Structure and Dynamics of Double Helices in Solution: Modes of DNA Bending

Abstract

The long range structure of DNA restriction fragments has been analysed by electro-optical measurements. The overall rotation time constants observed in a low salt buffer with monovalent ions is shown to decrease upon addition of Mg²⁺ or spermine. Since the circular dichroism and also the limiting value of the linear dichroism remain almost constant under these conditions, the effect is attributed to a change of the long range structure. According to a weakly bending rod model, the persistence length decreases from about 600 Å in the absence of Mg²⁺ or spermine to about 350 Å in the presence of these ions. The persistence length measured in the presence of Mg²⁺ is almost independent of temperature in the range of 10 to 40 °C. The nature of DNA bending is analysed by measurements of bending amplitudes and time constants from dichroism decay curves. The observed absence of changes in the bending amplitudes upon addition of Mg²⁺ or spermine, even though addition induces changes of the persistence length by a factor of 2, is hardly consistent with simple thermal bending. The combined results, including the remarkably small temperature dependence of persistence length and bending amplitude, can be explained by the existence of two bending effects: inherent curvature of DNA dominates at low temperature, whereas thermal bending prevails at high temperature. Analysis of bending amplitudes from dichroism decay curves according to an arc model provides an approximate measure for the degree of bending in restriction fragments. The model is consistent with the observed chain length dependence of bending amplitudes and provides an approximate curvature corresponding to a radius of ab out400Å. Thus the curvature observed in restriction fragments is similar to that observed for high molecular DNA condensed into toroids by addition of ions like spermine.

Particularly strong bending of DNA is induced by [CO(NH₃)₆]³⁺, indicated by an apparent persistence length of 200 Å and an increased bending amplitude together with a reduced limit value of the linear dichroism. This effect is attributed to the high charge density of this ion and potential site binding.     

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.     

Influence of DNA length on spermine-induced condensation. Importance of the bending and stiffening of DNA

Using DNA restriction fragments of 258 to 4362 base-pairs, we have investigated the influence of the DNA length on the condensation process induced by spermine, with the aid of electric dichroism measurements. The 258- and 436 bp fragments condensed into rod-like particles, while the fragments of 748 bp or more condensed into torus-shaped particles. Our results suggest that a DNA molecule longer than the circumference of the toroids observed previously (680 bp) is required to serve as a nucleus for the growth of the condensed particles. The toroids were more stable in the electric field than the rod-shaped particles, suggesting that rapid fluctuations of the bound spermine counterions can provide one of the main attractive forces yielding to the condensation process. Relaxation time data for the 436 bp fragment revealed that the structure of DNA was altered at a spermine concentration as low as one-tenth of that required for condensation: the DNA became bent in the presence of spermine. Moreover, the field strength dependence of the relaxation times, as well as the fitting of the decay curves at 12.5 kV/cm, showed an increase of the stiffness of the DNA double helix upon spermine addition. We estimated that, in the case of DNA condensation by spermine, a decrease in the measured persistence length may occur, irrespective of the DNA flexibility, owing to the bending of the DNA molecule.     

Influence of DNA length on spermine-induced condensation: Importance of the bending and stiffening of DNA

Using DNA restriction fragments of 258 to 4362 base-pairs, we have investigated the influence of the DNA length on the condensation process induced by spermine, with the aid of electric dichroism measurements. The 258- and 436 bp fragments condensed into rod-like particles, while the fragments of 748 bp or more condensed into torus-shaped particles. Our results suggest that a DNA molecule longer than the circumference of the toroids observed previously (680 bp) is required to serve as a nucleus for the growth of the condensed particles. The toroids were more stable in the electric field than the rod-shaped particles, suggesting that rapid fluctuations of the bound spermine counterions can provide one of the main attractive forces yielding to the condensation process. Relaxation time data for the 436 bp fragment revealed that the structure of DNA was altered at a spermine concentration as low as one-tenth of that required for condensation: the DNA became bent in the presence of spermine. Moreover, the field strength dependence of the relaxation times, as well as the fitting of the decay curves at 12.5 kV/cm, showed an increase of the stiffness of the DNA double helix upon spermine addition. We estimated that, in the case of DNA condensation by spermine, a decrease in the measured persistence length may occur, irrespective of the DNA flexibility, owing to the bending of the DNA molecule.     

Dynamic bending rigidity of DNA

Rapidly relaxing components in the decay of the transient electric dichroism of DNA restriction fragments were reported by Diekmann et al. [(1982) Biophys. Chem. 15, 263-270] and P?rschke et al. [(1987) Biopolymers 26, 1971-1974]. These are analyzed using a new normal mode theory for weakly bending rods and assigned to bending. The longest bending relaxation times for fragments with 95-250 base pairs coincide with the theoretical curve calculated for a dynamic bending rigidity corresponding to a dynamic persistence length Pd = 2100 A. Analysis of the relative amplitudes of fast and slow components following weak orienting pulses is also consistent with a rather large dynamic persistence length. The enhancement of the relative amplitude of the fast component in large electric fields is attributed to steady-state bending of initially perpendicular DNAs by the field. Several reasons are proposed why the dynamic bending rigidity is 4 times larger than the apparent static bending rigidity inferred from equilibrium persistence length measurements on the same fragments.     

A study of the molecular mechanism of DNA interaction with divalent metal ions

The interaction of DNA with divalent metal ions: Ba2+, Mg2+, Mn2+, Ni2+, Cu2+ in solutions at different ionic strengths mu was investigated. The combination of following methods: flow birefringence, viscometry, UV-spectroscopy and circular dichroism made possible to follow the state of the secondary and tertiary structure of the DNA molecule during its interaction with ions. The presence of divalent ions in solution affects the hydrodynamic properties of DNA only at low mu. At high mu the difference in the action of mono- and divalent ions disappears. The persistence length of DNA does not change during the experiment. It is shown that the Mg2+ and Ba2+ ions interact only with phosphate groups of DNA but Mn2+, Ni2+, Cu2+ ions interact also with the nitrogen bases of the macromolecule.     

Condensation and precipitation of chromatin by multivalent cations

The condensation and the precipitation of rat liver chromatin upon addition of spermine4+, spermidine3+, hexamminecobalt(III)3+ and Mg2+ cations have been studied using solubility, fluorescence, circular dichroism, melting curves, electric dichroism and spermidine binding measurements, made on both soluble and precipitated complexes. The soluble complexes obtained with tetra- and trivalent cations were depleted from all histones and enriched in other proteins, particularly high mobility group proteins 1 and 2, which brings about an important enhancement of tryptophan fluorescence without modification of its two lifetimes 5.1 and 1.2 ns. In the precipitates the non-histone proteins are eliminated. Under precipitation by Mg2+ ions, the distribution of proteins remains practically unchanged. The electric dichroism and the melting curves indicate that the soluble complexes between polyamines and chromatin undergo important condensation and, at high ratios of cation over phosphate, are constituted by heterogeneous assemblies of non-histone proteins and DNA. On the contrary, the insoluble complexes seem to retain the main features of original chromatin. Precipitation by Mg2+ ions reveal much less drastic changes than those produced by polyamines. Precipitation by spermidine occurs when one cation is bound per eight nucleotides, which in addition to the histone positive charges brings about a complete neutralization of chromatin phosphates.     

Bending and flexibility of kinetoplast DNA

We have evaluated the extent of bending at an anomalous locus in DNA restriction fragments from the kinetoplast body of Leishmania tarentolae using transient electric dichroism to measure the rate of rotational diffusion of DNA fragments in solution. We compare the rate of rotational diffusion of two fragments identical in sequence except for circular permutation, which places the bend near the center in one case and near one end of the molecule in the other. Hydrodynamic theory was used to conclude that the observed 20% difference in rotational relaxation times is a consequence of an overall average bending angle of 84 +/- 6 degrees between the end segments of the fragment that contains the bending locus near its center. If it is assumed that bending results from structural dislocations at the junctions between oligo(dA).oligo(dT) tracts and adjacent segments of B DNA, a bend angle of 9 +/- 0.5 degrees at each junction is required to explain the observations. The extent of bending is little affected by ionic conditions and is weakly dependent on temperature. Comparison of one of the anomalous fragments with an electrophoretically normal control fragment leads to the conclusion that they differ measurably in apparent stiffness, consistent with a significantly increased persistence length or contour length in the kinetoplast fragments.     

An unusual electrooptical effect observed for DNA fragments and its apparent relation to a permanent electric moment associated with bent DNA

Dichroism decay curves of DNA fragments with chain lengths in the range of 179-256 bp show an amplitude inversion suggesting the existence of a positive dichroism component, when these fragments are dissolved at monovalent salt concentrations above approx. 5 mM and are exposed to field pulses with amplitudes and/or lengths above critical values. At the critical values, the unusual dichroism is reflected by an apparent acceleration of the decay curves, which can be fitted by single exponentials with time constants much below the values expected from the DNA contour lengths. The critical pulse amplitudes and lengths decrease with increasing DNA chain length and increasing salt concentration. The experimental data are consistent with results obtained by hydrodynamic and electric model calculations on smoothly bent DNA double helices. The DNA is represented by a string of overlapping beads, which is used to calculate the rotational diffusion tensor and the center of diffusion. The distribution of phosphate charges is asymmetric with respect to this center and thus gives rise to a substantial permanent dipole moment. The magnitude of this dipole moment is calculated as a function of DNA curvature and is used together with experimental values of polarizabilities for simulations of dichroism decay curves. The curves simulated for bent DNA show the same phenomenon as observed experimentally. The ionic strength dependence of the unusual dichroism is explained by an independently observed strong decrease of the polarizability with increasing salt concentration. The field strength dependence is probably due to field-induced bending of double helices driven by the change of the dipole moment. Although our calculations are on rigid models of DNA and thus any flexibility of the double helix has not been considered, we conclude that the essential part of our experimental results can be explained by our model.     

Electro-optical analysis of 'curved' DNA fragments

The structure of some 'short' DNA fragments with 106-108 base-pairs and of a 'long' kinetoplast DNA fragment with 419 base-pairs has been analyzed by electro-optical procedures. According to their electrophoretic mobilities and circularization probabilities, it was concluded that two of our short fragments with clusters of four and five and six adenosines phased at the period of the double helix are inherently curved with an approximate curvature around 200 degrees. The dichroism decay curves of our short fragments exhibited two processes. A fast one with time constants of approx. 100 ns is attributed to bending; the bending amplitudes observed for the fragments with dA4 and dA5/6 clusters are slightly higher (23 and 29%, respectively) than those observed for control fragments (17-20%). The second process reflects the overall rotational diffusion of the whole fragments and shows some variation with the DNA sequence, but on average the rotation of fragments with dA4 and dA5/6 clusters corresponds to that observed for standard DNA. Since the rotational diffusion coefficients are very strongly dependent on the effective hydrodynamic lengths, we must conclude that the effective lengths of our fragments, including the 'curved' ones, are very similar under the conditions of our experiments. The rotation time constant for the long kinetoplast DNA is also rather close to those observed for the usual DNA fragments of corresponding length. One way to resolve the conflict of our results with conclusions obtained from other investigations would invoke the assumption that the curved fragments are not 'elastic'. According to this hypothesis, electric field pulses would stretch the curved fragments to an almost straight form and the stretched DNA would return to its equilibrium state with a time constant longer than the rotation time constant.     

Comparative study of the condensation of chicken erythrocyte and calf thymus chromatins by di- and multivalent cations

The condensation of chicken erythrocyte (CE) and calf thymus (CT) chromatins upon addition of di- and multivalent cations has been studied using turbidity, precipitation and electric dichroism measurements. For all the cations investigated (Mg2+, Tb3+, Co(NH3)6(3+), spermidine Spd2+ and spermine Sp4+) condensation of CE chromatin occurred before the onset of aggregation, while aggregation of CT chromatin started before condensation with all cations except Mg2+ and Tb3+. Precipitation of CE chromatin required lower di- and multivalent cations concentrations than CT chromatin. The electric dichroism data for both chromatins, at low ionic strength in the absence of di- or multivalent cations, indicated that the nucleoprotein molecules were not totally decondensed but that a "precondensed" state was already present. A positive electric dichroism was observed for the most condensed chromatin fibers, in agreement with the "cross-linker" models. Tb3+ led to less compact condensed particles as judged from the electric dichroism observations, but electron microscopy revealed that "30 nm fibers" were formed. Very little aggregation was produced by Tb3+. On the contrary, spermine produced very large networks of condensed molecules, but large spheroidal particles were also observed. The condensation of CE chromatin happened without changes of solution conductivity upon cation salt addition, regardless of the condensing cation, indicating a cooperative uptake of the ions during this process.     

Magnesium binding and conformational change of DNA in chromatin

The structure of chromatin in the presence of Mg2+ ions was examined by circular dichroism and equilibrium dialysis. Circular dichroism (CD) shows that above 260 nm the intensity of the spectrum of DNA in nucleoproteins decreases as the Mg2+ concentration increases. This change is an intrinsic characteristic of DNA since it is also observed in protein-free DNA and has been attributed to a change in the winding angle of base pairs around the DNA axis. Some structural elements of the DNA in the nucleosome core, therefore, are as movable as those of protein-free DNA. The basic organization of H1-depleted chromatin, 146 base pairs (bp) of DNA wound around core histones and a residual 49 bp in the linker region in the repeating unit, is maintained both in the presence and in the absence of Mg2+ ions, as shown by the fact that the CD spectrum of H1-depleted chromatin has the same type of linear combination between the spectrum of protein-free DNA and that of the nucleosome core in 0.2 mM MgCl2-10 mM triethanolamine (pH 7.8) as it has in 1 mM ethylenediaminetetraacetic acid-10 mM tris(hydroxymethyl) aminomethane (pH 7.8). The ellipticity of chromatin shows a smaller decrease relative to the other nucleoproteins and protein-free DNA upon the addition of Mg2+ ions. Therefore, some structural elements of chromatin are apparently somewhat protected against the conformational change induced by these ions. The spectrum of chromatin becomes almost indistinguishable from that of H1-depleted chromatin in 0.2 mM MgCl2.(ABSTRACT TRUNCATED AT 250 WORDS)     

Environmental influences on DNA curvature

DNA curvature plays an important role in many biological processes. To study environmental influences on DNA curvature we compared the anomalous migration on polyacrylamide gels of ligation ladders of 11 specifically-designed oligonucleotides. At low temperatures (25 degrees C and below) most of the sequences exhibited a degree of anomalous migration. Increased temperature had a significant effect on the anomalous migration (curvature) of some sequences but limited effects on others; at 50 degrees C only 1 sequence migrated anomalously. Mg2+ had a strong influence on the migration of certain sequences, whilst spermine enhanced the anomalous migration of a different set of sequences. Sequences with a GGC motif exhibited greater curvature than predicted by the presently-used angles for the nearest-neighbour wedge model and are especially sensitive to Mg2+. The data have implications for models for DNA curvature and for environmentally-sensitive DNA conformations in the regulation of gene expression.     

Structure of the Tet repressor and Tet repressor-operator complexes in solution from electrooptical measurements and hydrodynamic simulations

The Tet repressor protein and tet operator DNA fragments and their complexes have been analyzed by electrooptical procedures. The protein shows a positive linear dichroism at 280 nm, a negative linear dichroism at 248 nm, and a strong permanent dipole moment of 3.5 X 10(-27) C m, which is independent of the salt concentration within experimental accuracy. Its rotation time constant of 40 ns indicates an elongated structure, which is consistent with a prolate ellipsoid of 100 A for the long axis and 40 A for the short axis. The time constant can also be fitted by a cylinder of length 78 A and diameter 37 A, which is consistent with nuclease protection data reported on repressor-operator complexes, if the cylinder axis is aligned parallel to the DNA axis. Addition of tetracycline induces changes of the limit dichroism but very little change of the rotation time constant. The rotation time constants observed for the operator DNA fragments show some deviations from the values expected from their contour length; however, these deviations remain relatively small. Formation of repressor-operator complexes leads to some increase of the DNA rotation time constants. Simulations by bead models demonstrate that these time constants can be explained without any major change of the hydrodynamic dimension of the components. The data for the complexes are fitted by bead models with smooth bending of the DNA corresponding to a radius of curvature of 500 A, but at the given accuracy, we cannot rule out that the DNA in the complex remains straight or is bent to a smaller radius of approximately 400 A. Thus, binding of the Tet repressor, which is a helix-turn-helix protein as judged from its sequence, to its operator seems to induce minor bending but does not induce strong bending of the DNA double helix.     

Effect of Mn2+ and Mg2+ ions on DNA conformation

The DNA conformation was studied at different relation between Na+ and Me2+ (Mn2+ or Mg2+) ions in solution at the fixed total ionic strength mu. At low mu the intrinsic viscosity of DNA [eta] decreased to the limited fixed value with the increasing of Mn2+ or Mg2+ concentration (CMe2+). At higher mu greater than or equal to 0.1 M [eta] doesn't depend on CMe2+. The presence of Mn2+ in solution caused a decrease of the optical anisotropy of DNA and the value of epsilon 260 (p) independent on ionic strengths. In contrary, these parameters of DNA didn't change in solution with Mg2+-concentration. The observed differences in the effects of Mn2+ and Mg2+ on the optical properties of the macromolecule suggest that there are different modes of binding of these ions to DNA. It has been concluded, that Mn2+ interacts with bases and phosphate groups of DNA, but Mg2+--only with phosphates. The persistence length of DNA doesn't depend on Me2+ concentration under the conditions of the experiment (mu greater than or equal to 0.005 M).     

DNA persistence length revisited

DNA restriction fragments ranging from 79 to 789 base pairs in length have been characterized by transient electric birefringence (TEB) measurements at various temperatures between 4 and 43 degrees C. The DNA fragments do not contain runs of four or more adenine residues in a row and migrate with normal electrophoretic mobilities in polyacrylamide gels, indicating that they are not intrinsically curved or bent. The low ionic strength buffers used for the measurements contained 1 mM Tris Cl, pH 8.0, EDTA, and variable concentrations of Na(+) or Mg(2+) ions. The rotational relaxation times were obtained by fitting the TEB field-free decay signals with a nonlinear least-squared fitting program; the decay of the birefringence was monoexponential for fragments < or = 241 base pair (bp) in length and multiexponential for larger fragments. The terminal relaxation times, characteristic of the end-over-end rotation of the DNA molecules, were then used to determine the persistence length (p) and hydrodynamic radius (r) of DNA as a function of temperature and ionic strength, using several different hydrodynamic models. The specific values obtained for p and r are model dependent. The wormlike chain model of P. J. Hagerman and B. H. Zimm (Biopolymers 1981, Vol. 20, pp. 1481-1502) combined with the revised Broersma equation (J. Newman et al., Journal of Mol Biol 1997, Vol. 116, pp. 593-606) appears to be the most suitable for describing the flexibility of DNA in low ionic strength solutions. The values of p and r obtained from the global least squares fitting of this equation are independent of DNA length, and the deviations of the individual values from the average are reasonably small. The consensus r value calculated for DNA in various low ionic strength solutions containing 1 mM Tris buffer is 14.7 +/- 0.4 A at 20 degrees C. The consensus p values decrease from 814 approximately 564 A in solutions containing 1 mM Tris buffer plus 0.2-1 mM NaCl and decrease still further to 440 A in solutions containing 0.2 mM Mg(2+) ions. The persistence length exhibits a shallow maximum at 20 degrees C and decreases slowly upon either increasing or decreasing the temperature, regardless of the model used to fit the data. By contrast, the consensus values of the hydrodynamic radius are independent of temperature. The calculated persistence lengths and hydrodynamic radii are compared with other data in the literature.     

The conformation of single stranded oligonucleotides and of oligonucleotide-oligopeptide complexes from their rotation relaxation in the nanosecond time range

The conformation of single stranded oligonucleotides is analysed by measurements of their rotation time constants. The oligomers are aligned to some degree by short electric field pulses; after pulse termination the transition to a random orientation is followed by measurements of the linear dichroism. An efficient deconvolution procedure is developed for evaluation of the experimental data obtained in the ns-time range. The increase of rotation time constants observed for chain lengths in the range from 14 to 22 residues are interpreted according to a weakly bending rod model providing a persistence length and a Stokes' diameter. The Stokes' diameters obtained for ribo- and deoxyriboadenylates are about 13A, in approximate agreement with the expectation for a single stranded helix. The persistence length L = 53A corresponding to approximately 16 nucleotide residues found for riboadenylates at 2 degrees C appears to reflect relatively strong stacking interactions at this temperature. However, a comparison with the average length of stacked residues evaluated from available thermodynamic parameters of base stacking indicate that unstacked residues are not completely flexible. Apparently the ribose-phosphate chain provides an essential contribution to the stiffness of oligomers and polymers, even when the bases are unstacked. Addition of 100 microM Mg2+ leads to an increase of the persistence length to 88A. Corresponding measurements with deoxyriboadenylates show a slightly lower value of the persistence length than that found for riboadenylates. Addition of LysTrpLys and LysTyrLys to A(pA)19 leads to an increase of the rotation time constant, which corresponds approximately to a length increment by one residue per bound peptide. Since controls performed with LysLeuLys do not show any similar effect, the increase of the time constants induced by LysTrpLys and LysTyrLys is attributed to intercalation of the aromatic amino acids.     

Changes in conformation, stability and condensation of DNA by univalent and divalent cations in methanol-water mixtures

Circular dichroism spectroscopy, absorption spectroscopy, measurements of Tm values, sedimentation analysis and electron microscopy were used to study properties of calf thymus DNA in methanol-water mixtures as a function of monovalent cation (Na+ or Cs+) concentration and also in the presence of divalent cations Ca2+, Mg2+, and Mn2+. In the absence of divalent cations only slight conformational changes occurred and no condensation and/or aggregation could be detected. The Tm values depend on the amount of methanol and on the nature and concentration of cations. In methanol-water mixtures higher thermal stability was observed in solutions containing Cs+ ions. Up to 40% (v/v) methanol the addition of divalent ions leads to DNA stabilization. At methanol concentration higher than 50% the presence of divalent cations causes DNA condensation and denaturation even at room temperature. The denaturation is reversible with respect to EDTA addition indicating that no separation of complementary strands occurred and the resulting form of DNA is probably similar to the P form. DNA destacking appears to be a direct consequence of stronger cation binding by the condensed DNA in methanol-water mixtures.     

Thermodynamic stability theory for DNA doughnut shapes induced by charge neutralization

A thermodynamic analysis of the bending free energy of DNA, initiated previously, is extended. It was demonstrated that the ionic-strength dependence of the persistence length of DNA in aqueous NaCl can be understood by postulating (1) a rigid-rod 60 base-pair unit and (2) a negative contribution from nonelectrostatic sources to the bending free energy of this unit. On electrostatic neutralization of the phosphate charge, the latter term alone survives, and bending becomes a spontaneous process. In this paper it is demonstrated from a theoretical analysis of the ionic-strength dependence of the rate of denaturation of DNA that the above statements are restricted in their validity to radii of curvature greater than 170 Å; beyond this limit, the steeply rising repulsive energy of tightly packed atoms dominates the bending free energy. Therefore, spontaneous bending on charge neutralization proceeds up to, but not beyond, this barrier to curvature. It is noted that doughnut forms of DNA induced by binding of cationic spermidine, or mixtures of Mg²⁺ and polyamines are observed to possess maximum curvatures, corresponding to the radius of the “hole in the middle,” in the range 1/150–1/200 Å.