Calculating thermodynamic data for transitions of any molecularity from equilibrium melting curves

In this paper, we derive the general forms of the equations required to extract thermodynamic data from equilibrium transition curves on oligomeric and polymeric nucleic acids of any molecularity. Significantly, since the equations and protocols are general, they also can be used to characterize thermodynamically equilibrium processes in systems other than nucleic acids. We briefly review how the reduced forms of the general equations have been used by many investigators to evaluate mono- and bimolecular transitions, and then explain how these equations can be generalized to calculate thermodynamic parameters from common experimental observables for transitions of higher molecularities. We emphasize the strengths and weaknesses of each method of data analysis so that investigators can select the approach most appropriate for their experimental circumstances. We also describe how to analyze calorimetric heat capacity curves and noncalorimetric differentiated melting curves so as to extract both model-independent and model-dependent thermodynamic data for transitions of any molecularity. The general equations and methods of analysis described in this paper should be of particular interest to laboratories that currently are investigating association and dissociation processes in nucleic acids that exhibit molecularities greater than two.     

Analysis of thermal melting curves

T(m) is defined as Temperature of melting or, more accurately, as temperature of midtransition. This term is often used for nucleic acids (DNA and RNA, oligonucleotides and polynucleotides). A thermal denaturation experiment determines the stability of the secondary structure of a DNA or RNA and aids in the choice of the sequences for antisense oligomers or PCR primers. Beyond a simple numerical value (the T(m)), a thermal denaturation experiment, in which the folded fraction of a structure is plotted vs. temperature, yields important thermodynamic information. We present the classic problems encountered during these experiments and try to demonstrate that a number of useful pieces of information can be extracted from these experimental curves.     

Helix-coil transitions in nucleoprotein: Computer-generated curves for independent melting of double-stranded regions with fixed ends

Helix-coil transitions in nucleoprotein at low salt concentrations are known to be characterized by two phases of the process: independent melting of uncomplexed “naked” regions without rearrangement of proteins, followed, at higher temperatures, by melting of complexed DNA. Blocking at the ends of these regions increases their thermal stability and three is a shift of 10–20°C in t_m of the melting profiles. In this study the basic assumption is that the loop entropy effect is mainly responsible for such stabilization. Calculations are made using conventional h-c transition theory for a system of independently melted segments with fixed ends. Segments are either homosize or have randomly distributed lengths. Calculated melting curves are used to obtain t_m, and transition width-dependence on segment length (or average length when randomly distributed) and on the nucleation parameter σ. Base-pair heterogeneity is taken into account by averaging over different base-pair distributions in the individual segments, using Gaussian distribution around the overall (G+C)-content. It is shown that this causes only an additional widening of the transition but no additional t_m shift. Comparison is made with similar systems in the literature. The main conclusion drawn is that the treatment proposed may be useful for analysis of the lower temperature melting phase in nucleoprotein at low counterion concentrations. It may be used as an independent method to reveal the features of nucleoprotein structure.     

Theory of DNA melting curves

Exact algorithms for the calculation of melting curves of heterogeneous DNA with N base pairs apparently require computer time proportional to N². However, it is shown that a decomposition of the loop entropy factor into a sum of I exponential functions (1) gives an extremely accurate approximation to the loop entropy factor for small values of I and (2) makes the computer time for the exact algorithms proportional to I·N. In effect, exact results for melting curves and lengths of helix or coil stretches are obtained with computer time comparable to that required for the Frank-Kamenetskii approximation. The remarkable accuracy of the latter for the fraction of helical content (errors of 0.01–0.05) is confirmed, but appreciably larger errors are found for the lengths of helix or coil stretches (typical errors of 30–100%).     

Estimation of the diversity between DNA calorimetric profiles,differential melting curves and corresponding melting temperatures

The Poland–Fixman–Freire formalism was adapted for modeling of calorimetric DNA melting profiles, and applied to plasmid pBR 322 and long random sequences. We studied the influence of the difference (H_GC?H_AT) between the helix‐coil transition enthalpies of AT and GC base pairs on the calorimetric melting profile and on normalized calorimetric melting profile. A strong alteration of DNA calorimetrical profile with H_GC?H_AT was demonstrated. In contrast, there is a relatively slight change in the normalized profiles and in corresponding ordinary (optical) normalized differential melting curves (DMCs). For fixed H_GC?H_AT, the average relative deviation (S) between DMC and normalized calorimetric profile, and the difference between their melting temperatures (T_cal?T_m) are weakly dependent on peculiarities of the multipeak fine structure of DMCs. At the same time, both the deviation S and difference (T_cal?T_m) enlarge with the temperature melting range of the helix‐coil transition. It is shown that the local deviation between DMC and normalized calorimetric profile increases in regions of narrow peaks distant from the melting temperature.     

Determination of melting temperature and temperature melting range for DNA with multi-peak differential melting curves

Many factors that change the temperature position and interval of the DNA helix–coil transition often also alter the shape of multi-peak differential melting curves (DMCs). For DNAs with a multi-peak DMC, there is no agreement on the most useful definition for the melting temperature, T_m, and temperature melting width, ΔT, of the entire DNA transition. Changes in T_m and ΔT can reflect unstable variation of the shape of the DMC as well as alterations in DNA thermal stability and heterogeneity. Here, experiments and computer modeling for DNA multi-peak DMCs varying under different factors allowed testing of several methods of defining T_m and ΔT. Indeed, some of the methods give unreasonable “jagged” T_m and ΔT dependences on varying relative concentration of DNA chemical modifications (r_b), [Na⁺], and GC content. At the same time, T_m determined as the helix–coil transition average temperature, and ΔT, which is proportional to the average absolute temperature deviation from this temperature, are suitable to characterize multi-peak DMCs. They give smoothly varying theoretical and experimental dependences of T_m and ΔT on r_b, [Na⁺], and GC content. For multi-peak DMCs, T_m value determined in this way is the closest to the thermodynamic melting temperature (the helix–coil transition enthalpy/entropy ratio).     

Calculation of melting curves for DNA

A method is reported for calculating the melting curve of a DNA molecule of random base sequence, including in the formalism the dependence of the free energy of base pair formation on the size of a denatured section. Some explicit results are shown for a “typical” base sequence, in particular the probability of helix formation at individual base pairs in several different regions of the molecule and the amount of melting from the end of the chain. Particular attention is drawn to the variation of local melting behavior from one region of the molecule to another. It is found that sections rich in AT melt at relatively low temperatures with a fairly broad transition curve, whereas regions rich in GC pairs melt at higher temperatures (as expected) with a very abrupt, local transition curve. To account qualitatively for the results one may divide melting into two kinds of processes: (a) the nucleation and growth of denatured regions, and (b) the merging together of two denatured sections at the expense of the intervening helix. The first of these processes dominates in the first stages of melting, and leads to rather broad local melting curves, whereas the second process predominates in the later stages, and occurs, in a particular part of the molecule, over a very narrow temperature range. It is estimated that the average length of a helix plus adjacent coil section at the midpoint of the transition is approximately 600 base pairs. Since transition curves which measure the local melting behavior reflect local compositions fluctuations, these curves contain information about the broad outlines of base sequence in the molecule. Some suggestions are made concerning experiments by which this potential information source could be exploited. In particular, it is pointed out that one might hope to map AT or GC rich regions at particular genetic loci in a biologically active DNA molecule. Values of the relevant parameters found earlier for the transition of homopolymers produce melting curves for a DNA of random base sequence which are in good agreement with the experimental transition curve for T2 phage DNA. Hence the present theoretical picture of the melting of polynucleotides is at least internally self-consistent.     

Fine structure of DNA melting curves.

Theoretical calculations predict that the differential melting curves for random polynucleotide sequences having lengths up to several tens of thousands of base pairs have a clear-cut fine structure. This structure appears in the form of multiple narrow peaks 0.3–0.4°C wide on the bell shaped main curve. The differential melting curves have different shapes for different specific sequences. The theory also predicts the disappearance of the fine structure when the length of the sequence increases and when circular, covalently closed DNA is considered instead of the open structure. The predictions of the theory were confirmed by the measurements of differential melting curves for open and covalently closed circular forms of DNA for PM2 phage (N = 10⁴ base pairs) and also for other phage DNA's of different length: T7 (N = 3.8 × 10⁴); S_D (N = 9.2 × 10⁴); T2 (N = 17 × 10⁴). It was shown that the effect of fine structure results mainly from the cooperative melting out of DNA regions 300–500 base pairs long.     

Piecewise exponential survival curves with smooth transitions

Several models of a population survival curve composed of two piecewise exponential distributions are developed. In one formulation the hazard rate changes at a point that is an unobservable random variable that varies between individuals. The population hazard function may decrease with age even when all individuals' hazards are increasing. In a second formulation, the population hazard function is modeled directly. Several models are fit to the survival history of a cohort of 5751 highly inbred male Drosophila melanogaster and the British coal mining disaster data.     

Phase transitions in nuclei and chromatin

Changes in the volume of rat liver nuclei have been monitored as a function of modifications in ionic environment (from 0 to 20 mM), temperature (from 4 to 37°C) and pH (from 1 to 8). An abrupt reduction of nuclear volume occurred with increasing ion concentration, this contraction being more pronounced with bivalent (either Ca²⁺ or Mg²⁺) than with monovalent (either Na⁺ or K⁺) cations. The lowering of pH produced a similar effect. Parallel changes in chromatin structure took place at the same time as phase-like transitions. Atomic absorption spectroscopy allowed determination of free and nuclei-bound ions, pointing to the presence of a sizeable number of free binding sites for chromatin-DNA even within intact nuclei. DNA-phosphate sites appear to be neutralized by ions strictly according to the size of the electric charge and polyelectrolyte theory. Partial digestion (by micrococcal nuclease) or simple breaks (by chemical carcinogens) of the chromatin-DNA fiber caused respectively elimination or reduction of the abrupt volume changes in the intact nuclei. The apparent role of chromatin structure versus nuclear matrix in determining the shape and volume of intact nuclei is briefly discussed.     

Estimation of melting curves from enzymatic activity-temperature profiles

Measuring the reversible thermal unfolding of enzymes is valuable for quantifying the effects of environmental factors on the thermodynamic stability of proteins. The thermal unfolding behavior of enzymes is typically studied using calorimetry or optical techniques such as circular dichroism, fluorescence, or light scattering. These techniques often have practical limitations and usually require the protein to be electrophoretically pure. An alternative technique for analyzing the thermodynamic stability of enzymes is to estimate the melting curve from temperature-activity data. This technique does not require electrophoretically pure enzyme, provided the sample does not have competing enzymatic activities or proteins which can affect enzyme stability (e.g., proteases). Moreover, small amounts of contaminant proteins should not affect the results as long as enzymatic assays are performed at low protein concentrations where nonspecific protein-protein interactions are negligible. To illustrate this technique, the melting curve for beta-galactosidase from Escherichia coli in the presence of 1 mM EDTA, and the shift caused by adding 1 mM Mg(+2), were calculated from activity-temperature data. Melting temperatures predicted from activity-temperature data compared closely with those obtained using other techniques. Application of this analysis to multisubstrate enzymes is illustrated by estimating the melting profiles for partially purified hydrogenases from several thermophilic Methanococcii. Limitations and important considerations for estimating melting profiles from activity-temperature data are discussed. (c) 1993 John Wiley & Sons, Inc.     

Phase transitions in heterochromatin organization

Heterochromatin formation has been proposed to involve phase transitions on the level of the three-dimensional folding of heterochromatin regions and the liquid–liquid demixing of heterochromatin proteins. Here, I outline the hallmarks of such transitions and the current challenges to detect them in living cells. I further discuss the abundance and properties of prominent heterochromatin proteins and relate them to their potential role in driving phase transitions. Recent data from mouse fibroblasts indicate that pericentric heterochromatin is organized via a reordering transition on the level of heterochromatin regions that does not necessarily involve liquid–liquid demixing of heterochromatin proteins. Evaluating key hallmarks of the different candidate phase transition mechanisms across cell types and species will be needed to complete the current picture.     

Phase transitions of polymerizable phospholipids

Polymerizable lipids have received considerable attention in the last ten years as polymerization of lipids in vesicle systems is a possibility to increase the stability of lipid bilayers. Lipids with various polymerizable groups have been synthesized in the last years. This paper is focussed on those lipids which are closely related to natural phospholipids, i.e. molecules which have two hydrophobic chains and a head group containing a phosphate moiety. The phase behaviour of polymerizable phospholipids as lipid monomers and in the polymerized state is reviewed and discussed.     

Phase transitions in plant cuticles

The effect of temperature on wet plant cuticles has been investigated with the following techniques: Calorimetry, densitometry, spin-label electron-spin-resonance-(ESR)-spectroscopy, photo bleaching, and light and electron microscopy. At low temperatures cuticles ofCitrus aurantium L. andHedera helix show, at 16.3°C, a sharp transition (T0.5°C) with a latent heat of 4.7±0.5 J g^-1-cuticle. Below transition: The main orientation of the polymer matrix is parallel to the normal of the cuticle and the main orientation of the layer with soluble lipids is perpendicular to the normal. The cuticle is in a rigid state. Above transition (between 16.3°C and 38°C): Only the orientation of the polymer matrix has changed (tilted in parts). There exist several very sharp (T0.1°C) transitions (38°C, 41°C, 45°C, 49°C, ...) with a latent heat in the order of 0.4 J g^-1-cuticle. Above 38°C: The lamella of the soluble lipids is in a fluid state. Above 45°C there is a change in the molecular orientation of the soluble lipids as well as in the polymer matrix. The soluble lipids are mainly oriented parallel to the normal. The dry cuticles show no phase transition between 0°C and 200°C. At room temperature a dry/wet transition can be observed.Abbreviations ESR-spectroscopy electron-spin-resonance-spectroscopy     

Singular value decomposition of 3-D DNA melting curves reveals complexity in the melting process

The thermal denaturation of synthetic deoxypolynucleotides of defined sequence was studied by a three dimensional melting technique in which complete UV absorbance spectra were recorded as a function of temperature. The results of such an experiment defined a surface bounded by absorbance, wavelength, and temperature. A matrix of the experimental data was built, and analyzed by the method of singular value decomposition (SVD). SVD provides a rigorous, model-free analytical tool for evaluating the number of significant spectral species required to account for the changes in UV absorbance accompany-ing the duplex – to – single strand transition. For all of the polynucleotides studied (Poly dA – Poly dT; [Poly (dAdT)]₂; Poly dG – Poly dC; [Poly(dGdC)]₂), SVD indicated the existence of at least 4 – 5 significant spectral species. The DNA melting transition for even these simple repeating sequences cannot, therefore, be a simple two-state process. The basis spectra obtained by SVD analysis were found to be unique for each polynucleotide studied. Differential scanning calorimetry was used to obtain model free estimates for the enthalpy of melting for the polynucleotides studied, with results in good agreement with previously published values. Received: 16 April 1997 / Accepted: 9 July 1997     

Single nucleotide polymorphism genotyping using allele-specific PCR and fluorescence melting curves

We present a PCR method for identification of single nucleotide polymorphisms (SNPs), using allele-specific primers designed for selective amplification of each allele. Matching the SNP at the 3' end of the forward or reverse primer, and additionally incorporating a 3' mismatch to prevent amplification of the incorrect allele, results in selectivity of the allele-specific primers. DNA melting analysis with fluorescent SYBR Green affords detection of the PCR products. By incorporating a GC-rich sequence into one of the two allele-specific primers to increase the melting temperature, both alleles can be measured simultaneously at their respective melting temperatures. Applying the DNA melting analysis to SNPs in ApoE and ABCA1 yielded results identical to those obtained with other genotyping methods. This provides a cost-effective, high-throughput method for amplification and scoring of SNPs.