developed a quantitative method to characterize cell-cycle dynamics of the E. coli chromosome to probe the chromosomal steady-state mobility and segregation process. By tracking fluorescently labeled chromosomal loci in thousands of cells throughout the entire cell cycle, our method allows for the statistical analysis of locus position and motion, the step-size distribution for movement during segregation, and the locus drift velocity. The robust statistics of our detailed analysis of the wild-type E. coli nucleoid allow us to observe loci moving toward midcell before segregation occurs, consistent with a replication factory model. Then, as segregation initiates, we perform a detailed characterization of the average segregation velocity of loci. Contrary to origin-centric models of segregation, which predict distinct dynamics for oriC-proximal versus oriC-distal loci, we find that the dynamics of loci were universal and independent of genetic position.

Simulating the entropic collapse of coarse-grained chromosomes

Depletion forces play a role in the compaction and decompaction of chromosomal material in simple cells, but it has remained debatable whether they are sufficient to account for chromosomal collapse. We present coarse-grained molecular dynamics simulations, which reveal that depletion-induced attraction is sufficient to cause the collapse of a flexible chain of large structural monomers immersed in a bath of smaller depletants. These simulations use an explicit coarse-grained computational model that treats both the supercoiled DNA structural monomers and the smaller protein crowding agents as combinatorial, truncated Lennard-Jones spheres. By presenting a simple theoretical model, we quantitatively cast the action of depletants on supercoiled bacterial DNA as an effective solvent quality. The rapid collapse of the simulated flexible chromosome at the predicted volume fraction of depletants is a continuous phase transition. Additional physical effects to such simple chromosome models, such as enthalpic interactions between structural monomers or chain rigidity, are required if the collapse is to be a first-order phase transition.