On the application of polyelectrolyte limiting laws to the helix-coil transition of DNA. IV. Depedence of helix stability on the concentration of divalent metal ions.

The molecular theory of the previous paper in this series is extended to determine the effect of divalent metal ions on helix stability relative to coil at fixed ionic strength and nucleotide phosphate concentration. Specification of the state of condensed counterions, as well as their concentration, is essential for the solution of this problem, and it is assumed that they translate freely within a thin cylindrical shell close to the polynucleotide. As a function of divalent counterion concentration m the relative stability of the helix is highly nonlinear. Although the overall trend is that the helix stability increases with addition of divalent metal ion, there is a narrow concentration range for which it decreases slightly. The behavior of the relative stability as a function of m is determined by the translational degrees of freedom of the counterions, both univalent and divalent, both condensed and uncondensed. Detailed comparison of the theory with data is not given here, but it is pointed out that the calculated values of the relative stability are consistent with the order of magnitude of the observed effect Mg²⁺ on the melting temperature.     

On the application of polyelectrolyte “limiting laws” to the helix-coil transition of DNA. I. Excess univalent cations

A general theory of polyelectrolyte solutions is here used to calculate the differences in Gibbs free energy, enthalpy, and entropy between the coil and helix forms of DNA at any temperature and salt concentration. The salt has univalent cations and is assumed present in excess over the base concentration. The results are restricted to sufficiently dilute solutions. It is shown that the salt concentrations effect is entirely entropic in origin. When applied to the melting temperature, the calculations yield a relation between the enthalpy difference at the melting temperature and the slope of the plot of melting temperature vs. the logarithm of the salt concentration. In accord with observation, both the Gibbs free energy difference at any fixed temperature and the melting temperature are predicted to be linear functions of the log of the salt concentration. However, the theory is not in quantitative agreement with enthalpy data. Data on various colligative and transport properties of both helix and coil forms are reviewed in the text and in Appendix B, and good agreement is found with theory for both forms. No attempt is made to explain why the theory is quantitative for these properties but not for heat measurements. Finally, in Appendix A, an approximate calculation is made of the free energy contributions due to ionic effects not associated with the salt concentration.     

On the application of polyelectrolyte limiting laws to the helix–coil transition of DNA. V. Ionic effects on renaturation kinetics

The bimolecular rate constant k₂ for the association of complementary polynucleotide strands has been observed to increase strongly with increasing ionic strength—in fact, proportional to its third or fourth power. This effect is here interpreted quantitatively by means of polyelectrolyte theory starting with the Wetmur–Davidson postulate of a pre-equilibrium between separated strands and aligned segments close to one another but unbonded. The correct form, a power dependence of k₂ on ionic strength, is predicted. Comparison of the theoretical exponent with data allows the conclusion that each of the two single-stranded segments in the aligned but unbonded configuration consists of about 13–16 nucletides (not to be confused with the much smaller number of bonded base pairs in the nucleus), and that this number, denoted by Q, is possibly correlated either with a minimum length for duplex stability or with the persistence length of a single polynucleotide strand. It is suggested that experimental determination of the dependence of Q on (G+C)-content may distinguish between these possibilities. It is also suggested that addition of sufficient amounts of divalent metal ions such as Mg²⁺, Ca²⁺, or Co²⁺ may reverse the dependence of k₂ on ionic strength; under these conditions, k₂ is predicted to decrease with about the first power of ionic strength. At fixed ionic strength, k₂ should increase with increasing concentration of divalent metal ion, and, in fact, the published observation that the formation of poly(A)·2 poly(U) from poly(A)·poly(U) and poly(U) is second order in Mg²⁺ concentration is here correctly predicted from a priori molecular considerations. Finally, published association rate data for oligonucleotides are discussed in the present theoretical context.     

On the application of polyelectrolyte limiting laws to the helix-coil transition of DNA. III. TDependence of helix stability on excess univalent salt and on polynucleotide phosphate concentration for variable equivalent ratios of divalent metal ion to phosphate.

Using the free energy difference between double-helix and random-coil forms of DNA as a measure of the stability of the double helix, we calculate the dependence of the stability on excess univalent cation concentration and on polynucleotide phosphate concentration, both as functions of the equivalent ratio r of divalent cation-to-phosphate concentrations. The theoretical tool is merely to compare the free energy of one polyelectrolyte solution, characterized by the polyelectrolyte linear charge density, with the free energy of another, characterized by a different value of the charge density. It is assumed only that the charge density of the double helix is greater than that of the coil form. The calculation represents the only molecular theory given to date (for r ≠ O) for these aspects of helix stability. We find that, as excess univalent cation concentration increases, the helix stability increases if r is small but decreases if r is large (i.e., of the order of unity). Moreover, as the concentration of nucleotide phosphate increases, the helix stability does not change for small values of r but increases for large values. For both effects, a continuous transition as a function of r bridges the low-r and high-r behaviour.     

The application of polyelectrolyte limiting laws to the helix–coil transition of DNA. VI. The numerical value of the axial phosphate spacing for the coil form

It is concluded on the basis of comparison of polyelectrolyte theory with published data that the mean phosphate spacing b along the contour axis of an unfolded polynucleotide single strand is in the range 3–4 Å (polyelectrolyte parameter ξ ≈ 2), regardless of temperature, base composition, or extent of stacking. This result is consistent with the low-angle X-ray scattering measurements of Gulik, Inoue, and Luzzati on poly(C). No conclusion may be drawn from this value of b concerning the structure of the chain skeleton or the spatial arrangement of the bases other than that the chain is far from an all-trans local conformation (for which b would be about 6–7 Å, the length of a nucleotide unit). The structural implications, or lack thereof, are discussed in detail in the following paper.     

Magnesium ion effect on the helix-coil transition of DNA

The effect of magnesium ions on the parameters of the DNA helix-coil transition has been studied for the concentration range 10^?6–10^?1M at the ionic strengths of 10^?3M Na⁺. Special attention has been given to the region of low ion concentrations and to the effect of polyvalent metallic impurities present in DNA. It has been shown that binding with Mg⁺⁺ increases the DNA stability, the effect being observed mainly in the concentration range 10^?6–10^?4M. At[Mg⁺⁺]>10^?2M the thermal stability of DNA starts to decrease. The melting range extends to concentrations ～10^?5M and then decreases to 7–8°C at the ion content of 10^?3M. Asymmetry of the melting curves is observed at low ionic strengths ([Na⁺] = 10^?3M) and [Mg⁺⁺] ? 10^?5M. The results, analyzed in terms of the statistical thermodynamic theory of double-stranded homopolymers melting in the presence of ligands, suggest that the effects observed might be due to the ion redistribution from denatured to native DNA. An experimental DNA–Mg⁺⁺ phase diagram has been obtained which is in good agreement with the theory. It has been shown that thermal denaturation of the system may be an efficient method for determining the ion-binding constants for both native and denatured DNA.     

The effect of long-range interactions between adsorbed ligands on the DNA helix-coil transition

The influence of different types of long-range interaction of ligands adsorbed on DNA on the helix-coil transition was theoretically considered. The contact interaction was shown to differ significantly from the long-rang one. It was shown also that even weak dependence of a long-range potential on a degree of helicity resulted in the strong changes of a DNA melting curve. This result allowed to understand the different experimental data on DNA melting in the presence of different substances which reduced AT-and GC-base pairs thermostability difference.     

Kinetics of the helix-coil transition in DNA

The kinetics of the helix-coil transition have been investigated for T2 and T7 phage DNA in a formamide-water-salt mixed solvent using a slow temperature perturbation technique (applicable to kinetic processes with rate constants ? 3 min^?1). In this solvent degradation of the DNA is effectively suppressed. Complex kinetic curves are observed by absorbance and viscosity measurements for the response to denaturing perturbations in the transition region. Analysis of the decay curves indicates that the denaturation reaction in this time range can be treated as a first-order reaction with a variable first-order rate parameter, k, the derivative of the logarithm of the absorbance or viscosity change with respect to time. In the approach to denaturation equilibrium in the transition region, the rate parameter is determined only by the instantaneous extent of denaturation of the molecules. Near equilibrium, the rate parameter assumes a constant value characteristic of the equilibrium state. In this region, where the denaturation reaction proceeds as a simple first-order process, both the decay of absorbance (reflected local conformational change) and the decay of solution viscosity (reflecting macromolecular conformational change) are characterized by the same constant value of k. In 83% formamide, 0.3M Na⁺, the rate parameter k for T2 DNA decreases from an extrapolated value of 2.0 min^?1 at 0% denaturation to 0.11 min^?1 at 90% denaturation. Rate parameters determined for T7 DNA at the same counterion concentration and fraction of denaturation are approximately five times as large as those cited for T2 DNA, indicating an inverse proportionality of rate constant to molecular length. On the other hand, simple first-order kinetic responses with constant k are obtained for renaturing perturbations within the transition, indicating that the mechanism of rewinding differs, in most cases, from that of unwinding. Only in the limit of very small perturbations about a given equilibrium position are the rate constants k obtained from denaturing and renaturing perturbations equal. For perturbations of finite size, it appears possible that an intramolecular initiation or nucleation event may precede rewinding and limit the rate of this reaction. The rate parameters again are approximately inversely proportional to molecular weight. The one exception to the first-power dependence on molecular weight appears when temperature jumps are made upward into the post-transition region. Here the molecular-weight dependence is second power, but complications arising from the different strand-separation properties of T2 and T7 DNA's make interpretation difficult. The previously used model of friction-limited unwinding appears to fit all the observations except for the molecular-weight dependence.     

A study of the reversibility of helix-coil transition in DNA.

The reversibility of DNA melting has been thoroughly investigated at different ionic strengths. We concentrated on those stages of the process that do not involve a complete separation of the strands of the double helix. The differential melting curves of pBR 322 DNA and a fragment of T7 phage DNA in a buffer containing 0.02M Na+ have been shown to differ substantially from the differential curves of renaturation. Electron-microscopic mapping of pBR 322 DNA at different degrees of unwinding (by a previously elaborated technique) has shown that the irreversibility of melting under real experimental conditions is connected with the stage of forming new helical regions during renaturation. In a buffer containing 0.2M Na+ the melting curves of the DNAs used (pBR322, a fragment of T7 phage DNA, a fragment of phage Lambda DNA, a fragment of phiX174 phage DNA) coincide with the renaturation curves, i.e. the process is equilibrium. We have carried out a semi-quantitative analysis of the emergence of irreversibility in the melting of a double helix. The problem of comparing theoretical and experimental melting curves is discussed.     

DNA sequencing and helix-coil transition. I. Theory of DNA melting

An explicit analytic formula accurately describing the melting of a natural DNA is derived. For phage ?X-174 and virus SV-40, the nucleotide sequences of which are known, the formula fits experimental data for the differential melting curve almost within the experimental accuracy.     

Application of polyelectrolyte limiting laws to virial and asymptotic expansions for the Donnan equilibrium

We have applied a general polyelectrolyte theory to an analysis of the Donnan equilibrium. The polyelectrolyte concentration is measured by a dimensionless parameter x, equal to the ratio of the equivalent polyelectrolyte concentration to the concentration of salt in the external compartment. For small x, virial series - expansions in powers of x - are developed for the Donnan salt-exclusion, osmotic pressure, and electromotive force. For large x, asymptotic expansions for these effects are presented. Polyion-polyion interactions are explicitly neglected, so that the physical significance of the virial series differs from its meaning in neutral polymer chemistry. Numerical results illustrate large deviations from ideal Donnan behavior as well as satisfactory agreement with published data on the salt-exclusion and emf effects. However, results for the Donnan osmotic pressure disagree with the data, except in the case of zero salt concentration in the external compartment, for which agreement is almost exact.     

Changes in the polyelectrolyte-amphiphile interaction due to helix-coil transition induced by specific counterions or variations in temperature

For the system κ-carrageenan/amitriptyline it is shown that the degree of binding of amitriptyline is closely related to the carrageenan conformation as regulated by the counterions (Na⁺ or K⁺). The adsorption becomes much more pronounced when the carrageenan molecule is in the helix form (counterion K⁺) than when it has a coil conformation (counterion Na⁺). Furthermore, for the helical state the adsorption becomes strongly cooperative. It is also shown experimentally that the release from the adsorbed state has a conversion temperature at about 42°C (helix-coil transition). The effect is also related to the linear charge density. For κ-carrageenan with a higher charge density the adsorption is strong and cooperative both in the presence of Na⁺ and K⁺ ions. © 1996 John Wiley & Sons, Inc.     

Application of modified ising model to the helix-coil transition of DNA molecules

Equilibrium properties of heterogeneous DNA near the melting temperature T_m are investigated using the grand partition function. The present approach gives exact and analytical solutions. The algebraic expressions enhance a more thorough understanding of the correlation among many observed equilibrium phenomena. The following quantities have been examined: melting temperature T_m, transition width W, partial melting curves θ_AT and θ_GC, mean length of a helical segment h, and correlation length γ.     

On the controversy over "fast" and "slow" helix-coil transition rates in polypeptides

The controversy over “fast” and “slow” helix-coil transition rates in polypeptides is discussed. The “slow” results are derived from the assumption that multiple NMR spectra of α-CH and NH groups arise from chemical exchange. In this paper it is shown that such spectra may be obtained without invoking chemical exchange. The multiplicity arises from the difference in helicity of amino acid residues near the ends of the chain by comparison with amino acid residues nearer the middle, and from a polydispersity in molecular weight. As a consequence of this analysis, support is given to the “fast” transition rates.     

Thermodynamic parameters of helix-coil transition in polypeptide chains. II. Poly-L-lysine

The helix–coil transitions for poly-L -lysine (PL) were investigated by the methods of spectropolarimetry, viscometry and potentiometric titration in 0.2M NaCl at different temperatures as well as in 0.2MNaBr, 1MKCl, and in mixtures of 0.2MNaCl or NaBr with methanol at room temperature. The enthalpy and entropy differences between the helical and coillike states of uncharged PL molecules in 0.2.M NaCl were determined from the potentiometric titration curves. The cooperativity parameters σ for PL in different solvents were determined by two methods (from the sharpness of the transition and from the dependence of the intrinsic viscosity on the helical content in the transition region). In 0.2MNaCl σ has a value of (2.3 ± 0.5) × 10^?4 and does not depend on temperature, i.e., the cooperativity of the helix-coil transition, as for PGA, is mainly of an entropy origin (the initiating of the helical region is accompanied by the entropy decrease ΔS_i = ?12 eu/mole of helical regions). A comparison of the obtained results for PGA and PL with the molecular theories of the helix-coil transitions shows that the role of dipole-dipole interactions of nonneighboring peptide groups is greatly overestimated in these theories, leading to a considerable enthalpy contribution to the free energy of initiating helical regions which is not observed in the experiment.     

The effect of Mg ++ on the conformation of the Ca ++ -binding component of troponin

Mg⁺⁺ like Ca⁺⁺ induces a conformational change in the Ca⁺⁺-binding component of troponin. However, this change is only 36 % of the change in fluorescence intensity and 80 % of the change in optical rotation induced by Ca⁺⁺. The apparent binding constant of Mg⁺⁺ to the Ca⁺⁺-binding component is 5 × 10³ M⁻¹, much smaller than that of Ca⁺⁺. Circular dichroism measurements show that these changes are simple helix-coil transitions. Unlike the Ca⁺⁺-induced conformational change, the Mg⁺⁺-induced change cannot be propagated to other muscle proteins, and therefore has no physiological meaning.     

Cation radius effects on the helix-coil transition of DNA. Cryptates and other large cations.

Most polyelectrolyte theories of the effect of ions on the thermal melting of DNA assume that the predominant influence of the cations comes through their charge. Ion size and structure are treated, for analytic convenience, as negligible variables. We have examined the validity of this assumption by measuring the melting temperature of calf thymus DNA as a function of salt concentration with four univalent cations of different hydrated radii. These are K+ (3.3 A), (n-Pr)4N+ (4.5 A), (EtOH)4N+ (4.5 A), and C222-K+ (5 A). C222-K+ is a complex of cryptand C222 with K+. With K+ as the sole cation, Tm varies linearly with the log of ionic strength over the range 0.001-0.1 M. With all the K+ sequestered by an equimolar amount of C222, Tm is depressed by 10-20 degrees C and the slope of Tm vs. ionic strength is lower. At low ionic strength, an even greater reduction in Tm is achieved with (n-Pr)4N+; but the similar-sized (EtOH)4N+ gives a curve more similar to K+. Theoretical modeling, taking into account cation size through the Poisson-Boltzmann equation for cylindrical polyelectrolytes, predicts that larger cations should be less effective in stabilizing the double helix; but the calculated effect is less than observed experimentally. These results show that valence, cation size, and specific solvation effects are all important in determining the stability of the double-helical form of DNA.     

Effects of Na+ and Mg++ ions on the helix–coil transition of DNA

The effects of monovalent (Na⁺) and divalent (Mg⁺⁺) cations on the temperature and breadth of the helix–coil transition of phage DNA have been investigated. The experimental results confirm the findings of Dove and Davidson [J. Mol. Biol. 5 , 467–478 (1962)] for the limiting cases of zero divalent ion concentration and saturating levels of divalent ion, and extend their findings to the intermediate region of Mg⁺⁺ concentrations. A theory for the dependence of transition temperature on the ion concentrations is developed, utilizing the approach of Wyman [Adv. Protein Chem. 19 , 223–286 (1964)], modified to account for electrostatic nonideality of the polyelectrolytes. The theory is in agreement with Manning's treatment of the experiments of Dove and Davidson [Biopolymers 11 , 937–949, 951–955 (1972)] and is in fair agreement with experimental data over the entire range of ion concentrations. Further investigation of the structure and ion-binding properties of the denatured form will be required before a quantitative comparison between theory and experiment can be performed.