Marc Le Bret

Catastrophic variation of twist and writhing of circular DNAs with constraint

The number of constraint turns is defined as the number of bound unwinding ligands converted in turns necessary to relax the energy stored in a supercoiled DNA. It is equal to the sum of the twist and of the writhing and is interpreted geometrically. The twist is shown to be related to the derivative of the energy of superhelix formation with respect to the constraint turns. This leads to a semiempirical evaluation of the variation of the conformational energy with the writhing and of the writhing number. The bending contribution to the conformational energy is estimated independently using first‐order elasticity. The variation of the twist and of the writhing with the constraint is, in this model, catastrophic. In particular, the writhing number jumps between intervals of allowed values.

Electrostatic contribution to the persistence length of a polyelectrolyte

The electrostatic contribution to the persistence length is computed through the integration of the average electrostatic potential at the surface of a toroidal polyion over its charge parameter, when specific interaction between the mobile ions of the salt and the fixed charges of the polyion are ignored. When the radius of curvature of the polyion is very large, we assumed that the electrostatic potential may be developed in series as a function of the curvature. With this assumption, it may be computed from one‐dimensional integrations only. The assumption is then numerically verified by performing a painstaking two‐dimensional integration for finite values of the radius of curvature. Values for the electrostatic contribution to the persistence length are given in two limit cases: in the presence of an excess of added salt and in the total absence of added salt. The role of the dielectric constant and of the mobility of the ’’fixed’’ charges of the polyion is also determined.

Twist and writhing in short circular DNAs according to first‐order elasticity

The distribution of twist and writhing in a closed DNA shorter than its persistence length is examined. In this case, the only energy contribution is elastic. We have in tegrated the equations of elasticity for a homogeneous axially symmetric rod of undeformable infinitely small circular cross section with frictionless reactions, when there is no or only one self‐contact. In the absence of self‐contacts, the central line of the rod is drawn on a toroid. It makes ν turns around the axis of revolution of the toroid and m turns around its core. The integer, ν, is equal to one if the rod is unknotted. We prove that no infinitely thin rod with a positive Poisson ratio is stable in a toroidal conformation if there is no self‐contact. However, m‐leafed roses or rosettes, with but one multiple self‐contact, are shown to actually be stable when their writhing is not too great. When the integer, m, is equal to two, we have figure‐8 conformations. Buckling of the circle into a figure‐8 conformation occurs for the constraint such that the figure‐8 and the circular conformations have the same energy. This constraint is 1.845 turns for a bending‐to‐twisting elastic constants ratio of A/C = 1.5. For the same value of A/C, the figure‐8 conformation is unstable for a constraint greater than 2.4 turns. Corrections caused by a finite value of the radius ratio, a/L, of the cross section to the length of the rod, are estimated. For instance, both the circular conformation and the infinitesimally warped circle are simultaneous solutions for particular values of the β twist. β = A/C (m2 − ν2)½ [1 + (νπa/L)2/2]. The binding of ethidium to DNAs short enough to follow first‐order elasticity has been studied. Buckling occurs at an apparent average constraint of about 0.6. How the DNA molecules are distributed in figure‐8 conformations and circles has been determined as the ethidium concentration is varied.

Monte carlo computation of the supercoiling energy, the sedimentation constant, and the radius of gyration of unknotted and knotted circular DNA

Closed random Gaussian polygonal chains of N (6 < N < 150) bonds of equal length b and thickness d have been generated on a computer. The knot type, the writhing number w, the radius of gyration, and the average of the inverse of the distance between two apices have been determined for each chain. For all the studied knot types—0, 31, 41, 51, and 52—the probability density of finding a given w is Gaussian. The Gaussian is centered about 0 for the amphichiral knots. Therefore, for long circular DNAs, the contribution to the supercoiling energy, which depends on w only, may be considered as purely entropic and may be expressed as ARTw2/N, in agreement with previous semiempirical considerations. The parameter A increases with chain thickness, it decreases as N gets larger but rapidly reaches a plateau. Comparison with experimental data from the literature would suggest that the ratio of the writhing to the constraint increases with ionic strength. The ratio of sedimentation constant of the supercoiled DNA to the sedimentation constant of the nicked DNA varies as N1/4 (w/N)2, and therefore depends on the writhing density and on the length of the DNA.

Chromatin reconstitution on small DNA rings: II. DNA supercoiling on the nucleosome

DNA supercoiling on the nucleosome was investigated by relaxing with topoisomerase I mono- and dinucleosomes reconstituted on small DNA rings. Besides 359 base-pair (bp) rings whose linking differences were integers, two additional series of rings with fractional differences, 341 and 354 bp in size, were used. Mononucleosomes reconstituted on 359 bp rings were found to relax into a single mononucleosome form. In contrast, 341 and 354 bp mononucleosomes relaxed into a mixture of two forms, corresponding to two adjacent topoisomers. The observation that the ratio between these two forms was, within each ring series, virtually independent of the initial linking number of the topoisomer used for the reconstitution suggested that each partition reflected an equilibrium. Comparison with the equilibria observed for the same rings in the absence of histones showed that the formation of a single nucleosome is associated with a linking number change of -1.1(+/-0.1) turn. Dinucleosomes, in contrast, were not relaxed to completion and do not reach equilibria. The corresponding linking number change per nucleosome was, however, estimated to be similar to the above figure, in agreement with previous data from the literature obtained with circular chromatins containing larger numbers of nucleosomes. DNA structure in mononucleosomes was subsequently investigated by means of high-resolution electron microscopy and gel electrophoresis. It was found that the above linking number reduction could be ascribed to a particle with a large open extranucleosomal DNA loop and with no more than 1.5 turns of a superhelix around the histone core. A theoretical model of a nucleosome on a small ring was constructed in which one part of the DNA was wrapped around a cylinder and the other part was free to vary both in torsion and flexion. The linking number reduction predicted was found to be most consistent with experimental data when the twist of the DNA in the superhelix was between 10.5 and 10.65 pb per turn, suggesting that wrapping on the nucleosome does not alter the twist of the DNA significantly. A lower estimate of the linking number reduction associated with a two-turn nucleosome was also derived, based on an analysis of recent data obtained upon treatment of reconstituted minichromosomes with gyrase. The value, 1.6 turns, set a lower limit of 10.44 bp per turn for the twist of nucleosomal DNA, in agreement with the above estimate.(ABSTRACT TRUNCATED AT 400 WORDS)

Computation of the Helical Twist of Nucleosomal DNA

Abstract The aim of this Appendix is to stimulate the partial wrapping of a small DNA ring around the histone core in order to get a theoretical estimation of the topoisomer distribution that has been experimentally measured for mononucleosomes after treatment of the chromatin with topoisomerase I (see table I of the main paper).

Electronic structure of ethidium bromide

Ethidium bromide (3,8 diamino-6-phenyl-5-ethyl phenantridinium bromide) is a fluorescent compound widely used in molecular biology because of its physical properties and biological activities. It is a trypanotidal dye [l] with antiviral properties [2] which intercalates between adjacent base pairs in double helix regions of nucleic acids [3] . By intercalation, ethidium bromide has its fluorescence quantum yield strikingly enhanced [4], and the complex it forms with DNA displays optical activity in the visible [ 51 . In this paper we are concerned with the electronic structure, excitation energies, and direction of transition moments of ethidium bromide. These quantities are useful to interpret experimental data obtained by measurements of circular dichroism and intermolecular energy transfer [ 5,6] , thus investigating the structure of nucleic acids.

Object Command Language: a formalism to build molecular models and to analyze structural parameters in macromolecules, with applications to nucleic acids

We have written a programming language OCL (Object Command Language) to solve, in a general way, two recurring problems that arise during the construction of molecular models and during the geometrical characterization of macromolecules: how to move precisely and reproducibly any part of a molecular model in any user-defined local reference axes; and how to calculate standard or user-defined structural parameters that characterize the complex geometries of any macromolecule. OCL endows the user with three main capabilities: the definition of subsets of the macromolecule, called objects in OCL, with a formalism from elementary set theory or lexical analysis; the definition of sequences of elementary geometrical operations, called procedures in OCL, enabling one to build arbitrary three-dimensional (3D) orthonormal reference frames, to be associated with previously defined objects; and the transmission of these definitions to programs that allow one to display, to modify and to analyze interactively the molecular structure, or to programs that perform energy minimizations or molecular dynamics. Several applications to nucleic acids are presented.

Binding of Net-Fla, a netropsin-flavin hybrid molecule, to DNA: molecular mechanics and dynamics studies in vacuo and in water solution.

We have studied the binding of the hybrid netropsin-flavin (Net-Fla) molecule onto four sequences containing four A. T base pairs. Molecular mechanics minimizations in vacuo show numerous minimal conformations separated by one base pair. 400 ps molecular dynamics simulations in vacuo have been performed using the lowest minima as the starting conformations. During these simulations, the flavin moiety of the drug makes two hydrogen bonds with an amino group of a neighboring guanine. A 200 ps molecular dynamics simulation in explicit water solution suggests that the binding of Net-Fla upon the DNA substrate is enhanced by water bridges. A water molecule bridging the amidinium of Net-Fla to the N3 atom of an adenine seems to be stuck in the drug-DNA complex during the whole simulation. The fluctuations of the DNA helical parameters and of the torsion angles of the sugar-phosphate backbone are very similar in the simulations in vacuo and in water. The time auto-correlation functions for the DNA helical parameters decrease rapidly in the picosecond range in vacuo. The same functions computed from the water solution molecular dynamics simulations seem to have two modes: the rapid mode is similar to the behavior in vacuo, and is followed by a slower mode in the 10 ps range.

MORMIN: a quasi-Newtonian energy minimizer fitting the nuclear Overhauser data

In this article, we describe the program MORMIN, which can simultaneously minimize the mechanical energy of a given macromolecular structure, together with a weighted quadratic penalty function of the difference between the observed and computed nuclear Overhauser effect (nOe) peaks. The gradient of the nOe penalty function relatively to the proton coordinates is computed from an exact closed formula of a matrix exponential derivative. To cut CPU time, the molecular system is partitioned into nonoverlapping subsets containing the protons involved in the observed peaks. The algorithm is no longer exact, but if a 1% relative error is accepted it can be run, on a scalar computer, in about the same CPU time as needed for the calculation of the mechanical energy. We have successfully run the program in more than 1000 situations, including cases where the hybrid method failed because of the occurrence of negative eigenvalues. In some cases, the optimization of the Cartesian coordinates could be successfully extended to individual atomic diffusion times.