The trajectory of a stiff rod in a curved potential energy trough. An initial result for short nucleosomal rods

The equilibrium trajectory of the axis of a rod subject to an externally imposed curved potential energy trough tends to conform to the shape of the curved trough, but also tends to be straight because of elastic resistance to bending. The actual path of the axis is a balance between the two extremes. We consider a potential energy trough centered along a circular arc of radius R. For a rod of small length compared to R, we show that the axis at equilibrium forms an arc of a circle of radius greater than R. The value of the radius of the axial path depends on the relative values of the Hooke's Law bending constant for the rod and the depth and width of the trough. Motivation for the calculation is provided by nucleosomal DNA, which conforms to the surface of a roughly cylindrical histone core at physiological ionic strength, but is observed to unwind into a partially extended conformation at very low ionic strength. We suggest that the rigidity to bending of short DNA segments becomes sufficiently great at low ionic strength to overcome attractive interactions with the histone surface. Alternately, of course, if during the cell cycle mutually attractive forces between DNA and histone core are weakened at constant ionic strength, the same type of unfolding would be expected to occur as the strength of the DNA-histone contacts drops below the level required to overcome elastic resistance to bending of the DNA rod.     

Terminal Twist-Induced Writhe of DNA with Intrinsic Curvature

Supercoiling of a closed circular DNA rod may result from an application of terminal twist to the DNA rod by cutting the rod, rotating one of the cut faces as the other being fixed and then sealing the cut. According to White's formula, DNA supercoiling is probably accompanied by a writhe of the DNA axis. Deduced from the elastic rod model for DNA structure, an intrinsically straight closed circular DNA rod does not writhe as subject to a terminal twist, until the number of rotation exceeds a rod-dependent threshold. By contrast, a closed circular DNA rod with intrinsic curvature writhes instantly as subject to a terminal twist. This noteworthy character in fact belongs to many intrinsically curved DNA rods. By solving the dynamic equations, the linearization of the Euler–Lagrange equations governing intrinsically curved DNA rods, this paper shows that almost every clamped-end intrinsically curved DNA rod writhes instantly when subject to a terminal twist (clamped-end DNA rods include closed circular DNA rods and topological domains of open DNA rods). In terms of physical quantities, the exceptions are identified with points in ℝ⁶ whose projections onto ℝ⁵ (through ignoring the total energy density of a rod) form a subset of a quadratic hypersurface. This paper also suggests that the terminal twist induced writhe is due to the elasticity and the clamped-end boundary conditions of the DNA rods. To my sister for her 50th birthday.     

Modulating DNA configuration by interfacial traction: an elastic rod model to characterize DNA folding and unfolding

As a continuum model of DNA, a thin elastic rod subjected to interfacial interactions is used to investigate the equilibrium configuration of DNA in intracellular solution. The interfacial traction between the rod and the solution environment is derived in detail. Kirchhoff’s theory of elastic rods is used to analyze the equilibrium configuration of a DNA segment under the action of the interfacial traction. The influences of the interfacial energy factor and bending stiffness on the toroidal spool formation of the DNA segment are discussed. The results show that the equilibrium configuration of DNA is mainly determined by competition between the interfacial energy and elastic strain energy of the DNA itself, and the interfacial traction is one of the forces that drives DNA folding and unfolding.     

Anisotropic elastic bending models of DNA

Simplified elastic rod models of DNA were developed in which the rigidity of DNA is sequence dependent and asymmetrical, i.e. the bending is facilitated towards the major groove. By subjecting the models to bending load in various directions perpendicular to the longitudinal axis of DNA, the bending deformation and the average conformation of the models can be estimated using finite element methods. Intrinsically curved sequence motifs [(aaaattttgc)_n, (tctctaaaaaatatataaaaa)_n] are found to be curved by this modelling procedure whereas the average conformation of homopolymers and straight motifs [(a)_n, (atctaatctaacacaacaca)_n] show negligible or no curvature. This suggests that sequence dependent asymmetric rigidity of DNA can provide an explanation in itself for intrinsic DNA curvature. The average rigidity of various DNA sequences was calculated and a good correlation was found with such quantities as the free energy change upon the binding of the Cro repressor, the base stacking energy and the thermal fluctuations at room temperature.     

Evaluation of Elastic Rod Models with Long Range Interactions for Predicting Nucleosome Stability

Abstract

The ability of a dinucleotide-step based elastic-rod model of DNA to predict nucleosome binding free energies is investigated using four available sets of elastic parameters. We compare the predicted free energies to experimental values derived from nucleosome reconstitution experiments for 84 DNA sequences. Elastic parameters (conformation and stiffnessess) obtained from MD simulations are shown to be the most reliable predictors, as compared to those obtained from analysis of base-pair step melting temperatures, or from analysis of x-ray structures. We have also studied the effect of varying the folded conformation of nucleosomal DNA by means of our Fourier filtering knock-out and knock-in procedure. This study confirmed the above ranking of elastic parameters, and helped to reveal problems inherent in models using only a local elastic energy function. Long-range interactions were added to the elastic-rod model in an effort to improve its predictive ability. For this purpose a Debye-Huckel energy term with a single, homogenous point charge per base- pair was introduced. This term contains only three parameters,—its weight relative to the elastic energy, the Debye screening length, and a minimum sequence distance for including pairwise interactions between charges. After optimization of these parameters, our Debye-Huckel term is attractive, and yields the same level of correlation with experiment (R = 0.75) as was achieved merely by varying the nucleosomal shape in the elastic-rod model. We suggest this result indicates a linker DNA—histone attraction or, possibly, entropic effects, that lead to a stabilization of a nucleosome away from the ends of DNA segments longer than 147 bp. Such effects are not accounted for by a localized elastic energy model.     

The nature of forces stabilizing nucleosome structure. Dissociation of histone octamers from DNA

We have used the measurements of the histone fluorescence parameters to study the influence of the ionic strength on histone-DNA and histone-histone interactions in reconstructed nucleosomes. The ionic strength increase lead to the two-stage nucleosome dissociation. The dimer H2A-H2B dissociates at the first stage and the tetramer (H3-H4)2 at the second one. The dimer H2A-H2B dissociation from nucleosome is a two-stage process also. The ionic bonds between (H2A-H2B) histone dimer and DNA break at first and then the dissociation of dimer from histone tetramer (H3-H4)2 occurs. According to the proposed model the dissociation accompanying a nucleosome "swelling" and an increase of DNA curvature radius. It was shown that the energy of electrostatic interactions between histone dimer and DNA is sufficiently less than the energy of dimer-tetramer interaction. We propose that the nucleosome DNA ends interact with the dimer and tetramer simultaneously. The calculated number (approximately 30 divided by 40) of ionic bonds between DNA and histone octamer globular part practically coincides with the number of exposed cationic groups on the surface of octamer globular head. On this basis we have assumed that the spatial distribution of these groups is precisely determined, which explains the high evolutionary conservatism of the histone primary structure.     

Effect of the angular potential on the temperature control in dissipative particle dynamics simulations

Angular potentials in dissipative particle dynamics (DPD) simulation can describe the rigidity of rod segments in the rod–coil block copolymers and cause system instability monitored by the kinetic temperature. We studied the influence of the bending constant of angular potentials on the temperature control of the DPD simulation system by our defined two parameters, deviation temperature δ _T and standard deviation SD _T . Small bending constants have no obvious influence on the temperature control. For the large bending constants, a relation of δ _T  = a? ^b Δt ^c is constructed. Based on this relation, we can give the appropriate parameters to balance the requirement of the temperature control and the rigid formation of rod-like blocks correctly.     

Application of Polyelectrolyte Theory to the Study of the B-Z Transition in DNA (1)

Abstract

We have used the polyelectrolyte theory to study the ionic strength dependence of the B-Z equilibrium in DNA. A DNA molecule is molded as an infinitely long continuously charged cylinder of radius a with reduced linear charge density q. The parameters a and q for the B and Z forms were taken from X-ray data: a _B = 1nm, q _B = 4.2, a _z = 0.9 nm and q _z = 3.9. A simple theory shows that at low ionic strengths (when Debye screening length r _D>>a) the electrostatic free energy difference F ^el _Bz = F ^el _Z - F ^el _B increases with increasing ionic strength since q _B>q_z. At high ionic strengths (when r _D<<a) the F ^el _BZ would go on growing with increasing ionic strength if the inequality q _B/^a _B<q_z/a _z were valid. In the converse case when q _z/q _B<a_z/a _B the F ^el _BZ value decreases with increasing salt concentration at high ionic strength. Since X-ray data correspond to the latter case, theory predicts that the F ^el _BZ value reaches a maximum at an intermediate ionic strength of about 0.1 M (where r _D～a). We also performed rigorous calculations based on the Poisson-Boltzmann equation. These calculations have confirmed the above criterion of nonmonotonous behaviour of the F ^el _BZ value as a function of ionic strength. Different theoretical predictions for the B-Z transition in linear and superhelical molecules are discussed. Theory predicts specifically that at a very low ionic strength the Z form may prove to be more stable than the B form. Thus, one can observe the Z-B-Z transition with increasing ionic strength. In the light of our theoretical findings we discuss numerous experimental data on the B-Z transition in linear and superhelical DNA.     

An electrostatic model for the dielectric effects,the adsorption of multivalent ions,and the bending of B-DNA

We have studied electrostatic properties of DNA with a discrete charge model consisting of a cylindrical dielectric core with a radius of 8 Å and a dielectric constant D_i = 4, surrounded by two helical strings of phosphate point charges at 10 Å from the axis, immersed in an aqueous medium with dielectric constant D_w = 78.54. Eliminating the dielectric core makes potentials in the phosphate surface less negative by about 0.5 kT/e. Salt effects are evaluated for the model without a dielectric core, using the shielded Coulomb potential. Smearing the phosphate charges increases their potential by about 2.5 kT/e, due mostly to the self-potential of the smeared charge. Potentials in the center of the minor and major grooves vary less than 0.02 kT/e along their helical path. The potential in the center of the minor groove is from 1.0 to 1.7 kT/e, more negative than in the center of the major groove, depending on dielectric core and salt concentration. So multivalent cations and also larger cationic ligands, such as some antibiotics, are likely to adsorb in the minor groove, in agreement with earlier computations by A. and B. Pullman. Dielectric effects on the surface potential and the local potential variations are found to be relatively small. Bending of DNA is studied by placing a multivalent cation, M^Z+, in the center of the minor or major groove, curving DNA around it for a certain length, and calculating the free energy difference between the bent and the straight configuration. Boltzmann averaged bending angles, 〈β〉, are found to be maximal in 0.03M monovalent salt, for a length of about 50 or 25 Å of curved DNA when an M^Z+ ion is adsorbed in the minor or the major groove, respectively. When the dielectric constant of water is used throughout the calculation, we find maximal bends of 〈β〉 = 11° for M²⁺ and 〈β〉 = 16° for M³⁺ in the minor groove, 〈β〉 = 13° for M³⁺ in the major groove. The absence of bends in DNA adsorbed to mica in the presence of Mg salts supports the role of Mg²⁺ in “ion bridging” between DNA and mica. The treatment of the effective dielectric constant between two points outside a dielectric cylinder in water is appended. © 1998 John Wiley & Sons, Inc. Biopoly 46: 503–516, 1998     

The interaction of HMGB1 and linker histones occurs through their acidic and basic tails

H1 and HMGB1 bind to linker DNA in chromatin, in the vicinity of the nucleosome dyad. They appear to have opposing effects on the nucleosome, H1 stabilising it by "sealing" two turns of DNA around the octamer, and HMGB1 destabilising it, probably by bending the adjacent DNA. Their presence in chromatin might be mutually exclusive. Displacement/replacement of one by the other as a result of their highly dynamic binding in vivo might, in principle, involve interactions between them. Chemical cross-linking and gel-filtration show that a 1:1 linker histone/HMGB1 complex is formed, which persists at physiological ionic strength, and that complex formation requires the acidic tail of HMGB1. NMR spectroscopy shows that the linker histone binds, predominantly through its basic C-terminal domain, to the acidic tail of HMGB1, thereby disrupting the interaction of the tail with the DNA-binding faces of the HMG boxes. A potential consequence of this interaction is enhanced DNA binding by HMGB1, and concomitantly lowered affinity of H1 for DNA. In a chromatin context, this might facilitate displacement of H1 by HMGB1.     

Ionic strength-dependent conformational transitions of chromatin. Circular dichroism and thermal denaturation studies

High-molecular-weight chicken erythrocyte chromatin was prepared by mild digestion of nuclei with micrococcal nuclease. Samples of chromatin containing both core (H3, H4, H2A, H2B) and lysine-rich (H1, H5) histone proteins (whole chromatin) or only core histone proteins (core chromatin) were examined by CD and thermal denaturation as a function of ionic strength between 0.75 and 7.0 × 10^?3M Na⁺. CD studies at 21°C revealed a conformational transition over this range of ionic strengths in core chromatin, which indicated a partial unfolding of a segment of the core particle DNA at the lowest ionic strength studied. This transition is prevented by the presence of the lysine-rich histones in whole chromatin. Thermal-denaturation profiles of both whole and core chromatins, recorded by hyperchromicity at 260 nm, reproducibly and systematically varied with the ionic strength of the medium. Both materials displayed three resolvable thermal transitions, which represented the total DNA hyperchromicity on denaturation. The fractions of the total DNA which melted in each of these transitions were extremely sensitive to ionic strength. These effects are considered to result from intra- and/or internucleosomal electrostatic repulsions in chromatin studied at very low ionic strengths. Comparison of the whole and core chromatin melting profiles indicated substantial stabilization of the core-particle DNA by binding sites between the H1/H5 histones and the 140-base-pair core particle.     

Comparison between histones FV and F2a2 of chicken erythrocyte. II. Interaction with homologous DNA.

The conformation and stability of artificial complexes between chicken erythrocyte DNA and homologous histones FV and F2a2 was studied by circular dichroism (CD) and thermal denaturation followed by both absorbance and CD measurements. The complexes are made after a stepwise potassium fluoride gradient dialysis without urea and studied at low ionic strength (10-minus 3 M). 1) No structural changes of the DNA can be detected up to r equals 0.2 with FV and r equals 0.6 for F2a2. With FV at higher values of r the CD spectrum is altered, indicating the organization of DNA and histones in some kind of aggregate. 2) The conformation of histone molecules inside the complexes is not related to the ionic strength of the medium but to an effective ionic environment close to 0.1 M. This ionic strength would also correspond to the melting temperature of histone-covered DNA. 3) From the analysis of the absorbance melting profile the length of DNA covered with an histone molecule can be estimated. A good agreement is found between the negative charge of this piece of DNA and the net positive charge of the histone. 4) Since the CD transition at 227 nm occurs before the second absorbance transition at 280 nm, the DNA is stabilized no longer by native histone but partially or fully denatured histones. The helical regions of the histone molecule are not involved in the binding process, which appears to be almost purely coulombian and most likely related to some structural fit between the pattern of negative charges in the DNA helix and that of positive charges along the peptide chain.     

Writhe of DNA induced by a terminal twist

This paper considers the three-dimensional structure of B-form DNA. The molecule may be open or covalently closed. For the former, its two ends are not allowed to move or rotate freely in space unless the molecule is under the influence of rigid body motions of the ambient space. Implied by the elastic rod model for DNA, the molecule writhes immediately when subject to a terminal twist as long as its axis is none of the following curves: lines, circular arcs, circular helices. This result is remarkably different from well-known results about DNA of other conformations. For example, if a DNA is regarded as an elastic rod whose axis is a circle, then it has no induced writhe when subject to a terminal twist until the latter meets a critical extent. To my mother for her 70th birthday An erratum to this article is available at .     

Index to Subjects,Volume 5

Abstract

The interaction of histone H5 labelled with fluorescein isothiocyanate (FITC) with DNA has been studied by fluorescence titration, and diffusion-enhanced fluorescence energy transfer (DEFET) measurements with Tb(III) lanthanide chelates as donors.

Analysis of the binding data by the model of Schwarz and Watanabe (J. Mol. Biol. 163, 467-484 (1983)) yielded a mean stoichiometry of 60 nucleotides per H5 molecule, independently of ionic strength, in the range of 3 to 300 mM NaCl, at very low DNA concentration (6 μM in mononucleotide). It ensues an approximate electroneutrality of the saturated complexes. Histone H5 molecules appeared to be clustered along the DNA lattice in clusters containing on average 3 to 4 H5 molecules separated by about 79 base pairs, at mid-saturation of the binding sites. The interaction process was found highly cooperative but the cooperativity parameter was also insensitive to ionic strength in the above range.

DEFET experiments indicated an important decrease of accessibility of the FITC label to the TbHED3A° and TbEDTA^? chelates with ionic strength in the 0 to 100 mM NaCl range. In the presence of DNA, H5 appears already folded at low ionic strength so that the FITC probe is also not accessible to the donor chelate. The present study constitutes an indispensable preliminary step to further studies on the localization of histone H5 in condensed chromatin structures.     

Link, twist, energy, and the stability of DNA minicircles

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.     

Breathing and bending fluctuations in DNA modeled by an open-base-pair kink coupled to axial compression

We develop a model designed to show that flexibility in the DNA molecule can arise from relatively improbable transient opening of base pairs. The axial direction changes at the site of an open base pair. The region between open base pairs is a double helix of hydrogen-bonded base pairs with a slightly decreased rise per residue and a slightly increased helical winding angle. An analysis of the model yields several testable predictions. For example, we predict probability 0.026 for a base pair to be open at 25°C, a value close to that measured by hydrogen-exchange experiments. Other predictions involve matters like the variation of persistence length with ionic strength and temperature, the variation of helical winding angle with temperature, and the kinetics of heat denaturation. An additional result of the analysis is an explanation of the high degree of local stiffness of the DNA molecule. Strong resistance to bending fluctuations is provided from two sources: increased polyelectrolyte repulsion among phosphate groups in the axially compressed stacks between open base pairs and the tendency of stacking forces to oppose opening of a base pair. Stacking forces, however, also support compression of the stacks between open base pairs, so that the net effect of stacking forces on elastic bending of DNA is small relative to the polyelectrolyte effect. If the ionic charges on the phosphate groups were absent, DNA would spontaneously fold, driven by the entropy gained when about 1% of its base pairs open.     

DNA chain flexibility and the structure of chromatin nu-bodies.

The persistence length of high-molecular-weight, monodisperse-bihelical DNA has been evaluated from low-shear flow birefingence and viscosity data at several temperatures in 2.0 M Nacl neutral pH buffer. At these solvent conditions, both the DNA and histone components of chromatin nu-bodies have structural features similar to those in the intact nucleohistone complex at low ionic strength. The theory of Landau and Lifshitz is used to relate the experimental result to the thermodynamic functions for bending 140 nucleotide pairs of DNA into a plausible model structure: per nu-body, delta Gb=43.8 +/- 5.3 kcal/mole, delta Hb= 45.7 +/- 3.7 kcal/mole, and delta Sb = 6.2 +/- 12.4 entropy units. This bending free energy is comparable to or less than that estimated to be required for a kinked DNA configuration and appears to be well within the range of estimated electrostatic free energies available from DNA-histone interactions in a nu-body assembly.     

Histone H1: Ultracentrifugation studies of the effects of ionic strength and denaturants on the solution conformation

The conformation of histone H1 has been examined under native and denaturing conditions in the absence of DNA or chromatin. Sedimentation coefficients were determined for Histone H1 in 0.1 m KCl and in 6 m guanidine hydrochloride solutions at pH 7.4. The influence of ionic strength on the conformation of histone H1 has been determined by measurement of the sedimentation coefficient in tetramethylammonium chloride solutions of up to 2.5 m and extrapolated to infinite ionic strength. Results from these experiments suggest that the native conformation of histone H1 is very asymmetric in shape. The molecule is best described as a prolate ellipsoid with axes of 312 Å (2a) and 16 Å (2b) in low ionic strength media and also as a prolate ellipsoid with axes of 202 Å (2a) and 20 Å (2b) at high ionic strength or when associated with polyanions, e.g., DNA. Denaturation of histone H1 by guanidine hydrochloride was found to be completely reversible. In 6 m guanidine hydrochloride, the H1 molecule collapses to a sphere but the original extended conformation of the protein is readily restored on dialysis. This suggests rigid conformational requirements for the H1 molecule as incorporated into chromatin. The shape and dimensions for the H1 molecule at high ionic strength are not sufficiently conclusive to locate H1 in the chromatin structure. It is proposed, however, that viable models for chromatin architecture must be consistent with the histone H1 solution dimensions obtained here.     

Hydrolysis of DNA by a Branched Peptide without Aromatic Residues

A branched peptide Nα, Nɛ-di (l-leucyl)-l-lysine was found to efficiently cleave supercoiled double-strand DNA such as PUC19 DNA at optimum pH 4.0 in 40 mmol/l Britton–Robbinson buffer. The T4 ligase experiment implied that the DNA cleavage occurs via a hydrolytic path. The dependence of the cleavage reaction on the ionic strength indicated that the interaction of DNA with the branched peptide involve only electrostatic binding.