The influence of salt on the structure and energetics of supercoiled DNA.

We present a detailed computational study of the influence of salt on the configurations, energies, and dynamics of supercoiled DNA. A potential function that includes both elastic and electrostatic energy components is employed. Specifically, the electrostatic term, with salt-dependent coefficients, is modeled after Stigter's pioneering work on the effective diameter of DNA as a function of salt concentration. Because an effective charge per unit length is used, the electrostatic formulation does not require explicit modeling of phosphates and can be used to study long DNAs at any desired resolution of charge. With explicit consideration of the electrostatic energy, an elastic bending constant corresponding to the nonelectrostatic part of the bending contribution to the persistence length is used. We show, for a series of salt concentrations ranging from 0.005 to 1.0 M sodium, how configurations and energies of supercoiled DNA (1000 and 3000 base pairs) change dramatically with the simulated salt environment. At high salt, the DNA adopts highly compact and bent interwound states, with the bending energy dominating over the other components, and the electrostatic energy playing a minor role in comparison to the bending and twisting terms. At low salt, the DNA supercoils are much more open and loosely interwound, and the electrostatic components are dominant. Over the range of three decades of salt examined, the electrostatic energy changes by a factor of 10. The buckling transition between the circle and figure-8 is highly sensitive to salt concentration: this transition is delayed as salt concentration decreases, with a particularly sharp increase below 0.1 M. For example, for a bending-to-twisting force constant ratio of A/C = 1.5, the linking number difference (delta LK) corresponding to equal energies for the circle and figure-8 increases from 2.1 to 3.25 as salt decreases from 1.0 to 0.005 M. We also present in detail a family of three-lobed supercoiled DNA configurations that are predicted by elasticity theory to be stable at low delta Lk. To our knowledge, such three-dimensional structures have not been previously presented in connection with DNA supercoiling. These branched forms have a higher bending energy than the corresponding interwound configurations at the same delta Lk but, especially at low salt, this bending energy difference is relatively small in comparison with the total energy, which is dominated by the electrostatic contributions. Significantly, the electrostatic energies of the three-lobed and (straight) interwound forms are comparable at each salt environment.(ABSTRACT TRUNCATED AT 400 WORDS)     

Supercoiled DNA energetics and dynamics by computer simulation.

A new formulation is presented for investigating supercoiled DNA configurations by deterministic techniques. Thus far, the computational difficulties involved in applying deterministic methods to supercoiled DNA studies have generally limited computer simulations to stochastic approaches. While stochastic methods, such as simulated annealing and Metropolis-Monte Carlo sampling, are successful at generating a large number of configurations and estimating thermodynamic properties of topoisomer ensembles, deterministic methods offer an accurate characterization of the minima and a systematic following of their dynamics. To make this feasible, we model circular duplex DNA compactly by a B-spline ribbon-like model in terms of a small number of control vertices. We associate an elastic deformation energy composed of bending and twisting integrals and represent intrachain contact by a 6-12 Lennard Jones potential. The latter is parameterized to yield an energy minimum at the observed DNA-helix diameter inclusive of a hydration shell. A penalty term to ensure fixed contour length is also included. First and second partial derivatives of the energy function have been derived by using various mathematical simplifications. First derivatives are essential for Newton-type minimization as well as molecular dynamics, and partial second-derivative information can significantly accelerate minimization convergence through preconditioning. Here we apply a new large-scale truncated-Newton algorithm for minimization and a Langevin/implicit-Euler scheme for molecular dynamics. Our truncated-Newton method exploits the separability of potential energy functions into terms of differing complexity. It relies on a preconditioned conjugate gradient method that is efficient for large-scale problems to solve approximately for the search direction at every step. Our dynamics algorithm is numerically stable over large time steps. It also introduces a frequency-discriminating mechanism so that vibrational modes with frequencies greater than a chosen cutoff frequency are essentially frozen by the method. With these tools, we rapidly identify corresponding circular and interwound energy minima for small DNA rings for a series of imposed linking-number differences. These structures are consistent with available electron microscopy data. The energetic exchange of stability between the circle and the figure-8, in very good agreement with analytical results, is also detailed. Molecular dynamics trajectories at 100 femtosecond time steps then reveal the rapid folding of the unstable circular state into supercoiled forms. Significant bending and twisting motions of the interwound structures are also observed. Such information may be useful for understanding transition states along the folding pathway and the role of enzymes that regulate supercoiling.(ABSTRACT TRUNCATED AT 400 WORDS)     

On the toroidal condensed state of closed circular DNA

The influence of double helix torsional elasticity on the compaction and structure of circular DNA compact form is studied theoretically in the case when the compact (globular) form has torus shape. For closed circular DNA the topological invariant, the linking number, yields a strict connection between conformation of the double helix considered as unifilar homopolymer and elastic energy of torsional twisting. The contribution of torsional elasticity to the free energy of the toruslike globule is calculated. This contribution is shown to be proportional to the square of superhelical density. Allowance of the torsional elasticity decreases the equilibrium radius of the toruslike globule formed by circular DNA. Closure of linear DNA into a ring widens the stability range of the relatively short DNA compact form and tightens it for long DNA.     

Toroidal globular state of circular DNA

The influence of torsional elasticity of the double helix on compactization and structure of circular DNA in a compact form is studied in the case when the compact (globular) particle has a torus shape. For closed circular DNA the topological invariant, linking number of two strains, yields strict connection between conformation of double helix, considered as a unifilar homopolymer, and elastic energy of torsional twisting. The contribution of torsional elasticity to free energy of the toruslike globule is calculated. This contribution is shown to be proportional to the square of superturn's density. Torsional elasticity decreases the equilibrium radius of the toruslike globule formed by circular DNA in comparison with the case of linear DNA. Closure of linear DNA into a ring widens the stability range of the relatively short DNA compact form and tightens it for long DNA.     

On the Toroidal Condensed State of Closed Circular DNA

Abstract

The influence of double helix torsional elasticity on the compaction and structure of circular DNA compact form is studied theoretically in the case when the compact (globular) form has torus shape. For closed circular DNA the topological invariant, the linking number, yields a strict connection between conformation of the double helix considered as unifilar homopolymer and elastic energy of torsional twisting. The contribution of torsional elasticity to the free energy of the toruslike globule is calculated. This contribution is shown to be proportional to the square of superhelical density. Allowance of the torsional elasticity decreases the equilibrium radius of the toruslike globule formed by circular DNA. Closure of linear DNA into a ring widens the stability range of the relatively short DNA compact form and tightens it for long DNA.     

DNA Topological Context Affects Access to Eukaryotic DNA Topoisomerase I

Abstract

We have analyzed the reactivity of a 217 base pair segment of the intrinsically curved Crithidia fasciculata kinetoplast DNA towards eukaryotic DNA topoisomerase I. The substrates were open [linear fragment and nicked circle] and closed minidomains [closed relaxed circle and circles with linking differences of ?1 and ?2], We interpreted the results with the aid of a model that was used to predict the structures of the topoisomers. The modelling shows that the ΔLk(?l) form is unusually compact because of the curvature in the DNA. To determine the role of sequence-directed curvature in both the experimental and modeling studies, controls were examined in which the curved Crithidia sequence was replaced by an uncurved sequence obtained from the plasmid pBR322.

Reactivity of the Crithidia DNA [as analyzed both by the cleavage and the topoisomerization reactions] markedly varied among the DNA forms: (i) the hierarchy of overall reactivity observed is: linear fragment > nicked circular, closed circular [ΔLk(O)], interwound [ΔLk(?2)] > bent interwound [ΔLk(?l)]; (ii) the intensity of several cleavage positions differs among DNA forms.

The results show that eukaryotic DNA topoisomerase I is very sensitive to the conformation of the substrates and that its reactivity is modulated by the variation of the compactness of the DNA molecule. The C. fasciculata sequence contains a highly curved segment that determines the conformation of the closed circle in a complex way.     

Modeling protein-induced configurational changes in DNA minicircles

A method is offered for obtaining minimum energy configurations of DNA minicircles constrained by one or more DNA-binding proteins. The minicircles are modeled as elastic rods, while the presence of bound protein is implied by rigidly fixing portions of these chains. The configurations of the geometrically constrained circular rods are sampled stochastically and optimized according to a simple elastic energy model of nicked DNA. The shapes of the minimum energy structures identified after a simulated annealing process are analyzed in terms of relative protein orientation and writhing number. The procedure is applied to minicircles 500 base pairs in length, bound to two evenly spaced DNA-wrapping proteins. The presence of histone octamers is suggested by rigidly fixing the two protein-bound portions of each minicircle as small superhelices similar in dimension to nucleosomal DNA. The folded minimum energy forms of sample chains with different degrees of protein wrapping are noteworthy in themselves in that they offer a new resolution to the well-known minichromosome linking number paradox and point to future minicircle simulations of possible import. © 1997 John Wiley & Sons, Inc.     

Statistical mechanical theory of melting transition in supercoiled DNA

The melting curve for a covalently closed circular DNA has been analyzed on the basis of an expression for the supercoiling energy derived in terms of the elastic parameters of the macromolecule, treated as a homopolymer. The result obtained by applying the usual methods of statistical mechanics indicate close agreement with the available experimental data. It is found that the elevation of the melting temperature, as compared to that of the nicked circular or linear DNA, is a natural consequence of the fact that the supercoiled molecule is constrained by an invariant linking number. The flattening of the melting curve, on the other hand, arises as the closed circular duplex melts into a loose helix rather than random coils.     

Toroidal elastic supercoiling of DNA

Covalently closed circular DNA can exist in different configurations known as circular, toroidal, and interwound. Changes among these forms can be made in several ways, including the insertion of dye molecules between adjacent base pairs, which tends to untwist the double-helical structure. The aim of this paper is to discuss these configurations, and the changes among them, in the context of classical elastomechanics. The concepts of twisting, linkage and writhing are explained. Simple experiments on a twisted linear-elastic rod are described, and it is shown that although the circular and interwound forms may be modeled in this way, the toroidal form does not occur, being mechanically unstable. Theoretical energy calculations by Levitt on bent and twisted DNA show that DNA exhibits a particular kind of nonlinear elasticity in which there is an unusual coupling between bending and twisting. The aim of the paper is to show qualitatively that this special kind of elasticity can stabilize the toroidal form of closed circular DNA.     

Intramolecular compactization of circular DNA in interaction with dansyl hydrazide trivaline leading to a formation of a new type of structure--triple rings

Compactization of supercoiled circular plasmid pBR322 caused by interaction with synthetic oligopeptide dansyl hydrazide trivaline capable of beta-structure formation was studied by electron microscopy. The results show that at rising input peptide concentration circular DNA molecules undergo intramolecular structural transition with the formation of compact ring structures. The compact ring structures are formed by the fiber having the thickness of 60 A. The analysis of morphology of intermediate structures and the contour length measurements enable us to conclude that 60 A-fiber contains three lying side-by-side and interwound double-stranded DNA segments. Thus, the compact ring structures are addressed to as triple rings. The triple ring have one special point, where the triple region ends are locked by a duplex DNA segment. The mechanisms responsible for the triple ring formation may be of importance for DNA and chromatin compactization processes in vivo.     

The effect of a variable dielectric coefficient and finite ion size on Poisson-Boltzmann calculations of DNA-electrolyte systems.

The results of variable dielectric coefficient Poisson-Boltzmann calculations of the counter-ion concentration in the vicinity of an all-atom model of the B-form of DNA are presented with an emphasis on the importance of spatial variations in the dielectric properties of the solvent, particularly at the macro-ion-solvent interface. Calculations of the distribution of hard-sphere electrolyte ions of various dimensions are reported. The presence of a dielectric boundary significantly increases the magnitude of the electrostatic potential with a concomitant increase in the accumulation of small counter-ions in the groove regions of DNA. Because ions with radii greater than 2 A have restricted access to the minor groove, the effect there is less significant than it is within the major groove. Changes in the dielectric coefficient for the electrolyte solution, allowing variation from 10 to 25, 40, 60, and 78.5 within the first 7.4 A of the surface of DNA, substantially increases the calculated surface concentration of counter-ions of all sizes. A lower dielectric coefficient near the macro-ion surface also tends to increase the counter-ion density in regions where the electrostatic potential is more negative than -kT. Regardless of the choice of dielectric coefficient, the number of ions in regions where the electrostatic potential is less than -kT remains the same for 0.153 M added 1-1 monovalent electrolyte as for the case without added salt.(ABSTRACT TRUNCATED AT 250 WORDS)     

Direct visualization of supercoiled DNA molecules in solution.

The shape of supercoiled DNA molecules in solution is directly visualized by cryo-electron microscopy of vitrified samples. We observe that: (i) supercoiled DNA molecules in solution adopt an interwound rather than a toroidal form, (ii) the diameter of the interwound superhelix changes from about 12 nm to 4 nm upon addition of magnesium salt to the solution and (iii) the partition of the linking deficit between twist and writhe can be quantitatively determined for individual molecules.     

Configurational transitions in Fourier series-represented DNA supercoils.

A new Fourier series representation of supercoiled DNA is employed in Langevin dynamics simulations to study large-scale configurational motions of intermediate-length chains. The polymer is modeled as an ideal elastic rod subject to long-range van der Waals' interactions. The van der Waals' term prevents the self-contact of distant chain segments and also mimics attractive forces thought to stabilize the association of closely spaced charged rods. The finite Fourier series-derived polymer formulation is an alternative to the piecewise B-spline curves used in past work to describe the motion of smoothly deformed supercoiled DNA in terms of a limited number of independent variables. This study focuses on two large-scale configurational events: the interconversion between circular and figure-8 forms at a relatively low level of supercoiling, and the transformation between branched and interwound structures at a higher superhelical density.     

Elastic model of highly supercoiled DNA

The equilibrium shapes of supercoiled DNA are investigated by employing an elastic model. First, a set of Euler equations is derived to determine the equilibrium shapes under ring-closure conditions. Two exact solutions that describe circular and figure-8 shapes are obtained. Using these and their topological properties, the configuration change from the circular to the figure-8 form is discussed. Second, more intricate structures of supercoiling DNA are studied by a numerical analysis. Among a class of configurations, the shape that has the minimum elastic energy is explicitly determined. Poisson's ratio, the ratio of the self-avoiding radius to the total length, and the deficit (or excess) of the linking number ΔLk are found to be the important parameters. We conclude that the topology and the elastic theory of looped DNA explain the essential features of the supercoiling phenomena.     

Melting experiment concerning the topological structure of closed circular double-stranded DNA

A melting experiment was performed on the whole set of populations of the replicative form of ?X174 DNA, which can be obtained treating this DNA with rat liver nicking-closing enzyme in the presence of ethidium bromide. Gel electrophoresis performed by loading the DNA samples at neutral and alkaline pH allows separation of these populations in discrete sets of bands, which can then be compared. The outcome of the experiments indicates that in the range of electrophoretic mobilities which can be explored, no band is formed exclusively by circular complementary strands which can be separated by alkaline denaturation. These results are compared with what would be expected if double-stranded closed circular DNA had structures other than the canonical double helix. Under nonrestrictive hypotheses, the experiments reported allow one to obtain a minimum estimate of the absolute value of the linking number of a closed circular double-stranded DNA: for native ?X174 RF DNA, the linking number appears to be greater than 12 (in absolute value). Some data on the electrophoretic mobility of denatured closed circular duplexes are reported, which still wait for a physicochemical interpretation.     

Variance of writhe for wormlike DNA rings with excluded volume

We have calculated the variance of the equilibrium distribution of a circular wormlike polymer chain over the writhing number, less than (Wr)2 greater than, with allowance for the excluded volume effects. Within this model the less than (Wr)2 greater than value is a function of the number of Kuhn statistical segments, n, and the chain diameter, d measured in Kuhn statistical lengths, b. Simulated DNA chains varied from 200 to 10,000 base pairs and the d value varied from 0.02 to 0.2. Theory predicts a considerable ionic strength dependence of the DNA superhelix energy as a consequence of the change in the DNA diameter. A comparison with the available experimental data has yielded an estimate of the DNA torsional rigidity, the Kuhn statistical length, and the effective diameter of the double helix under conditions of the complete screening of the DNA electrostatic potential.     

Modulation of intramolecular interactions in superhelical DNA by curved sequences: a Monte Carlo simulation study.

A Monte Carlo model for the generation of superhelical DNA structures at thermodynamic equilibrium (Klenin et al., 1991; Vologodskii et al., 1992) was modified to account for the presence of local curvature. Equilibrium ensembles of a 2700-bp DNA chain at linking number difference delta Lk = -15 were generated, with one or two permanent bends up to 120 degrees inserted at different positions. The computed structures were then analyzed with respect to the number and positions of the end loops of the interwound superhelix, and the intramolecular interaction probability of different segments of the DNA. We find that the superhelix structure is strongly organized by permanent bends. A DNA segment with a 30 degrees bend already has a significantly higher probability of being at the apex of a superhelix than the control, and for a 120 degrees bend the majority of DNAs have one end loop at the position of the bend. The entropy change due to the localization of a 120 permanent bend in the end loop is estimated to be -17 kJ mol-1 K-1. When two bends are inserted, the conformation of the superhelix is found to be strongly dependent on their relative positions: the straight interwound form dominates when the two bends are separated by 50% of the total DNA length, whereas the majority of the superhelices are in a branched conformation when the bends are separated by 33%. DNA segments in the vicinity of the permanent bend are strongly oriented with respect to each other.     

Ethidium bromide-mediated renaturation of denatured closed circular DNAs. The nature of denaturation-resistant fractions of bacteriophage PM2 closed circular DNA

Addition of the intercalating dye ethidium bromide (EtdBr) to a solution of alkali-denatured double-stranded closed circular PM2, ΦX174, or λb₂b₅c phage DNAs, under conditions such that the solution remains strongly alkaline, can result in the renaturation of up to 100% of the DNA upon neutralization of the solution. For a fixed time of incubation of the alkaline dye-containing solution before neutralization, there exists a minimum concentration of the dye below which no EtdBr-mediated renaturation is observed for each species of closed circular DNA examined. These minimum concentrations increase, for a given DNA, with increasing ionic strength and temperature. The kinetics of accumulation of forms renaturing upon neutralization of alkaline solutions, at fixed concentrations of dye and DNA, are dependent upon the molecular weight and superhelix density of the starting DNA. After extended periods of incubation at a fixed ionic strength and temperature, however, the profiles of percentage of DNA renatured as a function of ethidium concentration become very similar for all the closed circular DNAs tested and display a transition from an absence of dye-mediated renaturation to virtually 100% renaturation upon neutralization over a small range of dye concentration. Circular DNA containing one or more strand scissions remains strand-separated under all the conditions used to effect the renaturation of closed circular DNA. These findings indicate that configurations of closed circular DNA, in which at least some of the complementary bases are apposed, can be selectively stabilized and accumulate in the presence of ethidium in solutions containing 0.19 N hydroxide ion.     

Geometry and mechanics of DNA superhelicity

This paper analyzes the elastic equilibrium conformations of duplex DNA constrained by the constancy of its molecular linking number, Lk. The DNA is regarded as having the mechanical properties of a homogeneous, linearly elastic substance with symmetric cross section. Integral representations of the writhing number Wr and of Lk are developed, in terms of which the equilibria are given as solutions to an isoperimetric problem. It is shown that the Euler angles defining equilibrium conformations must obey equations identical to those governing unconstrained equilibria. A scaling law is developed stating that molecules supercoiled the same amount ΔLk will have geometrically similar elastic equilibria regardless of their length. Thus, comparisons among molecules of properties related to their large-scale tertiary structure should be referred to differences in ΔLk rather than to their superhelix densities. Specific conditions on the elastic equilibrium conformations are developed that are necessary for ring closure. The equilibrium superhelical conformations accessible to closed-ring molecules are shown to approximate toroidal helices. Questions relating to the stability and nonuniqueness of equilibria are treated briefly. A comparison is made between these toroidal conformations and interwound configurations, which are shown to be stable, although they are not equilibria in the present sense. It is suggested that entropic factors are responsible for favouring the toroidal conformation in solution.