Equilibrium distributions of topological states in circular DNA: interplay of supercoiling and knotting

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.

Phase transition of DNA knotting in spherical space

Knots have been discovered in various biological systems, such as DNA. The knotting probability of DNA in free space depends non-monotonically on its bending rigidity and has a prominent peak. The current work aims to understand the underlying mechanism of the non-monotonic dependence of DNA knotting probability on bending rigidity. Monte Carlo simulations are performed on a closed DNA molecule confined in spherical space described by a worm-like chain model and a flexible kink model, respectively. The closed DNA's contour length and the spherical space radius both increase knotting probability, but also alter the unimodal dependence of knotting probability on bending rigidity. This is generalized using universal phase diagrams based on the two models. Under the flexible kink model, the total knotting probability of closed DNA is obviously increased at a relatively high excited energy. This supports the expectation that the entropy effect of knot size favours knot formation at a relatively low bending rigidity. In a given spherical space, the increasing contour length of closed DNA described by the worm-like chain model results in a visible shift in the knotting probability distribution. At the same time, the gyration radius of non-trivial closed DNA becomes comparable to that of trivial closed DNA, so that their ratio is not anti-correlated with average knot length. For closed DNA of various contour lengths, the relationship between average knot length and bending rigidity has a universal behaviour: the average knot length decreases to a local minimum at a bending rigidity of ∼5 and then gradually increases to a constant value. The existence of the local minimum is determined by the cut-off distance in repulsive Lennard-Jones potential. The bending rigidity corresponding to the beginning of the constant average knot length is consistent with that at the peak in the knotting distribution. At this point, the knot-size effect balances with the fragment free-energy effect and, at an even greater bending rigidity, knot length breathes around the average knot length value.

Keeping intracellular DNA untangled: A new role for condensin?

The DNA-passage activity of topoisomerase II accidentally produces DNA knots and interlinks within and between chromatin fibers. Fortunately, these unwanted DNA entanglements are actively removed by some mechanism. Here we present an outline on DNA knot formation and discuss recent studies that have investigated how intracellular DNA knots are removed. First, although topoisomerase II is able to minimize DNA entanglements in vitro to below equilibrium values, it is unclear whether such capacity performs equally in vivo in chromatinized DNA. Second, DNA supercoiling could bias topoisomerase II to untangle the DNA. However, experimental evidence indicates that transcriptional supercoiling of intracellular DNA boosts knot formation. Last, cohesin and condensin could tighten DNA entanglements via DNA loop extrusion (LE) and force their dissolution by topoisomerase II. Recent observations indicate that condensin activity promotes the removal of DNA knots during interphase and mitosis. This activity might facilitate the spatial organization and dynamics of chromatin.

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.

Bacterial Nucleoid: Interplay of DNA Demixing and Supercoiling

This work addresses the question of the interplay of DNA demixing and supercoiling in bacterial cells. Demixing of DNA from other globular macromolecules results from the overall repulsion between all components of the system and leads to the formation of the nucleoid, which is the region of the cell that contains the genomic DNA in a rather compact form. Supercoiling describes the coiling of the axis of the DNA double helix to accommodate the torsional stress injected in the molecule by topoisomerases. Supercoiling is able to induce some compaction of the bacterial DNA, although to a lesser extent than demixing. In this work, we investigate the interplay of these two mechanisms with the goal of determining whether the total compaction ratio of the DNA is the mere sum or some more complex function of the compaction ratios due to each mechanism. To this end, we developed a coarse-grained bead-and-spring model and investigated its properties through Brownian dynamics simulations. This work reveals that there actually exist different regimes, depending on the crowder volume ratio and the DNA superhelical density. In particular, a regime in which the effects of DNA demixing and supercoiling on the compaction of the DNA coil simply add up is shown to exist up to moderate values of the superhelical density. In contrast, the mean radius of the DNA coil no longer decreases above this threshold and may even increase again for sufficiently large crowder concentrations. Finally, the model predicts that the DNA coil may depart from the spherical geometry very close to the jamming threshold as a trade-off between the need to minimize both the bending energy of the stiff plectonemes and the volume of the DNA coil to accommodate demixing.

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.

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.

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

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

Controlling gene expression by DNA mechanics: emerging insights and challenges

Transcription initiation is a major control point for the precise regulation of gene expression. Our knowledge of this process has been mainly derived from protein-centric studies wherein cis-regulatory DNA sequences play a passive role, mainly in arranging the protein machinery to coalesce at the transcription start sites of genes in a spatial and temporal-specific manner. However, this is a highly dynamic process in which molecular motors such as RNA polymerase II (RNAPII), helicases, and other transcription factors, alter the level of mechanical force in DNA, rather than simply a set of static DNA-protein interactions. The double helix is a fiber that responds to flexural and torsional stress, which if accumulated, can affect promoter output as well as change DNA and chromatin structure. The relationship between DNA mechanics and the control of early transcription initiation events has been under-investigated. Genomic techniques to display topological stress and conformational variation in DNA across the mammalian genome provide an exciting new insight on the role of DNA mechanics in the early stages of the transcription cycle. Without understanding how torsional and flexural stresses are generated, transmitted, and dissipated, no model of transcription will be complete and accurate.

Controlling gene expression by DNA mechanics: emerging insights and challenges

Transcription initiation is a major control point for the precise regulation of gene expression. Our knowledge of this process has been mainly derived from protein-centric studies wherein cis-regulatory DNA sequences play a passive role, mainly in arranging the protein machinery to coalesce at the transcription start sites of genes in a spatial and temporal-specific manner. However, this is a highly dynamic process in which molecular motors such as RNA polymerase II (RNAPII), helicases, and other transcription factors, alter the level of mechanical force in DNA, rather than simply a set of static DNA-protein interactions. The double helix is a fiber that responds to flexural and torsional stress, which if accumulated, can affect promoter output as well as change DNA and chromatin structure. The relationship between DNA mechanics and the control of early transcription initiation events has been under-investigated. Genomic techniques to display topological stress and conformational variation in DNA across the mammalian genome provide an exciting new insight on the role of DNA mechanics in the early stages of the transcription cycle. Without understanding how torsional and flexural stresses are generated, transmitted, and dissipated, no model of transcription will be complete and accurate.

The polymer physics of single DNA confined in nanochannels

In recent years, applications and experimental studies of DNA in nanochannels have stimulated the investigation of the polymer physics of DNA in confinement. Recent advances in the physics of confined polymers, using DNA as a model polymer, have moved beyond the classic Odijk theory for the strong confinement, and the classic blob theory for the weak confinement. In this review, we present the current understanding of the behaviors of confined polymers while briefly reviewing classic theories. Three aspects of confined DNA are presented: static, dynamic, and topological properties. The relevant simulation methods are also summarized. In addition, comparisons of confined DNA with DNA under tension and DNA in semidilute solution are made to emphasize universal behaviors. Finally, an outlook of the possible future research for confined DNA is given.

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.

Two Adhesive Sites Can Enhance the Knotting Probability of DNA

Self-entanglement, or knotting, is entropically favored in long polymers. Relatively short polymers such as proteins can knot as well, but in this case the entanglement is mainly driven by fine-tuned, sequence-specific interactions. The relation between the sequence of a long polymer and its topological state is here investigated by means of a coarse-grained model of DNA. We demonstrate that the introduction of two adhesive regions along the sequence of a self-avoiding chain substantially increases the probability of forming a knot.

Topological Behavior of Plasmid DNA

The discovery of the B-form structure of DNA by Watson and Crick led to an explosion of research on nucleic acids in the fields of biochemistry, biophysics, and genetics. Powerful techniques were developed to reveal a myriad of different structural conformations that change B-DNA as it is transcribed, replicated, and recombined and as sister chromosomes are moved into new daughter cell compartments during cell division. This article links the original discoveries of superhelical structure and molecular topology to non-B form DNA structure and contemporary biochemical and biophysical techniques. The emphasis is on the power of plasmids for studying DNA structure and function. The conditions that trigger the formation of alternative DNA structures such as left-handed Z-DNA, inter- and intra-molecular triplexes, triple-stranded DNA, and linked catenanes and hemicatenanes are explained. The DNA dynamics and topological issues are detailed for stalled replication forks and for torsional and structural changes on DNA in front of and behind a transcription complex and a replisome. The complex and interconnected roles of topoisomerases and abundant small nucleoid association proteins are explained. And methods are described for comparing in vivo and in vitro reactions to probe and understand the temporal pathways of DNA and chromosome chemistry that occur inside living cells.

Free-energy calculations for semi-flexible macromolecules: applications to DNA knotting and looping

We present a method to obtain numerically accurate values of configurational free energies of semiflexible macromolecular systems, based on the technique of thermodynamic integration combined with normal-mode analysis of a reference system subject to harmonic constraints. Compared with previous free-energy calculations that depend on a reference state, our approach introduces two innovations, namely, the use of internal coordinates to constrain the reference states and the ability to freely select these reference states. As a consequence, it is possible to explore systems that undergo substantially larger fluctuations than those considered in previous calculations, including semiflexible biopolymers having arbitrary ratios of contour length L to persistence length P. To validate the method, high accuracy is demonstrated for free energies of prime DNA knots with L/P = 20 and L/P = 40, corresponding to DNA lengths of 3000 and 6000 base pairs, respectively. We then apply the method to study the free-energy landscape for a model of a synaptic nucleoprotein complex containing a pair of looped domains, revealing a bifurcation in the location of optimal synapse (crossover) sites. This transition is relevant to target-site selection by DNA-binding proteins that occupy multiple DNA sites separated by large linear distances along the genome, a problem that arises naturally in gene regulation, DNA recombination, and the action of type-II topoisomerases.

Efficient deformation algorithm for plasmid DNA simulations

BACKGROUND

Plasmid DNA molecules are closed circular molecules that are widely used in life sciences, particularly in gene therapy research. Monte Carlo methods have been used for several years to simulate the conformational behavior of DNA molecules. In each iteration these simulation methods randomly generate a new trial conformation, which is either accepted or rejected according to a criterion based on energy calculations and stochastic rules. These simulation trials are generated using a method based on crankshaft motion that, apart from some slight improvements, has remained the same for many years.

RESULTS

In this paper, we present a new algorithm for the deformation of plasmid DNA molecules for Monte Carlo simulations. The move underlying our algorithm preserves the size and connectivity of straight-line segments of the plasmid DNA skeleton. We also present the results of three experiments comparing our deformation move with the standard and biased crankshaft moves in terms of acceptance ratio of the trials, energy and temperature evolution, and average displacement of the molecule. Our algorithm can also be used as a generic geometric algorithm for the deformation of regular polygons or polylines that preserves the connections and lengths of their segments.

CONCLUSION

Compared with both crankshaft moves, our move generates simulation trials with higher acceptance ratios and smoother deformations, making it suitable for real-time visualization of plasmid DNA coiling. For that purpose, we have adopted a DNA assembly algorithm that uses nucleotides as building blocks.

New biologically motivated knot table

The knot nomenclature in common use, summarized in Rolfsen's knot table [Rolfsen (1990) Knots and Links, American Mathematical Society], was not originally designed to distinguish between mirror images. This ambiguity is particularly inconvenient when studying knotted biopolymers such as DNA and proteins, since their chirality is often significant. In the present article, we propose a biologically meaningful knot table where a representative of a chiral pair is chosen on the basis of its mean writhe. There is numerical evidence that the sign of the mean writhe is invariant for each knot in a chiral pair. We review numerical evidence where, for each knot type K, the mean writhe is taken over a large ensemble of randomly chosen realizations of K. It has also been proposed that a chiral pair can be distinguished by assessing the writhe of a minimal or ideal conformation of the knot. In all cases examined to date, the two methods produce the same results.

On the mean and variance of the writhe of random polygons

We here address two problems concerning the writhe of random polygons. First, we study the behavior of the mean writhe as a function length. Second, we study the variance of the writhe. Suppose that we are dealing with a set of random polygons with the same length and knot type, which could be the model of some circular DNA with the same topological property. In general, a simple way of detecting chirality of this knot type is to compute the mean writhe of the polygons; if the mean writhe is non-zero then the knot is chiral. How accurate is this method? For example, if for a specific knot type K the mean writhe decreased to zero as the length of the polygons increased, then this method would be limited in the case of long polygons. Furthermore, we conjecture that the sign of the mean writhe is a topological invariant of chiral knots. This sign appears to be the same as that of an "ideal" conformation of the knot. We provide numerical evidence to support these claims, and we propose a new nomenclature of knots based on the sign of their expected writhes. This nomenclature can be of particular interest to applied scientists. The second part of our study focuses on the variance of the writhe, a problem that has not received much attention in the past. In this case, we focused on the equilateral random polygons. We give numerical as well as analytical evidence to show that the variance of the writhe of equilateral random polygons (of length n) behaves as a linear function of the length of the equilateral random polygon.

Interplay between writhe and knotting for swollen and compact polymers

The role of the topology and its relation with the geometry of biopolymers under different physical conditions is a nontrivial and interesting problem. Aiming at understanding this issue for a related simpler system, we use Monte Carlo methods to investigate the interplay between writhe and knotting of ring polymers in good and poor solvents. The model that we consider is interacting self-avoiding polygons on the simple cubic lattice. For polygons with fixed knot type, we find a writhe distribution whose average depends on the knot type but is insensitive to the length N of the polygon and to solvent conditions. This "topological contribution" to the writhe distribution has a value that is consistent with that of ideal knots. The standard deviation of the writhe increases approximately as square root(N) in both regimes, and this constitutes a geometrical contribution to the writhe. If the sum over all knot types is considered, the scaling of the standard deviation changes, for compact polygons, to approximately N(0.6). We argue that this difference between the two regimes can be ascribed to the topological contribution to the writhe that, for compact chains, overwhelms the geometrical one, thanks to the presence of a large population of complex knots at relatively small values of N. For polygons with fixed writhe, we find that the knot distribution depends on the chosen writhe, with the occurrence of achiral knots being considerably suppressed for large writhe. In general, the occurrence of a given knot thus depends on a nontrivial interplay between writhe, chain length, and solvent conditions.

Group theory and biomolecular conformation: I. Mathematical and computational models

Biological macromolecules, and the complexes that they form, can be described in a variety of ways ranging from quantum mechanical and atomic chemical models, to coarser grained models of secondary structure and domains, to continuum models. At each of these levels, group theory can be used to describe both geometric symmetries and conformational motion. In this survey, a detailed account is provided of how group theory has been applied across computational structural biology to analyze the conformational shape and motion of macromolecules and complexes.

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.

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.

Coupling of twist and writhe in short DNA loops

While bending and twist can be treated as independent degrees of freedom for linear DNA molecules, the loop closure constraint introduces a coupling between these variables in circular DNA. We performed Monte Carlo simulations of wormlike rods with both bending and twist rigidity in order to study the coupling between the writhe and twist distributions for various DNA lengths. We find that for sufficiently short DNA, the writhe distribution differs from that of a model with bending energy only. We show that the factorization approximation introduced by previous researchers coincides, within numerical accuracy, with our simulation results, and conclude that the closure constraint is fully accounted for by the White-Fuller relation. Experimental tests of our results for short DNA plasmids are proposed.

Topoisomerase I-mediated DNA relaxation as a tool to study intercalation of small molecules into supercoiled DNA

Several biochemical and biophysical methods are available to study the intercalation of a small molecule between two consecutive base pairs of DNA. Among them, the topoisomerase I-mediated DNA relaxation assay has proved highly efficient, relatively easy to handle and very informative to investigate drug binding to DNA. The test relies on the use of a supercoiled plasmid to mimic the topological constraints of genomic DNA. The three main components of the assay - the topoisomerase I enzyme, DNA helix and intercalating small molecules - are presented here in a structural context. The principle of the assay is described in detail, along with a typical experimental protocol.

DNA-DNA interactions in bacteriophage capsids are responsible for the observed DNA knotting

Recent experiments showed that the linear double-stranded DNA in bacteriophage capsids is both highly knotted and neatly structured. What is the physical basis of this organization? Here we show evidence from stochastic simulation techniques that suggests that a key element is the tendency of contacting DNA strands to order, as in cholesteric liquid crystals. This interaction favors their preferential juxtaposition at a small twist angle, thus promoting an approximately nematic (and apolar) local order. The ordering effect dramatically impacts the geometry and topology of DNA inside phages. Accounting for this local potential allows us to reproduce the main experimental data on DNA organization in phages, including the cryo-EM observations and detailed features of the spectrum of DNA knots formed inside viral capsids. The DNA knots we observe are strongly delocalized and, intriguingly, this is shown not to interfere with genome ejection out of the phage.

First-principles calculation of DNA looping in tethered particle experiments

We calculate the probability of DNA loop formation mediated by regulatory proteins such as Lac repressor (LacI), using a mathematical model of DNA elasticity. Our model is adapted to calculating quantities directly observable in tethered particle motion (TPM) experiments, and it accounts for all the entropic forces present in such experiments. Our model has no free parameters; it characterizes DNA elasticity using information obtained in other kinds of experiments. It assumes a harmonic elastic energy function (or wormlike chain type elasticity), but our Monte Carlo calculation scheme is flexible enough to accommodate arbitrary elastic energy functions. We show how to compute both the 'looping J factor' (or equivalently, the looping free energy) for various DNA construct geometries and LacI concentrations, as well as the detailed probability density function of bead excursions. We also show how to extract the same quantities from recent experimental data on TPM, and then compare to our model's predictions. In particular, we present a new method to correct observed data for finite camera shutter time and other experimental effects. Although the currently available experimental data give large uncertainties, our first-principles predictions for the looping free energy change are confirmed to within about 1 k(B)T, for loops of length around 300 basepairs. More significantly, our model successfully reproduces the detailed distributions of bead excursion, including their surprising three-peak structure, without any fit parameters and without invoking any alternative conformation of the LacI tetramer. Indeed, the model qualitatively reproduces the observed dependence of these distributions on tether length (e.g., phasing) and on LacI concentration (titration). However, for short DNA loops (around 95 basepairs) the experiments show more looping than is predicted by the harmonic-elasticity model, echoing other recent experimental results. Because the experiments we study are done in vitro, this anomalously high looping cannot be rationalized as resulting from the presence of DNA-bending proteins or other cellular machinery. We also show that it is unlikely to be the result of a hypothetical 'open' conformation of the LacI tetramer.

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.

Temperature dependence of circular DNA topological states

Circular double-stranded DNA has different topological states which are defined by their linking numbers. Equilibrium distribution of linking numbers can be obtained by closing a linear DNA into a circle by ligase. Using Monte Carlo simulation, we predict the temperature dependence of the linking number distribution of small circular DNAs. Our predictions are based on flexible defect excitations that resulted from local melting or unstacking of DNA base pairs. We found that the reduced bending rigidity alone can lead to measurable changes of the variance of linking number distribution of short circular DNAs. If the defect is accompanied by local unwinding, the effect becomes much more prominent. The predictions can be easily investigated in experiments, providing a new method to study the micromechanics of sharply bent DNAs and the thermal stability of specific DNA sequences. Furthermore, the predictions are directly applicable to the studies of binding of DNA-distorting proteins that can locally reduce DNA rigidity, form DNA kinks, or introduce local unwinding.

DNA topoisomerases: harnessing and constraining energy to govern chromosome topology

DNA topoisomerases are a diverse set of essential enzymes responsible for maintaining chromosomes in an appropriate topological state. Although they vary considerably in structure and mechanism, the partnership between topoisomerases and DNA has engendered commonalities in how these enzymes engage nucleic acid substrates and control DNA strand manipulations. All topoisomerases can harness the free energy stored in supercoiled DNA to drive their reactions; some further use the energy of ATP to alter the topology of DNA away from an enzyme-free equilibrium ground state. In the cell, topoisomerases regulate DNA supercoiling and unlink tangled nucleic acid strands to actively maintain chromosomes in a topological state commensurate with particular replicative and transcriptional needs. To carry out these reactions, topoisomerases rely on dynamic macromolecular contacts that alternate between associated and dissociated states throughout the catalytic cycle. In this review, we describe how structural and biochemical studies have furthered our understanding of DNA topoisomerases, with an emphasis on how these complex molecular machines use interfacial interactions to harness and constrain the energy required to manage DNA topology.

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.

Production of highly knotted DNA by means of cosmid circularization inside phage capsids

BACKGROUND

The formation of DNA knots is common during biological transactions. Yet, functional implications of knotted DNA are not fully understood. Moreover, potential applications of DNA molecules condensed by means of knotting remain to be explored. A convenient method to produce abundant highly knotted DNA would be highly valuable for these studies.

RESULTS

We had previously shown that circularization of the 11.2 kb linear DNA of phage P4 inside its viral capsid generates complex knots by the effect of confinement. We demonstrate here that this mechanism is not restricted to the viral genome. We constructed DNA cosmids as small as 5 kb and introduced them inside P4 capsids. Such cosmids were then recovered as a complex mixture of highly knotted DNA circles. Over 250 mug of knotted cosmid were typically obtained from 1 liter of bacterial culture.

CONCLUSION

With this biological system, DNA molecules of varying length and sequence can be shaped into very complex and heterogeneous knotted forms. These molecules can be produced in preparative amounts suitable for systematic studies and applications.

Hin-mediated DNA knotting and recombining promote replicon dysfunction and mutation

Background

The genetic code imposes a dilemma for cells. The DNA must be long enough to encode for the complexity of an organism, yet thin and flexible enough to fit within the cell. The combination of these properties greatly favors DNA collisions, which can knot and drive recombination of the DNA. Despite the well-accepted propensity of cellular DNA to collide and react with itself, it has not been established what the physiological consequences are.

Results

Here we analyze the effects of recombined and knotted plasmids in E. coli using the Hin site-specific recombination system. We show that Hin-mediated DNA knotting and recombination (i) promote replicon loss by blocking DNA replication; (ii) block gene transcription; and (iii) cause genetic rearrangements at a rate three to four orders of magnitude higher than the rate for an unknotted, unrecombined plasmid.

Conclusion

These results show that DNA reactivity leading to recombined and knotted DNA is potentially toxic and may help drive genetic evolution.

Evaluation of elastic properties of atomistic DNA models

A number of intriguing aspects in dynamics of double-helical DNA is related to the coupling between its macroscopic and microscopic states. A link between the elastic properties of long DNA chains and their atom-level dynamics can be established by comparing the worm-like chain model of polymer DNA with the conformational ensembles produced by molecular dynamics simulations. This problem is complicated by the complexity of the DNA structure, the small size of DNA fragments, and relatively short trajectory durations accessible in computer simulations of microscopic DNA dynamics. A careful study of all these aspects has been performed by using longer DNA fragments and increased durations of MD trajectories as compared to earlier such investigations. Special attention is paid to the necessary conditions and criteria of time convergence, and the possibility to increase the sampling by using constrained DNA models and simplified simulation conditions. It is found that dynamics of 25-mer duplexes with regular sequences agrees well with the worm-like chain theory and that accurate evaluation of DNA elastic parameters requires at least two turns of the double helix and approximately 20-ns duration of trajectories. Bond length and bond-angle constraints affect the estimates within numerical errors. In contrast, simplified treatment of solvation can strongly change the observed elastic parameters of DNA. The elastic parameters evaluated for AT- and GC-alternating duplexes reasonably agree with experimental data and suggest that, in different basepair sequences, the torsional and stretching elasticities vary stronger than the bending stiffness.

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

Topoisomerases may unknot by recognizing specific DNA juxtapositions. The physical basis of this hypothesis is investigated by considering single-loop conformations in a coarse-grained polymer model. We determine the statistical relationship between the local geometry of a juxtaposition of two chain segments and whether the loop is knotted globally, and ascertain how the knot/unknot topology is altered by a topoisomerase-like segment passage at the juxtaposition. Segment passages at a "free" juxtaposition tend to increase knot probability. In contrast, segment passages at a "hooked" juxtaposition cause more transitions from knot to unknot than vice versa, resulting in a steady-state knot probability far lower than that at topological equilibrium. The reduction in knot population by passing chain segments through a hooked juxtaposition is more prominent for loops of smaller sizes, n, but remains significant even for larger loops: steady-state knot probability is only approximately 2%, and approximately 5% of equilibrium, respectively, for n=100 and 500 in the model. An exhaustive analysis of approximately 6000 different juxtaposition geometries indicates that the ability of a segment passage to unknot correlates strongly with the juxtaposition's "hookedness". Remarkably, and consistent with experiments on type-2 topoisomerases from different organisms, the unknotting potential of a juxtaposition geometry in our polymer model correlates almost perfectly with its corresponding decatenation potential. These quantitative findings suggest that it is possible for topoisomerases to disentangle by acting selectively on juxtapositions with "hooked" geometries.

Conformational Analysis of Stiff Chiral Polymers with End-Constraints

We present a Lie-group-theoretic method for the kinematic and dynamic analysis of chiral semi-flexible polymers with end constraints. The first is to determine the minimum energy conformations of semi-flexible polymers with end constraints, and the second is to perform normal mode analysis based on the determined minimum energy conformations. In this paper, we use concepts from the theory of Lie groups and principles of variational calculus to model such polymers as inextensible or extensible chiral elastic rods with coupling between twisting and bending stiffnesses, and/or between twisting and extension stiffnesses. This method is general enough to include any stiffness and chirality parameters in the context of elastic filament models with the quadratic elastic potential energy function. As an application of this formulation, the analysis of DNA conformations is discussed. We demonstrate our method with examples of DNA conformations in which topological properties such as writhe, twist, and linking number are calculated from the results of the proposed method. Given these minimum energy conformations, we describe how to perform the normal mode analysis. The results presented here build both on recent experimental work in which DNA mechanical properties have been measured, and theoretical work in which the mechanics of non-chiral elastic rods has been studied.

DNA knots reveal a chiral organization of DNA in phage capsids

Icosahedral bacteriophages pack their double-stranded DNA genomes to near-crystalline density and achieve one of the highest levels of DNA condensation found in nature. Despite numerous studies, some essential properties of the packaging geometry of the DNA inside the phage capsid are still unknown. We present a different approach to the problems of randomness and chirality of the packed DNA. We recently showed that most DNA molecules extracted from bacteriophage P4 are highly knotted because of the cyclization of the linear DNA molecule confined in the phage capsid. Here, we show that these knots provide information about the global arrangement of the DNA inside the capsid. First, we analyze the distribution of the viral DNA knots by high-resolution gel electrophoresis. Next, we perform Monte Carlo computer simulations of random knotting for freely jointed polygons confined to spherical volumes. Comparison of the knot distributions obtained by both techniques produces a topological proof of nonrandom packaging of the viral DNA. Moreover, our simulations show that the scarcity of the achiral knot 4(1) and the predominance of the torus knot 5(1) over the twist knot 5(2) observed in the viral distribution of DNA knots cannot be obtained by confinement alone but must include writhe bias in the conformation sampling. These results indicate that the packaging geometry of the DNA inside the viral capsid is writhe-directed.

Monte Carlo simulations of supercoiled DNAs confined to a plane

Recent advances in atomic force microscopy (AFM) have enabled researchers to obtain images of supercoiled DNAs deposited on mica surfaces in buffered aqueous milieux. Confining a supercoiled DNA to a plane greatly restricts its configurational freedom, and could conceivably alter certain structural properties, such as its twist and writhe. A program that was originally written to perform Monte Carlo simulations of supercoiled DNAs in solution was modified to include a surface potential. This potential flattens the DNAs to simulate the effect of deposition on a surface. We have simulated transfers of a 3760-basepair supercoiled DNA from solution to a surface in both 161 and 10 mM ionic strength. In both cases, the geometric and thermodynamic properties of the supercoiled DNAs on the surface differ significantly from the corresponding quantities in solution. At 161 mM ionic strength, the writhe/twist ratio is 1.20-1.33 times larger for DNAs on the surface than for DNAs in solution and significant differences in the radii of gyration are also observed. Simulated surface structures in 161 mM ionic strength closely resemble those observed by AFM. Simulated surface structures in 10 mM ionic strength are similar to a minority of the structures observed by AFM, but differ from the majority of such structures for unknown reasons. In 161 mM ionic strength, the internal energy (excluding the surface potential) decreases substantially as the DNA is confined to the surface. Evidently, supercoiled DNAs in solution are typically deformed farther from the minimum energy configuration than are the corresponding surface-confined DNAs. Nevertheless, the work (Delta A(int)) done on the internal coordinates, which include uniform rotations at constant configuration, during the transfer is positive and 2.6-fold larger than the decrease in internal energy. The corresponding entropy change is negative, and its contribution to Delta A(int) is positive and exceeds the decrease in internal energy by 3.6 fold. The work done on the internal coordinates during the solution-to-surface transfer is directed primarily toward reducing their entropy. Evidently, the number of configurations available to the more deformed solution DNA is vastly greater than for the less deformed surface-confined DNA.

DNA topology on an increase in positive writhing number of DNA: conformation changes in the time course of cis-diamminedichloroplatinum(II)-DNA adducts

We show that the topological significance of the gel mobility of cis-diamminedichloroplatinum(II) (DDP)-closed circular DNA (ccDNA) adducts decreases with reaction time, until a point at which it joins relaxed DNA, and that the mobility of the adducts increases again. There is no relationship between the relative length of the adducts and the gel mobility. Although the significance of the decrease of gel mobility is due to the unwinding of cis-DDP-DNA (or trans-DDP-DNA) adducts, the conformational significance of the subsequent increase in mobility is unclear. To elucidate the conformational significance for unwinding of the adducts, we measured the writhing number (Wk) of the adducts using electron microscopy and analyzed the topological states of cis-DDP (or trans-DDP) adducts based on the White rule, Lk=Wk+Tk. Where, Lk and Tk represent the linking and twisting number in the ring, respectively. From the data, we found that the Wk of cis-DDP-ccDNA adducts in comparison with trans-DDP-ccDNA adducts increases from a negative to a positive number with time. This suggests that cis-DDP plays a role in the change of the topological state of ccDNA. In the abstraction of platinum from the adducts with CN- ion, the differences in both topological states may explain why Pt in trans-DDP is abstracted more easily than in cis-DDP. To explain the abstraction of Pt ion, we also discuss the findings based on the thermodynamic cycle in a intermolecular crosslink model Pt(NH3)2(guanine)2(2+)-->Pt(CN)4(2-) using the Pt parametrized PM3 method.

Pkd1 unusual DNA conformations are recognized by nucleotide excision repair

The 2.5-kilobase pair poly(purine.pyrimidine) (poly(R.Y)) tract present in intron 21 of the polycystic kidney disease 1 (PKD1) gene has been proposed to contribute to the high mutation frequency of the gene. To evaluate this hypothesis, we investigated the growth rates of 11 Escherichia coli strains, with mutations in the nucleotide excision repair, SOS, and topoisomerase I and/or gyrase genes, harboring plasmids containing the full-length tract, six 5'-truncations of the tract, and a control plasmid (pSPL3). The full-length poly(R.Y) tract induced dramatic losses of cell viability during the first few hours of growth and lengthened the doubling times of the populations in strains with an inducible SOS response. The extent of cell loss was correlated with the length of the poly(R.Y) tract and the levels of negative supercoiling as modulated by the genotype of the strains or drugs that specifically inhibited DNA gyrase or bound to DNA directly, thereby affecting conformations at specific loci. We conclude that the unusual DNA conformations formed by the PKD1 poly(R.Y) tract under the influence of negative supercoiling induced the SOS response pathway, and they were recognized as lesions by the nucleotide excision repair system and were cleaved, causing delays in cell division and loss of the plasmid. These data support a role for this sequence in the mutation of the PKD1 gene by stimulating repair and/or recombination functions.

Topoisomerase IV, alone, unknots DNA in E. coli

Knotted DNA has potentially devastating effects on cells. By using two site-specific recombination systems, we tied all biologically significant simple DNA knots in Escherichia coli. When topoisomerase IV activity was blocked, either with a drug or in a temperature-sensitive mutant, the knotted recombination intermediates accumulated whether or not gyrase was active. In contrast to its decatenation activity, which is strongly affected by DNA supercoiling, topoisomerase IV unknotted DNA independently of supercoiling. This differential supercoiling effect held true regardless of the relative sizes of the catenanes and knots. Finally, topoisomerase IV unknotted DNA equally well when DNA replication was blocked with hydroxyurea. We conclude that topoisomerase IV, not gyrase, unknots DNA and that it is able to access DNA in the cell freely. With these results, it is now possible to assign completely the topological roles of the topoisomerases in E. coli. It is clear that the topoisomerases in the cell have distinct and nonoverlapping roles. Consequently, our results suggest limitations in assigning a physiological function to a protein based upon sequence similarity or even upon in vitro biochemical activity.