The trajectory of a stiff rod in a curved potential energy trough

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 radiusR. For a rod of small length compared toR, we show that the axis at equilibrium forms an arc of a circle of radius greater thanR. 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 roughtly 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.     

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.     

An elastic rod model to evaluate effects of ionic concentration on equilibrium configuration of DNA in salt solution

As a coarse-gained model, a super-thin elastic rod subjected to interfacial interactions is used to investigate the condensation of DNA in a multivalent salt solution. The interfacial traction between the rod and the solution environment is determined in terms of the Young–Laplace equation. Kirchhoff’s theory of elastic rod is used to analyze the equilibrium configuration of a DNA chain under the action of the interfacial traction. Two models are established to characterize the change of the interfacial traction and elastic modulus of DNA with the ionic concentration of the salt solution, respectively. The influences of the ionic concentration on the equilibrium configuration of DNA are discussed. The results show that the condensation 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 forces that drive DNA condensation. With the change of concentration, the DNA segments will undergo a series of alteration from the original configuration to the condensed configuration, and the spiral-shape appearing in the condensed configuration of DNA is independent of the original configuration.     

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.     

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.     

Polyelectrolyte theory and chromatin-DNA quaternary structure: Role of ionic strength and Hl histone

Under the assumptions outlined in this paper and those of Manning's theory of polyelectrolyte screening the electrostatic interaction of histone 1 and DNA is examined and shown to lead to spontaneous bending of DNA. Critical variables for this bending are the ionic strength, the length of DNA interacting with each histone 1 molecule, and the fraction of DNA phosphate charges neutralized by the histone 1 molecule. This interaction is postulated as accounting for the observed folding of polynucleosomes into the “solenoid” regular structure. Wherever possible comparison is made between theoretical predictions and available experimental results. Finally, biological implications are briefly discussed.     

Self-attraction and natural curvature in null DNA

Forces of self-attraction inherent in DNA are unmasked when its ionic charge is neutralized. On the global level, self-attraction operates between segments to condense null (charge-neutralized) DNA into a segment-rich particle. Locally, self-attraction tends to contract an individual segment along its axis. If certain conditions are satisfied, the compressed segment buckles outward from the original line of the axis. Its most stable shape is then curved, or, as an extreme case, even completely folded. Buckling conditions are derived and shown to be met by DNA, thus explaining the high degree of ordered curvature and folding in the observed morphologies of condensed null DNA. The central concept employed is the buckling persistence length. It is evaluated for null DNA (40-50 bp) and agrees with experimental data (less than 60 bp). It helps in understanding the observed cooperative unit in the condensation/decondensation equilibrium (about 60 bp) and the observed size of digestion fragments unstable in the condensed phase (about 80 bp). The root-mean-square thermal compression/extension fluctuation in DNA is estimated at about 0.1 A/bp.     

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.     

Condensation of DNA by trivalent cations. 1. Effects of DNA length and topology on the size and shape of condensed particles

In vitro condensation of DNA by multivalent cations can provide useful insights into the physical factors governing folding and packaging of DNA in vivo. We have made a detailed study of hexammine cobalt (III) induced condensation of 2700 and 1350 base pair (bp) fragments of plasmid pUC12 DNA by electron microscopy and laser light scattering. The condensed DNA takes the form of toroids and rods. Both are present in all condensates, but the proportion of toroids is higher with the larger fragments. The intact, closed circular plasmid produces smaller particles than the linear fragments. The size of a particle is independent of DNA fragment length. Two hours after adding the condensing agent, a typical toroid is about 800 A in diameter; the outer radius (R1) is approximately 400 A, and the inner radius (R2) is approximately 140 A for both sets of fragments. These dimensions are relatively stable, but there is sufficient change in both R1 and R2 to produce approximately 50% increase in volume from 2 to 24 h. A typical rod at 2 h is about 1800 A long and 300 A wide. The distribution of rod lengths is similar to that of mean toroid circumferences pi (R1 + R2), and the distribution of rod widths is similar to that of toroidal widths (R1-R2). The 2700-bp fragments show a significantly higher ratio of toroids to rods than the 1350-bp fragments. Both types of particle are multimolecular. The average number of molecules/particle, calculated from the above dimensions, assuming hexagonally packed B-form DNA with a center-to-center spacing of 27 A, is 13 +/- 4 for condensates of 2700-bp fragments and 26 +/- 11 for those of 1350-bp fragments. Monomolecular condensates of much larger DNAs have similar dimensions, suggesting that size is governed primarily by the balance of attractive and repulsive intermolecular forces rather than by the entropic factors associated with incorporation of a number of small particles into a larger one. The similar dimensions and volumes of toroids and rods indicate that the free energy cost of continual bending in toroids, minus that gained by extra net attraction in a cyclic particle, is comparable to that of abrupt bending or kinking in rods. Although the condensed particles are multimeric, their distinct toroidal or rodlike shapes distinguish them from the random aggregates that would be generally expected from the multimolecular association of large, flexible polymers.     

Determination of bilayer membrane bending stiffness by tether formation from giant, thin-walled vesicles.

The curvature elastic modulus (bending stiffness) of stearoyloleoyl phosphatidylcholine (SOPC) bilayer membrane is determined from membrane tether formation experiments. R. E. Waugh and R. M. Hochmuth 1987. Biophys. J. 52:391-400) have shown that the radius of a bilayer cylinder (tether) is inversely related to the force supported along its axis. The coefficient that relates the axial force on the tether to the tether radius is the membrane bending stiffness. Thus, the bending stiffness can be calculated directly from measurements of the tether radius as a function of force. Giant (10-50-microns diam) thin-walled vesicles were aspirated into a micropipette and a tether was pulled out of the surface by gravitational forces on small glass beads that had adhered to the vesicle surface. Because the vesicle keeps constant surface area and volume, formation of the tether requires displacement of material from the projection of the vesicle in the pipette. Tethers can be made to grow longer or shorter or to maintain equilibrium by adjusting the aspiration pressure in the micropipette at constant tether force. The ratio of the change in the length of the tether to the change in the projection length is proportional to the ratio of the pipette radius to the tether radius. Thus, knowing the density and diameter of the glass beads and measuring the displacement of the projection as a function of tether length, independent determinations of the force on the tether and the tether radius were obtained. The bending stiffness for an SOPC bilayer obtained from these data is approximately 2.0 x 10(-12) dyn cm, for tether radii in the range of 20-100 nm.(ABSTRACT TRUNCATED AT 250 WORDS)     

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 .     

荧光能量转移和偏振研究小牛胸腺DNA与组蛋白在高低离子盐溶液中的离合

本文研究小牛胸腺DNA和组蛋白在体外低、高离子强度盐溶液中的动态缔合与解离。 实验结果是低离子溶液中荧光给体DANsyl-Cl-组蛋白在发射峰位上荧光强度降低,荧光受体吖啶橙-DNA在发射峰位上的荧光强度增高。此两峰位上荧光受体的荧光增量比值是2.7/1(大于1),有能量转移发生,DNA和组蛋白缔合。高离子溶液中两峰位上受体的荧光增量比值降到1.6/1,能量转移减少,DNA和组蛋白解离。 低离子溶液中所得的吖啶橙-DNA的荧光偏振度P值小而高离子溶液中的P值大。说明解离的DNA硷基排列比缔合的DNA硷基排列有序程度强,进一步证明低离子溶液中DNA和组蛋白是缔合的,而高离子溶液中它们是解离的。     

The osmotic pressure of chondroitin sulphate solutions: experimental measurements and theoretical analysis

We used equilibrium dialysis to measure the osmotic pressure of chondroitin sulphate (CS) solutions as a function of their concentration and fixed charge density (FCD) and the ionic strength and composition of the solution. Osmotic pressure varied nonlinearly with the concentration of chondroitin sulphate and in 0.15 M NaCl at FCDs typical of uncompressed cartilage (approximately 0.4 mmol/g extrafibrillar H2O) was approximately 3 atmospheres. Osmotic pressure fell by 60% as solution ionic strength increased up to about 1 M, but remained relatively constant at higher ionic strengths. The ratio of Ca2+ to Na+ in the medium was a minor determinant of osmotic pressure. The data are compared with a theoretical model of the electrostatic contribution to osmotic pressure calculated from the Poisson-Boltzmann equation using a rod-in-cell model for CS. The effective radius of the polyelectrolyte rod is taken as a free parameter. The model qualitatively reproduces the non-linear concentration dependence, but underestimates the osmotic pressure by an amount that is independent of ionic strength. This difference, presumably arising from oncotic and entropic effects, is approximately 1/3 of the total osmotic pressure at physiological polymer concentrations and ionic strength.     

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.     

Electric dichroism and bending amplitudes of DNA fragments according to a simple orientation function for weakly bent rods

The linear dichroism is calculated for DNA fragments in their thermal bending equilibrium. These calculations are given for relatively short fragments, where bent molecules can be described by an arc model. Using the measured value of 350 A for the persistence length, the limit dichroism (corresponding to complete alignment) decreases due to thermal bending, e.g., for a fragment with 100 base pairs to 80% of the value expected for straight molecules. Thermal bending should lead to a strong continuous decrease of the dichroism with increasing chain length, which is not observed, however, in electric dichroism experiments due to electric stretching. The influence of the electric field on the bending equilibrium is described by a contribution to the bending energy, which is calculated from the movement of charge equivalents against the potential gradient upon bending. The charge equivalents, which are assigned to the helix ends, are derived from the dipole moments causing the stationary degree of orientation. By this procedure the energy term inducing DNA stretching is given for induced, permanent, and saturating induced dipole models without introduction of any additional parameter. The stationary dichroism at a given electric field strength is then calculated according to an arc model by integration over all angles of orientation of helix axes or chords with respect to the field vector, and at each of these angles the contribution to the dichroism is calculated by integration over all helices with different degrees of bending. Orientation functions obtained by this procedure are fitted to dichroism data measured for various restriction fragments. Optimal fits are found for an induced dipole model with saturation of the polarizability. The difference between orientation functions with and without electric stretching is used to evaluate dichroism bending amplitudes. Both chain length and field strength dependence of bending amplitudes are consistent with experimental amplitudes derived from the dichroism decay in low salt buffers containing multivalent ions like Mg2+, spermine, or [CoNH3)6]3+. Bending amplitudes can be used to evaluate the persistence length from electrooptical data obtained for a single DNA restriction fragment. Bending and stretching effects are considerable already at relatively low chain length, and thus should not be neglected in any quantitative evaluation of experimental data.     

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

Elasticity theory of naturally curved rods is employed to study the effects of intrinsic curvature on the properties of the equilibrium conformations of supercoiled DNA. The results stand in sharp contrast to those obtained when the molecule is viewed as being straight in its relaxed form. Starting from very fundamental principles of the theory, we show that the torsion of an open segment with a curved duplex axis can vary when the temperature, and along with it, the intrinsic twist is changed. Conversely, an imposed helicity, such as might be associated with binding to a histone, can change the intrinsic twist. It is also shown that another consequence of the presence of naturally curved sequences is that the twist density will, in general, vary with position along the chain in all equilibrium states. Then portions of the molecule will be more or less susceptible to interaction with other agents sensitive to such a variation. Finally, some closed equilibrium global structures uniquely associated with intrinsic curvature are discussed. © 1993 John Wiley & Sons, Inc.