Topological information embodied in local juxtaposition geometry provides a statistical mechanical basis for unknotting by type-2 DNA topoisomerases

DNA-Topology Simplification by Topoisomerases

The topological properties of DNA molecules, supercoiling, knotting, and catenation, are intimately connected with essential biological processes, such as gene expression, replication, recombination, and chromosome segregation. Non-trivial DNA topologies present challenges to the molecular machines that process and maintain genomic information, for example, by creating unwanted DNA entanglements. At the same time, topological distortion can facilitate DNA-sequence recognition through localized duplex unwinding and longer-range loop-mediated interactions between the DNA sequences. Topoisomerases are a special class of essential enzymes that homeostatically manage DNA topology through the passage of DNA strands. The activities of these enzymes are generally investigated using circular DNA as a model system, in which case it is possible to directly assay the formation and relaxation of DNA supercoils and the formation/resolution of knots and catenanes. Some topoisomerases use ATP as an energy cofactor, whereas others act in an ATP-independent manner. The free energy of ATP hydrolysis can be used to drive negative and positive supercoiling or to specifically relax DNA topologies to levels below those that are expected at thermodynamic equilibrium. The latter activity, which is known as topology simplification, is thus far exclusively associated with type-II topoisomerases and it can be understood through insight into the detailed non-equilibrium behavior of type-II enzymes. We use a non-equilibrium topological-network approach, which stands in contrast to the equilibrium models that are conventionally used in the DNA-topology field, to gain insights into the rates that govern individual transitions between topological states. We anticipate that our quantitative approach will stimulate experimental work and the theoretical/computational modeling of topoisomerases and similar enzyme systems.

Grid diagrams as tools to investigate knot spaces and topoisomerase-mediated simplification of DNA topology

Grid diagrams with their relatively simple mathematical formalism provide a convenient way to generate and model projections of various knots. It has been an open question whether these 2D diagrams can be used to model a complex 3D process such as the topoisomerase-mediated preferential unknotting of DNA molecules. We model here topoisomerase-mediated passages of double-stranded DNA segments through each other using the formalism of grid diagrams. We show that this grid diagram-based modeling approach captures the essence of the preferential unknotting mechanism, based on topoisomerase selectivity of hooked DNA juxtapositions as the sites of intersegmental passages. We show that the grid diagram-based approach provides an important, new, and computationally convenient framework for investigating entanglement in biopolymers.

Kinetic pathways of topology simplification by Type-II topoisomerases in knotted supercoiled DNA

The topological state of covalently closed, double-stranded DNA is defined by the knot type $K$ and the linking-number difference $\Delta Lk$ relative to unknotted relaxed DNA. DNA topoisomerases are essential enzymes that control the topology of DNA in all cells. In particular, type-II topoisomerases change both $K$ and $\Delta Lk$ by a duplex-strand-passage mechanism and have been shown to simplify the topology of DNA to levels below thermal equilibrium at the expense of ATP hydrolysis. It remains a key question how small enzymes are able to preferentially select strand passages that result in topology simplification in much larger DNA molecules. Using numerical simulations, we consider the non-equilibrium dynamics of transitions between topological states $(K,\Delta Lk)$ in DNA induced by type-II topoisomerases. For a biological process that delivers DNA molecules in a given topological state $(K,\Delta Lk)$ at a constant rate we fully characterize the pathways of topology simplification by type-II topoisomerases in terms of stationary probability distributions and probability currents on the network of topological states $(K,\Delta Lk)$. In particular, we observe that type-II topoisomerase activity is significantly enhanced in DNA molecules that maintain a supercoiled state with constant torsional tension. This is relevant for bacterial cells in which torsional tension is maintained by enzyme-dependent homeostatic mechanisms such as DNA-gyrase activity.

Coarse-grained residue-based models of disordered protein condensates: utility and limitations of simple charge pattern parameters

Biomolecular condensates undergirded by phase separations of proteins and nucleic acids serve crucial biological functions. To gain physical insights into their genetic basis, we study how liquid-liquid phase separation (LLPS) of intrinsically disordered proteins (IDPs) depends on their sequence charge patterns using a continuum Langevin chain model wherein each amino acid residue is represented by a single bead. Charge patterns are characterized by the "blockiness" measure κ and the "sequence charge decoration" (SCD) parameter. Consistent with random phase approximation (RPA) theory and lattice simulations, LLPS propensity as characterized by critical temperature Tcr* increases with increasingly negative SCD for a set of sequences showing a positive correlation between κ and -SCD. Relative to RPA, the simulated sequence-dependent variation in Tcr* is often-though not always-smaller, whereas the simulated critical volume fractions are higher. However, for a set of sequences exhibiting an anti-correlation between κ and -SCD, the simulated Tcr*'s are quite insensitive to either parameter. Additionally, we find that blocky sequences that allow for strong electrostatic repulsion can lead to coexistence curves with upward concavity as stipulated by RPA, but the LLPS propensity of a strictly alternating charge sequence was likely overestimated by RPA and lattice models because interchain stabilization of this sequence requires spatial alignments that are difficult to achieve in real space. These results help delineate the utility and limitations of the charge pattern parameters and of RPA, pointing to further efforts necessary for rationalizing the newly observed subtleties.

Dynamics of supercoiled DNA with complex knots: large-scale rearrangements and persistent multi-strand interlocking

Knots and supercoiling are both introduced in bacterial plasmids by catalytic processes involving DNA strand passages. While the effects on plasmid organization has been extensively studied for knotting and supercoiling taken separately, much less is known about their concurrent action. Here, we use molecular dynamics simulations and oxDNA, an accurate mesoscopic DNA model, to study the kinetic and metric changes introduced by complex (five-crossing) knots and supercoiling in 2 kbp-long DNA rings. We find several unexpected results. First, the conformational ensemble is dominated by two distinct states, differing in branchedness and knot size. Secondly, fluctuations between these states are as fast as the metric relaxation of unknotted rings. In spite of this, certain boundaries of knotted and plectonemically-wound regions can persist over much longer timescales. These pinned regions involve multiple strands that are interlocked by the cooperative action of topological and supercoiling constraints. Their long-lived character may be relevant for the simplifying action of topoisomerases.

Disentangling DNA molecules

The widespread circular form of DNA molecules inside cells creates very serious topological problems during replication. Due to the helical structure of the double helix the parental strands of circular DNA form a link of very high order, and yet they have to be unlinked before the cell division. DNA topoisomerases, the enzymes that catalyze passing of one DNA segment through another, solve this problem in principle. However, it is very difficult to remove all entanglements between the replicated DNA molecules due to huge length of DNA comparing to the cell size. One strategy that nature uses to overcome this problem is to create the topoisomerases that can dramatically reduce the fraction of linked circular DNA molecules relative to the corresponding fraction at thermodynamic equilibrium. This striking property of the enzymes means that the enzymes that interact with DNA only locally can access their topology, a global property of circular DNA molecules. This review considers the experimental studies of the phenomenon and analyzes the theoretical models that have been suggested in attempts to explain it. We describe here how various models of enzyme action can be investigated computationally. There is no doubt at the moment that we understand basic principles governing enzyme action. Still, there are essential quantitative discrepancies between the experimental data and the theoretical predictions. We consider how these discrepancies can be overcome.

How topoisomerase IV can efficiently unknot and decatenate negatively supercoiled DNA molecules without causing their torsional relaxation

Freshly replicated DNA molecules initially form multiply interlinked right-handed catenanes. In bacteria, these catenated molecules become supercoiled by DNA gyrase before they undergo a complete decatenation by topoisomerase IV (Topo IV). Topo IV is also involved in the unknotting of supercoiled DNA molecules. Using Metropolis Monte Carlo simulations, we investigate the shapes of supercoiled DNA molecules that are either knotted or catenated. We are especially interested in understanding how Topo IV can unknot right-handed knots and decatenate right-handed catenanes without acting on right-handed plectonemes in negatively supercoiled DNA molecules. To this end, we investigate how the topological consequences of intersegmental passages depend on the geometry of the DNA-DNA juxtapositions at which these passages occur. We observe that there are interesting differences between the geometries of DNA-DNA juxtapositions in the interwound portions and in the knotted or catenated portions of the studied molecules. In particular, in negatively supercoiled, multiply interlinked, right-handed catenanes, we detect specific regions where DNA segments belonging to two freshly replicated sister DNA molecules form left-handed crossings. We propose that, due to its geometrical preference to act on left-handed crossings, Topo IV can specifically unknot supercoiled DNA, as well as decatenate postreplicative catenanes, without causing their torsional relaxation.

Consistent rationalization of type-2 topoisomerases' unknotting, decatenating, supercoil-relaxing actions and their scaling relation

How type-2 topoisomerases discern global topology from local properties of DNA is not known precisely but the hypothesis that the enzymes selectively pass double-helix strands at hook-like juxtapositions is promising. Building upon an investigation of unknotting and decatenating using an improved wormlike DNA model, here we focus primarily on the enzymes' action in narrowing the distribution of linking number (Lk) in supercoiled DNA. Consistent with experiments, with selective passage at a hooked juxtaposition, the simulated narrowing factor RLk diminishes with decreasing DNA circle size but approaches an asymptotic RLk ≈ 1.7-1.8 for circle size ≳3.5 kb. For the larger DNA circles, we found that (RLk - 1) ≈ 0.42log10RK ≈ 0.68log10RL and thus RK ≈ (RL)(1.6) holds for the computed RLk and knot and catenane reduction factors RK and RL attained by selective passage at different juxtaposition geometries. Remarkably, this general scaling relation is essentially identical to that observed experimentally for several type-2 topoisomerases from a variety of organisms, indicating that the different disentangling powers of the topoisomerases likely arise from variations in the hooked geometries they select. Taken together, our results suggest strongly that type-2 topoisomerases recognize not only the curvature of the G-segment but also that of the T-segment.

Generation of supercoils in nicked and gapped DNA drives DNA unknotting and postreplicative decatenation

Due to the helical structure of DNA the process of DNA replication is topologically complex. Freshly replicated DNA molecules are catenated with each other and are frequently knotted. For proper functioning of DNA it is necessary to remove all of these entanglements. This is done by DNA topoisomerases that pass DNA segments through each other. However, it has been a riddle how DNA topoisomerases select the sites of their action. In highly crowded DNA in living cells random passages between contacting segments would only increase the extent of entanglement. Using molecular dynamics simulations we observed that in actively supercoiled DNA molecules the entanglements resulting from DNA knotting or catenation spontaneously approach sites of nicks and gaps in the DNA. Type I topoisomerases, that preferentially act at sites of nick and gaps, are thus naturally provided with DNA–DNA juxtapositions where a passage results in an error-free DNA unknotting or DNA decatenation.

Extracting physics of life at the molecular level: A review of single-molecule data analyses

Studying individual biomolecules at the single-molecule level has proved very insightful recently. Single-molecule experiments allow us to probe both the equilibrium and nonequilibrium properties as well as make quantitative connections with ensemble experiments and equilibrium thermodynamics. However, it is important to be careful about the analysis of single-molecule data because of the noise present and the lack of theoretical framework for processes far away from equilibrium. Biomolecular motion, whether it is free in solution, on a substrate, or under force, involves thermal fluctuations in varying degrees, which makes the motion noisy. In addition, the noise from the experimental setup makes it even more complex. The details of biologically relevant interactions, conformational dynamics, and activities are hidden in the noisy single-molecule data. As such, extracting biological insights from noisy data is still an active area of research. In this review, we will focus on analyzing both fluorescence-based and force-based single-molecule experiments and gaining biological insights at the single-molecule level. Inherently nonequilibrium nature of biological processes will be highlighted. Simulated trajectories of biomolecular diffusion will be used to compare and validate various analysis techniques.

Topoisomerase II minimizes DNA entanglements by proofreading DNA topology after DNA strand passage

By transporting one DNA double helix (T-segment) through a double-strand break in another (G-segment), topoisomerase II reduces fractions of DNA catenanes, knots and supercoils to below equilibrium values. How DNA segments are selected to simplify the equilibrium DNA topology is enigmatic, and the biological relevance of this activity is unclear. Here we examined the transit of the T-segment across the three gates of topoisomerase II (entry N-gate, DNA-gate and exit C-gate). Our experimental results uncovered that DNA transport probability is determined not only during the capture of a T-segment at the N-gate. When a captured T-segment has crossed the DNA-gate, it can backtrack to the N-gate instead of exiting by the C-gate. When such backtracking is precluded by locking the N-gate or by removing the C-gate, topoisomerase II no longer simplifies equilibrium DNA topology. Therefore, we conclude that the C-gate enables a post-DNA passage proofreading mechanism, which challenges the release of passed T-segments to either complete or cancel DNA transport. This proofreading activity not only clarifies how type-IIA topoisomerases simplify the equilibrium topology of DNA in free solution, but it may explain also why these enzymes are able to solve the topological constraints of intracellular DNA without randomly entangling adjacent chromosomal regions.

Crossing-sign discrimination and knot-reduction for a lattice model of strand passage

By performing strand-passages on DNA, type II topoisomerases are known to resolve topological constraints that impede normal cellular functions. The full details of this enzyme–DNA interaction mechanism are, however, not completely understood. To better understand this mechanism, researchers have proposed and studied a variety of random polygon models of enzyme-induced strand-passage. In the present article, we review results from one such model having the feature that it is amenable to combinatorial and asymptotic analysis (as polygon length goes to infinity). The polygons studied, called Θ-SAPs, are on the simple-cubic lattice and contain a specific strand-passage structure, called Θ, at a fixed site. Another feature of this model is the availability of Monte Carlo methods that facilitate the estimation of crossing-sign-dependent knot-transition probabilities. From such estimates, it has been possible to investigate how knot-reduction depends on the crossing-sign and the local juxtaposition geometry at the strand-passage site. A strong relationship between knot-reduction and a crossing-sign-dependent crossing-angle has been observed for this model. In the present article, we review these results and present heuristic geometrical arguments to explain this crossing-sign and angle-dependence. Finally, we discuss potential implications for other models of type II topoisomerase action on DNA.

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.

Bullied no more: when and how DNA shoves proteins around

The predominant protein-centric perspective in protein-DNA-binding studies assumes that the protein drives the interaction. Research focuses on protein structural motifs, electrostatic surfaces and contact potentials, while DNA is often ignored as a passive polymer to be manipulated. Recent studies of DNA topology, the supercoiling, knotting, and linking of the helices, have shown that DNA has the capability to be an active participant in its transactions. DNA topology-induced structural and geometric changes can drive, or at least strongly influence, the interactions between protein and DNA. Deformations of the B-form structure arise from both the considerable elastic energy arising from supercoiling and from the electrostatic energy. Here, we discuss how these energies are harnessed for topology-driven, sequence-specific deformations that can allow DNA to direct its own metabolism.

Human topoisomerase II-DNA interaction study by using atomic force microscopy

Type II topoisomerases (Topo II) are unique enzymes that change the DNA topology by catalyzing the passage of two double-strands across each other by using the energy from ATP hydrolysis. In vitro, human Topo II relaxes positive supercoiled DNA around 10-fold faster than negative supercoiled DNA. By using atomic force microscopy (AFM) we found that human Topo II binds preferentially to DNA cross-overs. Around 50% of the DNA crossings, where Topo II was bound to, presented an angle in the range of 80-90°, suggesting a favored binding geometry in the chiral discrimination by Topo II. Our studies with AFM also helped us visualize the dynamics of the unknotting action of Topo II in knotted molecules.

The ancestral role of ATP hydrolysis in type II topoisomerases: prevention of DNA double-strand breaks

Type II DNA topoisomerases (topos) catalyse changes in DNA topology by passing one double-stranded DNA segment through another. This reaction is essential to processes such as replication and transcription, but carries with it the inherent danger of permanent double-strand break (DSB) formation. All type II topos hydrolyse ATP during their reactions; however, only DNA gyrase is able to harness the free energy of hydrolysis to drive DNA supercoiling, an energetically unfavourable process. A long-standing puzzle has been to understand why the majority of type II enzymes consume ATP to support reactions that do not require a net energy input. While certain type II topos are known to 'simplify' distributions of DNA topoisomers below thermodynamic equilibrium levels, the energy required for this process is very low, suggesting that this behaviour is not the principal reason for ATP hydrolysis. Instead, we propose that the energy of ATP hydrolysis is needed to control the separation of protein-protein interfaces and prevent the accidental formation of potentially mutagenic or cytotoxic DSBs. This interpretation has parallels with the actions of a variety of molecular machines that catalyse the conformational rearrangement of biological macromolecules.

Local sensing of global DNA topology: from crossover geometry to type II topoisomerase processivity

Type II topoisomerases are ubiquitous enzymes that control the topology and higher order structures of DNA. Type IIA enzymes have the remarkable property to sense locally the global DNA topology. Although many theoretical models have been proposed, the molecular mechanism of chiral discrimination is still unclear. While experimental studies have established that topoisomerases IIA discriminate topology on the basis of crossover geometry, a recent single-molecule experiment has shown that the enzyme has a different processivity on supercoiled DNA of opposite sign. Understanding how cross-over geometry influences enzyme processivity is, therefore, the key to elucidate the mechanism of chiral discrimination. Analysing this question from the DNA side reveals first, that the different stability of chiral DNA cross-overs provides a way to locally sense the global DNA topology. Second, it shows that these enzymes have evolved to recognize the G- and T-segments stably assembled into a right-handed cross-over. Third, it demonstrates how binding right-handed cross-overs across their large angle imposes a different topological link between the topoIIA rings and the plectonemes of opposite sign thus directly affecting the enzyme freedom of motion and processivity. In bridging geometry and kinetic data, this study brings a simple solution for type IIA topoisomerase chiral discrimination.

Direct measurement of DNA bending by type IIA topoisomerases: implications for non-equilibrium topology simplification

Type IIA topoisomerases modify DNA topology by passing one segment of duplex DNA (transfer or T-segment) through a transient double-strand break in a second segment of DNA (gate or G-segment) in an ATP-dependent reaction. Type IIA topoisomerases decatenate, unknot and relax supercoiled DNA to levels below equilibrium, resulting in global topology simplification. The mechanism underlying this non-equilibrium topology simplification remains speculative. The bend angle model postulates that non-equilibrium topology simplification scales with the bend angle imposed on the G-segment DNA by the binding of a type IIA topoisomerase. To test this bend angle model, we used atomic force microscopy and single-molecule Förster resonance energy transfer to measure the extent of bending imposed on DNA by three type IIA topoisomerases that span the range of topology simplification activity. We found that Escherichia coli topoisomerase IV, yeast topoisomerase II and human topoisomerase IIα each bend DNA to a similar degree. These data suggest that DNA bending is not the sole determinant of non-equilibrium topology simplification. Rather, they suggest a fundamental and conserved role for DNA bending in the enzymatic cycle of type IIA topoisomerases.

Action at hooked or twisted-hooked DNA juxtapositions rationalizes unlinking preference of type-2 topoisomerases

The mathematical basis of the hypothesis that type-2 topoisomerases recognize and act at specific DNA juxtapositions has been investigated by coarse-grained lattice polymer models, showing that selective segment passages at hooked juxtapositions can result in dramatic reductions in catenane and knot populations. The lattice modeling approach is here extended to account for the narrowing of variance of linking number (Lk) of DNA circles by type-2 topoisomerases. In general, the steady-state variance of Lk resulting from selective segment passages at a specific juxtaposition geometry j is inversely proportional to the average linking number, Lk(j), of circles with the given juxtaposition. Based on this formulation, we demonstrate that selective segment passages at hooked juxtapositions reduce the variance of Lk. The dependence of this effect on model DNA circle size is remarkably similar to that observed experimentally for type-2 topoisomerases, which appear to be less capable in narrowing Lk variance for small DNA circles than for larger DNA circles. This behavior is rationalized by a substantial cancellation of writhe in small circles with hook-like juxtapositions. During our simulations, we uncovered a twisted variation of the hooked juxtaposition that has an even more dramatic effect on Lk variance narrowing than the hooked juxtaposition. For an extended set of juxtapositions, we detected a significant correlation between the Lk narrowing potential and the logarithmic decatenating and unknotting potentials for a given juxtaposition, a trend reminiscent of scaling relations observed with experimental measurements on type-2 topoisomerases from a variety of organisms. The consistent agreement between theory and experiment argues for type-2 topoisomerase action at hooked or twisted-hooked DNA juxtapositions.

Biophysics of knotting

Knots appear in a wide variety of biophysical systems, ranging from biopolymers, such as DNA and proteins, to macroscopic objects, such as umbilical cords and catheters. Although significant advancements have been made in the mathematical theory of knots and some progress has been made in the statistical mechanics of knots in idealized chains, the mechanisms and dynamics of knotting in biophysical systems remain far from fully understood. We report on recent progress in the biophysics of knotting-the formation, characterization, and dynamics of knots in various biophysical contexts.

The role of ATP in the reactions of type II DNA topoisomerases

Type II DNA topoisomerases catalyse changes in DNA topology in reactions coupled to the hydrolysis of ATP. In the case of DNA gyrase, which can introduce supercoils into DNA, the requirement for free energy is clear. However, the non-supercoiling type II enzymes carry out reactions that are apparently energetically favourable, so their requirement for ATP hydrolysis is not so obvious. It has been shown that many of these enzymes (the type IIA family) can simplify the topology of their DNA substrates to a level beyond that expected at equilibrium. Although this seems to explain their usage of ATP, we show that the free energies involved in topology simplification are very small (<0.2% of that available from ATP) and we argue that topology simplification may simply be an evolutionary relic.

DNA supercoiling and its role in DNA decatenation and unknotting

Chromosomal and plasmid DNA molecules in bacterial cells are maintained under torsional tension and are therefore supercoiled. With the exception of extreme thermophiles, supercoiling has a negative sign, which means that the torsional tension diminishes the DNA helicity and facilitates strand separation. In consequence, negative supercoiling aids such processes as DNA replication or transcription that require global- or local-strand separation. In extreme thermophiles, DNA is positively supercoiled which protects it from thermal denaturation. While the role of DNA supercoiling connected to the control of DNA stability, is thoroughly researched and subject of many reviews, a less known role of DNA supercoiling emerges and consists of aiding DNA topoisomerases in DNA decatenation and unknotting. Although DNA catenanes are natural intermediates in the process of DNA replication of circular DNA molecules, it is necessary that they become very efficiently decatenated, as otherwise the segregation of freshly replicated DNA molecules would be blocked. DNA knots arise as by-products of topoisomerase-mediated intramolecular passages that are needed to facilitate general DNA metabolism, including DNA replication, transcription or recombination. The formed knots are, however, very harmful for cells if not removed efficiently. Here, we overview the role of DNA supercoiling in DNA unknotting and decatenation.

Local site preference rationalizes disentangling by DNA topoisomerases

To rationalize the disentangling action of type II topoisomerases, an improved wormlike DNA model was used to delineate the degree of unknotting and decatenating achievable by selective segment passage at specific juxtaposition geometries and to determine how these activities were affected by DNA circle size and solution ionic strength. We found that segment passage at hooked geometries can reduce knot populations as dramatically as seen in experiments. Selective segment passage also provided theoretical underpinning for an intriguing empirical scaling relation between unknotting and decatenating potentials.

Helical chirality: a link between local interactions and global topology in DNA

DNA supercoiling plays a major role in many cellular functions. The global DNA conformation is however intimately linked to local DNA-DNA interactions influencing both the physical properties and the biological functions of the supercoiled molecule. Juxtaposition of DNA double helices in ubiquitous crossover arrangements participates in multiple functions such as recombination, gene regulation and DNA packaging. However, little is currently known about how the structure and stability of direct DNA-DNA interactions influence the topological state of DNA. Here, a crystallographic analysis shows that due to the intrinsic helical chirality of DNA, crossovers of opposite handedness exhibit markedly different geometries. While right-handed crossovers are self-fitted by sequence-specific groove-backbone interaction and bridging Mg(2+) sites, left-handed crossovers are juxtaposed by groove-groove interaction. Our previous calculations have shown that the different geometries result in differential stabilisation in solution, in the presence of divalent cations. The present study reveals that the various topological states of the cell are associated with different inter-segmental interactions. While the unstable left-handed crossovers are exclusively formed in negatively supercoiled DNA, stable right-handed crossovers constitute the local signature of an unusual topological state in the cell, such as the positively supercoiled or relaxed DNA. These findings not only provide a simple mechanism for locally sensing the DNA topology but also lead to the prediction that, due to their different tertiary intra-molecular interactions, supercoiled molecules of opposite signs must display markedly different physical properties. Sticky inter-segmental interactions in positively supercoiled or relaxed DNA are expected to greatly slow down the slithering dynamics of DNA. We therefore suggest that the intrinsic helical chirality of DNA may have oriented the early evolutionary choices for DNA topology.

The size of knots in polymers

Circular DNA in viruses and bacteria is often knotted. While mathematically problematic, the determination of the knot size is crucial for the study of the physical and biological behaviour of long macromolecules. Here, we review work on the size distribution of these knots under equilibrium conditions. We discuss knot localization in good and poor solvents, or in polymers that are adsorbed on a surface. We also discuss recent evidence that knot size is a crucial quantity in relaxation processes of knotted polymers.

Theoretical models of DNA topology simplification by type IIA DNA topoisomerases

It was discovered 12 years ago that type IIA topoisomerases can simplify DNA topology--the steady-state fractions of knots and links created by the enzymes are many times lower than the corresponding equilibrium fractions. Though this property of the enzymes made clear biological sense, it was not clear how small enzymes could selectively change the topology of very large DNA molecules, since topology is a global property and cannot be determined by a local DNA-protein interaction. A few models, suggested to explain the phenomenon, are analyzed in this review. We also consider experimental data that both support and contravene these models.

The why and how of DNA unlinking

The nucleotide sequence of DNA is the repository of hereditary information. Yet, it is now clear that the DNA itself plays an active role in regulating the ability of the cell to extract its information. Basic biological processes, including control of gene transcription, faithful DNA replication and segregation, maintenance of the genome and cellular differentiation are subject to the conformational and topological properties of DNA in addition to the regulation imparted by the sequence itself. How do these DNA features manifest such striking effects and how does the cell regulate them? In this review, we describe how misregulation of DNA topology can lead to cellular dysfunction. We then address how cells prevent these topological problems. We close with a discussion on recent theoretical advances indicating that the topological problems, themselves, can provide the cues necessary for their resolution by type-2 topoisomerases.

How do type II topoisomerases use ATP hydrolysis to simplify DNA topology beyond equilibrium? Investigating the relaxation reaction of nonsupercoiling type II topoisomerases

DNA topoisomerases control the topology of DNA (e.g., the level of supercoiling) in all cells. Type IIA topoisomerases are ATP-dependent enzymes that have been shown to simplify the topology of their DNA substrates to a level beyond that expected at equilibrium (i.e., more relaxed than the product of relaxation by ATP-independent enzymes, such as type I topoisomerases, or a lower-than-equilibrium level of catenation). The mechanism of this effect is currently unknown, although several models have been suggested. We have analyzed the DNA relaxation reactions of type II topoisomerases to further explore this phenomenon. We find that all type IIA topoisomerases tested exhibit the effect to a similar degree and that it is not dependent on the supercoil-sensing C-terminal domains of the enzymes. As recently reported, the type IIB topoisomerase, topoisomerase VI (which is only distantly related to type IIA enzymes), does not exhibit topology simplification. We find that topology simplification is not significantly dependent on circle size in the range approximately 2-9 kbp and is not altered by reducing the free energy available from ATP hydrolysis by varying the ADP:ATP ratio. A direct test of one model (DNA tracking; i.e., sliding of a protein clamp along DNA to trap supercoils) suggests that this is unlikely to be the explanation for the effect. We conclude that geometric selection of DNA segments by the enzymes is likely to be a primary source of the effect, but that it is possible that other kinetic factors contribute. We also speculate whether topology simplification might simply be an evolutionary relic, with no adaptive significance.

Simulations of knotting in confined circular DNA

The packing of DNA inside bacteriophages arguably yields the simplest example of genome organization in living organisms. As an assay of packing geometry, the DNA knot spectrum produced upon release of viral DNA from the P4 phage capsid has been analyzed, and compared to results of simulation of knots in confined volumes. We present new results from extensive stochastic sampling of confined self-avoiding and semiflexible circular chains with volume exclusion. The physical parameters of the chains (contour length, cross section, and bending rigidity) have been set to match those of P4 bacteriophage DNA. By using advanced sampling techniques, involving multiple Markov chain pressure-driven confinement combined with a thermodynamic reweighting technique, we establish the knot spectrum of the circular chains for increasing confinement up to the highest densities for which available algorithms can exactly classify the knots. Compactified configurations have an enclosing hull diameter approximately 2.5 times larger than the P4 caliper size. The results are discussed in relation to the recent experiments on DNA knotting inside the capsid of a P4 tailless mutant. Our investigation indicates that confinement favors chiral knots over achiral ones, as found in the experiments. However, no significant bias of torus over twist knots is found, contrary to the P4 results. The result poses a crucial question for future studies of DNA packaging in P4: is the discrepancy due to the insufficient confinement of the equilibrium simulation or does it indicate that out-of-equilibrium mechanisms (such as rotation by packaging motors) affect the genome organization, hence its knot spectrum in P4?

DNA supercoiling inhibits DNA knotting

Despite the fact that in living cells DNA molecules are long and highly crowded, they are rarely knotted. DNA knotting interferes with the normal functioning of the DNA and, therefore, molecular mechanisms evolved that maintain the knotting and catenation level below that which would be achieved if the DNA segments could pass randomly through each other. Biochemical experiments with torsionally relaxed DNA demonstrated earlier that type II DNA topoisomerases that permit inter- and intramolecular passages between segments of DNA molecules use the energy of ATP hydrolysis to select passages that lead to unknotting rather than to the formation of knots. Using numerical simulations, we identify here another mechanism by which topoisomerases can keep the knotting level low. We observe that DNA supercoiling, such as found in bacterial cells, creates a situation where intramolecular passages leading to knotting are opposed by the free-energy change connected to transitions from unknotted to knotted circular DNA molecules.

Efficient chain moves for Monte Carlo simulations of a wormlike DNA model: excluded volume, supercoils, site juxtapositions, knots, and comparisons with random-flight and lattice models

We develop two classes of Monte Carlo moves for efficient sampling of wormlike DNA chains that can have significant degrees of supercoiling, a conformational feature that is key to many aspects of biological function including replication, transcription, and recombination. One class of moves entails reversing the coordinates of a segment of the chain along one, two, or three axes of an appropriately chosen local frame of reference. These transformations may be viewed as a generalization, to the continuum, of the Madras-Orlitsky-Shepp algorithm for cubic lattices. Another class of moves, termed T+/-2, allows for interconversions between chains with different lengths by adding or subtracting two beads (monomer units) to or from the chain. Length-changing moves are generally useful for conformational sampling with a given site juxtaposition, as has been shown in previous lattice studies. Here, the continuum T+/-2 moves are designed to enhance their acceptance rate in supercoiled conformations. We apply these moves to a wormlike model in which excluded volume is accounted for by a bond-bond repulsion term. The computed autocorrelation functions for the relaxation of bond length, bond angle, writhe, and branch number indicate that the new moves lead to significantly more efficient sampling than conventional bead displacements and crankshaft rotations. A close correspondence is found in the equilibrium ensemble between the map of writhe computed for pair of chain segments and the map of site juxtapositions or self-contacts. To evaluate the more coarse-grained freely jointed chain (random-flight) and cubic lattice models that are commonly used in DNA investigations, twisting (torsional) potentials are introduced into these models. Conformational properties for a given superhelical density sigma may then be sampled by computing the writhe and using White's formula to relate the degree of twisting to writhe and sigma. Extensive comparisons of contact patterns and knot probabilities of the more coarse-grained models with the wormlike model show that the behaviors of the random-flight model are similar to that of DNA molecules in a solution environment with high ionic strengths, whereas the behaviors of the cubic lattice model with excluded volume are akin to that of DNA molecules under low ionic strengths.

3D visualization software to analyze topological outcomes of topoisomerase reactions

The action of various DNA topoisomerases frequently results in characteristic changes in DNA topology. Important information for understanding mechanistic details of action of these topoisomerases can be provided by investigating the knot types resulting from topoisomerase action on circular DNA forming a particular knot type. Depending on the topological bias of a given topoisomerase reaction, one observes different subsets of knotted products. To establish the character of topological bias, one needs to be aware of all possible topological outcomes of intersegmental passages occurring within a given knot type. However, it is not trivial to systematically enumerate topological outcomes of strand passage from a given knot type. We present here a 3D visualization software (TopoICE-X in KnotPlot) that incorporates topological analysis methods in order to visualize, for example, knots that can be obtained from a given knot by one intersegmental passage. The software has several other options for the topological analysis of mechanisms of action of various topoisomerases.

Local selection rules that can determine specific pathways of DNA unknotting by type II DNA topoisomerases

We performed numerical simulations of DNA chains to understand how local geometry of juxtaposed segments in knotted DNA molecules can guide type II DNA topoisomerases to perform very efficient relaxation of DNA knots. We investigated how the various parameters defining the geometry of inter-segmental juxtapositions at sites of inter-segmental passage reactions mediated by type II DNA topoisomerases can affect the topological consequences of these reactions. We confirmed the hypothesis that by recognizing specific geometry of juxtaposed DNA segments in knotted DNA molecules, type II DNA topoisomerases can maintain the steady-state knotting level below the topological equilibrium. In addition, we revealed that a preference for a particular geometry of juxtaposed segments as sites of strand-passage reaction enables type II DNA topoisomerases to select the most efficient pathway of relaxation of complex DNA knots. The analysis of the best selection criteria for efficient relaxation of complex knots revealed that local structures in random configurations of a given knot type statistically behave as analogous local structures in ideal geometric configurations of the corresponding knot type.