DNA Knots Research Articles

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.

Dynamics of a rigid body in a Stokes fluid

We demonstrate that the dynamics of a rigid body falling in an infinite viscous fluid can, in the Stokes limit, be reduced to the study of a three-dimensional system of ordinary differential equations can be approximated using the Rotne–Prager theory, and we present various examples corresponding to certain ideal shapes of knots which illustrate the various possible multiplicities of steady states. Our simulations of rigid ideal knots in a Stokes fluid predict an approximate linear relation between sedimentation speed and average crossing number, as has been observed experimentally for the much more complicated system of real DNA knots in gel electrophoresis.

A mathematical model for enzymatic action on DNA knots and links

This paper deals with some studies pertaining te, nonprocessive recombinase viz. Topoisomerases III, IV. The mathematics of tangles is found to be very useful in studying topoisomerases and recombinases (processive and nonprocessive). It has been seen that the enzyme acts on the DNA if it is in a certain configuration. Electron micrographs of the enzyme-DNA complex show the enzyme as a blob with DNA looping out of it, but they are unable to determine the configuration of the DNA within the blob. By using knot theory and tangle calculus, the configuration of DNA within the enzyme blob as well as the enzyme action has been determined in saine cases.

Topological analysis of Hin-catalysed DNA recombination in vivo and in vitro.

In vitro studies have demonstrated that Hin-catalysed site-specific DNA inversion occurs within a tripartite invertasome complex assembled at a branch on a supercoiled DNA molecule. Multiple DNA exchanges within a recombination complex (processive recombination) have been found to occur with particular substrates or reaction conditions. To investigate the mechanistic properties of the Hin recombination reaction in vivo, we have analysed the topology of recombination products generated by Hin catalysis in growing cells. Recombination between wild-type recombination sites in vivo is primarily limited to one exchange. However, processive recombination leading to knotted DNA products is efficient on substrates containing recombination sites with non-identical core nucleotides. Multiple exchanges are limited by a short DNA segment between the Fis-bound enhancer and closest recombination site and by the strength of Fis-Hin interactions, implying that the enhancer normally remains associated with the recombining complex throughout a single exchange reaction, but that release of the enhancer leads to multiple exchanges. This work confirms salient mechanistic aspects of the reaction in vivo and provides strong evidence for the propensity of plectonemically branched DNA in prokaryotic cells. We also demonstrated that a single DNA exchange resulting in inversion in vitro is accompanied by a loss of four negative supercoils.

Model carbyne knots vs ideal knots.

The structure and stability of model carbyne knots built from 60 to 120 carbon atoms with 0, 3, 4,., 7 crossings have been estimated by semiempirical AM1 calculations. The calculations have shown an increase of the knot-cycle energy difference (deltaE) with an increasing number of knot crossings and a decrease of deltaE with an increasing number of atoms constituting the molecule. The deltaE changes nonlinearly with the characteristics of the corresponding ideal knots such as the average crossing number (ACN) and the length-to-diameter ratio (L/D). The molecular mechanic strain energy of carbyne knots correlates similarly with ACN and L/D of ideal knots. The calculated energy of the model carbyne knots correlates also with the electrophoretic mobility or sedimentation coefficient of DNA knots. Thus, similarly to characteristics of ideal knots, the energy of carbyne knots is a rather easily available parameter which can be used for further correlations with some characteristics of DNA knots.

Wezly i sploty DNA

A review of DNA knots and links (catenanes) is presented. Knots are interlaced cyclic structures while links are at least two cyclic structures mutually interlaced, as chain links are, which no projection on a plane without crossing points exists for. The examples of DNA knots and links, that are produced by replication and recombination processes, have been given as well as the methods of their investigations have been brought. Model research works focused on determining the probability of topological structures creations, with using molecular mechanics methods, have been discussed. An idea of ideal knots and links, allowing to find the correlations between electrophoretic mobility and sedimentation coefficient of real DNA knots and links and parameters of ideal ones, was discussed.

Topology matters: Some aspects of DNA physics

ABSTRACTBiological cells, in some sense, are all about topology: Biomembranes separating different volumes from one another, ions or even macromolecules having to cross these membranes in controlled fashion through membrane pores; or certain proteins moving along the DNA to find their target sequence instead of searching for this site in the full 3-dimensional cell volume. Even on the single biopolymer level, topology is an essential ingredient: Intriguingly, in bacteria DNA occurs knotted, i.e., in a state topologically different from a simply connected ring. It is a key question to understand the statistical behaviour of such knotted DNA to understand a number of physiological processes having to overcome this knottedness, or to quantify results from DNA separation techniques such as electrophoresis, in which the knottedness influences the mobility. At the same time, double-stranded DNA continuously opens up floppy single-stranded bubbles, which fluctuate in size, exposing the single Watson-Crick bases to binding proteins. Again, statistical mechanical tools can be employed to examine the bubble dynamics. Here, we introduce some recent results on DNA knots and bubble fluctuations.

Construction and electrophoretic migration of single-stranded DNA knots and catenanes.

In recent years there has been growing interest in the question of how the particular topology of polymeric chains affects their overall dimensions and physical behavior. The majority of relevant studies are based on numerical simulation methods or analytical treatment; however, both these approaches depend on various assumptions and simplifications. Experimental verification is clearly needed but was hampered by practical difficulties in obtaining preparative amounts of knotted or catenated polymers with predefined topology and precisely set chain length. We introduce here an efficient method of production of various single-stranded DNA knots and catenanes that have the same global chain length. We also characterize electrophoretic migration of the produced single-stranded DNA knots and catenanes with increasing complexity.

Novel display of knotted DNA molecules by two-dimensional gel electrophoresis.

We describe a two-dimensional agarose gel electrophoresis procedure that improves the resolution of knotted DNA molecules. The first gel dimension is run at low voltage, and DNA knots migrate according to their compactness. The second gel dimension is run at high voltage, and DNA knots migrate according to other physical parameters such as shape and flexibility. In comparison with one-dimensional gel electrophoresis, this procedure segregates the knotted DNA molecules from other unknotted forms of DNA, and partially resolves populations of knots that have the same number of crossings. The two-dimensional display may allow quantitative and qualitative characterization of different types of DNA knots simply by gel velocity.

Topological Entropy of a Stiff Ring Polymer and Its Connection to DNA Knots

We discuss the entropy of a circular polymer under a topological constraint. We call it the {\it topological entropy} of the polymer, in short. A ring polymer does not change its topology (knot type) under any thermal fluctuations. Through numerical simulations using some knot invariants, we show that the topological entropy of a stiff ring polymer with a fixed knot is described by a scaling formula as a function of the thickness and length of the circular chain. The result is consistent with the viewpoint that for stiff polymers such as DNAs, the length and diameter of the chains should play a central role in their statistical and dynamical properties. Furthermore, we show that the new formula extends a known theoretical formula for DNA knots.

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.

BIOLOGICAL DISTANCES ON DNA KNOTS AND LINKS: Applications to XER recombination

The mathematics of tangles has been very useful in studying recombinases which act processively and which require DNA to be in a certain configuration in order for the enzyme to act. Electron micrographs of the enzyme-DNA complex show the enzyme as a blob with DNA looping out of it. The configuration of the DNA within the blob cannot be determined form the electron micrographs. However, mathematics can in some cases determine the configuration of the DNA within the enzyme blob as well as the enzyme action. In this paper, several theorems used to analyze recombinase experiments are summarized. In particular Xer recombinase, an enzyme which does not act processively is analyzed. Unfortunately, for enzymes which do not act processively, infinitely many possibilities exist. Several experiments are proposed to reduce this number and to emphasize both the usefulness and limitations of tangle analysis. Although the local action cannot be mathematically determined without more biological assumptions, it is possible to determine the topology of the synaptic complex through additional biolgical experiments.

The Biot–Savart operator for application to knot theory, fluid dynamics, and plasma physics

The writhing number of a curve in 3-space is the standard measure of the extent to which the curve wraps and coils around itself; it has proved its importance for molecular biologists in the study of knotted DNA and of the enzymes which affect it. The helicity of a vector field defined on a domain in 3-space is the standard measure of the extent to which the field lines wrap and coil around one another; it plays important roles in fluid dynamics and plasma physics. The Biot–Savart operator associates with each current distribution on a given domain the restriction of its magnetic field to that domain. When the domain is simply connected, the divergence-free fields which are tangent to the boundary and which minimize energy for given helicity provide models for stable force-free magnetic fields in space and laboratory plasmas; these fields appear mathematically as the extreme eigenfields for an appropriate modification of the Biot–Savart operator. Information about these fields can be converted into bounds on the writhing number of a given piece of DNA. The purpose of this paper is to reveal new properties of the Biot–Savart operator which are useful in these applications.

Plasmid DNA supercoiling by DNA gyrase.

DNA gyrase catalyzes DNA supercoiling in a reaction coupled to ATP hydrolysis (1). The enzyme has been found in many eubacterial species and is unique among the topoisomerases in promoting negative supercoiling of DNA (2,3). A variety of studies have shown that gyrase is essential for bacterial growth with roles in DNA replication, transcription, and recombination. Moreover, in combination with the relaxing activity of DNA topoisomerase I, gyrase is responsible for the homeostatic regulation of DNA supercoiling in bacteria (reviewed in refs. 3,4). The enzyme from Escherichia coli has been the most extensively characterized, and is a tetramer made up of two GyrA and two GyrB subunits encoded by the gyrA and gyrB genes, respectively. DNA supercoiling takes place by the directional crossing of a DNA duplex through a transient enzyme-bridged double-strand break in a 120–150 bp segment of DNA wrapped on the enzyme (5–7). This process changes the linking number of DNA in steps of two, and together with an ability to form and resolve DNA knots and catenanes, establishes gyrase as a type II topoisomerase.

DNA loops and semicatenated DNA junctions

BackgroundAlternative DNA conformations are of particular interest as potential signals to mark important sites on the genome. The structural variability of CA microsatellites is particularly pronounced; these are repetitive poly(CA) · poly(TG) DNA sequences spread in all eukaryotic genomes as tracts of up to 60 base pairs long. Many in vitro studies have shown that the structure of poly(CA) · poly(TG) can vary markedly from the classical right handed DNA double helix and adopt diverse alternative conformations. Here we have studied the mechanism of formation and the structure of an alternative DNA structure, named Form X, which was observed previously by polyacrylamide gel electrophoresis of DNA fragments containing a tract of the CA microsatellite poly(CA) · poly(TG) but had not yet been characterized.ResultsFormation of Form X was found to occur upon reassociation of the strands of a DNA fragment containing a tract of poly(CA) · poly(TG), in a process strongly stimulated by the nuclear proteins HMG1 and HMG2. By inserting Form X into DNA minicircles, we show that the DNA strands do not run fully side by side but instead form a DNA knot. When present in a closed DNA molecule, Form X becomes resistant to heating to 100°C and to alkaline pH.ConclusionsOur data strongly support a model of Form X consisting in a DNA loop at the base of which the two DNA duplexes cross, with one of the strands of one duplex passing between the strands of the other duplex, and reciprocally, to form a semicatenated DNA junction also called a DNA hemicatenane.

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

Two variables define the topological state of closed double-stranded DNA: the knot type, K, and DeltaLk, the linking number difference from relaxed DNA. The equilibrium distribution of probabilities of these states, P(DeltaLk, K), is related to two conditional distributions: P(DeltaLk|K), the distribution of DeltaLk for a particular K, and P(K|DeltaLk) and also to two simple distributions: P(DeltaLk), the distribution of DeltaLk irrespective of K, and P(K). We explored the relationships between these distributions. P(DeltaLk, K), P(DeltaLk), and P(K|DeltaLk) were calculated from the simulated distributions of P(DeltaLk|K) and of P(K). The calculated distributions agreed with previous experimental and theoretical results and greatly advanced on them. Our major focus was on P(K|DeltaLk), the distribution of knot types for a particular value of DeltaLk, which had not been evaluated previously. We found that unknotted circular DNA is not the most probable state beyond small values of DeltaLk. Highly chiral knotted DNA has a lower free energy because it has less torsional deformation. Surprisingly, even at |DeltaLk| > 12, only one or two knot types dominate the P(K|DeltaLk) distribution despite the huge number of knots of comparable complexity. A large fraction of the knots found belong to the small family of torus knots. The relationship between supercoiling and knotting in vivo is discussed.

Tying a molecular knot with optical tweezers.

Filamentous structures are abundant in cells. Relatively rigid filaments, such as microtubules and actin, serve as intracellular scaffolds that support movement and force, and their mechanical properties are crucial to their function in the cell. Some aspects of the behaviour of DNA, meanwhile, depend critically on its flexibility-for example, DNA-binding proteins can induce sharp bends in the helix. The mechanical characterization of such filaments has generally been conducted without controlling the filament shape, by the observation of thermal motions or of the response to external forces or flows. Controlled buckling of a microtubule has been reported, but the analysis of the buckled shape was complicated. Here we report the continuous control of the radius of curvature of a molecular strand by tying a knot in it, using optical tweezers to manipulate the strand's ends. We find that actin filaments break at the knot when the knot diameter falls below 0.4 microm. The pulling force at breakage is around 1 pN, two orders of magnitude smaller than the tensile stress of a straight filament. The flexural rigidity of the filament remained unchanged down to this diameter. We have also knotted a single DNA molecule, opening up the possibility of studying curvature-dependent interactions with associated proteins. We find that the knotted DNA is stronger than actin.

Formation of knots in partially replicated DNA molecules

Bacterial plasmids with two origins of replication in convergent orientation are frequently knottedin vivo. The knots formed are localised within the newly replicated DNA regions. Here, we analyse DNA knots tied within replication bubbles of such plasmids, and observe that the knots formed show predominantly positive signs of crossings. We propose that helical winding of replication bubbles in vivo leads to topoisomerase-mediated formation of knots on partially replicated DNA molecules.

Analysis of DNA knots and catenanes by agarose-gel electrophoresis.

Supercoiling, knotting, and catenation are three common higher-order structures involving coiling of the axis of double-stranded DNA. These forms appear as a result of a number of important biological activities, including topoisomerase action, DNA replication, and genetic recombination (-). All of these species have mobilities in agarose gels that are distinct from those of normal open circular and linear DNA molecules of the same size. The electrophoretic properties of linking number topoisomers are dealt with elsewhere in this volume; this chapter focuses on the separation and characterization of mixtures of knotted or catenated forms.

Solving tangle equations arising in a DNA recombination model

In the tangle model for DNA site-specific recombination, one is required to solve simultaneous equations for unknown tangles which are summands of observed DNA knots and links. For 0[les ]i[les ]3, given fixed 4-plats Ki where the set {K1, K2, K3} contains at least two distinct 4-plats, let O and R denote unknown 2-string tangles such that {O, R} are the variables in the system of four tangle equations N(O+iR)=Ki, where N is the numerator construction, and nR denotes the tangle sum of n copies of R. Then there is at most one simultaneous solution {O, R} and this solution must be of the form R an integral tangle and O either a rational tangle or the sum of two rational tangles. In addition, if there exists a solution, then at least one of the 4-plats is chiral. We exhibit an algorithm for solving simultaneous tangle equations of this form.