Global force-torque phase diagram for the DNA double helix: structural transitions, triple points, and collapsed plectonemes

The Free Energy of Nucleosomal DNA Based on the Landau Model and Topology

The free energy of nucleosomal DNA plays a key role in the formation of nucleosomes in eukaryotes. Some work on the free energy of nucleosomal DNA have been carried out in experiments. However, the relationships between the free energy of nucleosomal DNA and its conformation, especially its topology, remain unclear in theory. By combining the Landau theory, the Hopfion model and experimental data, we find that the free energy of nucleosomal DNA is at the lower level. With the help of the energy minimum principle, we conclude that nucleosomal DNA stays in a stable state. Moreover, we discover that small perturbations on nucleosomal DNA have little effect on its free energy. This implies that nucleosomal DNA has a certain redundancy in order to stay stable. This explains why nucleosomal DNA will not change significantly due to small perturbations.

Traveling Perversion as Constant Torque Actuator

Mechanical stress and conformation of helical elastic rods clamped at both ends were studied upon unwinding. By axial rotation of one end, the winding number was progressively changed from the natural one (n=n_{0}) to complete chirality inversion (n=-n_{0}) while keeping the total elongation fixed and monitoring the applied torque M and tension T. Along the unwinding process, the system crosses three distinct states: natural helix (+), mixed state (+/-), and inverted helix (-). The mixed state involves two helices with opposite chiralities spatially connected by a perversion (helicity inversion). Upon unwinding, the perversion is "injected" (nucleated) from one side and travels toward the opposite side where it is eventually "absorbed" (annihilated), leaving the system in the (-) state. In the mixed state, the profile of M(n) is almost flat: the system behaves as a constant torque actuator. The three states are quantitatively well described in the framework of a biphasic model which neglects the perversion energy and finite size effects. The latter are taken into account in a numerical simulation based on the Kirchhoff theory of elastic rods. The traveling perversion in helical elastic rods and related topological phenomena are universal, with applications from condensed matter to biological and bioinspired systems, including in particular mechanical engineering and soft robotics.

The Origin of Left-Handed Poly[d(G-C)]

The discovery of a reversible transition in the helical sense of a double-helical DNA was initiated by the first synthesis in 1967 of the alternating sequence poly[d(G-C)]. In 1968, exposure to high salt concentration led to a cooperative isomerization of the double helix manifested by an inversion in the CD spectrum in the 240-310 nm range and in an altered absorption spectrum. The tentative interpretation, reported in 1970 and then in detailed form in a 1972 publication by Pohl and Jovin, was that the conventional right-handed B-DNA structure (R) of poly[d(G-C)] transforms at high salt concentration into a novel, alternative left-handed (L) conformation. The historical course of this development and its aftermath, culminating in the first crystal structure of left-handed Z-DNA in 1979, is described in detail. The research conducted by Pohl and Jovin after 1979 is summarized, ending with an assessment of "unfinished business": condensed Z*-DNA; topoisomerase IIα (TOP2A) as an allosteric ZBP (Z-DNA-binding protein); B-Z transitions of phosphorothioate-modified DNAs; and parallel-stranded poly[d(G-A)], a double helix with high stability under physiological conditions and potentially also left-handed.

Structural domains of SARS-CoV-2 nucleocapsid protein coordinate to compact long nucleic acid substrates

The SARS-CoV-2 nucleocapsid (N) protein performs several functions including binding, compacting, and packaging the ∼30 kb viral genome into the viral particle. N protein consists of two ordered domains, with the N terminal domain (NTD) primarily associated with RNA binding and the C terminal domain (CTD) primarily associated with dimerization/oligomerization, and three intrinsically disordered regions, an N-arm, a C-tail, and a linker that connects the NTD and CTD. We utilize an optical tweezers system to isolate a long single-stranded nucleic acid substrate to measure directly the binding and packaging function of N protein at a single molecule level in real time. We find that N protein binds the nucleic acid substrate with high affinity before oligomerizing and forming a highly compact structure. By comparing the activities of truncated protein variants missing the NTD, CTD, and/or linker, we attribute specific steps in this process to the structural domains of N protein, with the NTD driving initial binding to the substrate and ensuring high localized protein density that triggers interprotein interactions mediated by the CTD, which forms a compact and stable protein-nucleic acid complex suitable for packaging into the virion.

DNA fluctuations reveal the size and dynamics of topological domains

DNA supercoiling is a key regulatory mechanism that orchestrates DNA readout, recombination, and genome maintenance. DNA-binding proteins often mediate these processes by bringing two distant DNA sites together, thereby inducing (transient) topological domains. In order to understand the dynamics and molecular architecture of protein-induced topological domains in DNA, quantitative and time-resolved approaches are required. Here, we present a methodology to determine the size and dynamics of topological domains in supercoiled DNA in real time and at the single-molecule level. Our approach is based on quantifying the extension fluctuations-in addition to the mean extension-of supercoiled DNA in magnetic tweezers (MT). Using a combination of high-speed MT experiments, Monte Carlo simulations, and analytical theory, we map out the dependence of DNA extension fluctuations as a function of supercoiling density and external force. We find that in the plectonemic regime, the extension variance increases linearly with increasing supercoiling density and show how this enables us to determine the formation and size of topological domains. In addition, we demonstrate how the transient (partial) dissociation of DNA-bridging proteins results in the dynamic sampling of different topological states, which allows us to deduce the torsional stiffness of the plectonemic state and the kinetics of protein-plectoneme interactions. We expect our results to further the understanding and optimization of magnetic tweezer measurements and to enable quantification of the dynamics and reaction pathways of DNA processing enzymes in the context of physiologically relevant forces and supercoiling densities.

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.

Mechanical response to tension and torque of molecular chains via statistically interacting particles associated with extension, contraction, twist, and supercoiling

A methodology for the statistical mechanical analysis of polymeric chains under tension introduced previously is extended to include torque. The response of individual bonds between monomers or of entire groups of monomers to a combination of tension and torque involves, in the framework of this method of analysis, the (thermal or mechanical) activation of a specific mix of statistically interacting particles carrying quanta of extension or contraction and quanta of twist or supercoiling. The methodology, which is elucidated in applications of increasing complexity, is capable of describing the conversion between twist chirality and plectonemic chirality in quasistatic processes. The control variables are force or extension and torque or linkage (a combination of twist and writhe). The versatility of this approach is demonstrated in two applications relevant and promising for double-stranded DNA under controlled tension and torque. One application describes conformational transformations between (native) B-DNA, (underwound) S-DNA, and (overwound) P-DNA in accord with experimental data. The other application describes how the conversion between a twisted chain and a supercoiled chain accommodates variations of linkage and excess length in a buckling transition.

DNA supercoiling-mediated collective behavior of co-transcribing RNA polymerases

Multiple RNA polymerases (RNAPs) transcribing a gene have been known to exhibit collective group behavior, causing the transcription elongation rate to increase with the rate of transcription initiation. Such behavior has long been believed to be driven by a physical interaction or ‘push’ between closely spaced RNAPs. However, recent studies have posited that RNAPs separated by longer distances may cooperate by modifying the DNA segment under transcription. Here, we present a theoretical model incorporating the mechanical coupling between RNAP translocation and the DNA torsional response. Using stochastic simulations, we demonstrate DNA supercoiling-mediated long-range cooperation between co-transcribing RNAPs. We find that inhibiting transcription initiation can slow down the already recruited RNAPs, in agreement with recent experimental observations, and predict that the average transcription elongation rate varies non-monotonically with the rate of transcription initiation. We further show that while RNAPs transcribing neighboring genes oriented in tandem can cooperate, those transcribing genes in divergent or convergent orientations can act antagonistically, and that such behavior holds over a large range of intergenic separations. Our model makes testable predictions, revealing how the mechanical interplay between RNAPs and the DNA they transcribe can govern transcriptional dynamics.

Writhing and hockling instabilities in twisted elastic fibers

The buckling and twisting of slender, elastic fibers is a deep and well-studied field. A slender elastic rod that is twisted with respect to a fixed end will spontaneously form a loop, or hockle, to relieve the torsional stress that builds. Further twisting results in the formation of plectonemes-a helical excursion in the fiber that extends with additional twisting. Here we use an idealized, micron-scale experiment to investigate the energy stored, and subsequently released, by hockles and plectonemes as they are pulled apart, in analogy with force spectroscopy studies of DNA and protein folding. Hysteresis loops in the snapping and unsnapping inform the stored energy in the twisted fiber structures.

Nanomechanics of negatively supercoiled diaminopurine-substituted DNA

Single molecule experiments have demonstrated a progressive transition from a B- to an L-form helix as DNA is gently stretched and progressively unwound. The particular sequence of a DNA segment defines both base stacking and hydrogen bonding that affect the partitioning and conformations of the two phases. Naturally or artificially modified bases alter H-bonds and base stacking and DNA with diaminopurine (DAP) replacing adenine was synthesized to produce linear fragments with triply hydrogen-bonded DAP:T base pairs. Both unmodified and DAP-substituted DNA transitioned from a B- to an L-helix under physiological conditions of mild tension and unwinding. This transition avoids writhing and the ease of this transition may prevent cumbersome topological rearrangements in genomic DNA that would require topoisomerase activity to resolve. L-DNA displayed about tenfold lower persistence length than B-DNA. However, left-handed DAP-substituted DNA was twice as stiff as unmodified L-DNA. Unmodified DNA and DAP-substituted DNA have very distinct mechanical characteristics at physiological levels of negative supercoiling and tension.

Force spectroscopy with electromagnetic tweezers

Force spectroscopy using magnetic tweezers (MTs) is a powerful method to probe the physical characteristics of single polymers. Typically, molecules are functionalized for specific attachment to a glass surface at one end and a micrometer-scale paramagnetic bead at the other end. By applying an external magnetic field, multiple molecules can be stretched and twisted simultaneously without exposure to potentially damaging radiation. The majority of MTs utilize mobile, permanent magnets to produce forces on the beads (and the molecule under test). However, translating and rotating the permanent magnets may require expensive precision actuators, limit the rate at which force can be changed, and may induce vibrations that disturb tether dynamics and bead tracking. Alternatively, the magnetic field can be produced with an electromagnet, which allows fast force modulation and eliminates motor-associated vibration. Here, we describe a low-cost quadrapolar electromagnetic tweezer design capable of manipulating DNA-tethered MyOne paramagnetic beads with forces as high as 15 pN. The solid-state nature of the generated B-field modulated along two axes is convenient for accessing the range of forces and torques relevant for studying the activity of DNA motor enzymes like polymerases and helicases. Our design specifically leverages technology available at an increasing number of university maker spaces and student-run machine shops. Thus, it is an accessible tool for undergraduate education that is applicable to a wide range of biophysical research questions.

Single-molecule insights into torsion and roadblocks in bacterial transcript elongation

During transcription, RNA polymerase (RNAP) translocates along the helical template DNA while maintaining high transcriptional fidelity. However, all genomes are dynamically twisted, writhed, and decorated by bound proteins and motor enzymes. In prokaryotes, proteins bound to DNA, specifically or not, frequently compact DNA into conformations that may silence genes by obstructing RNAP. Collision of RNAPs with these architectural proteins, may result in RNAP stalling and/or displacement of the protein roadblock. It is important to understand how rapidly transcribing RNAPs operate under different levels of supercoiling or in the presence of roadblocks. Given the broad range of asynchronous dynamics exhibited by transcriptional complexes, single-molecule assays, such as atomic force microscopy, fluorescence detection, optical and magnetic tweezers, etc. are well suited for detecting and quantifying activity with adequate spatial and temporal resolution. Here, we summarize current understanding of the effects of torsion and roadblocks on prokaryotic transcription, with a focus on single-molecule assays that provide real-time detection and readout.

Nanomechanics of G-quadruplexes within the promoter of the KIT oncogene

G-quadruplexes (G4s) are tetrahelical DNA structures stabilized by four guanines paired via Hoogsteen hydrogen bonds into quartets. While their presence within eukaryotic DNA is known to play a key role in regulatory processes, their functional mechanisms are still under investigation. In the present work, we analysed the nanomechanical properties of three G4s present within the promoter of the KIT proto-oncogene from a single-molecule point of view through the use of magnetic tweezers (MTs). The study of DNA extension fluctuations under negative supercoiling allowed us to identify a characteristic fingerprint of G4 folding. We further analysed the energetic contribution of G4 to the double-strand denaturation process in the presence of negative supercoiling, and we observed a reduction in the energy required for strands separation.

Anomalous Laterally Stressed Kinetically Trapped DNA Surface Conformations

DNA kinking is inevitable for the highly anisotropic 1D-1D electrostatic interaction with the one-dimensionally periodically charged surface. The double helical structure of the DNA kinetically trapped on positively charged monomolecular films comprising the lamellar templates is strongly laterally stressed and extremely perturbed at the nanometer scale. The DNA kinetic trapping is not a smooth 3D-> 2D conformational flattening but is a complex nonlinear in-plane mechanical response (bending, tensile and unzipping) driven by the physics beyond the scope of the applicability of the linear worm-like chain approximation. Up to now, the DNA molecule adsorbed on a surface was believed to always preserve its native structure. This belief implies a negligible contribution of lateral surface forces during and after DNA adsorption although their impact has never been elucidated. High-resolution atomic force microscopy was used to observe that stiff DNA molecules kinetically trapped on monomolecular films comprising one-dimensional periodically charged lamellar templates as a single layer or as a sublayer are oversaturated by sharp discontinuous kinks and can also be locally melted and supercoiled. We argue that kink/anti-kink pairs are induced by an overcritical lateral bending stress (> 30 pNnm) inevitable for the highly anisotropic 1D-1D electrostatic interaction of DNA and underlying rows of positive surface charges. In addition, the unexpected kink-inducing mechanical instability in the shape of the template-directed DNA confined between the positively charged lamellar sides is observed indicating the strong impact of helicity. The previously reported anomalously low values of the persistence length of the surface-adsorbed DNA are explained by the impact of the surface-induced low-scale bending. The sites of the local melting and supercoiling are convincingly introduced as other lateral stress-induced structural DNA anomalies by establishing a link with DNA high-force mechanics. The results open up the study in the completely unexplored area of the principally anomalous kinetically trapped DNA surface conformations in which the DNA local mechanical response to the surface-induced spatially modulated lateral electrostatic stress is essentially nonlinear. The underlying rich and complex in-plane nonlinear physics acts at the nanoscale beyond the scope of applicability of the worm-like chain approximation.

Energetics of twisted DNA topologies

Our goal is to review the main theoretical models used to calculate free energy changes associated with common, torsion-induced conformational changes in DNA and provide the resulting equations hoping to facilitate quantitative analysis of both in vitro and in vivo studies. This review begins with a summary of work regarding the energy change of the negative supercoiling-induced B- to L-DNA transition, followed by a discussion of the energetics associated with the transition to Z-form DNA. Finally, it describes the energy changes associated with the formation of DNA curls and plectonemes, which can regulate DNA-protein interactions and promote cross talk between distant DNA elements, respectively. The salient formulas and parameters for each scenario are summarized in table format to facilitate comparison and provide a concise, user-friendly resource.

Single-molecule micromanipulation studies of methylated DNA

Cytosine methylated at the five-carbon position is the most widely studied reversible DNA modification. Prior findings indicate that methylation can alter mechanical properties. However, those findings were qualitative and sometimes contradictory, leaving many aspects unclear. By applying single-molecule magnetic force spectroscopy techniques allowing for direct manipulation and dynamic observation of DNA mechanics and mechanically driven strand separation, we investigated how CpG and non-CpG cytosine methylation affects DNA micromechanical properties. We quantitatively characterized DNA stiffness using persistence length measurements from force-extension curves in the nanoscale length regime and demonstrated that cytosine methylation results in longer contour length and increased DNA flexibility (i.e., decreased persistence length). In addition, we observed the preferential formation of plectonemes over unwound single-stranded "bubbles" of DNA under physiologically relevant stretching forces and supercoiling densities. The flexibility and high structural stability of methylated DNA is likely to have significant consequences on the recruitment of proteins recognizing cytosine methylation and DNA packaging.

Molecular chains under tension: Thermal and mechanical activation of statistically interacting extension and contraction particles

This work introduces a methodology for the statistical mechanical analysis of polymeric chains under tension controlled by optical or magnetic tweezers at thermal equilibrium with an embedding fluid medium. The response of single bonds between monomers or of entire groups of monomers to tension is governed by the activation of statistically interacting particles representing quanta of extension or contraction. This method of analysis is capable of describing thermal unbending of the freely jointed or wormlike chain kind, linear or nonlinear contour elasticity, and structural transformations including effects of cooperativity. The versatility of this approach is demonstrated in an application to double-stranded DNA undergoing torsionally unconstrained stretching across three regimes of mechanical response including an overstretching transition. The three-regime force-extension characteristic, derived from a single free-energy expression, accurately matches empirical evidence.

Influence of Nucleoid-Associated Proteins on DNA Supercoiling

DNA supercoiling, where the DNA strand forms a writhe to relieve torsional stress, plays a vital role in packaging the genetic material in cells. Experiment, simulation, and theory have all demonstrated how supercoiling emerges due to the over- or underwinding of the DNA strand. Nucleoid-associated proteins (NAPs) help structure DNA in prokaryotes, yet the role that they play in the supercoiling process has not been as thoroughly investigated. We develop a coarse-grained simulation to model DNA supercoiling in the presence of proteins, providing a rigorous physical understanding of how NAPs affect supercoiling behavior. Specifically, we demonstrate how the force and torque necessary to form supercoils are affected by the presence of NAPs. NAPs that bend DNA stabilize the supercoil, thus shifting the transition between extended and supercoiled DNAs. We develop a theory to explain how NAP binding affects DNA supercoiling. This provides insight into how NAPs modulate DNA compaction via a combination of supercoiling and local protein-dependent deformations.

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.

Protein-mediated loops in supercoiled DNA create large topological domains

Supercoiling can alter the form and base pairing of the double helix and directly impact protein binding. More indirectly, changes in protein binding and the stress of supercoiling also influence the thermodynamic stability of regulatory, protein-mediated loops and shift the equilibria of fundamental DNA/chromatin transactions. For example, supercoiling affects the hierarchical organization and function of chromatin in topologically associating domains (TADs) in both eukaryotes and bacteria. On the other hand, a protein-mediated loop in DNA can constrain supercoiling within a plectonemic structure. To characterize the extent of constrained supercoiling, 400 bp, lac repressor-secured loops were formed in extensively over- or under-wound DNA under gentle tension in a magnetic tweezer. The protein-mediated loops constrained variable amounts of supercoiling that often exceeded the maximum writhe expected for a 400 bp plectoneme. Loops with such high levels of supercoiling appear to be entangled with flanking domains. Thus, loop-mediating proteins operating on supercoiled substrates can establish topological domains that may coordinate gene regulation and other DNA transactions across spans in the genome that are larger than the separation between the binding sites.

Defect-facilitated buckling in supercoiled double-helix DNA

We present a statistical-mechanical model for stretched twisted double-helix DNA, where thermal fluctuations are treated explicitly from a Hamiltonian without using any scaling hypotheses. Our model applied to defect-free supercoiled DNA describes the coexistence of multiple plectoneme domains in long DNA molecules at physiological salt concentrations (≈0.1M Na^{+}) and stretching forces (≈1pN). We find a higher (lower) number of domains at lower (higher) ionic strengths and stretching forces, in accord with experimental observations. We use our model to study the effect of an immobile point defect on the DNA contour that allows a localized kink. The degree of the kink is controlled by the defect size, such that a larger defect further reduces the bending energy of the defect-facilitated kinked end loop. We find that a defect can spatially pin a plectoneme domain via nucleation of a kinked end loop, in accord with experiments and simulations. Our model explains previously reported magnetic tweezer experiments [A. Dittmore et al., Phys. Rev. Lett. 119, 147801 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.147801] showing two buckling signatures: buckling and "rebuckling" in supercoiled DNA with a base-unpaired region. Comparing with experiments, we find that under 1 pN force, a kinked end loop nucleated at a base-mismatched site reduces the bending energy by ≈0.7 k_{B}T per unpaired base. Our model predicts the coexistence of three states at the buckling and rebuckling transitions, which warrants new experiments.

Transfer-matrix calculations of the effects of tension and torque constraints on DNA-protein interactions

Organization and maintenance of the chromosomal DNA in living cells strongly depends on the DNA interactions with a plethora of DNA-binding proteins. Single-molecule studies show that formation of nucleoprotein complexes on DNA by such proteins is frequently subject to force and torque constraints applied to the DNA. Although the existing experimental techniques allow to exert these type of mechanical constraints on individual DNA biopolymers, their exact effects in regulation of DNA-protein interactions are still not completely understood due to the lack of systematic theoretical methods able to efficiently interpret complex experimental observations. To fill this gap, we have developed a general theoretical framework based on the transfer-matrix calculations that can be used to accurately describe behaviour of DNA-protein interactions under force and torque constraints. Potential applications of the constructed theoretical approach are demonstrated by predicting how these constraints affect the DNA-binding properties of different types of architectural proteins. Obtained results provide important insights into potential physiological functions of mechanical forces in the chromosomal DNA organization by architectural proteins as well as into single-DNA manipulation studies of DNA-protein interactions.

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.

Annealing helicase HARP closes RPA-stabilized DNA bubbles non-processively

We investigate the mechanistic nature of the Snf2 family protein HARP, mutations of which are responsible for Schimke immuno-osseous dysplasia. Using a single-molecule magnetic tweezers assay, we construct RPA-stabilized DNA bubbles within torsionally constrained DNA to investigate the annealing action of HARP on a physiologically relevant substrate. We find that HARP closes RPA-stabilized bubbles in a slow reaction, taking on the order of tens of minutes for ∼600 bp of DNA to be re-annealed. The data indicate that DNA re-anneals through the removal of RPA, which is observed as clear steps in the bubble-closing traces. The dependence of the closing rate on both ionic strength and HARP concentration indicates that removal of RPA occurs via an association-dissociation mechanism where HARP does not remain associated with the DNA. The enzyme exhibits classical Michaelis–Menten kinetics and acts cooperatively with a Hill coefficient of 3 ± 1. Our work also allows the determination of some important features of RPA-bubble structures at low supercoiling, including the existence of multiple bubbles and that RPA molecules are mis-registered on the two strands.

Large-Scale Conformational Transitions in Supercoiled DNA Revealed by Coarse-Grained Simulation

Topological constraints, such as those associated with DNA supercoiling, play an integral role in genomic regulation and organization in living systems. However, physical understanding of the principles that underlie DNA organization at biologically relevant length scales remains a formidable challenge. We develop a coarse-grained simulation approach for predicting equilibrium conformations of supercoiled DNA. Our methodology enables the study of supercoiled DNA molecules at greater length scales and supercoiling densities than previously explored by simulation. With this approach, we study the conformational transitions that arise due to supercoiling across the full range of supercoiling densities that are commonly explored by living systems. Simulations of ring DNA molecules with lengths at the scale of topological domains in the Escherichia coli chromosome (∼10 kilobases) reveal large-scale conformational transitions elicited by supercoiling. The conformational transitions result in three supercoiling conformational regimes that are governed by a competition among chiral coils, extended plectonemes, and branched hyper-supercoils. These results capture the nonmonotonic relationship of size versus degree of supercoiling observed in experimental sedimentation studies of supercoiled DNA, and our results provide a physical explanation of the conformational transitions underlying this behavior. The length scales and supercoiling regimes investigated here coincide with those relevant to transcription-coupled remodeling of supercoiled topological domains, and we discuss possible implications of these findings in terms of the interplay between transcription and topology in bacterial chromosome organization.

Platinum-Based Drugs and DNA Interactions Studied by Single-Molecule and Bulk Measurements

Platinum-containing molecules are widely used as anticancer drugs. These molecules exert cytotoxic effects by binding to DNA through various mechanisms. The binding between DNA and platinum-based drugs hinders the opening of DNA, and therefore, DNA duplication and transcription are severely hampered. Overall, impeding the above-mentioned important DNA mechanisms results in irreversible DNA damage and the induction of apoptosis. Several molecules, including multinuclear platinum compounds, belong to the family of platinum drugs, and there is a body of research devoted to developing more efficient and less toxic versions of these compounds. In this study, we combined different biophysical methods, including single-molecule assays (magnetic tweezers) and bulk experiments (ultraviolet absorption for thermal denaturation) to analyze the differential stability of double-stranded DNA in complex with either cisplatin or multinuclear platinum agents. Specifically, we analyzed how the binding of BBR3005 and BBR3464, two representative multinuclear platinum-based compounds, to DNA affects its stability as compared with cisplatin binding. Our results suggest that single-molecule approaches can provide insights into the drug-DNA interactions that underlie drug potency and provide information that is complementary to that generated from bulk analysis; thus, single-molecule approaches have the potential to facilitate the selection and design of optimized drug compounds. In particular, relevant differences in DNA stability at the single-molecule level are demonstrated by analyzing nanomechanically induced DNA denaturation. On the basis of the comparison between the single-molecule and bulk analyses, we suggest that transplatinated drugs are able to locally destabilize small portions of the DNA chain, whereas other regions are stabilized.

Structural transitions in torsionally constrained DNA and their dependence on solution electrostatics

Experimental studies on single molecules of DNA have reported a rich variety of structural transitions, including coexistence of three phases, in a torsionally constrained molecule. A comprehensive knowledge of these structural transitions is useful for unraveling the in vivo and in vitro behavior of DNA. Our objective is to understand the structural transitions in a torsionally constrained DNA molecule when it is pulled using optical or magnetic tweezers. We use foundational concepts from the Zimm-Bragg helix-coil transition theory and merge them with ideas from the theory of fluctuating elastic rods to model the mechanics of DNA. We also account for the electrostatic interactions between the ions and the negatively charged phosphate backbone of DNA. Using our model, we calculate the force and torque corresponding to the overstretching transition characterized by a 70% jump in the contour length of the molecule and examine the effect of salt concentration on this transition. We also deduce conditions under which the co-existence of B-, S- and P-DNA is possible. We examine how the cooperativity parameter for each transition affects the force-extension curve or torque-rotation curve. We attempt to rationalize the non-monotonic dependence of external work done on the ion concentration by connecting it to the electrostatic dependence of the interfacial energy between two phases of DNA. Our theoretical results are in agreement with multiple experiments documented in the literature and they generate falsifiable predictions that can be tested in new experiments.

STATEMENT OF SIGNIFICANCE

The overarching objective of this paper is to explore the implications of variation in ion concentration on the structural transitions driven by external forces in a torsionally constrained DNA molecule. A comprehensive understanding of the phase behavior of torsionally constrained DNA is useful because DNA in cells is tightly packaged and is acted upon by molecular machines in different ionic environments. We examine the mechanics of the overstretching transition, characterized by a 70% jump in contour length, wherein a mixture of B- and S-DNA converts into a mixture of S- and P-DNA through a triple point in the phase diagram. Our results are corroborated by experimental data at every step and we make predictions that are experimentally verifiable.

Statistical mechanical model for a closed loop plectoneme with weak helix specific forces

We develop a statistical mechanical framework, based on a variational approximation, to describe closed loop plectonemes. This framework incorporates weak helix structure dependent forces into the determination of the free energy and average structure of a plectoneme. Notably, due to their chiral nature, helix structure dependent forces break the symmetry between left and right handed supercoiling. The theoretical approach, presented here, also provides a systematic way of enforcing the topological constraint of closed loop supercoiling in the variational approximation. At large plectoneme lengths, by considering correlation functions in an expansion in terms of the spatial mean twist density about its thermally averaged value, it can be argued that topological constraint may be approximated by replacing twist and writhe by their thermal averages. A Lagrange multiplier, containing the sum of average twist and writhe, can be added to the free energy to conveniently inforce this result. The average writhe can be calculated through the thermal average of the Gauss' integral in the variational approximation. Furthermore, this approach allows for a possible way to calculate finite size corrections due to the topological constraint. Using interaction energy terms from the mean-field Kornyshev-Leikin theory, for parameter values that correspond to weak helix dependent forces, we calculate the free energy, fluctuation magnitudes and mean geometric parameters for the plectoneme. We see a slight asymmetry, where interestingly, left handed supercoils have a looser structure than right handed ones, although with a lower free energy, unlike what the previous ground state calculations would suggest.

Theoretical Methods for Studying DNA Structural Transitions under Applied Mechanical Constraints

Recent progress in single-molecule manipulation technologies has made it possible to exert force and torque on individual DNA biopolymers to probe their mechanical stability and interaction with various DNA-binding proteins. It was revealed in these experiments that the DNA structure and formation of nucleoprotein complexes by DNA-architectural proteins can be strongly modulated by an intricate interplay between the entropic elasticity of DNA and its global topology, which is closely related to the mechanical constraints applied to the DNA. Detailed understanding of the physical processes underlying the DNA behavior observed in single-molecule experiments requires the development of a general theoretical framework, which turned out to be a rather challenging task. Here, we review recent advances in theoretical methods that can be used to interpret single-molecule manipulation experiments on DNA.

Transfer-matrix calculations of DNA polymer micromechanics under tension and torque constraints

Recent development of single-molecule manipulation technologies has made it possible to exert constant force and torque on individual DNA biopolymers to probe their elastic characteristics and structural stability. It has been previously shown that depending on the nature of applied mechanical constraints, DNA can exist in several forms including B-, L-, and P-DNA. However, there is still a lack of understanding of how structural heterogeneity of DNA, which may naturally arise due to sequence-dependent DNA properties, protein binding, or DNA damage, influences local stability of the above DNA states. To provide a more complete and detailed description of the DNA mechanics, we developed a theoretical framework based on transfer-matrix calculations and demonstrated how it can be used to predict the DNA behavior upon application of a wide range of force and torque constraints. The resulting phase diagram shows DNA structural transitions that are in good agreement with previous experimental and theoretical studies. We further discuss how the constructed formalism can be extended to include local inhomogeneities in the DNA physical properties, thus making it possible to investigate the effect of DNA sequence as well as protein binding on DNA structural stability.

Supercoiling biases the formation of loops involved in gene regulation

The function of DNA as a repository of genetic information is well-known. The post-genomic effort is to understand how this information-containing filament is chaperoned to manage its compaction and topological states. Indeed, the activities of enzymes that transcribe, replicate, or repair DNA are regulated to a large degree by access. Proteins that act at a distance along the filament by binding at one site and contacting another site, perhaps as part of a bigger complex, create loops that constitute topological domains and influence regulation. DNA loops and plectonemes are not necessarily spontaneous, especially large loops under tension for which high energy is required to bring their ends together, or small loops that require accessory proteins to facilitate DNA bending. However, the torsion in stiff filaments such as DNA dramatically modulates the topology, driving it from extended and genetically accessible to more looped and compact, genetically secured forms. Furthermore, there are accessory factors that bias the response of the DNA filament to supercoiling. For example, small molecules like polyamines, which neutralize the negative charge repulsions along the phosphate backbone, enhance flexibility and promote writhe over twist in response to torsion. Such increased flexibility likely pushes the topological equilibrium from twist toward writhe at tensions thought to exist in vivo. A predictable corollary is that stiffening DNA antagonizes looping and bending. Certain sequences are known to be more or less flexible or to exhibit curvature, and this may affect interactions with binding proteins. In vivo all of these factors operate simultaneously on DNA that is generally negatively supercoiled to some degree. Therefore, in order to better understand gene regulation that involves protein-mediated DNA loops, it is critical to understand the thermodynamics and kinetics of looping in DNA that is under tension, negatively supercoiled, and perhaps exposed to molecules that alter elasticity. Recent experiments quantitatively reveal how much negatively supercoiling DNA lowers the free energy of looping, possibly biasing the operation of genetic switches.

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.

Unravelling the structural plasticity of stretched DNA under torsional constraint

Regions of the genome are often held under torsional constraint. Nevertheless, the influence of such constraint on DNA-protein interactions during genome metabolism is still poorly understood. Here using a combined optical tweezers and fluorescence microscope, we quantify and explain how torsional constraint influences the structural stability of DNA under applied tension. We provide direct evidence that concomitant basepair melting and helical unwinding can occur in torsionally constrained DNA at forces >∼50 pN. This striking result indicates that local changes in linking number can be absorbed by the rest of the DNA duplex. We also present compelling new evidence that an overwound DNA structure (likely P-DNA) is created (alongside underwound structures) at forces >∼110 pN. These findings substantiate previous theoretical predictions and highlight a remarkable structural plasticity of torsionally constrained DNA. Such plasticity may be required in vivo to absorb local changes in linking number in DNA held under torsional constraint.

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.

Competition between B-Z and B-L transitions in a single DNA molecule: Computational studies

Under negative torsion, DNA adopts left-handed helical forms, such as Z-DNA and L-DNA. Using the random copolymer model developed for a wormlike chain, we represent a single DNA molecule with structural heterogeneity as a helical chain consisting of monomers which can be characterized by different helical senses and pitches. By Monte Carlo simulation, where we take into account bending and twist fluctuations explicitly, we study sequence dependence of B-Z transitions under torsional stress and tension focusing on the interaction with B-L transitions. We consider core sequences, (GC)_{n} repeats or (TG)_{n} repeats, which can interconvert between the right-handed B form and the left-handed Z form, imbedded in a random sequence, which can convert to left-handed L form with different (tension dependent) helical pitch. We show that Z-DNA formation from the (GC)_{n} sequence is always supported by unwinding torsional stress but Z-DNA formation from the (TG)_{n} sequence, which are more costly to convert but numerous, can be strongly influenced by the quenched disorder in the surrounding random sequence.

Experimental phase diagram of negatively supercoiled DNA measured by magnetic tweezers and fluorescence

The most common form of DNA is the well-known B-structure of double-helix DNA. Many processes in the cell, however, exert force and torque, inducing structural changes to the DNA that are vital to biological function. Virtually all DNA in cells is in a state of negative supercoiling, with a DNA structure that is complex. Using magnetic tweezers combined with fluorescence imaging, we here study DNA structure as a function of negative supercoiling at the single-molecule level. We classify DNA phases based on DNA length as a function of supercoiling, down to a very high negative supercoiling density σ of -2.5, and forces up to 4.5 pN. We characterize plectonemes using fluorescence imaging. DNA bubbles are visualized by the binding of fluorescently labelled RPA, a eukaryotic single-strand-binding protein. The presence of Z-DNA, a left-handed form of DNA, is probed by the binding of Zα77, the minimal binding domain of a Z-DNA-binding protein. Without supercoiling, DNA is in the relaxed B-form. Upon going toward negative supercoiling, plectonemic B-DNA is being formed below 0.6 pN. At higher forces and supercoiling densities down to about -1.9, a mixed state occurs with plectonemes, multiple bubbles and left-handed L-DNA. Around σ = -1.9, a buckling transition occurs after which the DNA end-to-end length linearly decreases when applying more negative turns, into a state that we interpret as plectonemic L-DNA. By measuring DNA length, Zα77 binding, plectoneme and ssDNA visualisation, we thus have mapped the co-existence of many DNA structures and experimentally determined the DNA phase diagram at (extreme) negative supercoiling.

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.

Equilibrium and kinetics of DNA overstretching modeled with a quartic energy landscape

It is well known that the dsDNA molecule undergoes a phase transition from B-DNA into an overstretched state at high forces. For some time, the structure of the overstretched state remained unknown and highly debated, but recent advances in experimental techniques have presented evidence of more than one possible phase (or even a mixed phase) depending on ionic conditions, temperature, and basepair sequence. Here, we present a theoretical model to study the overstretching transition with the possibility that the overstretched state is a mixture of two phases: a structure with portions of inner strand separation (melted or M-DNA), and an extended phase that retains the basepair structure (S-DNA). We model the double-stranded DNA as a chain composed of n segments of length l, where the transition is studied by means of a Landau quartic potential with statistical fluctuations. The length l is a measure of cooperativity of the transition and is key to characterizing the overstretched phase. By analyzing the different values of l corresponding to a wide spectrum of experiments, we find that for a range of temperatures and ionic conditions, the overstretched form is likely to be a mix of M-DNA and S-DNA. For a transition close to a pure S-DNA state, where the change in extension is close to 1.7 times the original B-DNA length, we find l ? 25 basepairs regardless of temperature and ionic concentration. Our model is fully analytical, yet it accurately reproduces the force-extension curves, as well as the transient kinetic behavior, seen in DNA overstretching experiments.