Elastic model of DNA supercoiling in the infinite‐length limit

Circular Polycatenanes: Supramolecular Structures with Topologically Tunable Properties

Polycatenanes, macrochains of topologically interlocked rings with unique physical properties have recently gained considerable interest in supramolecular chemistry, biology, and soft matter. Most of the work has been, so far, focused on linear chains and on their variety of conformational properties compared to standard polymers. Here we go beyond the linear case and show that, by circularizing such macrochains, one can exploit the topology of the local interlockings to store twist in the system, significantly altering its metric and local properties. Moreover, by properly defining the twist (Tw) and writhe (Wr) of these macrorings we show the validity of a relation equivalent to the Călugăreanu-White-Fuller theorem Tw+Wr=const, originally proved for ribbonlike structures such as double stranded DNA. Our results suggest that circular polycatenanes with storable and tunable twist can form a new category of highly designable multiscale structures with potential applications in supramolecular chemistry and material science.

Pack, unpack, bend, twist, pull, push: the physical side of gene expression

Molecular motors such as polymerases produce physical constraints on DNA and chromatin. Recent techniques, in particular single-molecule micromanipulation, provide estimation of the forces and torques at stake. These biophysical approaches have improved our understanding of chromatin behaviour under physiological physical constraints and should, in conjunction with genome wide and in vivo studies, help to build more realistic mechanistic models of transcription in the context of chromatin. Here, we wish to provide a brief overview of our current knowledge in the field, and emphasize at the same time the importance of DNA supercoiling as a major parameter in gene regulation.

Elasticity and electrostatics of plectonemic DNA

We present a self-contained theory for the mechanical response of DNA in single molecule experiments. Our model is based on a one-dimensional continuum description of the DNA molecule and accounts both for its elasticity and for DNA-DNA electrostatic interactions. We consider the classical loading geometry used in experiments where one end of the molecule is attached to a substrate and the other one is pulled by a tensile force and twisted by a given number of turns. We focus on configurations relevant to the limit of a large number of turns, which are made up of two phases, one with linear DNA and the other one with superhelical DNA. The model takes into account thermal fluctuations in the linear phase and electrostatic interactions in the superhelical phase. The values of the torsional stress, of the supercoiling radius and angle, and key features of the experimental extension-rotation curves, namely the slope of the linear region and thermal buckling threshold, are predicted. They are found in good agreement with experimental data.

Electron and scanning force microscopy studies of alterations in supercoiled DNA tertiary structure

The configuration of supercoiled DNA (scDNA) was investigated by electron microscopy and scanning force microscopy. Changes in configuration were induced by varying monovalent/divalent salt concentrations and manifested by variation in the number of nodes (crossings of double helical segments). A decrease in the concentration of monovalent cations from 50 mM to approximately 1 mM resulted in a significant change of apparent configuration of negatively supercoiled DNA from a plectonemic form with virtually approximately 15 nodes (the value expected for molecules of approximately 3000 bp) to one or two nodes. This result was in good agreement with values calculated using an elastic rod model of DNA and salt concentration in the range of 5-50 mM. The effect did not depend on the identity of the monovalent cation (Na(+), K(+)) or the nature of the support used for electron microscopy imaging (glow-discharged carbon film, polylysine film). At very low salt concentrations, a single denatured region several hundred base-pairs in length was often detected. Similarly, at low concentrations of divalent cations (Mg(2+), Ca(2+), Zn(2+)), scDNA was apparently relaxed, although the effect was slightly dependent on the nature of the cation. Positively supercoiled DNA behaved in a manner different from that of its negative counterpart when the ion concentration was varied. As expected for these molecules, an increase in salt concentration resulted in an apparent relaxation; however, a decrease in salt concentration also led to an apparent relaxation manifested by a slight decrease in the number of nodes. Scanning force microscopy imaging of negatively scDNA molecules deposited onto a mica surface under various salt conditions also revealed an apparent relaxation of scDNA molecules. However, due to weak interactions with the mica surface in the presence of a mixture of mono/divalent cations, the effect occurred under conditions differing from those used for electron microscopy. We conclude that the observed changes in scDNA configuration are inherent to the DNA structure and do not reflect artifacts arising from the method(s) of sample preparation.

Electrostatic-undulatory theory of plectonemically supercoiled DNA

We present an analytical calculation of the electrostatic interaction in a plectonemic supercoil within the Poisson-Boltzmann approximation. Undulations of the supercoil strands arising from thermal motion couple nonlinearly with the electrostatic interaction, giving rise to a strong enhancement of the bare interaction. In the limit of fairly tight winding, the free energy of a plectonemic supercoil may be split into an elastic contribution containing the bending and torsional energies and an electrostatic-undulatory free energy. The total free energy of the supercoil is minimized according to an iterative scheme, which utilizes the special symmetry inherent in the usual elastic free energy of the plectoneme. The superhelical radius, opening angle, and undulation amplitudes in the radius and pitch are obtained as a function of the specific linking difference and the concentration of monovalent salt. Our results compare favorably with the experimental values for these parameters of Boles et al. (1990. J. Mol. Biol. 213:931-951). In particular, we confirm the experimental observation that the writhe is a virtually constant fraction of the excess linking number over a wide range of superhelical densities. Another important prediction is the ionic strength dependence of the plectonemic parameters, which is in reasonable agreement with the results from computer simulations.

Monovalent cations differ in their effects on transcription initiation from a sigma-70 promoter of Escherichia coli

Initiation of transcription from the sigma-70 rep promoter of plasmid pBR322 was measured by abortive transcription assays at various concentrations of potassium, rubidium, and sodium acetate. When linear and negatively supercoiled templates were compared, each salt generated a characteristic response. Increasing the salt concentration decreased transcription from a linear template but produced an increase (potassium) or a bell-shaped response (rubidium) with a supercoiled template. In the case of sodium ions, increasing concentration inhibited transcription initiation from both linear and supercoiled templates. These results are discussed with respect to effects of monovalent cations on DNA twist.

The effect of ionic conditions on the conformations of supercoiled DNA. I. Sedimentation analysis

We studied the conformations of supercoiled DNA as a function of superhelicity and ionic conditions by determining its sedimentation coefficient both experimentally and by calculation. To cancel out unknown parameters from both calculations and experiments, we determined the ratio of the sedimentation coefficient, s, to that of open circular DNA, s(oc). Calculations of the sedimentation coefficient were based on direct solution of the Burgers-Oseen problem for an equilibrium set of DNA conformations generated for each condition by the Metropolis Monte Carlo procedure. There were no adjustable parameters in the Monte Carlo simulations because all three parameters of the DNA model used, bending and torsional elasticity of DNA and DNA effective diameter specifying electrostatic interactions, were known from independent data. The good agreement between measured and calculated values of s/s(oc) allowed us to interpret the sedimentation results in terms of DNA conformations, with particular emphasis on the marked effect of ionic conditions. As NaCl concentration decreases, s/s(oc) increases because the superhelix becomes less regular and more compact. In the presence of just 10 mM MgCl(2), supercoiled DNA adopts essentially the same set of conformations as in moderate to high concentrations of NaCl. Our simulations showed that s is a strong function of the superhelix branching frequency. At near physiological ionic conditions, there are about four branches in the 7 kb DNA molecule used in this work. We found no indication of superhelix collapse in any ionic conditions even remotely approaching physiological ones. For all ionic conditions studied, we conclude that the electrostatic interaction of DNA segments specified by the DNA effective diameter is the primary determinant of supercoiled DNA conformations.

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.

On higher buckling transitions in supercoiled DNA

A combination of detailed energy minimization and molecular dynamics studies of closed circular DNA offers here new information that may be relevant to the dynamics of short DNA chains and/or low superhelical densities. We find a complex dependence of supercoiled DNA energies and geometries on the linking number difference delta Lk as physiological superhelical densities (magnitude of sigma approximately 0.06) are approached. The energy minimization results confirm and extend predictions of classical elasticity theory for the equilibria of elastic rods. The molecular dynamics results suggest how these findings may affect the dynamics of supercoiled DNA. The minimization reveals sudden higher order configurational transitions in addition to the well-known catastrophic buckling from the circle to the figure-8. The competition among the bending, twisting, and self-contact forces leads to different families of supercoiled forms. Some of those families begin with configurations of near-zero twist. This offers the intriguing possibility that nicked DNA may relax to low-twist forms other than the circle, as generally assumed. Furthermore, for certain values of delta Lk, more than one interwound DNA minimum exists. The writhing number as a function of delta Lk is discontinuous in some ranges; it exhibits pronounced jumps as delta Lk is increased from zero, and it appears to level off to a characteristic slope only at higher values of delta Lk. These findings suggest that supercoiled DNA may undergo systematic rapid interconversions between different minima that are both close in energy and geometry. Our molecular dynamics simulations reveal such transitional behavior. We observe the macroscopic bending and twisting fluctuations of interwound forms about the global helix axis as well as the end-over-end tumbling of the DNA as a rigid body. The overall mobility can be related to magnitude of sigma and to the bending, twisting, and van der Waals energy fluctuations. The general character of molecular motions is thus determined by the types of energy minima found at a given delta Lk. Different time scales may be attributed to each type of motion: The overall chain folding occurs on a time scale almost an order of magnitude faster than the end-over-end tumbling. The local bending and twisting of individual chain residues occur at an even faster rate, which in turn correspond to several cycles of local variations for each large-scale bending and straightening motion of the DNA.

Torsional rigidity of positively and negatively supercoiled DNA

Time-correlated single-photon counting of intercalated ethidium bromide was used to measure the torsion constants of positively supercoiled, relaxed, and negatively supercoiled pBR322 DNA, which range in superhelix density from +0.042 to -0.123. DNA behaves as coupled, nonlinear torsional pendulums under superhelical stress, and the anharmonic term in the Hamiltonian is approximately 15 percent for root-mean-square fluctuations in twist at room temperature. At the level of secondary structure, positively supercoiled DNA is significantly more flexible than negatively supercoiled DNA. These results exclude certain models that account for differential binding affinity of proteins to positively and negatively supercoiled DNA.