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We describe how the stability properties of DNA minicircles can be directly read from plots of various biologically intuitive quantities along families of equilibrium configurations. Our conclusions follow from extensions of the mathematical theory of distinguished bifurcation diagrams that are applied within the specific context of an elastic rod model of minicircles. Families of equilibria arise as a twisting angle alpha is varied. This angle is intimately related to the continuously varying linking number Lk for nicked DNA configurations that is defined as the sum of Twist and Writhe. We present several examples of such distinguished bifurcation diagrams involving plots of the energy E, linking number Lk, and a twist moment m3, along families of cyclized equilibria of both intrinsically straight and intrinsically curved DNA fragments.

We describe four previously unpublished methods allowing the computation of the writhe for a supercoiled DNA molecule modeled by a polymer chain consisting of straight segments. These methods are compared with each other in terms of computational efficiency and the scope of their applicability is discussed.

The supercoiling free energy of pUC19 DNA [2686 base pairs (bp)] was measured in various concentrations of PEG 8000 (polyethylene glycol; molecular weight 8000) by the topoisomer distribution method. The effective twist energy parameter (E(T)) that governs the supercoiling free energy declined linearly by 1.9-fold with increasing w/v % PEG from 0 to 7.5%, which lies below the threshold for intermolecular condensation. In principle, PEG could affect E(T) either via an osmotic exclusion mechanism or by altering the torsion elastic constant, bending rigidity, or self-repulsions of the DNA. Possible alterations of the DNA secondary structure and torsion elastic constant were assessed by CD spectroscopy and time-resolved fluorescence polarization anisotropy of intercalated ethidium. Up to 7.5% PEG, the secondary structure of the DNA remained largely unaltered, as evidenced by (1) the absence of any significant change in the CD spectrum, (2) an extremely small relative decrease (-0.0013) in intrinsic twist, and (3) a negligibly small change in the torsion elastic constant. The observed reduction in E(T) cannot be ascribed primarily to a decrease in torsion elastic constant, and most likely does not stem from a decrease in bending rigidity either. The decrease in medium dielectric constant due to PEG should increase the self-repulsions, and thereby increase E(T), which is opposite to the observed trend. Instead, the observed decline in E(T) is attributed to an osmotic exclusion mechanism. The change in molar volume excluded to the PEG (Delta V(ex)), when the linking difference converts from Delta l = 0 to Delta l = +/-1, was determined from the observed E(T) value and PEG osmotic pressure at each concentration. The experimental Delta V(ex) values agree well with theoretical estimates reckoned for a simple osmotic exclusion model, in which PEG is excluded by hard-core interactions from a concentric cylindrical volume around every duplex segment. The difference in volume excluded to PEG between the Delta l = 0 and the Delta l = +/-1 topoisomers is attributed entirely to the approximately 0.7 additional writhe "crossing" of two duplex strands at roughly 90 degrees, which is known to occur in the latter species. When the separation between the duplex centers at the "crossing" was adjusted so that the theoretical estimate of Delta V(ex) matched the experimental value at each PEG concentration, a value near 5.7 nm was obtained in each case. The invariance and plausible magnitude of this mean separation at the crossing provide strong support for this simple osmotic exclusion model. An alternative model, in which the PEG is excluded from the entire coil envelope of the DNA out to its radius of gyration, perhaps because it decreases the local dielectric constant, was also considered. The estimated difference in excluded volume in that case exceeds the experimental value by a factor of nearly 10(4), and could be ruled out on that basis.

Closed DNA loops containing one or more bent regions are important structures that occur in the regulation of gene expression. We analyze the response of structures of this type to a change in applied rotation (change in linking deficiency, delta Lk). Our results apply to a closed loop formed from an elastic rod that is intrinsically bent in Nb discrete, 20 degrees steps up to a maximum of 240 degrees, the bent regions being initially coplanar with the plane of the relaxed DNA loop. We determine the effect of changing the intrinsic elastic resistance of the DNA loop to bending and torsional deformations. This relative resistance is expressed by Poisson's ratio v, which depends upon the ratio of bending stiffness to torsional rigidity. Poisson's ratio is primarily a function of salt type and concentration. We find that the tertiary structure of DNA loops changes with delta Lk, but that the geometric response can be either of two quite different types, depending upon the precise (Nb, v) pair. For combinations of Nb and v that are above a critical curve (the Fickel curve), the response to increasing delta Lk is nonmonotonic (NMT region): the distance between the loop closure point and its diametric opposite first decreases, then increases, as delta Lk increases. For combinations of Nb and v that are below the Fickel curve (NMT region), the corresponding diameter never increases, but always decreases with increasing delta Lk. In addition to these results, we define and implement a new measure of tertiary structure in closed DNA: the absolute writhe, AWr.

The formation of a topologically closed DNA loop is important in many biological processes, including the regulation of transcription, recombination, and replication. Modeling DNA as an isotropic elastic rod, we use finite element analysis to show that the dependence of the twist (delta Tw) and the writhe (Wr) upon the linking number deficit (delta Lk) is strongly influenced by intrinsic bends. We determine how the geometry of a DNA loop changes as a function of the number of uniformly spaced coplanar 20 degrees bends, oriented so as to open toward the center of the loop. We also calculate the geometry of DNA rods that are smoothly bent to the same extent. The response of both delta Tw and Wr of a bent DNA to changes in delta Lk falls into one of three categories, depending upon the number of bends. For a single bend of 20 degrees, Wr increases monotonically with delta Lk and the change in delta Tw with distance is constant along the entire DNA axis. For two to ten 20 degrees bends, Wr passes first through a local maximum, then through a local minimum, and finally increases monotonically as delta Lk increases. For eleven to eighteen 20 degrees bends, Wr again varies monotonically with delta Lk. For all numbers of bends greater than two, the delta Tw per unit length depends upon the distribution of intrinsic bends, being constant between any two adjoining bends but varying with their position relative to the cut location. Accompanying these delta Lk-associated changes in Wr and delta Tw per unit length are characteristic changes in geometry that are specific for each category. The results of these calculations raise the possibility that intrinsic bends can serve as a control factor in the biological functions associated with loop formation in DNA.

Monte Carlo simulations are employed to investigate the thermodynamics of the first transition in writhe of a circular model filament corresponding to a 468 base-pair DNA. Parameters employed in these simulations are the torsional rigidity, C = 2.0 x 10(-19) dyne cm2, and persistence length, P = 500 A. Intersubunit interactions are modeled by a screened Coulomb potential. For a straight line of subunits this accurately approximates the nonlinear Poisson-Boltzmann potential of a cylinder with the linear charge density of DNA. Curves of relative free energy vs writhe at fixed linking difference (delta l) exhibit two minima, one corresponding to slightly writhed circles and one to slightly underwrithed figure-8's, whenever delta l lies in the transition region. The free energies of the two minima are equal when delta lc = 1.35, which defines the midpoint of the transition. At this midpoint, the free energy barrier between the two minima is found to be delta Gbar = (0.20) kBT at 298 K. Curves of mean potential energy vs writhe at fixed linking difference similarly exhibit two minima for delta l values in the transition region, and the two minimum mean potential energies are equal when delta l = 1.50. At the midpoint writhe, delta lc = 1.35, the difference in mean potential energy between the minimum free energy figure-8 and circle states is (1.3) kBT, and the difference in their entropies is 1.3 kB. Thus, the entropy of the minimum free energy figure-8 state significantly exceeds that of the circle at the midpoint of the transition. The first transition in writhe is found to occur over a rather broad range of delta l values from 0.85 to 1.85. The twist energy parameter (ET), which governs the overall free energy of supercoiling, undergoes a sigmoidal decrease, while the translational diffusion coefficient undergoes a sigmoidal increase, over this same range. The static structure factor exhibits an increase, which reflects a decrease in radius of gyration associated with the circle to figure-8 transition.

Buckling transitions in superhelical DNA are sudden changes in shape that accompany a smooth variation in a key parameter, such as superhelical density. Here we explore the dependence of these transitions on the elastic constants for bending and twisting. A and C, important characteristics of DNA's bending and twisting persistence lengths. The large range we explore extends to other elastic materials with self-contact interactions, modeled here by a Debye-Huckel electrostatic potential. Our collective description of DNA shapes and energies over a wide range of p = A/C reveals a dramatic dependence of DNA shape and associated configurational transitions on p: transitions are sharp for large p but masked for small p. In particular, at small p, a nonplanar circular family emerges, in agreement with Julicher's recent analytical predictions: a continuum of forms (and associated writhing numbers) is also observed. The relevance of these buckling transitions to DNA in solution is examined through studies of size dependence and thermal effects. Buckling transitions smooth considerably as size increases, and this can be explained in part by the lower curvature in larger plasmids. This trend suggests that buckling transitions should not be detectable for isolated (i.e., unbound) DNA plasmids of biological interest, except possibly for very large p. Buckling phenomena would nonetheless be relevant for small DNA loops, particularly for higher values of p, and might have a role in regulatory mechanisms: a small change in superhelical stress could lead to a large configurational change. Writhe distributions as a function of p, generated by Langevin dynamics simulations, reveal the importance of thermal fluctuations. Each distribution range (and multipeaked shape) can be interpreted by our buckling profiles. Significantly, the distributions for moderate to high superhelical densities are most sensitive to p, isolating different distribution patterns. If this effect could be captured experimentally for small plasmids by currently available imaging techniques, such results suggest a slightly different experimental procedure for estimating the torsional stiffness of supercoiled DNA than considered to date.

Normal vectors perpendicular to individual base pairs are a powerful tool for studying the bending behavior of B-DNA, both in the form of normal vector plots and in matrices that list angles between vectors for all possible base pair combinations. A new analysis program, FREEHELIX, has been written for this purpose, and applied to 86 examples of sequence-specific protein/DNA complexes whose coordinates are on deposit in the Nucleic Acid Data Base. Bends in this sample of 86 structures almost invariably follow from roll angles between adjacent base pairs; tilt makes no net contribution. Roll in a direction compressing the broad major groove is much more common than that which compresses the minor groove. Three distinct types of B-DNA bending are observed, each with a different molecular origin: (1) Localized kinking is produced by large roll at single steps or at two steps separated by one turn of helix. (2) Smooth, planar curvature is produced by positive and negative roll angles spaced a half-turn apart, with random side-to-side zigzag roll at intermediate points, rather than a tilt contribution that might have been expected theoretically. (3) Three-dimensional writhe results from significant roll angles at a continuous series of steps. Writhe need not change the overall direction of helix axis, if it is continued indefinitely or for an integral number of helical turns. A-DNA itself can be formally considered as possessing uniform, continuous writhe that yields no net helix bending. Smooth curvature is the most intricate deformation of the three, and is least common. Writhe is the simplest deformation and is most common; indeed, a low level of continuous writhe is the normal condition of an otherwise unbent B-DNA helix of general sequence. With one exception, every example of major kinking in this sample of 86 structures involves a pyrimidine–purine step: C–A/T–G, T–A, or C–G. Purine–purine steps, especially A–A, show the least tendency toward roll deformations. © 1998 John Wiley & Sons, Inc. Biopoly 44: 361–403, 1997

The mean-square radius 〈S2〉N of gyration of the DNA topoisomer with the linking number N is evaluated as a function of N and chain length L on the basis of a (circular) twisted wormlike chain, i.e., a special case of the helical wormlike chain. Evaluation of 〈S2〉N and also of the moment 〈Wr2〉 of the writhe Wr is carried out over a wide range of L, following the Monte Carlo procedure of Frank-Kamenetskii et al. It is found that the present Monte Carlo values of 〈Wr2〉 for large L are appreciably larger than the known Monte Carlo values for freely jointed chains. Thus, the empirical interpolation formula for 〈Wr2〉 previously constructed on the basis of the theoretical values for small L along with the latter Monte Carlo values for large L is revised with the present Monte Carlo values. By the use of the revised formula, a reanalysis of the experimental data for the distribution of topoisomers is made, and it is found that the present estimates of the torsional constant and the stiffness parameter are equal to and somewhat larger than the previous ones, respectively. It is shown that 〈S2〉N decreases with increasing |ΔN|, where ΔN = N − N, with N the number of helix turns in the linear DNA chain in its undeformed state. The mean-square radii of gyration 〈S2〉Wr and 〈S2〉 of the original circular Kratky–Porod wormlike chain with and without Wr fixed are also evaluated.

A method is offered to model the complex trajectories of closed circular DNA supercoils and knots. The trajectories are approximated by polygons and analytical expressions of the curves are generated from the polygons with B-spline functions. The resulting curves are used to evaluate the writhe and elastic energy of a series of interrelated supercoils, and to generate detailed atomic models of the deformed double helix.