DNA Knots Research Articles

Geometry and Physics of Catenanes Applied to the Study of DNA Replication

The concept of ideal geometric configurations was recently applied to the classification and characterization of various knots. Different knots in their ideal form (i.e., the one requiring the shortest length of a constant-diameter tube to form a given knot) were shown to have an overall compactness proportional to the time-averaged compactness of thermally agitated knotted polymers forming corresponding knots. This was useful for predicting the relative speed of electrophoretic migration of different DNA knots. Here we characterize the ideal geometric configurations of catenanes (called links by mathematicians), i.e., closed curves in space that are topologically linked to each other. We demonstrate that the ideal configurations of different catenanes show interrelations very similar to those observed in the ideal configurations of knots. By analyzing literature data on electrophoretic separations of the torus-type of DNA catenanes with increasing complexity, we observed that their electrophoretic migration is roughly proportional to the overall compactness of ideal representations of the corresponding catenanes. This correlation does not apply, however, to electrophoretic migration of certain replication intermediates, believed up to now to represent the simplest torus-type catenanes. We propose, therefore, that freshly replicated circular DNA molecules, in addition to forming regular catenanes, may also form hemicatenanes.

Topological chirality and achirality of DNA knots

In this paper, the novel topology of DNA knots is discussed on the basis of the Seifert construction in knot theory. Our result shows that T 4 k -DNA knots possess the point symmetry group S 1 and T 2 k + 1 -DNA knots possess the point symmetry group C 1. Hence, C 1 represents the most chiral and S 1the least achiral. We say that a structure with C 1 symmetry is topologically chiral, while a structure with S 1 symmetry is shown to have the properties of a topological rubber glove, and that the orientability of the Seifert construction is a primary tool for the left-right classification of molecular knots.

Tight knot values deviate from linear relations

Applications of knots to the study of polymers have emphasized geometric measures on curves such as ‘energy’1,2,3,4 and ‘rope length’5,6,7, which, when minimized over different configurations of a knot, give computable knot invariants related to physical quantities8. In DNA knots, electrophoretic mobility appears to be correlated with the average crossing number of rope-length-minimizing configurations9, and a roughly linear empirical relation has been observed between the crossing number and rope length10. Here we show that a linear relation cannot hold in general, and we construct infinite families of knots whose rope length grows as the 3/4 power of the crossing number11. It can be shown that no smaller power is possible12,13,14.

Properties of ideal composite knots.

The shortest tube of constant diameter that can form a given knot represents the 'ideal' form of the knot. Ideal knots provide an irreducible representation of the knot, and they have some intriguing mathematical and physical features, including a direct correspondence with the time-averaged shapes of knotted DNA molecules in solution. Here we describe the properties of ideal forms of composite knots-knots obtained by the sequential tying of two or more independent knots (called factor knots) on the same string. We find that the writhe (related to the handedness of crossing points) of composite knots is the sum of that of the ideal forms of the factor knots. By comparing ideal composite knots with simulated configurations of knotted, thermally fluctuating DNA, we conclude that the additivity of writhe applies also to randomly distorted configurations of composite knots and their corresponding factor knots. We show that composite knots with several factor knots may possess distinct structural isomers that can be interconverted only by loosening the knot.

T7 RNA polymerase cannot transcribe through a highly knotted DNA template.

The ability of T7 RNA polymerase to transcribe a plasmid DNA in vitro in its linear, supercoiled, relaxed and knotted forms was analysed. Similar levels of transcription were found on each template with the exception of plasmids showing varying degrees of knotting (obtained using stoichiometric amounts of yeast topoisomerase II). A purified fraction of knotted DNA with a high number of nodes (crosses) was found to be refractory to transcription. The unknotting of the knotted plasmids, using catalytic amounts of topoisomerase II, restored their capacity as templates for transcription to levels similar to those obtained for the other topological forms. These results demonstrate that highly knotted DNA is the only topological form of DNA that is not a template for transcription. We suggest that the regulation of transcription, which depends on the topological state of the template, might be related to the presence of knotted DNA with different number of nodes.

The design and engineering of nucleic acid nanoscale assemblies

It is possible to design DNA molecules that can form unusual structures and topologies. Stable DNA-branched junctions have been used to construct polyhedral catenated molecules with the connectivities of a cube and of a truncated octahedron. The truncated octahedron has been constructed following a solid-support-based methodology. Branched-DNA molecules are flexible, suggesting that triangular and deltahedral DNA objects should be favored as the components of two- and three-dimensional nucleic acid arrays. DNA polyhedra are complex catenanes. The engineering of single-stranded DNA knots and catenanes exploits the fact that a node can be equated with a half-turn of DNA.

DNA Knotting Abolishes in Vitro Chromatin Assembly

Topological knots can be formed in vitro by incubating covalently closed double stranded DNA and purified topoisomerase II from the yeast Saccharomyces cerevisiae in an ATP-dependent reaction. Knotting production requires a starting enzyme/DNA mass ratio of 1. Analysis of knotted DNA was carried out by using both one- and two-dimensional agarose gel electrophoresis. The knots generated are efficiently untied, and give relaxed DNA rings, by catalytic amounts of topoisomerase II, but not by topoisomerase I. Time course analysis shows the knotting formation over relaxed and supercoiled DNA. When supercoiled DNA was used as a susbtrate, knots appear immediately whereas no transient relaxed rings were observed. The cell-free extract from Xenopus oocytes S-150 cannot assemble nucleosomes on knotted DNA templates as revealed by topological and micrococcal nuclease analysis. Nevertheless, the presence of knotted DNA templates does not inhibit the assembly over the relaxed plasmid. Finally, a pretreatment of knotted DNA with trace amounts of topoisomerase II before the addition of the S-150 yields a canonical minichromosome assembled in vitro. Taking into account these results, I suggest a mechanism of chromatin assembly regulation directed by topoisomerase II.

P-B1-04 Ideal forms of knots allow prediction of electrophoretic mobility of real DNA knots: Stasiak A, Katritch V, Bednar J, Michoud D, & Dubochet J. Laboratoire d'Analyse Ultrastructurale, Université de Lausanne, Switzerland

P-B1-04 Ideal forms of knots allow prediction of electrophoretic mobility of real DNA knots: Stasiak A, Katritch V, Bednar J, Michoud D, & Dubochet J. Laboratoire d'Analyse Ultrastructurale, Université de Lausanne, Switzerland

DNA topology

Two topological levels are inherent in closed circular (cc) DNA. First, the double helix as a whole can form knots of different types, or form the trivial knot (to be unknotted). Secondly, the two complementary strands are interlinked in ccDNA, which results in a biologically very important phenomenon of DNA supercoiling. During the past 20 years we have elaborated a theoretical model, which makes it possible to treat the DNA topological properties at equilibrium in a rigorous way by the Monte Carlo method. Within the framework of the model, the double helix is considered as a homogeneous, isotropic elastic rod with given values of bending and torsional rigidity parameters and the effective diameter. We have predicted the equilibrium fraction of knotted DNA molecules for different DNA lengths and for different values of DNA effective diameter. These theoretical predictions have recently been fully confirmed by experiment. Comparison of theory with experiment also gave the value of DNA effective diameter under a variety of ambient conditions. Calculations of equilibrium distribution of ccDNA over topoisomers yielded a reliable estimation of the DNA torsional rigidity. As a result, all three parameters of the DNA model have been reliably estimated. This makes possible unambiguous predictions of various properties of DNA under topological constraints. In particular, extensive computer simulations of supercoiled DNA have been carried out.

Monte Carlo simulations of supercoiling free energies for unknotted and trefoil knotted DNAs

A new Monte Carlo (MC) algorithm is proposed for simulating inextensible circular chains with finite twisting and bending rigidity. This new algorithm samples the relevant Riemann volume elements in a uniform manner, when the constraining potential vanishes. Simulations are performed for filaments comprising 170 subunits, each containing approximately 28 bp, which corresponds to a DNA length of 4770 bp. The bending rigidity is chosen to yield a persistence length, P = 500 A, and the intersubunit potential is taken to be a hard-cylinder potential with diameter d = 50 A. This value of d yields the same second virial coefficient as the electrostatic potential obtained by numerical solution of the Poisson-Boltzmann equation for 150 mM salt. Simulations are performed for unknotted circles and also for trefoil knotted circles using two different values of the torsional rigidity, C = (2.0 and 3.0) x 10(-19) dyne cm2. In the case of unknotted circles, the simulated supercoiling free energy varies practically quadratically with linking difference delta l. The simulated twist energy parameter ET, its slope dET/dT, and the mean reduced writhe <w>/delta l for C = 3 x 10(-19) dyne cm2 all agree well with recent simulations for unknotted circles using the polygon-folding algorithm with identical P, d, and C. The simulated ET vs. delta l data for C = 2.0 x 10(-19) dyne cm2 agree rather well with recent experimental data for p30 delta DNA (4752 bp), for which the torsional rigidity, C = 2.07 x 10(-19) dyne cm2, was independently measured. The experimental data for p30 delta are enormously more likely to have arisen from C = 2.0 x 10(-19) than from C = 3.0 x 10(-19) dyne cm2. Serious problems with the reported experimental assessments of ET for pBR322 and their comparison with simulated data are noted. In the case of a trefoil knotted DNA, the simulated value, (ET)tre, exceeds that of the unknotted DNA, (ET)unk, by approximately equal to 1.40-fold at magnitude of delta l = 1.0, but declines to a plateau about 1.09-fold larger than (ET)unk when magnitude of delta l > or = 15. Although the predicted ratio, (ET)tre/(ET)unk approximately equal to 1.40, agrees fairly well with recent experimental measurements on a 5600-bp DNA, the individual measured ET values, like some of those reported for pBR322, are so large that they cannot be simulated using P = 500 A, d = 50 A, and any previous experimental estimate of C.

Topological transformations of synthetic DNA knots.

Two synthetic DNA molecules that can be knotted have been employed as substrates for E. coli DNA topoisomerases I and III. Both molecules contain 104 nucleotides, including sequences that can form two single-turn helical domains, connected by single-stranded oligo(dT) linkers in an X-Y-X'-Y' pairing motif. One of the knots can be ligated to form cyclic molecules with the topologies of a circle, a trefoil knot with negative nodes, or a figure-8 knot. Cyclic molecules constructed from the other molecule can form a circle, a figure-8 knot, and trefoil knots with either positive or negative nodes. The topologically negative nodes in these knots are derived from right-handed B-DNA, and the positive nodes are derived from left-handed Z-DNA. The topoisomerases can catalyze the interconversion of the different topological forms of these molecules, as a function of solution conditions and the extent to which they favor B-DNA or Z-DNA. The enzymes appear to catalyze a single strand-passage event at a time. The topoisomerases can catalyze strand passage events involving both positive and negative nodes as substrates. Gel retention experiments show that both knots can bind up to four molecules of E. coli DNA topoisomerase I. The thermal denaturation of the domains of a trefoil knot closely related to these knots suggests that the two helical domains are uncoupled, so the single-stranded linkers in the knots are not taut. Chemical ligation experiments yield a distribution of products similar to those of enzymatic ligation, showing that the ATP cofactor in DNA knot ligation does not appear to skew the products markedly. Knots that are stressed by being placed in unfavorable solution conditions have been shown to be a highly sensitive system for detecting topoisomerase activity.

Purification and properties of DNA topoisomerase II from Daucus carota cells

Topoisomerase II was partially purified from Daucus carota cells by a procedure including ammonium sulphate fractionation, ion-exchange, and affinity chromatography steps. The type II enzyme, identified for its ability to unknot knotted P4 DNA and decatenate Trypanosoma cruzi kDNA, requires ATP and Mg 2+ for activity. The unknotting activity was sensitive to an inhibitor of the mammalian type II enzyme, the drug VP16 (IC 50 32 mmol m −3 ), whereas inhibitors of DNA gyrase showed a limited effect on activity. The SDS-PAGE analysis of the dsDNA cellulose fraction revealed the presence of four polypeptides of apparent molecular masses of 72, 71, 34, and 33 kDa among which only a polypeptide of about 70 kDa crossreacted with antibodies against yeast topoisomerase II. Immunoprecipitation experiments with monoclonal antibodies to the α and β isoforms of the human enzyme confirmed the recognition of a polypeptide of 70 kDa. The sedimentation coefficient (S) of the topoisomerase II in the phosphocellulose fraction, calculated by analytical glycerol gradient, was 6.1 corresponding to a molecular mass of about 123 kDa. Results suggest the presence in carrot of a protein of molecular mass of 70 kDa having the typical properties of an eukaryotic topoisomerase II and carrying epitopes recognized by MoAbs to both human α and β enzymes. The 70 kDa polypeptide might then represent the monomer of a homodimer enzyme of 123 kDa

DNA KNOT FORMATION IN AQUEOUS SOLUTIONS

The knotting probability of a closed chain has been calculated as a function of chain dimensions and solvent properties in a number of studies. We have measured the probability of DNA knot formation upon random cyclization of linear DNA in vitro to provide an experimental test of the various theoretical treatments of the problem; parameters of these models, such as the effective chain diameter of DNA, were calculated in different concentrations of counterions. Our results in the presence of sodium ions agree well with theoretical treatments of DNA as a polyelectrolyte; knotting data in the presence of divalent magnesium ions indicate that moderate concentrations of magnesium ions can induce an attractive potential between DNA segments, resulting in negative values of the calculated effective DNA helix diameter. We discuss structures in which the divalent magnesium counterion facilitates the close apposition of two DNA segments, and review the effect of chemical- and protein-mediated crosslinks between DNA segments on DNA knot formation. Finally, we consider DNA knot formation in vivo.

The construction of a trefoil knot from a DNA branched junction motif

A DNA trefoil (3(1)) knot has been constructed from a 104-nucleotide molecule whose strands form a 3-arm branched junction motif. This construction tests the notion that a node in a DNA knot can be equated with a half-turn of double-helical DNA, and is consistent with that concept. Of five 104-mer sequences tested, only one produces high yields of the target knot. The other molecules produce larger quantities of circular material and of a knot containing more nodes. The key features that differentiate the successful design from the others are (1) the ligation takes place in the linker region between helical domains and (2) only six nucleotide pairs are used for each of the double-helical arms of the junction. The successful design separates the double-helical regions from each other by a spacer containing two deoxythymidine nucleotides at the site of the branched junction.

On the simultaneous binding of eukaryotic DNA topoisomerase II to a pair of double-stranded DNA helices

Stabilization of crossings of pairs of DNA helices by binding of eukaryotic DNA topoisomerase II was studied by two types of experiments. In one, mixtures of yeast DNA topoisomerase II and supercoiled DNA were incubated with vaccinia virus topoisomerase, and the linking numbers of the DNA products were measured to quantitate supercoils that were constrained by the stoichiometrically bound yeast enzyme molecules; in parallel, the same yeast enzyme-supercoiled DNA mixtures were incubated with a nonhydrolyzable ATP analog AMPPNP (adenosine 5'-(beta, gamma-imido)triphosphate) instead of the vaccinia enzyme, and DNA linking number changes following the addition of AMPPNP were measured to monitor DNA transport mediated by the yeast enzyme and AMPPNP. In the second type of experiments, formation of knotted DNA rings by the addition of AMPPNP to mixtures of yeast DNA topoisomerase II and different topological forms of DNA rings was studied. These experiments indicate that binding of yeast DNA topoisomerase II to DNA crossings is significant, especially in low salt media containing Mg(II), and that this mode of binding strongly affects DNA knotting. It appears, however, that stabilization of DNA crossovers by the eukaryotic type II enzyme is not directly related to its DNA transport activity.

Knotting of a DNA chain during ring closure.

The formation of knotted species on random ring closure of two DNAs that are 5.6 kilobase pairs (kbp) and 8.6 kbp in length was measured, and these data were used to calculate the effective DNA helix diameter as a function of sodium ion and magnesium ion concentration. In the presence of more than 50 mM magnesium ion, interactions between DNA segments appear to be attractive rather than repulsive. The free energy of formation of relaxed trefoil and figure-eight DNA knots and of supercoiled trefoil DNA knots was also evaluated.

Tight single-stranded DNA knots.

Trefoil (3(1)) and figure-8 (4(1)) knots have been synthesized from DNA molecules containing two single-turn helical domains, linked by four oligodeoxythymidine linkers. Both topologies are derived from the same DNA molecule. The tightest knots are fashioned by minimizing the lengths of the linkers. The shortest equal-length linkers from which a trefoil knot can be made readily are seven nucleotides long, in a 74-nucleotide molecule, whereas those in the shortest figure-8 knot are six nucleotides long, in a 70-nucleotide molecule. In addition to these limiting knots, other knots containing 80, 88, 96 and 104 nucleotides have been constructed. The mobilities of these molecules on denaturing gels show the conventional logarithmic dependence on length. Ferguson analysis of their mobilities indicates a linear dependence of surface area on length. The 80-mer trefoil knot is the tightest molecule that can be restricted in both domains.

Synthesis of a DNA knot containing both positive and negative nodes

A figure-eight (4 1 ) knot, staining two positive and two negative nodes, has been synthesized by ligating a single-stranded DNA molecule into a closed cycle. The molecule consists of two single-turn double-helical domains connected by four dT 14 or dT 15 linkers. One of the domains contains the sequence (dCpdGp) 6 , which is capable of forming left-handed Z-DNA. Ligation yields a trefoil (3 1 ) knot or a 4 1 knot, depending upon the conditions in solution

The chirality of ground DNA knots and links

The chirality of ground DNA knots and links is described and characterized in terms of color symmetry groups (CSG), i.e. color symmetry groups I and II, which correspond to topochirality (topological chirality) and topoachirality (topological achirality) which bear an uncanny resemblance to point groups I (proper) and point groups II (improper) used for testing geochirality (geometrical chirality) and geoachirality (geometrical achirality), respectively. By regarding these two crossing modes in mirror images as white and black vertices, DNA knots and links with minimal crossings can be mapped to vertex-bicolored graphs under a working hypothesis that DNA knots and links exist in ground states with minimal energy m0. The color symmetry group of a vertex-bicolored graph G is defined as the set of all permutations and permutation asymmetrizations of the vertices of G that preserve its topology (connectivity), where asymmetrization, denoted as (a), is the operation of changing vertices' colors, and a permutation followed by an (a) is a permutation asymmetrization. The color symmetry groups I contains only permutations, whereas color symmetry groups II comprise permutation asymmetrizations as well as permutations. Four DNA knots and links in nature are analyzed and tabulated consisely. In addition, the well-known figure-of-eight knot and Borromean rings are discussed in much the same way.

DNA supercoiling and relaxation by ATP-dependent DNA topoisomerases.

Bacterial DNA gyrase and the eukaryotic type II DNA topoisomerases are ATPases that catalyse the introduction or removal of DNA supercoils and the formation and resolution of DNA knots and catenanes. Gyrase is unique in using ATP to drive the energetically unfavourable negative supercoiling of DNA, an example of mechanochemical coupling: in contrast, eukaryotic topoisomerase II relaxes DNA in an ATP-requiring reaction. In each case, the enzyme-DNA complex acts as a 'gate' mediating the passage of a DNA segment through a transient enzyme-bridged double-strand DNA break. We are using a variety of genetic and enzymic approaches to probe the nature of these complexes and their mechanism of action. Recent studies will be described focusing on the role of DNA wrapping on the A2B2 gyrase complex, subunit activities uncovered by using ATP analogues and the coumarin and quinolone inhibitors, and the identification and functions of discrete subunit domains. Homology between gyrase subunits and the A2 homodimer of eukaryotic topo II suggests functional conservation between these proteins. The role of ATP hydrolysis by these topoisomerases will be discussed in regard to other energy coupling systems.