Braiding DNA: experiments, simulations, and models

Magnetic tweezers principles and promises

Magnetic tweezers have become popular with the outbreak of single molecule micromanipulation: catching a single molecule of DNA, RNA or a single protein and applying mechanical constrains using micron-size magnetic beads and magnets turn out to be easy. Various factors have made this possible: the fact that manufacturers have been preparing these beads to catch various biological entities-the ease of use provided by magnets which apply a force or a torque at a distance thus inside a flow cell-some chance: since the forces so generated are in the right range to stretch a single molecule. This is a little less true for torque. Finally, one feature which also appears very important is the simplicity of their calibration using Brownian motion. Here we start by describing magnetic tweezers used routinely in our laboratory where we have tried to develop a device as simple as possible so that the experimentalist can really focus on the biological aspect of the biomolecules that he/she is interested in. We discuss the implications of the various components and their important features. Next, we summarize what is easy to achieve and what is less easy. Then we refer to contributions by other groups who have brought valuable insights to improve magnetic tweezers.

Energetics of twisted elastic filament pairs

We investigate the elastic energy stored in a filament pair as a function of applied twist by measuring torque under prescribed end-to-end separation conditions. We show that the torque increases rapidly to a peak with applied twist when the filaments are initially separate, then decreases to a minimum as the filaments cross and come into contact. The torque then increases again while the filaments form a double helix with increasing twist. A nonlinear elasto-geometric model that combines the effect of geometrical nonlinearities with large stretching and self-twist is shown to capture the evolution of the helical geometry, torque profile, and stored energy with twist. We find that a large fraction of the total energy is stored in stretching the filaments, which increases with separation distance and applied tension. We find that only a small fraction of energy is stored in the form of bending energy, and that the contribution due to contact energy is negligible. Further, we provide analytical formulas for the torque observed as a function of the applied twist and the inverse relation of the observed angle for a given applied torque in the Hookean limit. Our study highlights the consequences of stretchablility on filament twisting, which is a fundamental topological transformation relevant to making ropes, tying shoelaces, actuating robots, and the physical properties of entangled polymers.

DNA superhelicity

Closing each strand of a DNA duplex upon itself fixes its linking number L. This topological condition couples together the secondary and tertiary structures of the resulting ccDNA topoisomer, a constraint that is not present in otherwise identical nicked or linear DNAs. Fixing L has a range of structural, energetic and functional consequences. Here we consider how L having different integer values (that is, different superhelicities) affects ccDNA molecules. The approaches used are primarily theoretical, and are developed from a historical perspective. In brief, processes that either relax or increase superhelicity, or repartition what is there, may either release or require free energy. The energies involved can be substantial, sufficient to influence many events, directly or indirectly. Here two examples are developed. The changes of unconstrained superhelicity that occur during nucleosome attachment and release are examined. And a simple theoretical model of superhelically driven DNA structural transitions is described that calculates equilibrium distributions for populations of identical topoisomers. This model is used to examine how these distributions change with superhelicity and other factors, and applied to analyze several situations of biological interest.

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.

Efficient golden gate assembly of DNA constructs for single molecule force spectroscopy and imaging

Single-molecule techniques such as optical tweezers and fluorescence imaging are powerful tools for probing the biophysics of DNA and DNA-protein interactions. The application of these methods requires efficient approaches for creating designed DNA structures with labels for binding to a surface or microscopic beads. In this paper, we develop a simple and fast technique for making a diverse range of such DNA constructs by combining PCR amplicons and synthetic oligonucleotides using golden gate assembly rules. We demonstrate high yield fabrication of torsionally-constrained duplex DNA up to 10 kbp in length and a variety of DNA hairpin structures. We also show how tethering to a cross-linked antibody substrate significantly enhances measurement lifetime under high force. This rapid and adaptable fabrication method streamlines the assembly of DNA constructs for single molecule biophysics.

Modeling multiple duplex DNA attachments in a force-extension experiment

Optical tweezers-based DNA stretching often relies on tethering a single end-activated DNA molecule between optically manipulated end-binding beads. Measurement success can depend on DNA concentration. At lower DNA concentrations tethering is less common, and many trials may be required to observe a single-molecule stretch. At higher DNA concentrations tethering is more common; however, the resulting force-extensions observed are more complex and may vary from measurement to measurement. Typically these more complex results are attributed to the formation of multiple tethers between the beads; however, to date there does not appear to have been a critical examination of this hypothesis or the potential usefulness of such data. Here we examine stretches at a higher DNA concentration and use analysis and simulation to show how the more complex force-extensions observed can be understood in terms of multiple DNA attachments.

Topoisomerase VI is a chirally-selective, preferential DNA decatenase

DNA topoisomerase VI (topo VI) is a type IIB DNA topoisomerase found predominantly in archaea and some bacteria, but also in plants and algae. Since its discovery, topo VI has been proposed to be a DNA decatenase; however, robust evidence and a mechanism for its preferential decatenation activity was lacking. Using single-molecule magnetic tweezers measurements and supporting ensemble biochemistry, we demonstrate that Methanosarcina mazei topo VI preferentially unlinks, or decatenates DNA crossings, in comparison to relaxing supercoils, through a preference for certain DNA crossing geometries. In addition, topo VI demonstrates a significant increase in ATPase activity, DNA binding and rate of strand passage, with increasing DNA writhe, providing further evidence that topo VI is a DNA crossing sensor. Our study strongly suggests that topo VI has evolved an intrinsic preference for the unknotting and decatenation of interlinked chromosomes by sensing and preferentially unlinking DNA crossings with geometries close to 90°.

Investigating site-selection mechanisms of retroviral integration in supercoiled DNA braids

We theoretically study the integration of short viral DNA in a DNA braid made up by two entwined double-stranded DNA molecules. We show that the statistics of single integration events substantially differ in the straight and buckled, or plectonemic, phase of the braid and are more likely in the latter. We further discover that integration is most likely close to plectoneme tips, where the larger bending energy helps overcome the associated energy barrier and that successive integration events are spatio-temporally correlated, suggesting a potential mechanistic explanation of clustered integration sites in host genomes. The braid geometry we consider provides a novel experimental set-up to quantify integration in a supercoiled substrate in vitro, and to better understand the role of double-stranded DNA topology during this process.

Single-molecule measurements reveal that PARP1 condenses DNA by loop stabilization

Poly(ADP-ribose) polymerase 1 (PARP1) is an abundant nuclear enzyme that plays important roles in DNA repair, chromatin organization and transcription regulation. Although binding and activation of PARP1 by DNA damage sites has been extensively studied, little is known about how PARP1 binds to long stretches of undamaged DNA and how it could shape chromatin architecture. Here, using single-molecule techniques, we show that PARP1 binds and condenses undamaged, kilobase-length DNA subject to sub-piconewton mechanical forces. Stepwise decondensation at high force and DNA braiding experiments show that the condensation activity is due to the stabilization of DNA loops by PARP1. PARP inhibitors do not affect the level of condensation of undamaged DNA but act to block condensation reversal for damaged DNA in the presence of NAD⁺ Our findings suggest a mechanism for PARP1 in the organization of chromatin structure.

Synergistic Coordination of Chromatin Torsional Mechanics and Topoisomerase Activity

DNA replication in eukaryotes generates DNA supercoiling, which may intertwine (braid) daughter chromatin fibers to form precatenanes, posing topological challenges during chromosome segregation. The mechanisms that limit precatenane formation remain unclear. By making direct torque measurements, we demonstrate that the intrinsic mechanical properties of chromatin play a fundamental role in dictating precatenane formation and regulating chromatin topology. Whereas a single chromatin fiber is torsionally soft, a braided fiber is torsionally stiff, indicating that supercoiling on chromatin substrates is preferentially directed in front of the fork during replication. We further show that topoisomerase II relaxation displays a strong preference for a single chromatin fiber over a braided fiber. These results suggest a synergistic coordination-the mechanical properties of chromatin inherently suppress precatenane formation during replication elongation by driving DNA supercoiling ahead of the fork, where supercoiling is more efficiently removed by topoisomerase II. VIDEO ABSTRACT.

Direct observation of helicase-topoisomerase coupling within reverse gyrase

Reverse gyrases (RGs) are the only topoisomerases capable of generating positive supercoils in DNA. Members of the type IA family, they do so by generating a single-strand break in substrate DNA and then manipulating the two single strands to generate positive topology. Here, we use single-molecule experimentation to reveal the obligatory succession of steps that make up the catalytic cycle of RG. In the initial state, RG binds to DNA and unwinds ∼2 turns of the double helix in an ATP-independent fashion. Upon nucleotide binding, RG then rewinds ∼1 turn of DNA. Nucleotide hydrolysis and/or product release leads to an increase of 2 units of DNA writhe and resetting of the enzyme, for a net change of topology of +1 turn per cycle. Final dissociation of RG from DNA results in rewinding of the 2 turns of DNA that were initially disrupted. These results show how tight coupling of the helicase and topoisomerase activities allows for induction of positive supercoiling despite opposing torque.

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.

Plectoneme dynamics and statistics in braided polymers

Braids composed of two interwoven polymer chains exhibit a "buckling" transition whose origin has been explained through the onset of plectonemic structures. Here we study, by a combination of simulation and analytics, the dynamics of plectoneme formation and their statistics in steady state. The introduction of an order parameter-the plectonemic fraction-allows us to map out the phase boundary between the straight-braid phase and the plectonemic one. We then monitor the formation and the growth of plectonemes, observing events typical of phase separation kinetics for liquid-gas systems (fusion, fission, and one-dimensional Ostwald ripening) but also of DNA supercoiling dynamics (plectonemic hopping). Finally, we propose a stochastic field theory for the coupled dynamics of twist and local writhe which explains the phenomenology found with Brownian dynamics simulations as well as the power laws underlying the coarsening of plectonemes.

Topological entanglement of interlocked knotted-unknotted polymer rings

Topological entanglements in biopolymers could drive them to certain internal statics and dynamics with important implications for biological functions. In this study, by means of molecular dynamics simulations, we demonstrate that the minimal crossing pattern of a braid plays a major role in its structural and dynamical properties; the braid consists of a knotted ring and an interlocked entwined unknotted polymer ring. In particular, we show that depending on the bending rigidity of the chains, the conformational energy of the braid can be either lower or higher than the unlocked polymer rings. Additionally, we find that a non-identical crossing pattern in the braid could distinctly enforce concerted internal conformational fluctuations between the interlocked rings.

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.

Quantifying Local Molecular Tension Using Intercalated DNA Fluorescence

The ability to measure mechanics and forces in biological nanostructures, such as DNA, proteins and cells, is of great importance as a means to analyze biomolecular systems. However, current force detection methods often require specialized instrumentation. Here, we present a novel and versatile method to quantify tension in molecular systems locally and in real time, using intercalated DNA fluorescence. This approach can report forces over a range of at least ∼0.5-65 pN with a resolution of 1-3 pN, using commercially available intercalating dyes and a general-purpose fluorescence microscope. We demonstrate that the method can be easily implemented to report double-stranded (ds)DNA tension in any single-molecule assay that is compatible with fluorescence microscopy. This is particularly useful for multiplexed techniques, where measuring applied force in parallel is technically challenging. Moreover, tension measurements based on local dye binding offer the unique opportunity to determine how an applied force is distributed locally within biomolecular structures. Exploiting this, we apply our method to quantify the position-dependent force profile along the length of flow-stretched DNA and reveal that stretched and entwined DNA molecules-mimicking catenated DNA structures in vivo-display transient DNA-DNA interactions. The method reported here has obvious and broad applications for the study of DNA and DNA-protein interactions. Additionally, we propose that it could be employed to measure forces in any system to which dsDNA can be tethered, for applications including protein unfolding, chromosome mechanics, cell motility, and DNA nanomachines.

Nucleation of Multiple Buckled Structures in Intertwined DNA Double Helices

We study the statistical-mechanical properties of intertwined double-helical DNAs (DNA braids). In magnetic tweezers experiments, we find that torsionally stressed stretched braids supercoil via an abrupt buckling transition, which is associated with the nucleation of a braid end loop, and that the buckled braid is characterized by a proliferation of multiple domains. Differences between the mechanics of DNA braids and supercoiled single DNAs can be understood as an effect of the increased bulkiness in the structure of the former. The experimental results are in accord with the predictions of a statistical-mechanical model.

Torque and buckling in stretched intertwined double-helix DNAs

We present a statistical-mechanical model for the behavior of intertwined DNAs, with a focus on their torque and extension as a function of their catenation (linking) number and applied force, as studied in magnetic tweezers experiments. Our model produces results in good agreement with available experimental data and predicts a catenation-dependent effective twist modulus distinct from what is observed for twisted individual double-helix DNAs. We find that buckling occurs near the point where experiments have observed a kink in the extension versus linking number, and that the subsequent "supercoiled braid" state corresponds to a proliferation of multiple small plectoneme structures. We predict a discontinuity in extension at the buckling transition corresponding to nucleation of the first plectoneme domain. We also find that buckling occurs for lower linking number at lower salt; the opposite trend is observed for supercoiled single DNAs.

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.

Exploring Condensins with Magnetic Tweezers

The signature activity of condensins as DNA reshaping machines is their ability to impose the giant loop architecture onto the chromosome. At the heart of this activity lies the propensity of the proteins to assemble into macromolecular clusters that bring distant DNA segments together. This gives rise to a rich dynamic behavior when the proteins are presented with the DNA substrate. The protocols in this section describe how the interaction between Escherichia coli condensin MukB and DNA proceeds in real time as observed using magnetic tweezers.

The dynamic interplay between DNA topoisomerases and DNA topology

Topological properties of DNA influence its structure and biochemical interactions. Within the cell, DNA topology is constantly in flux. Transcription and other essential processes, including DNA replication and repair, not only alter the topology of the genome but also introduce additional complications associated with DNA knotting and catenation. These topological perturbations are counteracted by the action of topoisomerases, a specialized class of highly conserved and essential enzymes that actively regulate the topological state of the genome. This dynamic interplay among DNA topology, DNA processing enzymes, and DNA topoisomerases is a pervasive factor that influences DNA metabolism in vivo. Building on the extensive structural and biochemical characterization over the past four decades that has established the fundamental mechanistic basis of topoisomerase activity, scientists have begun to explore the unique roles played by DNA topology in modulating and influencing the activity of topoisomerases. In this review we survey established and emerging DNA topology-dependent protein-DNA interactions with a focus on in vitro measurements of the dynamic interplay between DNA topology and topoisomerase activity.

A HaloTag Anchored Ruler for Week-Long Studies of Protein Dynamics

Under physiological conditions, protein oxidation and misfolding occur with very low probability and on long times scales. Single-molecule techniques provide the ability to distinguish between properly folded and damaged proteins that are otherwise masked in ensemble measurements. However, at physiological conditions these rare events occur with a time constant of several hours, inaccessible to current single-molecule approaches. Here we present a magnetic-tweezers-based technique that allows, for the first time, the study of folding of single proteins during week-long experiments. This technique combines HaloTag anchoring, sub-micrometer positioning of magnets, and an active correction of the focal drift. Using this technique and protein L as a molecular template, we generate a magnet law by correlating the distance between the magnet and the measuring paramagnetic bead with unfolding/folding steps. We demonstrate that, using this magnet law, we can accurately measure the dynamics of proteins over a wide range of forces, with minimal dispersion from bead to bead. We also show that the force calibration remains invariant over week-long experiments applied to the same single proteins. The approach demonstrated in this Article opens new, exciting ways to examine proteins on the "human" time scale and establishes magnetic tweezers as a valuable technique to study low-probability events that occur during protein folding under force.

The Dynamic Interplay Between DNA Topoisomerases and DNA Topology

Topological properties of DNA influence its structure and biochemical interactions. Within the cell DNA topology is constantly in flux. Transcription and other essential processes including DNA replication and repair, alter the topology of the genome, while introducing additional complications associated with DNA knotting and catenation. These topological perturbations are counteracted by the action of topoisomerases, a specialized class of highly conserved and essential enzymes that actively regulate the topological state of the genome. This dynamic interplay among DNA topology, DNA processing enzymes, and DNA topoisomerases, is a pervasive factor that influences DNA metabolism in vivo. Building on the extensive structural and biochemical characterization over the past four decades that established the fundamental mechanistic basis of topoisomerase activity, the unique roles played by DNA topology in modulating and influencing the activity of topoisomerases have begun to be explored. In this review we survey established and emerging DNA topology dependent protein-DNA interactions with a focus on in vitro measurements of the dynamic interplay between DNA topology and topoisomerase activity.

Collapse and coexistence for a molecular braid with an attractive interaction component subject to mechanical forces

Dual mechanical braiding experiments provide a useful tool with which to investigate the nature of interactions between rod-like molecules, for instance actin and DNA. In conditions close to molecular condensation, one would expect an appearance of a local minimum in the interaction potential between the two molecules. We investigate this situation, introducing an attractive component into the interaction potential, using a model developed for describing such experiments. We consider both attractive interactions that do not depend on molecular structure and those which depend on a DNA-like helix structure. In braiding experiments, an attractive term may lead to certain effects. A local minimum may cause molecules to collapse from a loosely braided configuration into a tight one, occurring at a critical value of the moment applied about the axis of the braid. For a fixed number of braid pitches, this may lead to coexistence between the two braiding states, tight and loose. Coexistence implies certain proportions of the braid are in each state, their relative size depending on the number of braid pitches. This manifests itself as a linear dependence in numerically calculated quantities as functions of the number of braid pitches. Also, in the collapsed state, the braid radius stays roughly constant. Furthermore, if the attractive interaction is helix dependent, the left-right handed braid symmetry is broken. For a DNA like charge distribution, using the Kornyshev-Leikin interaction model, our results suggest that significant braid collapse and coexistence only occurs for left handed braids. Regardless of the interaction model, the study highlights the possible qualitative physics of braid collapse and coexistence; and the role helix specific forces might play, if important. The model could be used to connect other microscopic theories of interaction with braiding experiments.

Single-molecule analysis uncovers the difference between the kinetics of DNA decatenation by bacterial topoisomerases I and III

Escherichia coli topoisomerases I and III can decatenate double-stranded DNA (dsDNA) molecules containing single-stranded DNA regions or nicks as well as relax negatively supercoiled DNA. Although the proteins share a mechanism of action and have similar structures, they participate in different cellular processes. Whereas topoisomerase III is a more efficient decatenase than topoisomerase I, the opposite is true for DNA relaxation. In order to investigate the differences in the mechanism of these two prototypical type IA topoisomerases, we studied DNA decatenation at the single-molecule level using braids of intact dsDNA and nicked dsDNA with bulges. We found that neither protein decatenates an intact DNA braid. In contrast, both enzymes exhibited robust decatenation activity on DNA braids with a bulge. The experiments reveal that a main difference between the unbraiding mechanisms of these topoisomerases lies in the pauses between decatenation cycles. Shorter pauses for topoisomerase III result in a higher decatenation rate. In addition, topoisomerase III shows a strong dependence on the crossover angle of the DNA strands. These real-time observations reveal the kinetic characteristics of the decatenation mechanism and help explain the differences between their activities.

Invincible DNA tethers: covalent DNA anchoring for enhanced temporal and force stability in magnetic tweezers experiments

Magnetic tweezers are a powerful single-molecule technique that allows real-time quantitative investigation of biomolecular processes under applied force. High pulling forces exceeding tens of picoNewtons may be required, e.g. to probe the force range of proteins that actively transcribe or package the genome. Frequently, however, the application of such forces decreases the sample lifetime, hindering data acquisition. To provide experimentally viable sample lifetimes in the face of high pulling forces, we have designed a novel anchoring strategy for DNA in magnetic tweezers. Our approach, which exploits covalent functionalization based on heterobifunctional poly(ethylene glycol) crosslinkers, allows us to strongly tether DNA while simultaneously suppressing undesirable non-specific adhesion. A complete force and lifetime characterization of these covalently anchored DNA-tethers demonstrates that, compared to more commonly employed anchoring strategies, they withstand 3-fold higher pulling forces (up to 150 pN) and exhibit up to 200-fold higher lifetimes (exceeding 24 h at a constant force of 150 pN). This advance makes it possible to apply the full range of biologically relevant force scales to biomolecular processes, and its straightforward implementation should extend its reach to a multitude of applications in the field of single-molecule force spectroscopy.

Undulations in a weakly interacting mechanically generated molecular braid under tension

We consider mechanically generated molecular braids composed of two molecules where long range interactions between them can be considered to be very weak. We describe a model that takes account of the thermal fluctuations of the braid, steric interactions between the molecules, and external mechanical forces. In this model, both sets of ends, of the two molecules, are considered to be separated by a fixed distance much larger than the radius of the braid. One set of ends is rotated to generate a braid of a certain number of pitches (or turns), while the other set remains fixed. This model may describe the situation in which the ends of each molecule are attached to a substrate and a magnetic bead; to the latter a pulling force and rotational torque can be applied. We discuss various aspects of our model. Most importantly, an expression for the free energy is given, from which equations, determining the various geometric parameters of the braid, can be obtained. By numerically solving these equations, we give predictions from the model for the external torque needed to produce a braid with a certain number of turns per bending persistence length, as well as the end to end extension of the two molecules for a given pulling force. Other geometric parameters, as well as the lateral force required to keep the ends of the two molecules apart, are also calculated.

Self-consistent treatment of electrostatics in molecular DNA braiding through external forces

In this paper we consider a physical system in which two DNA molecules braid about each other. The distance between the two molecular ends, on either side of the braid, is held at a distance much larger than supercoiling radius of the braid. The system is subjected to an external pulling force, and a moment that induces the braiding. In a model, developed for understanding such a system, we assume that each molecule can be divided into a braided and unbraided section. We also suppose that the DNA is nicked so that there is no constraint of the individual linking numbers of the molecules. Included in the model are steric and electrostatic interactions, thermal fluctuations of the braided and unbraided sections of the molecule, as well as the constraint on the braid linking (catenation) number. We compare two approximations used in estimating the free energy of the braided section. One is where the amplitude of undulations of one molecule with respect to the other is determined only by steric interactions. The other is a self-consistent determination of the mean-squared amplitude of these undulations. In this second approximation electrostatics should play an important role in determining this quantity, as suggested by physical arguments. We see that if the electrostatic interaction is sufficiently large there are indeed notable differences between the two approximations. We go on to test the self-consistent approximation-included in the full model-against experimental data for such a system, and we find good agreement. However, there seems to be a slight left-right-handed braid asymmetry in some of the experimental results. We discuss what might be the origin of this small asymmetry.

A highly processive topoisomerase I: studies at the single-molecule level

Amongst enzymes which relieve torsional strain and maintain chromosome supercoiling, type IA topoisomerases share a strand-passage mechanism that involves transient nicking and re-joining of a single deoxyribonucleic acid (DNA) strand. In contrast to many bacterial species that possess two type IA topoisomerases (TopA and TopB), Actinobacteria possess only TopA, and unlike its homologues this topoisomerase has a unique C-terminal domain that lacks the Zn-finger motifs characteristic of type IA enzymes. To better understand how this unique C-terminal domain affects the enzyme's activity, we have examined DNA relaxation by actinobacterial TopA from Streptomyces coelicolor (ScTopA) using real-time single-molecule experiments. These studies reveal extremely high processivity of ScTopA not described previously for any other topoisomerase of type I. Moreover, we also demonstrate that enzyme processivity varies in a torque-dependent manner. Based on the analysis of the C-terminally truncated ScTopA mutants, we propose that high processivity of the enzyme is associated with the presence of a stretch of positively charged amino acids in its C-terminal region.

Relationships between the winding angle, the characteristic radius, and the torque for a long polymer chain wound around a cylinder: implications for RNA winding around DNA during transcription

Long polymer chains are ubiquitous in biological systems and their mechanical properties have significant impact upon biological processes. Of particular interest is the situation in which polymer chains are wound around each other or around other objects. We have analyzed the parameters of a long Gaussian polymer chain wound around a cylinder as a function of the torque applied to the ends of the chain. We have shown that for sufficiently long polymer chains, an average winding angle and a characteristic radius of the chain can be determined from a modified Bessel function of purely imaginary order, in which the value of the order is equivalent to the applied torque, normalized to the product of the absolute temperature and the Boltzmann constant. The obtained results are consistent with a simplified interpretation in terms of "torsional blobs," and this could be extended to nonideal chains with excluded volumes. We have also extended our results to the case of a polymer chain rotating in viscous medium. Our results could be used to estimate the mechanical strains that appear in DNA and RNA during transcription, as these might initiate formation of unusual DNA structures, invasion of RNA into the DNA duplex (R-loop formation), and modulation of the interactions of DNA and RNA with proteins.

Effect of undulations on spontaneous braid formation

This paper is an extension of a recent study where it was shown that forces dependent on molecular helical structure may cause two DNA molecules to spontaneously braid [R. Cortini et al., Biophys. J. 101, 875 (2011)]. Here, bending fluctuations of molecular center lines about the braid axis are incorporated into the braiding theory, which may be generalized to other helix-dependent interactions and other helical molecules. The free energy of the pair of molecules is recalculated and compared to its value without incorporating undulations. We find that the loss of configurational entropy due to confinement of the molecules in the braid is quite high. This contribution to the free energy increases the amount of attraction needed for spontaneous braiding due to helix-dependent forces. The theory will be further developed for plectonemes and braids under mechanical forces in later work.

Chiral discrimination and writhe-dependent relaxation mechanism of human topoisomerase IIα

BACKGROUND

Human topoisomerase IIα unlinks catenated chromosomes and preferentially relaxes positive supercoils.

RESULTS

Supercoil chirality, twist density, and tension determine topoisomerase IIα relaxation rate and processivity.

CONCLUSION

Strand passage rate is determined by the efficiency of transfer segment capture that is modulated by the topoisomerase C-terminal domains.

SIGNIFICANCE

Single-molecule measurements reveal the mechanism of chiral discrimination and tension dependence of supercoil relaxation by human topoisomerase IIα. Type IIA topoisomerases (Topo IIA) are essential enzymes that relax DNA supercoils and remove links joining replicated chromosomes. Human topoisomerase IIα (htopo IIα), one of two human isoforms, preferentially relaxes positive supercoils, a feature shared with Escherichia coli topoisomerase IV (Topo IV). The mechanistic basis of this chiral discrimination remains unresolved. To address this important issue, we measured the relaxation of individual supercoiled and "braided" DNA molecules by htopo IIα using a magnetic tweezers-based single-molecule assay. Our study confirmed the chiral discrimination activity of htopo IIα and revealed that the strand passage rate depends on DNA twist, tension on the DNA, and the C-terminal domain (CTD). Similar to Topo IV, chiral discrimination by htopo IIα results from chiral interactions of the CTDs with DNA writhe. In contrast to Topo IV, however, these interactions lead to chiral differences in relaxation rate rather than processivity. Increasing tension or twist disrupts the CTD-DNA interactions with a subsequent loss of chiral discrimination. Together, these results suggest that transfer segment (T-segment) capture is the rate-limiting step in the strand passage cycle. We propose a model for T-segment capture that provides a mechanistic basis for chiral discrimination and provides a coherent explanation for the effects of DNA twist and tension on eukaryotic type IIA topoisomerases.

Comparison of DNA decatenation by Escherichia coli topoisomerase IV and topoisomerase III: implications for non-equilibrium topology simplification

Type II topoisomerases are essential enzymes that regulate DNA topology through a strand-passage mechanism. Some type II topoisomerases relax supercoils, unknot and decatenate DNA to below thermodynamic equilibrium. Several models of this non-equilibrium topology simplification phenomenon have been proposed. The kinetic proofreading (KPR) model postulates that strand passage requires a DNA-bound topoisomerase to collide twice in rapid succession with a second DNA segment, implying a quadratic relationship between DNA collision frequency and relaxation rate. To test this model, we used a single-molecule assay to measure the unlinking rate as a function of DNA collision frequency for Escherichia coli topoisomerase IV (topo IV) that displays efficient non-equilibrium topology simplification activity, and for E. coli topoisomerase III (topo III), a type IA topoisomerase that unlinks and unknots DNA to equilibrium levels. Contrary to the predictions of the KPR model, topo IV and topo III unlinking rates were linearly related to the DNA collision frequency. Furthermore, topo III exhibited decatenation activity comparable with that of topo IV, supporting proposed roles for topo III in DNA segregation. This study enables us to rule out the KPR model for non-equilibrium topology simplification. More generally, we establish an experimental approach to systematically control DNA collision frequency.

Non-bias-limited tracking of spherical particles, enabling nanometer resolution at low magnification

We present a three-dimensional tracking routine for nondiffraction-limited particles, which significantly reduces pixel bias. Our technique allows for increased resolution compared to that of previous methods, especially at low magnification or at high signal/noise ratio. This enables tracking with nanometer accuracy in a wide field of view and tracking of many particles. To reduce bias induced by pixelation, the tracking algorithm uses interpolation of the image on a circular grid to determine the x-, y-, and z-positions. We evaluate the proposed algorithm by tracking simulated images and compare it to well-known center-of-mass and cross-correlation methods. The final resolution of the described method improves up to an order of magnitude in three dimensions compared to conventional tracking methods. We show that errors in x,y-tracking can seriously affect z-tracking if interpolation is not used. We validate our results with experimental data obtained for conditions matching those used in the simulations. Finally, we show that the increased performance of the proposed algorithm uniquely enables it to extract accurate data for the persistence length and end-to-end distance of 107 DNA tethers in a single experiment.

Single-molecule studies using magnetic traps

In recent years, techniques have been developed to study and manipulate single molecules of DNA and other biopolymers. In one such technique, the magnetic trap, a single DNA molecule is bound at one end to a glass surface and at the other to a magnetic microbead. Small magnets, whose position and rotation can be controlled, pull on and rotate the microbead. This provides a simple method to stretch and twist the molecule. The system allows one to apply and measure forces ranging from 10(-3) to >100 pN. In contrast to other techniques, the force measurement is absolute and does not require calibration of the sensor. In this article, we describe the principle of the magnetic trap, as well as its use in the measurement of the elastic properties of DNA and the study of DNA-protein interactions.

Highly parallel magnetic tweezers by targeted DNA tethering

Single-molecule force-spectroscopy methods such as magnetic and optical tweezers have emerged as powerful tools for the detailed study of biomechanical aspects of DNA-enzyme interactions. As typically only a single molecule of DNA is addressed in an individual experiment, these methods suffer from a low data throughput. Here, we report a novel method for targeted, nonrandom immobilization of DNA-tethered magnetic beads in regular arrays through microcontact printing of DNA end-binding labels. We show that the increase in density due to the arrangement of DNA-bead tethers in regular arrays can give rise to a one-order-of-magnitude improvement in data-throughput in magnetic tweezers experiments. We demonstrate the applicability of this technique in tweezers experiments where up to 450 beads are simultaneously tracked in parallel, yielding statistical data on the mechanics of DNA for 357 molecules from a single experimental run. Our technique paves the way for kilo-molecule force spectroscopy experiments, enabling the study of rare events in DNA-protein interactions and the acquisition of large statistical data sets from individual experimental runs.

Electrostatic braiding and homologous pairing of DNA double helices

Homologous pairing and braiding (supercoiling) have crucial effects on genome organization, maintenance, and evolution. Generally, the pairing and braiding processes are discussed in different contexts, independently of each other. However, analysis of electrostatic interactions between DNA double helices suggests that in some situations these processes may be related. Here we present a theory of DNA braiding that accounts for the elastic energy of DNA double helices as well as for the chiral nature of the discrete helical patterns of DNA charges. This theory shows that DNA braiding may be affected, stabilized, or even driven by chiral electrostatic interactions. For example, electrostatically driven braiding may explain the surprising recent observation of stable pairing of homologous double-stranded DNA in solutions containing only monovalent salt. Electrostatic stabilization of left-handed braids may stand behind the chiral selectivity of type II topoisomerases and positive plasmid supercoiling in hyperthermophilic bacteria and archea.

Single-molecule measurements of DNA topology and topoisomerases

Topological properties of DNA influence its mechanical and biochemical interactions. Genomic DNA is maintained in a state of topological homeostasis by topoisomerases and is subjected to mechanical stress arising from replication and segregation. Despite their fundamental roles, the effects of topology and force have been difficult to ascertain. Developments in single-molecule manipulation techniques have enabled precise control and measurement of the topology of individual DNA molecules under tension. This minireview provides an overview of these single-molecule techniques and illustrates their unique capabilities through a number of specific examples of single-molecule measurements of DNA topology and topoisomerase activity.

Mechanics of DNA bridging by bacterial condensin MukBEF in vitro and in singulo

Structural maintenance of chromosome (SMC) proteins comprise the core of several specialized complexes that stabilize the global architecture of the chromosomes by dynamically linking distant DNA fragments. This reaction however remains poorly understood giving rise to numerous proposed mechanisms of the proteins. Using two novel assays, we investigated real-time formation of DNA bridges by bacterial condensin MukBEF. We report that MukBEF can efficiently bridge two DNAs and that this reaction involves multiple steps. The reaction begins with the formation of a stable MukB-DNA complex, which can further capture another protein-free DNA fragment. The initial tether is unstable but is quickly strengthened by additional MukBs. DNA bridging is modulated but is not strictly dependent on ATP and MukEF. The reaction revealed high preference for right-handed DNA crossings indicating that bridging involves physical association of MukB with both DNAs. Our data establish a comprehensive view of DNA bridging by MukBEF, which could explain how SMCs establish both intra- and interchromosomal links inside the cell and indicate that DNA binding and bridging could be separately regulated.

Simulation of DNA catenanes

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

Mechanisms of chiral discrimination by topoisomerase IV

Topoisomerase IV (Topo IV), an essential ATP-dependent bacterial type II topoisomerase, transports one segment of DNA through a transient double-strand break in a second segment of DNA. In vivo, Topo IV unlinks catenated chromosomes before cell division and relaxes positive supercoils generated during DNA replication. In vitro, Topo IV relaxes positive supercoils at least 20-fold faster than negative supercoils. The mechanisms underlying this chiral discrimination by Topo IV and other type II topoisomerases remain speculative. We used magnetic tweezers to measure the relaxation rates of single and multiple DNA crossings by Topo IV. These measurements allowed us to determine unambiguously the relative importance of DNA crossing geometry and enzymatic processivity in chiral discrimination by Topo IV. Our results indicate that Topo IV binds and passes DNA strands juxtaposed in a nearly perpendicular orientation and that relaxation of negative supercoiled DNA is perfectly distributive. Together, these results suggest that chiral discrimination arises primarily from dramatic differences in the processivity of relaxing positive and negative supercoiled DNA: Topo IV is highly processive on positively supercoiled DNA, whereas it is perfectly distributive on negatively supercoiled DNA. These results provide fresh insight into topoisomerase mechanisms and lead to a model that reconciles contradictory aspects of previous findings while providing a framework to interpret future results.