The effect of solvent viscosity and temperature on DNA viscoelastic behavior

The effect of solvent viscosity (η_s) and temperature (T) on the shape of the concentration dependence of the principal and total recoils in creep-recovery viscoelastometry experiments has been studied for T4 DNA solutions. The range of DNA concentration (c) was 2 – 40 μg/ml; glycerol, 70–80% v/v, sucrose, 60% v/v; NaCl, 5 mM – 1M; and T, 275 – 323 K. A linear proportionality between recoil and c was obtained at high η_s/T. At low η_s/T, the c-dependence was nonlinear, approaching saturation at higher c. At low c, the slope of both curves was the same. Transition between “linear” and “nonlinear” values occurred over a narrow range of η_s/T (a width of 1–5 K if η_s/T was changed by varying T). (η_s/T)_tr, the midpoint of the transition, was independent of solvent properties other than viscosity. Also, (η_s/T)_tr increased with c. For a given c, η_s/T values above this transitional value yield linear behavior; below this, nonlinear behavior. The ratio of linear to nonlinear recoil values is a linear function of c with K_c, the slope of this dependence, independent of η_s and T. A kinetic model for the observed nonlinearity of recoil with c is presented. It explains the independence of K_c on η_s and T. An attempt has been made to explain the linear–nonlinear transitions by comparison of τ₁ and T_R, the lifetime of the contact points of the polymer network in the de Gennes theory. The nonlinear values are consistent with a pseudogel that exists when τ₁ < T_R. At τ₁ > T_R, the DNA behavior is similar to that in dilute solutions (linear values). Thus, the condition for transition is τ₁ = T_R. However, some unsolved problems remain.     

Light scattering studies of supercoiled and nicked DNA

Static and dynamic light scattering measurements were made of solutions of pGem1a plasmids (3730 base pairs) in the relaxed circular (nicked) and supercoiled forms. The static structure factor and the spectrum of decay modes in the autocorrelation function were examined in order to determine the salient differences between the behaviors of nicked DNA and supercoiled DNA. The concentrations studied are within the dilute regime, which is to say that the structure and dynamics of an isolated DNA molecule were probed. Static light scattering measurements yielded estimates for the molecular weight M, second virial coefficient A₂, and radius of gyration R_G. For the nicked DNA, M = (2.8 ± 0.4) × 10⁶g/mol, A₂ = (0.9 ± 0.2) × 10⁻³ mol cm³/g², and R_G = 90 ± 3 nm were obtained. For the supercoiled DNA, M = (2.5 ± 0.4) × 10⁶ g/mol, A₂ = (1.2 ± 0.2) × 10⁻³ mol cm³/g², and R_G = 82 ± 2.5 nm were obtained. The static structure factors for the nicked and supercoiled DNA were found to superpose when they were scaled by the radius of gyration. The intrinsic stiffness of DNA was evident in the static light scattering data. Homodyne intensity autocorrelation functions were collected for both DNAs at several angles, or scattering vectors. At the smallest scattering vectors the probe size was comparable to the longest intramolecular distance, while at the largest scattering vectors the probe size was smaller than the persistence length of the DNA. Values of the self-diffusion coefficients D were obtained from the low-angle data. For the nicked DNA, D = (2.9 ± 0.3) × 10⁻⁸ cm²/s, and for the supercoiled DNA, D = (4.11 ± 0.21) × 10⁻⁸ cm²/s. The contribution to the correlation function from the internal dynamics of the DNA was seen to result in a strictly bimodal decay function. The rates of the faster mode Γ_int, reached plateau values at low angles. For the nicked DNA, Γ_int = 2500 ± 500 s⁻¹, and for the supercoiled DNA, Γ_int = 5000 ± 500 s⁻¹. These rates correspond to the slowest internal relaxation modes of the DNAs. The dependence of the relaxation rates on scattering vector was monitored with the aid of cumulants analysis and compared with theoretical predictions for the semiflexible ring molecule. The internal mode rates and the dependence of the cumulants moments reflected the difference between the nicked DNA and the supercoiled DNA dynamical behavior. The supercoiled DNA behavior seen here indicates that conformational dynamics might play a larger role in DNA behavior than is suggested by the notion of a branched interwound structure. © 1996 John Wiley & Sons, Inc.     

Histone-induced conformational changes in DNA as probed by quasi-elastic light scattering.

Quasi-elastic light scattering as measured by intensity fluctuation (self-beat) spectroscopy in the time domain can be profitably used to follow both the translational diffusion D and the dominant internal flexing mode τ_int of DNA and its complexes with various histones in aqueous salt solutions. Without histones, DNA is found to have D = 1.6 × 10^?8 cm²/sec and τ_int ? 5 × 10^?4 sec in 0.8 M NaCl, 2 M urea at 20°C. Total histone as well as fraction F2A induce supercoiling (D = 2.6 × 10^?8 cm²/sec, τ_int ? 2.8 × 10^?4 sec) whereas fraction F1 induces uncoiling (D = 1.0 × 10^?8 cm²/sec, τ_int ? 9.4 × 10^?4 sec). Upon increasing the salt concentration to 1.5 M the DNA–histone complex dissociates (D = 1.8 × 10^?8 cm²/sec). Upon decreasing the salt concentration to far below 0.8 M, the DNA–histone complex eventually precipitates as a chromatin gel.     

A critical evaluation of models for complex molecular dynamics: application of NMR studies of double- and single-stranded DNA

The nature of internal and overall motions in native (double-stranded) and denatured (single-stranded) DNA fragments 120–160 base pairs (bp) long is examined by molecular-dynamics modeling using ¹³C-nmr spin-relaxation data obtained over the frequency range of 37–125 MHz. The broad range of ¹³C frequencies is required to differentiate among various models. Relatively narrow linewidths, large nuclear Overhauser enhancements (NOEs), and short T₁ values all vary significantly with frequency and indicate the presence of rapid, restricted internal motions on the nanosecond time scale. For double-stranded DNA monomer fragments (147 bp, 24 Å diam at 32°C), the overall motion is that of an axially symmetric cylinder (τ_x = ～10^?6 s;τ_Z = ～1.8 × 10^?8s), which is in good agreement with values calculated from hydrodynamic theory (τ_x = ～1.8 × 10^?6 s; τ_Z = ～2.7 × 10^?8 s). The DNA internal motion can be modeled as restricted amplitude internal diffusion of individual C? H vectors of deoxyribose methine carbons C1′, C3′, and C4′, either with conic boundary conditions (τ_w = ～4 × 10^?9 s, θ_cone = ～21°) or as a bistable jump (τ_A = τ_B = ～2 × 10^?9 s, θ = ～15°). We discuss the critical role in molecular-dynamics modeling played by the angle (β) that individual C? H vectors make with the long axis of the DNA helix. Heat denaturation brings about increases in both the rate and amplitude of the internal motion (described by the wobble model with τ_W = ～0.2 × 10^?9 s, θ_cone = ～50°), and overall motion is affected by becoming essentially isotropic (τ_x = τ_Z = ～5 × 10^?8 s) for the single-stranded molecules. Since ¹³C-nmr data obtained at various DNA concentrations for C2′ of the deoxyribose ring is not described well by the above models, a new model incorporating an additional internal motion is proposed to take into account the rapid, extensive, and weakly coupled motion of C2′.     

Prediction of membrane fouling in MBR systems using empirically estimated specific cake resistance

The focus of this study was to empirically estimate the specific cake resistance (SCR) by the variation in shear intensity (G) in four laboratory-scale MBRs. The control reactor (MBR₀) was operated with aeration only while other MBRs (MBR₁₅₀, MBR₃₀₀ and MBR₄₅₀) were operated with aeration and mechanical mixing intensities of 150, 300 and 450 rpm, respectively. It was found that the SCR was strongly correlated (R² = 0.99) with the fouling rates in the MBRs. Moreover, the contribution of cake resistance (R_c) to the total hydraulic resistance (R_t) was predominant compared to the irreversible fouling resistance (R_f). On this basis, the cake filtration model was selected as a predictive tool for membrane fouling. This model was modified by replacing the SCR with its empirical shear intensity relationship. The modified model can predict the fouling rate for a given shear intensity (G) within 80 and 250 s⁻¹ in a MBR system.     

Photon correlation spectroscopy, total intensity light scattering with laser radiation, and hydrodynamic studies of a well fractionated DNA sample.

A Malvern laser light-scattering instrument has been modified for use at scattering angles down to 5° and both total intensity and quasi-elastic scattering experiments. A sample of sheared, length-fractionated calf-thymus DNA was characterized by sedimentation, viscosity and electron microscopy. Quasi-elastic scattering and absolute intensity determinations were performed with the laser instrument and intensity determinations only with a Fica conventional light-scattering photometer. The total intensity experiments gave M?_w = (3.75 ± 0.15) × 10⁶ and 〈R²〉^1/2_z = (206.9 ± 10.3) nm which yielded a value for the persistence length, allowing for polydispersity, of 66 ± 6nm. The quasi-elastic experiments at scattering angles below 20° gave D⁰_{20, w} = (2.23 ± 0.06) × 10^?8 cm²/sec which combined with S⁰_{20, w} = 15.6 in the Svedberg equation gave M?_w = (3.73 ± 0.18) × 10⁶. In addition, from the higher angle data we extracted a value of the longest intramolecular relaxation time, τ1 of 17.5 msec. This is not in particularly good agreement with τ1 predicted by the Zimm–Rouse theory using our other experimental parameters. The disagreement may be due to the restricted applicability of the Zimm–Rouse spring-bead model as a quantitative representation of DNA molecules. Alternatively, it may be due to present difficulties in the unambiguous interpretation of molecular motions from the experimental autocorrelation functions.     

NMR study on molecular mobility of DNA molecules in solution

The molecular mobility of calf thymus DNA molecules in solution has been discussed in terms of correlation time τ calculated from measurements of longitudinal T₁ and transverse T₂ magnetic relaxation times. The influence of DNA concentration and ionic strength of the solution upon freedom of movement of DNA molecules was studied for native and denatured DNA and also during thermal helix-coil transition. The dependence of τ values on temperature was carried out by comparing the values of correlation times τ_tat given temperature with the correlation time τ₂₀ at 20°C. The molecular rotation of DNA at 20°C and at higher ionic strength at 0.15 and 1.0.M NaCl is described by τ values of the order of 1.0–1.2 × 10^?8 and was reduced slightly with increase of temperature below the helix-coil transition. The molecular rotation of DNA in 0.02MNaCl was lower at 20°C as compared to DNA in solvents with higher NaCl concentrations and increases rapidly with increase of temperature in the range 20–60°C. The values of correlation time are characterized by fast increase at temperatures above the spectrophotometrically determined beginning of melting curve. The beginning of this increase is observed at about 65, 80, and 85°C for DNA in 0.02, 0.15, and 1.0MNaCl, respectively. Values of correlation time for denatured DNA are in all cases about 1.1–1.4 times that for native DNA. The obtained results are discussed in terms of conformation of DNA molecules in solution as well as in terms of water dipole binding in DNA hydration shells.     

How to compare fluctuating asymmetry of different traits

Comparing fluctuating asymmetry (FA) between different traits can be difficult because traits vary at different scales. FA is generally quantified either as the variance of the difference between left and right (σ²_L?R) or the mean of the absolute value of this difference (μ_|R?L|). Corrections for scale differences are obtained by dividing by trait size mean. We show that a third index, one minus the correlation coefficient between left and right (1 ? r_L,R), is equivalent to σ²_L?R standardized by trait size variance. The indices are compared with Monte‐Carlo simulations. All achieve the expected correction for scale differences. High type I error rates (false indication of differences) occur only for σ²_L?R and μ_|R?L| if trait sizes close to or below 0 occur. 1 ? r_L,R with a bootstrap test has always low error rates. Recommendation of the index to be used should be based on whether standardization of FA by trait size mean or trait size variance is preferred. A survey of 36 traits in the Speckled Wood Butterfly (Pararge aegeria) indicated that σ²_L?R is slightly higher correlated to trait size variance than to trait size mean. Thus 1 ? r_L,R seems to be the superior index and should be reported when FA of different traits is compared.     

Light scattering and viscosity study of heat aggregation of insulin

Aggregation behavior and hydrodynamic parameters of insulin have been determined from static and dynamic light scattering experiments and intrinsic viscosity measurements carried out at pH 4.0, 7.5, and 9.0 in the temperature range 20–40°C in aqueous solutions. The protein aggregated extensively at elevated temperatures in the acidic solutions. Intermolecular interactions were found to be attractive and to increase with temperature. The measured intrinsic viscosity [η], diffusion coefficient D₀, molecular weight M, and radius of gyration R_g exhibited the universal behavior: M[η] = (2.4 ± 02) × 10⁻²⁷ (R_e,η/R_e,D)³(D_0η/T)⁻³ and (D₀√n)₋₁ ≃ (√6 πη₀ζβ/k_BT) [1 + 0.201)(v/β³)√n], where n is the number of segments in the polypeptide. The effective hydrodynamic radii deduced from [η], (R_e, η) and the same deduced from D₀, (R_e,D) showed a constant ratio, (R_e,η/R_e,D = 1.1 ± 0.1). R_e,D/R_g = ξ was found to be (0.76 ± 0.07). From the known solvent viscosity η₀, the segment length β was deduced to be (10 ± 1) Å. The excluded volume was deduced to be (5 Å)³ regardless of pH. The Flory-Huggins interaction parameter was found to be χ = 0.45 ± 0.04, independent of pH and temperature. © 1998 John Wiley & Sons, Inc. Biopoly 45: 1–8, 1998     

Dielectric relaxation of DNA in aqueous solutions.

Using a four-electrode cell and a new electronic system for direct detection of the frequency differences specturm of solution impedance, the complex dielectric constant of calf thymus DNA (M_r = 4 × 10⁶) in aqueous NaCl at 10°C is measured at frequencies ranging from 0.2 Hz to 30 kHz. The DNA concentrations are C_p = 0.01% and 0.05%, and the NaCl concentrations are varied from C_s = 10^?4 M to 10^?3 M. A single relaxation regions is found in this frequency range, the relaxation frequency being 10 Hz at C_p = 0.01% and C_s = 10^?3 M. At C_p = 0.05% it is evidenced that the DNA chains have appreciable intermolecular interactions. The dielectric relaxaton time τ_d at C_p = 0.01% agrees well with the rotational relaxation time estimated from the reduced visocisty on the assumption that the DNA is not representable as a rigid rod but a coiled chain. It is concluded that the dielectric relaxiatioinis ascribed to the rotation of the molecule. Observed values of dielectric increment and other experimental findings are reasonably explained by assuming that the dipole moment of DNA results from the slow counterion fluctuation which has a longer relaxation time than τ_d.     

Hydrodynamic shear breakage of DNA

The rate of breakage of duplex DNA molecules by laminar flow through a capillary has been studied. For λb2b5c DNA (molecular wt., M = 25 × 10⁶) the point at which breakage occurs is normally distributed around the center of the molecule with a standard deviation of 12.5% of the molecular length. At constant shear stress or shear rate, the breakage rate is independent of ionic strength. Thus, shear induced local denaturation is not a rate limiting, preliminary step in breakage. In experiments at constant temperature with varying solvent viscosity (controlled by added sucrose) the breakage rate is a function of shear rate, not of shear stress. The rate of opening of hydrogenbonded circles into linear molecules by hydrodynamic shear is also shown to be a function of shear rate and not of shear stress. The breakage rate at constant shear rate is not greatly dependent on temperature. The shear rate required to achieve breakage is inversely proportional to M^1,2. The breakage rate constant, k varies as a very high power of the shear rate; at 25°C, d In k/d In Gm ～ 15; at 10°C, d In k/d In Gm ～ 26, where Gm is the maximum shear rate at the capillary wall. The unexpected result that breakage rate is mainly dependent on shear rate, not shear stress, supports a model in which the DNA molecule is distorted with a driving force which depends on the hydrodynamic shear stress, ηG, but the rate limiting step is segment diffusion into a highly extended configuration. The characteristic time to achieve this configuration is proportional to solvent viscosity, η, hence the breakage rate is dependent on ηG/η or G, the shear rate.     

Life history traits of Aphanius ginaonis Holly, 1929 (Cyprinodontidae) and potential risks of extinction in the Geno hot spring (Iran) population

This study presents some traits of the Aphanius ginaonis life history in the Geno hot spring and explains the potential risks of its extinction. Sampling was from March 2009 to February 2010. A total of 61 males and 71 females were measured (total length) and weighed, with data on reproductive biometry also taken. Growth parameters were determined in addition to weight–length relationships: W (t) = 0.012TL^3.42, R² = 0.96 for females and W (t) = 0.0101TL^3.38, R² = 0.94 for males. A. ginaonis females showed an asymptotic total length (TL_∞) of 53.03 mm; the growth coefficient, K (year^?1) 0.15, t₀ (year) 1.01; and natural mortality coefficient M (year^?1) 0.62. In males the value for TL_∞ was 48.83 mm; for K (year^?1) 0.2, t₀ (year) 0.44; and M (year^?1) 0.49. The relationship between absolute fecundity and fish size (total length, body weight or age) showed a strong correlation to body weight. The A. ginaonis population is threatened with extinction – and a co‐management in cooperation with the local population for strong protection measures is urgently needed.     

Quasielastic light scattering: Effect of ionic strength on the internal dynamics of DNA

Quasielastic light scattering is used to study the effect of ionic strength on the dynamic behaviour of DNA. In a first approach the spectrum of scattered light is analyzed in terms of a single relaxation process. The large difference between the observed behaviour and that expected according to a pure diffusional process reflects the contribution associated with internal modes, which increases with decreasing ionic strength. Such behaviour is better analyzed in terms of a double relaxation process by using two relaxation times, the reciprocals of which are equal to DK² and DK² + τ_i^?1 (K), respectively, where τ_i (K) is an average value describing the set of modes observed at a given K value. Relative intensity and relaxation times, which are the more accurate parameters, were used to interpret the results. The observed increase of the relative contribution of internal modes with decreasing ionic strength is actually a relative decrease of the diffusional contribution induced by a corresponding increase of the radius of gyration R_G. On the other hand, the reciprocal τ_i^?1 (K) of the relaxation time is a linear function of K² in the analyzed KR_G range and is insensitive to ionic strength between 10^?2M and 1M. These results, when discussed according to Rouse's model, lead to define for each value of τ_i^?1 (K) a corresponding mean-squared equilibrium length 〈μ〉 which is found to be a linear function of K^?2.     

Dielectric relaxation of DNA solutions. II

Real and imaganiry parts of complex dielectric constant of dilute solutions of DNA in 10^?3M NaCl with molecular weight ranging from 0.4 × 10⁶ to 4 × 10⁶ were measured at frequencies from 0.2 Hz to 30 kHz. Dielectric increments Δε were obtained from Cole-Cole plots and relaxation times τ_D from the loss maximum frequency. The τ_D of all samples agrees well with twice of the maximum viscoelastic relexation time in the Zimm theory, indicating that the low-frequency dielectric relaxiation should be ascribed to be the rotation of DNA. The rms dipole moment, which was obtained from Δε, agree well with that calculated from the counterion fluctuation theory. The dielectric increment was found to be greatly depressed in MgCl₂, which is resonably interpreted in terms of a strong binding of Mg⁺⁺ ions with DNA.     

Allometric equations for yield predictions of enset (Ensete ventricosum) and khat (Catha edulis) grown in home gardens of southern Ethiopia

Enset is a large, single‐stemmed perennial herbaceous plant domesticated as a staple food crop only in Ethiopia. Khat is a perennial plant cultivated for its economically important leaves and twigs that are the sources of stimulant when chewed. We address the issue of yield estimation of both crops, as they are important for the livelihoods of smallholders in the home garden systems in Southern Ethiopia and have received little attention so far. The objective of this study was to develop linear allometric models for estimating the edible (food and feed) and commercial yields of enset and khat plants, respectively. Data were collected from 20 enset and 100 khat plants. Diameter at 50‐cm height (d₅₀), pseudostem height (h_p) and their combination were good predictor variables for the food products of enset with adjusted R² values above 0.85, while d₅₀, h_p, edible pseudostem height (h_ep), total height (h_t) and their combination were good predictor variables for the feed products of enset with adjusted R² values above 0.70. For dwarf khat plants crown area (ca) combined with total height (h_t) resulted in the best prediction with an adjusted R² of 0.77, while the leaf and twig dry weight for tall khat plants was best predicted by ca with adjusted R² of 0.43. In all cases linear models were used.     

Conformation of mucous glycoproteins in aqueous solvents

Light-scattering techniques have been used to measure the z-average radius of gyration R_g z-average translational diffusion coefficient D_t and weight–average molecular weight M_w of porcine submaxillary mucin (PSM) in solution. PSM isolated at low shear in the presence of protease inhibitors has a M_w about twice as large as a sample prepared without these precautions. The former sample has a M_w of 17 × 10⁶ in 0.1M NaCl, which decreases to 8 × 10⁶ in 6M guanidine hydrochloride (GdnHCl) and then to 2 × 10⁶ on addition of 0.1M mercaptoethanol to the 6M GdnHCl solution. The R_g or D values obtained for PSM in this work superimpose with those of other authors for different mucin glycoproteins, leading to linear log–log relationships to the molecular weight of the protein core. Comparison of these results with those in the literature for denatured proteins suggest that mucins are linear random coils in which the protein core is stiffened by the presence of the oligosaccharide side chains. The length of the oligosaccharides and the nature of the solvent have little effect on the extension of the protein core. This suggests that the stiffness of the protein core is maintained by steric repulsion of the residues at the beginning of the oligosaccharide chains.     

Orientational interactions in low-concentration DNA solutions

The rotational relaxation tiem τ₃ of DNA molecules (M_w ? 5 × 10⁶) in solution has been determined by the transient electric birefringence method. The analysis of the birefringence decay makes it possible to study only the higher-molecular-weight fraction, the molecules being considered as rigid elongated particles in a short time scale. A marked concentration dependence of the relaxation time has been observed for DNA in low ionic strengths. Above a critical concentration c*, τ₃ increases with the DNA concentration, c. The value of c* increases with the ionic strength. For 10^?3 ionic strength (with NaCl), c* is about 10 μg/mL; then we observe the same strong concentration dependence of rotational relaxation times as recently reported for rodlike M-13 viruses [Maguire, J. F., McTague, J. P. & Rondelez, F. (1980) Phys. Rev. Lett. 45 , 1891–1894]. These results may be discussed in terms of the Doi-Edwards theory for rotational relaxation time of rigid macromolecules [Doi, M. (1975) J. Phys. 36 , 607–611; Doi, M. & Edwards, S. F. (1978) J. Chem. Soc. Faraday Trans. 74 , 918–932] and the critical concentration above which the interactions between the molecules begin to appear allows determining the corresponding molecular length. We observe a very good agreement between the DNA lengths obtained from the c* values and by using the infinite dilution value of τ₃ and Broersma's equation. Therefore, only highly diluted solutions can be used if intrinsic molecular properties based on the rotational diffusion of high-molecular-weight elongated molecules are studied.     

ALGORITHM FOR APPLICATION OF FOURIER ANALYSIS FOR BIORHYTHMIC BASELINES OF PHARMACODYNAMIC INDIRECT RESPONSE MODELS

The change of an indirect pharmacological response R(t) can be described by a periodic time-dependent production rate k_in (t) and a first-order loss constant k_out. If k_in(t) follows some biological rhythm (e.g., circadian), then the response R(t) also displays a periodic behavior. A new approach for describing the input function in indirect response models with biorhythmic baselines of physiologic substances is introduced. The present approach uses the baseline (placebo) response R_b(t) to recover the equation for k_in(t). Fourier analysis provides an approximate equation for R_b(t) that consists of terms (usually two or three) of the Fourier series (harmonics) that contribute most to the overall sum. The model differential equation is solved backward for k_in(t), yielding the equation involving R_b(t). A computer program was developed to perform the square L²-norm approximation technique. Fourier analysis was also performed based on nonlinear regression. Cortisol suppression after inhalation of fluticasone propionate (FP) was modeled based on the inhibition of the secretion rate k_in(t) using ADAPT II. The pharmacodynamic parameters k_out and IC₅₀ were estimated from the model equation with k_in(t) derived by the new approach. The proposed method of describing the input function needs no assumption about the behavior of k_in(t), is as efficient as methods used previously, and is more flexible in describing the baseline data than the nonlinear regression method. (Chronobiology International, 17(1), 77–93, 2000)     

On the possibility of true phase transition in heterogeneous DNA during thermal denaturation under conditions close to equal stability of A+T and G+C pairs

The DNA helix–coil transition has been studied in the presence of high concentrations of manganese ions (about 10^?3M), which corresponds to the conditions close to equal stability of the A+T and G+C pairs, at the ionic strengths of 10^?1, 10^?2, and 1.6 × 10^?3M Na⁺. With the Mn²⁺ ion effect, the transition range is significantly reduced to not more than 0.2°C at 1.2 × 10^?3M Mn²⁺ and 1.6 × 10^?3M Na⁺. The melting curves display a sharp kink at the end of the helix–coil transition, which is interpreted as an indication of the second-order phase transition. It is shown that the melting curves obtained can be approximated by a simple analytical expression 1 – θ = exp[–a(t_c - t)], where θ is the DNA helix fraction, t_c is the phase transition temperature, and a is an empirical parameter characterizing the breadth of the melting range and responsible for the magnitude of a jump of the helicity derivative with respect to the temperature at the phase transition point.