The effect of intrinsic curvature on supercoiling: predictions of elasticity theory

An analytical method to connect open curves for modeling protein-bound DNA minicircles

We introduce an analytical method to generate the pathway of a closed protein-bound DNA minicircle. We develop an analytical equation to connect two open curves smoothly and use the derived expressions to join the ends of two helical pathways and form models of nucleosome-decorated DNA minicircles. We find that the simplest smooth connector which satisfies the boundary conditions at the end points and the length requirement for such connections to be a quartic function on the xy-plane and linear along the z-direction. This is a general method which can be used to connect any two open curves with well defined mathematical definitions as well as pairs of discrete systems found experimentally. We used this method to describe the configurations of torsionally relaxed, 360-base pair DNA rings with two evenly-spaced, ideal nucleosomes. We considered superhelical nucleosomal pathways with different levels of DNA wrapping and allowed for different inter-nucleosome orientations. We completed the DNA circles with the smooth connectors and studied the associated bending and electrostatic energies for different configurations in the absence and presence of salt. The predicted stable states bear close resemblance to reconstituted minicircles observed under low and high salt conditions.

Reshaping of a Janus ring

We consider the reshaping of closed Janus filaments acquiring intrinsic curvature upon actuation of a driven component-a nematic elastomer elongating upon phase transition. Linear stability analysis establishes instability thresholds of circles with no imposed twist, dependent on the ratio q of the intrinsic curvature to the inverse radius of the original circle. Twisted circles are proven to be absolutely unstable but the linear analysis well predicts the dependence of the looping number of the emerging configurations on the imposed twist. Modeling stable configurations by relaxing numerically the overall elastic energy detects multiple stable and metastable states with different looping numbers. The bifurcation of untwisted circles turns out to be subcritical, so that nonplanar shapes with a lower energy exist at q below the critical value. The looping number of stable shapes generally increases with q.

Normal-Mode Analysis of Circular DNA at the Base-Pair Level. 2. Large-Scale Configurational Transformation of a Naturally Curved Molecule

Fine structural and energetic details embedded in the DNA base sequence, such as intrinsic curvature, are important to the packaging and processing of the genetic material. Here we investigate the internal dynamics of a 200 bp closed circular molecule with natural curvature using a newly developed normal-mode treatment of DNA in terms of neighboring base-pair "step" parameters. The intrinsic curvature of the DNA is described by a 10 bp repeating pattern of bending distortions at successive base-pair steps. We vary the degree of intrinsic curvature and the superhelical stress on the molecule and consider the normal-mode fluctuations of both the circle and the stable figure-8 configuration under conditions where the energies of the two states are similar. To extract the properties due solely to curvature, we ignore other important features of the double helix, such as the extensibility of the chain, the anisotropy of local bending, and the coupling of step parameters. We compare the computed normal modes of the curved DNA model with the corresponding dynamical features of a covalently closed duplex of the same chain length constructed from naturally straight DNA and with the theoretically predicted dynamical properties of a naturally circular, inextensible elastic rod, i.e., an O-ring. The cyclic molecules with intrinsic curvature are found to be more deformable under superhelical stress than rings formed from naturally straight DNA. As superhelical stress is accumulated in the DNA, the frequency, i.e., energy, of the dominant bending mode decreases in value, and if the imposed stress is sufficiently large, a global configurational rearrangement of the circle to the figure-8 form takes place. We combine energy minimization with normal-mode calculations of the two states to decipher the configurational pathway between the two states. We also describe and make use of a general analytical treatment of the thermal fluctuations of an elastic rod to characterize the motions of the minicircle as a whole from knowledge of the full set of normal modes. The remarkable agreement between computed and theoretically predicted values of the average deviation and dispersion of the writhe of the circular configuration adds to the reliability in the computational approach. Application of the new formalism to the computed modes of the figure-8 provides insights into macromolecular motions which are beyond the scope of current theoretical treatments.

Intrinsic curvature of DNA influences LacR-mediated looping

Protein-mediated DNA looping is a common mechanism for regulating gene expression. Loops occur when a protein binds to two operators on the same DNA molecule. The probability of looping is controlled, in part, by the basepair sequence of inter-operator DNA, which influences its structural properties. One structural property is the intrinsic or stress-free curvature. In this article, we explore the influence of sequence-dependent intrinsic curvature by exercising a computational rod model for the inter-operator DNA as applied to looping of the LacR-DNA complex. Starting with known sequences for the inter-operator DNA, we first compute the intrinsic curvature of the helical axis as input to the rod model. The crystal structure of the LacR (with bound operators) then defines the requisite boundary conditions needed for the dynamic rod model that predicts the energetics and topology of the intervening DNA loop. A major contribution of this model is its ability to predict a broad range of published experimental data for highly bent (designed) sequences. The model successfully predicts the loop topologies known from fluorescence resonance energy transfer measurements, the linking number distribution known from cyclization assays with the LacR-DNA complex, the relative loop stability known from competition assays, and the relative loop size known from gel mobility assays. In addition, the computations reveal that highly curved sequences tend to lower the energetic cost of loop formation, widen the energy distribution among stable and meta-stable looped states, and substantially alter loop topology. The inclusion of sequence-dependent intrinsic curvature also leads to nonuniform twist and necessitates consideration of eight distinct binding topologies from the known crystal structure of the LacR-DNA complex.

Thermal fluctuations of small rings of intrinsically helical DNA treated like an elastic rod

It is shown that the writhe distribution in a collection of thermally fluctuating rings formed from intrinsically helical DNA treated like an elastic rod can be determined completely from a knowledge of the dependence of the configuration integral on the intrinsic geometric torsion of the rods. When the length of the rods is of the order of the persistence length or less, the configuration integral can be expressed in terms of the frequencies of the normal modes of the rings, and so these torsion-dependent frequencies then become the quantities of interest. The nature of the normal modes of rings formed from intrinsically helical elastic rods, as deduced from the equations of motion in the Kirchhoff-Clebsch theory of the dynamics of elastic rods, is described. Examples of computed writhe distributions are given. The observed broadening of the writhe distribution as the intrinsic curvature of the rods is increased is shown to be related to the stability properties of the circular equilibrium configuration of the elastic rods. These results thus imply that thermal fluctuations in small rings of intrinsically curved DNA will be more pronounced than in rings of the same size formed from intrinsically straight DNA.

DNA rings with multiple energy minima

Within the context of DNA rings, we analyze the relationship between intrinsic shape and the existence of multiple stable equilibria, either nicked or cyclized with the same link. A simple test, based on a perturbation expansion of symmetry breaking within a continuum elastic rod model, provides good predictions of the occurrence of such multiple equilibria. The reliability of these predictions is verified by direct computation of nicked and cyclized equilibria for several thousand DNA minicircles with lengths of 200 and 900 bp. Furthermore, our computations of equilibria for nicked rings predict properties of the equilibrium distribution of link, as calculated by much more computationally intensive Monte Carlo simulations.

Terminal twist induced continuous writhe of a circular rod with intrinsic curvature

It is well known that a large linking number induces an abrupt writhing of a circular rod with zero intrinsic curvature, i.e., the stress-free state of the rod is straight. We show here that for any rod with a uniform natural curvature, no matter how small the intrinsic curvature is, a twist will induce a continuous writhing from the circular configuration and the abrupt writhing is only the limiting case when the intrinsic curvature is absolutely zero. The implication of this result on elastic models of circular DNA is discussed.

Supercoiling-dependent flexibility of adenosine-tract-containing DNA detected by a topological method

Intrinsically bent DNA sequences have been implicated in the activation of transcription by promoting juxtaposition of DNA sequences near the terminal loop of a superhelical domain. We have developed a novel topological assay for DNA looping based on lambda integrative recombination to study the effects of intrinsically bent DNA sequences on the tertiary structure of negatively supercoiled DNA. Remarkably, the localization of adenosine-tract (A-tract) sequences in the terminal loop of a supercoiled plasmid is independent of the extent of intrinsic bending. The results suggest that A-tract-containing sequences have other properties that organize the structure of superhelical domains apart from intrinsic bending and may explain the lack of conservation in the degree of A-tract-dependent bending among DNA sequences located upstream of bacterial promoters.

Buckling transitions in superhelical DNA: dependence on the elastic constants and DNA size

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-Hückel 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 Jülicher'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.

Molecular dynamics simulations of small DNA plasmids: effects of sequence and supercoiling on intramolecular motions

Small (600 base pair) DNA plasmids were modeled with a simplified representation (3DNA) and the intramolecular motions were studied using molecular mechanics and molecular dynamics techniques. The model is detailed enough to incorporate sequence effects. At the same time, it is simple enough to allow long molecular dynamics simulations. The simulations revealed that large-scale slithering occurs in a homogeneous sequence. In a heterogeneous sequence, containing numerous small intrinsic curves, the centers of the curves are preferentially positioned at the tips of loops. With more curves than loop tips (two in unbranched supercoiled DNA), the heterogeneous sequence plasmid slithers short distances to reposition other curves into the loop tips. However, the DNA is immobilized most of the time, with the loop tips positioned over a few favored curve centers. Branching or looping also appears in the heterogeneous sequence as a new method of repositioning the loop tips. Instead of a smooth progression of increasing writhing with increasing linking difference, theoretical studies have predicted that there is a threshold between unwrithed and writhed DNA at a linking difference between one and two. This has previously been observed in simulations of static structures and is demonstrated here for dynamic homogeneous closed DNA. Such an abrupt transition is not found in the heterogeneous sequence in both the static and dynamic cases.

Light scattering studies of supercoiled and nicked DNA

Static and dynamic light scattering measurements were made of solutions of pGem1a plasmids (3730 base pairs) in the relaxed circular (nicked) and supercoiled forms. The static structure factor and the spectrum of decay modes in the autocorrelation function were examined in order to determine the salient differences between the behaviors of nicked DNA and supercoiled DNA. The concentrations studied are within the dilute regime, which is to say that the structure and dynamics of an isolated DNA molecule were probed. Static light scattering measurements yielded estimates for the molecular weight M, second virial coefficient A2, and radius of gyration RG. For the nicked DNA, M = (2.8 +/- 0.4) x 10(6) g/mol, A2 = (0.9 +/- 0.2) x 10(-3) mol cm3/g2, and RG = 90 +/- 3 nm were obtained. For the supercoiled DNA, M = (2.5 +/- 0.4) x 10(6) g/mol, A2 = (1.2 +/- 0.2) x 10(-3) mol cm3/g2, and RG = 82 +/- 2.5 nm were obtained. The static structure factors for the nicked and supercoiled DNA were found to superpose when they were scaled by the radius of gyration. The intrinsic stiffness of DNA was evident in the static light scattering data. Homodyne intensity autocorrelation functions were collected for both DNAs at several angles, or scattering vectors. At the smallest scattering vectors the probe size was comparable to the longest intramolecular distance, while at the largest scattering vectors the probe size was smaller than the persistence length of the DNA. Values of the self-diffusion coefficients D were obtained from the low-angle data. For the DNA, D = (2.9 +/- 0.3) x 10(-8) cm2/s, and for the supercoiled DNA, D = (4.11 +/- 0.21) x 10(-8) cm2/s. The contribution to the correlation function from the internal dynamics of the DNA was seen to result in a strictly bimodal decay function. The rates of the faster mode gamma int, reached plateau values at low angles. For the nicked DNA, gamma int = 2500 +/- 500 s-1, and for the supercoiled DNA, gamma int = 5000 +/- 500 s-1. These rates correspond to the slowest internal relaxation modes of the DNAs. The dependence of the relaxation rates on scattering vector was monitored with the aid of cumulants analysis and compared with theoretical predictions for the semiflexible ring molecule. The internal mode rates and the dependence of the cumulants moments reflected the difference between the nicked DNA and the supercoiled DNA dynamical behavior. The supercoiled DNA behavior seen here indicates that conformational dynamics might play a larger role in DNA behavior than is suggested by the notion of a branched interwound structure.

Effects of localized bending on DNA supercoiling

The DNA double helix is straight only in the idealized case. In reality, it bends, twists and stretches in response to local base sequence and to specific interactions with proteins and other bound ligands. Naturally occurring bends appear to promote the assembly of nucleosomes, and in some cases can effectively replace regulatory DNA-binding proteins in vivo. Recently, a computational method known as 'finite element analysis', which is used routinely by engineers to analyse the stability of buildings and bridges, has been applied to the quantitative assessment of natural curvature in supercoiled DNA structures, providing new insight into the relationship between local, sequence-dependent features and the overall topology of these chains.

Flexing and folding double helical DNA

DNA base sequence, once thought to be interesting only as a carrier of the genetic blueprint, is now recognized as playing a structural role in modulating the biological activity of genes. Primary sequences of nucleic acid bases describe real three-dimensional structures with properties reflecting those structures. Moreover, the structures are base sequence dependent with individual residues adopting characteristic spatial forms. As a consequence, the double helix can fold into tertiary arrangements, although the deformation is much more gradual and spread over a larger molecular scale than in proteins. As part of an effort to understand how local structural irregularities are translated at the macromolecular level in DNA and recognized by proteins, a series of calculations probing the structure and properties of the double helix have been performed. By combining several computational techniques, complementary information as well as a series of built-in checks and balances for assessing the significance of the findings are obtained. The known sequence dependent bending, twisting, and translation of simple dimeric fragments have been incorporated into computer models of long open DNAs of varying length and chemical composition as well as in closed double helical circles and loops. The extent to which the double helix can be forced to bend and twist is monitored with newly parameterized base sequence dependent elastic energy potentials based on the observed configurations of adjacent base pairs in the B-DNA crystallographic literature.

Modeling superhelical DNA: recent analytical and dynamic approaches

During the past year, a variety of diverse and complementary approaches have been presented for modeling superhelical DNA, offering new physical and biological insights into fundamental functional processes of DNA. Analytical approaches have probed deeper into the effects of entropy and thermal fluctuations on DNA structure and on various topological constraints induced by DNA-binding proteins. In tandem, new kinetic approaches--by molecular, Langevin and Brownian dynamics, as well as extensions of elastic-rod theory--have begun to offer dynamic information associated with supercoiling. Such dynamic approaches, along with other equilibrium studies, are refining the basic elastic-rod and polymer framework and incorporating more realistic treatments of salt and sequence-specific features. These collective advances in modeling large DNA molecules, in concert with technological innovations, are pointing to an exciting interplay between theory and experiment on the horizon.

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.