Thermodynamics of long supercoiled molecules: insights from highly efficient Monte Carlo simulations

Statistical mechanics of multiplectoneme phases in DNA

A stretched DNA molecule that is also under- or overwound undergoes a buckling transition, forming intertwined looped domains called plectonemes. Here we develop a simple theory that extends the two-phase model of stretched supercoiled DNA, allowing for the coexistence of multiple plectonemic domains by including positional and length distribution entropies. Such a multiplectoneme phase is favored in long DNA molecules in which the gain of positional entropy compensates for the cost of nucleating a plectoneme along a stretched DNA segment. Despite its simplicity, the developed theory is shown to be in excellent agreement with Monte Carlo simulations of the twistable wormlike chain model. The theory predicts more plectonemes than experimentally observed, which we attribute to the limited resolution of experimental data. Since plectonemes are detected through fluorescence signals, those shorter than the observable threshold are likely missed.

DNA supercoiling in bacteria: state of play and challenges from a viewpoint of physics based modeling

DNA supercoiling is central to many fundamental processes of living organisms. Its average level along the chromosome and over time reflects the dynamic equilibrium of opposite activities of topoisomerases, which are required to relax mechanical stresses that are inevitably produced during DNA replication and gene transcription. Supercoiling affects all scales of the spatio-temporal organization of bacterial DNA, from the base pair to the large scale chromosome conformation. Highlighted in vitro and in vivo in the 1960s and 1970s, respectively, the first physical models were proposed concomitantly in order to predict the deformation properties of the double helix. About fifteen years later, polymer physics models demonstrated on larger scales the plectonemic nature and the tree-like organization of supercoiled DNA. Since then, many works have tried to establish a better understanding of the multiple structuring and physiological properties of bacterial DNA in thermodynamic equilibrium and far from equilibrium. The purpose of this essay is to address upcoming challenges by thoroughly exploring the relevance, predictive capacity, and limitations of current physical models, with a specific focus on structural properties beyond the scale of the double helix. We discuss more particularly the problem of DNA conformations, the interplay between DNA supercoiling with gene transcription and DNA replication, its role on nucleoid formation and, finally, the problem of scaling up models. Our primary objective is to foster increased collaboration between physicists and biologists. To achieve this, we have reduced the respective jargon to a minimum and we provide some explanatory background material for the two communities.

Equilibrium fluctuations of DNA plectonemes

Plectonemes are intertwined helically looped domains which form when a DNA molecule is supercoiled, i.e., over- or underwound. They are ubiquitous in cellular DNA, and their physical properties have attracted significant interest both from the experimental side and from the modeling side. In this paper, we investigate fluctuations of the end-point distance z of supercoiled linear DNA molecules subject to external stretching forces. Our analysis is based on a two-phase model, which describes the supercoiled DNA as composed of a stretched phase and a plectonemic phase. A variety of mechanisms are found to contribute to extension fluctuations, characterized by the variance 〈Δz^{2}〉. We find the dominant contribution to 〈Δz^{2}〉 to originate from phase-exchange fluctuations, the transient shrinking and expansion of plectonemes, which is accompanied by an exchange of molecular length between the two phases. We perform Monte Carlo simulations of the twistable wormlike chain and analyze the fluctuation of various quantities, the results of which are found to agree with the two-phase model predictions. Furthermore, we show that the extension and its variance at high forces are very well captured by the two-phase model, provided that one goes beyond quadratic approximations.

Monte Carlo simulation of a lattice model for the dynamics of randomly branching double-folded ring polymers

Supercoiled DNA, crumpled interphase chromosomes, and topologically constrained ring polymers often adopt treelike, double-folded, randomly branching configurations. Here we study an elastic lattice model for tightly double-folded ring polymers, which allows for the spontaneous creation and deletion of side branches coupled to a diffusive mass transport, which is local both in space and on the connectivity graph of the tree. We use Monte Carlo simulations to study systems falling into three different universality classes: ideal double-folded rings without excluded volume interactions, self-avoiding double-folded rings, and double-folded rings in the melt state. The observed static properties are in good agreement with exact results, simulations, and predictions of Flory theory for randomly branching polymers. For example, in the melt state rings adopt compact configurations and exhibit territorial behavior. In particular, we show that the emergent dynamics is in excellent agreement with a recent scaling theory and illustrate the qualitative differences with the familiar reptation dynamics of linear chains.

Supercoiled DNA and non-equilibrium formation of protein complexes: A quantitative model of the nucleoprotein ParBS partition complex

ParABS, the most widespread bacterial DNA segregation system, is composed of a centromeric sequence, parS, and two proteins, the ParA ATPase and the ParB DNA binding proteins. Hundreds of ParB proteins assemble dynamically to form nucleoprotein parS-anchored complexes that serve as substrates for ParA molecules to catalyze positioning and segregation events. The exact nature of this ParBS complex has remained elusive, what we address here by revisiting the Stochastic Binding model (SBM) introduced to explain the non-specific binding profile of ParB in the vicinity of parS. In the SBM, DNA loops stochastically bring loci inside a sharp cluster of ParB. However, previous SBM versions did not include the negative supercoiling of bacterial DNA, leading to use unphysically small DNA persistences to explain the ParB binding profiles. In addition, recent super-resolution microscopy experiments have revealed a ParB cluster that is significantly smaller than previous estimations and suggest that it results from a liquid-liquid like phase separation. Here, by simulating the folding of long (≥ 30 kb) supercoiled DNA molecules calibrated with realistic DNA parameters and by considering different possibilities for the physics of the ParB cluster assembly, we show that the SBM can quantitatively explain the ChIP-seq ParB binding profiles without any fitting parameter, aside from the supercoiling density of DNA, which, remarkably, is in accord with independent measurements. We also predict that ParB assembly results from a non-equilibrium, stationary balance between an influx of produced proteins and an outflux of excess proteins, i.e., ParB clusters behave like liquid-like protein condensates with unconventional “leaky” boundaries.

In bacteria, faithful genome inheritance requires the two replicated DNA molecules to be segregated at the opposite halves of the cell. ParABS, the most widespread bacterial DNA segregation system, is composed of a centromere sequence, parS, and two proteins, the ParA ATPase and the ParB DNA binding protein. Hundreds of ParB assemble dynamically to form clusters around parS, which then serve as substrates for ParA molecules to catalyze the positioning and segregation events. The nature of these clusters and their interaction with DNA have remained elusive. Here, we propose a realistic minimal model that captures quantitatively the peculiar DNA binding profile of ParB in the vicinity of parS in Escherichia coli. From the viewpoint of DNA, the only fitting parameter is the in vivo supercoiling density resulting from the removal of DNA helices by toposiomerases, which is in accord with previous independent estimations. From the viewpoint of ParB clusters, we predict that they behave like liquid-like protein condensates with unconventional boundaries. Namely, we predict boundaries to be leaky (i.e. not sharp) as a result of the non-equilibrium protein production, diffusion and dilution. Altogether, our work provides novel insights into bacterial DNA organization and intracellular liquid-liquid phase separation.

WASP: a software package for correctly characterizing the topological development of ribbon structures

We introduce the Writhe Application Software Package (WASP) which can be used to characterisze the topology of ribbon structures, the underlying mathematical model of DNA, Biopolymers, superfluid vorticies, elastic ropes and magnetic flux ropes. This characterization is achieved by the general twist–writhe decomposition of both open and closed ribbons, in particular through a quantity termed the polar writhe. We demonstrate how this decomposition is far more natural and straightforward than artificial closure methods commonly utilized in DNA modelling. In particular, we demonstrate how the decomposition of the polar writhe into local and non-local components distinctly characterizes the local helical structure and knotting/linking of the ribbon. This decomposition provides additional information not given by alternative approaches. As example applications, the WASP routines are used to characterise the evolving topology (writhe) of DNA minicircle and open ended plectoneme formation magnetic/optical tweezer simulations, and it is shown that the decomponsition into local and non-local components is particularly important for the detection of plectonemes. Finally it is demonstrated that a number of well known alternative writhe expressions are actually simplifications of the polar writhe measure.

Coarse-grained modelling of DNA plectoneme pinning in the presence of base-pair mismatches

Damaged or mismatched DNA bases result in the formation of physical defects in double-stranded DNA. In vivo, defects in DNA must be rapidly and efficiently repaired to maintain cellular function and integrity. Defects can also alter the mechanical response of DNA to bending and twisting constraints, both of which are important in defining the mechanics of DNA supercoiling. Here, we use coarse-grained molecular dynamics (MD) simulation and supporting statistical-mechanical theory to study the effect of mismatched base pairs on DNA supercoiling. Our simulations show that plectoneme pinning at the mismatch site is deterministic under conditions of relatively high force (>2 pN) and high salt concentration (>0.5 M NaCl). Under physiologically relevant conditions of lower force (0.3 pN) and lower salt concentration (0.2 M NaCl), we find that plectoneme pinning becomes probabilistic and the pinning probability increases with the mismatch size. These findings are in line with experimental observations. The simulation framework, validated with experimental results and supported by the theoretical predictions, provides a way to study the effect of defects on DNA supercoiling and the dynamics of supercoiling in molecular detail.

DNA elasticity from coarse-grained simulations: The effect of groove asymmetry

It is well established that many physical properties of DNA at sufficiently long length scales can be understood by means of simple polymer models. One of the most widely used elasticity models for DNA is the twistable worm-like chain (TWLC), which describes the double helix as a continuous elastic rod with bending and torsional stiffness. An extension of the TWLC, which has recently received some attention, is the model by Marko and Siggia, who introduced an additional twist-bend coupling, expected to arise from the groove asymmetry. By performing computer simulations of two available versions of oxDNA, a coarse-grained model of nucleic acids, we investigate the microscopic origin of twist-bend coupling. We show that this interaction is negligible in the oxDNA version with symmetric grooves, while it appears in the oxDNA version with asymmetric grooves. Our analysis is based on the calculation of the covariance matrix of equilibrium deformations, from which the stiffness parameters are obtained. The estimated twist-bend coupling coefficient from oxDNA simulations is G=30±1 nm. The groove asymmetry induces a novel twist length scale and an associated renormalized twist stiffness κ_t≈80 nm, which is different from the intrinsic torsional stiffness C≈110 nm. This naturally explains the large variations on experimental estimates of the intrinsic stiffness performed in the past.

Probing the salt dependence of the torsional stiffness of DNA by multiplexed magnetic torque tweezers

The mechanical properties of DNA fundamentally constrain and enable the storage and transmission of genetic information and its use in DNA nanotechnology. Many properties of DNA depend on the ionic environment due to its highly charged backbone. In particular, both theoretical analyses and direct single-molecule experiments have shown its bending stiffness to depend on salt concentration. In contrast, the salt-dependence of the twist stiffness of DNA is much less explored. Here, we employ optimized multiplexed magnetic torque tweezers to study the torsional stiffness of DNA under varying salt conditions as a function of stretching force. At low forces (<3 pN), the effective torsional stiffness is ∼10% smaller for high salt conditions (500 mM NaCl or 10 mM MgCl₂) compared to lower salt concentrations (20 mM NaCl and 100 mM NaCl). These differences, however, can be accounted for by taking into account the known salt dependence of the bending stiffness. In addition, the measured high-force (6.5 pN) torsional stiffness values of C = 103 ± 4 nm are identical, within experimental errors, for all tested salt concentration, suggesting that the intrinsic torsional stiffness of DNA does not depend on salt.

Twist-Bend Coupling and the Torsional Response of Double-Stranded DNA

Recent magnetic tweezers experiments have reported systematic deviations of the twist response of double-stranded DNA from the predictions of the twistable wormlike chain model. Here we show, by means of analytical results and computer simulations, that these discrepancies can be resolved if a coupling between twist and bend is introduced. We obtain an estimate of 40±10  nm for the twist-bend coupling constant. Our simulations are in good agreement with high-resolution, magnetic-tweezers torque data. Although the existence of twist-bend coupling was predicted long ago [J. Marko and E. Siggia, Macromolecules 27, 981 (1994)MAMOBX0024-929710.1021/ma00082a015], its effects on the mechanical properties of DNA have been so far largely unexplored. We expect that this coupling plays an important role in several aspects of DNA statics and dynamics.

Modeling Bacterial DNA: Simulation of Self-Avoiding Supercoiled Worm-Like Chains Including Structural Transitions of the Helix

Under supercoiling constraints, naked DNA, such as a large part of bacterial DNA, folds into braided structures called plectonemes. The double-helix can also undergo local structural transitions, leading to the formation of denaturation bubbles and other alternative structures. Various polymer models have been developed to capture these properties, with Monte-Carlo (MC) approaches dedicated to the inference of thermodynamic properties. In this chapter, we explain how to perform such Monte-Carlo simulations, following two objectives. On one hand, we present the self-avoiding supercoiled Worm-Like Chain (ssWLC) model, which is known to capture the folding properties of supercoiled DNA, and provide a detailed explanation of a standard MC simulation method. On the other hand, we explain how to extend this ssWLC model to include structural transitions of the helix.

Probing the mechanical properties, conformational changes, and interactions of nucleic acids with magnetic tweezers

Nucleic acids are central to the storage and transmission of genetic information. Mechanical properties, along with their sequence, both enable and fundamentally constrain the biological functions of DNA and RNA. For small deformations from the equilibrium conformations, nucleic acids are well described by an isotropic elastic rod model. However, external forces and torsional strains can induce conformational changes, giving rise to a complex force-torque phase diagram. This review focuses on magnetic tweezers as a powerful tool to precisely determine both the elastic parameters and conformational transitions of nucleic acids under external forces and torques at the single-molecule level. We review several variations of magnetic tweezers, in particular conventional magnetic tweezers, freely orbiting magnetic tweezers and magnetic torque tweezers, and discuss their characteristic capabilities. We then describe the elastic rod model for DNA and RNA and discuss conformational changes induced by mechanical stress. The focus lies on the responses to torque and twist, which are crucial in the mechanics and interactions of nucleic acids and can directly be measured using magnetic tweezers. We conclude by highlighting several recent studies of nucleic acid-protein and nucleic acid-small-molecule interactions as further applications of magnetic tweezers and give an outlook of some exciting developments to come.