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3.3 Thermodynamic Analysis for the Formation of Duplex
ОглавлениеIn the case of canonical duplex structures, the base stacking decreases the transition dipole moment of bases, which makes UV absorbance at 260 nm of duplex smaller than that of single-stranded state. Heating of nucleic acids causes the strands to be denatured by disrupting the ordered stacking of the bases and breaking hydrogen bonds. The process can be conveniently monitored by an increase in UV absorbance as the duplex unwinds to single strands owing to hyperchromicity.
The example in Figure 3.4a shows a melting curve (UV absorption as a function of temperature). Slow heating of duplex causes the unwinding of the ordered helical structure into the two single-stranded constituents. The unwinding can be seen as a sigmoidal curve of increasing UV absorption. The midpoint corresponding to the precise melting temperature (Tm) of the duplex is indicated.
Methods to obtain the thermodynamic parameters of enthalpy (ΔH°), entropy (ΔS°), and free energy changes at 25 °C (Δ) for the formation of a nucleic acid structure are described below; DNA duplex formation is taken as an example. Data are typically analyzed with a two-state model, which assumes that each strand is either completely paired or unpaired. The equilibrium for the duplex formation is represented as either a self-complementary or non-self-complementary association as follows [4]:
where A, B, and C indicate the single strands of DNA and A2 and B·C indicate the double-stranded DNA.
Figure 3.4 UV melting curves of the self-complementary duplex (5′-ATGCGCAT-3′) at 5 μM strand concentration with (a) Tm value and (b) upper and lower baselines. Upper and lower baselines can be represented as εds = mdsT + bds and εss = mssT + bss, respectively, where εds and εss indicate the absorbance for the double-stranded and single-stranded DNA, respectively. The mds and bds or mss and bss represent the slope and intercept of the upper baseline or lower baseline for the UV melting curve, respectively.
The equilibrium for an intramolecular transition, for example, a hairpin, is represented as
where Dss and Df indicate the single-stranded (unfolded) and folded DNA structures, respectively.
For self-complementary (Eq. 3.1) or non-self-complementary equilibria (Eq. 3.2) with equal concentrations of B and C, the observed equilibrium constant Kobs is given by
(3.4)
where Ct is the total strand concentration, s has a value of 1 for self-complementary duplexes and 4 for non-self-complementary duplexes, and α is the fraction of strands in a duplex:
(3.5)
(3.6)
For a unimolecular transition,
where
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)
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)
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:
where R is the gas constant and mds and bds or mss and bss represent the slope and intercept of the upper baseline or lower baseline for the melting curve of a duplex dissociation (Figure 3.4b), respectively. The six variables (εds, εss, bds, bss, ΔH°, and ΔS°) can be calculated by the curve fitting using Eqs. (3.11, 3.12, 3.13, 3.14). These calculations, which are performed using a PC equipped with the curve fitting software such as IGOR Pro, KaleidaGraph, or ORIGIN, derive the thermodynamic parameters from the shape of the melting curve.
Estimated errors in the thermodynamic values (σΔH°, σΔS°, and ) derived from the curve fitting procedure are calculated from the standard deviations among data points of each melting curve at different Ct values. Estimated errors in ΔH°(σΔH°) and ΔS°(σΔS°) obtained from the versus ln(Ct/s) plots are calculated from the linearity of the plots, and those for Δ () are calculated using Eq. (3.15):
where RΔH°,ΔS° is the correlation coefficient between ΔH° and ΔS°. The final thermodynamic parameters are the average values obtained from curve fitting and the versus ln(Ct/s) plots. The errors of σΔH°, σΔS°, and are usually within several percent of each thermodynamic value.
