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duplexes and 4 for non-self-complementary duplexes, and α is the fraction of strands in a duplex:

      (3.5)upper C Subscript normal t Baseline equals left-bracket normal upper A right-bracket plus 2 left-bracket normal upper A right-bracket left-parenthesis self minus complementary duplex right-parenthesis

      (3.6)upper C Subscript normal t Baseline equals left-bracket normal upper B right-bracket plus left-bracket normal upper C right-bracket plus left-bracket normal upper B dot normal upper C right-bracket left-parenthesis non minus self minus complementary duplex right-parenthesis

      For a unimolecular transition,

      Figure 3.4 shows a UV melting curve for the self-complementary duplex (5′-ATGCGCAT-3′). At low temperatures, the strands are in duplex form and the absorbance is low. As the temperature is increased, the duplex dissociates into single strands. The UV absorbance of the duplex is increased by dissociation of the duplex, and the increment of absorbance is referred to as hyperchromicity. (The opposite, a decrement of absorbance, is called hypochromicity.) For self-complementary or non-self-complementary duplexes with equal concentrations of each strand, the melting temperature, Tm (in degrees Kelvin), is the point at which the concentrations of strands in duplex and in single strands are equal (Figure 3.4a). The steepness of the transition indicates the cooperativity of the transition. The width and maximum of the first derivative of the melting curve can also indicate the cooperativity and melting temperature, although the peak of the derivative curve only occurs at the Tm only when the transition is unimolecular [5]. Tm is most accurately measured by fitting the lower and upper baselines. The melting temperature is measured at several concentrations over a 100-fold range and then plotted versus the concentration in a van't Hoff plot. The van't Hoff equation relates the Tm (in degrees Kelvin), Ct, ΔH°, and ΔS°:

      (3.9)upper T Subscript normal m Superscript negative 1 Baseline equals upper R ln left-parenthesis upper C Subscript normal t Baseline slash s right-parenthesis slash upper Delta upper H Superscript degree Baseline plus upper Delta upper S Superscript degree Baseline slash upper Delta upper H Superscript degree

      where R is the ideal gas constant, 1.987 cal K−1 mol−1 or 8.314 J K−1 mol−1. The slope of the van't Hoff plot gives the ΔH°, and the y-intercept gives the ratio of ΔH° to ΔS°. The free energy and equilibrium constant at any temperature can then be calculated using Gibb's relation:

      (3.10)upper Delta upper G Subscript normal upper T Superscript degree Baseline equals upper Delta upper H Superscript degree Baseline minus upper T dot upper Delta upper S Superscript degree Baseline comma upper K Subscript o b s Baseline equals normal e Superscript negative upper Delta Baseline upper G Subscript normal upper T Superscript degree Superscript slash italic upper R upper T

      To increase the accuracy of these parameters, data analysis can be performed by curve fitting as shown below. When the ratio of the double-stranded DNA is represented by α, absorbance (A) at a temperature (T) is calculated as

      where εds and εss indicate the absorbance for the single-stranded and double-stranded DNA, respectively, and l and Ct represent the length of the light pass (or the path length of the cuvette used) and the total concentration of DNA strands, respectively. The εds, εss, and observed equilibrium constant (Kobs) for the duplex formation can be represented as follows with the assumption that absorbance is directly proportional to temperature: