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upper D Superscript 2 Baseline EndFraction StartFraction 3 Over upper C Subscript normal upper T Baseline EndFraction equals exp left-parenthesis StartFraction minus upper Delta upper H Subscript normal upper D Baseline plus upper T upper Delta upper S Subscript normal upper D Baseline Over italic upper R upper T EndFraction right-parenthesis"/>

      where ΔH_D and ΔS_D are the van't Hoff enthalpy and entropy of duplex formation in the 1 : 1 : 1 mixture of the three strands, respectively, and α_D is the molar fraction of the coiled strands in the structured duplex form. For this case, the maximum temperature should take place at α_D = 0.50 [18], and therefore, the van't Hoff equation can be written as

      (3.24)

      3.5.3 Thermodynamic Analysis for the Tetraplex

The unfolding of the tetraplex-stranded DNAs is generally not in equilibrium (Figure 3.11a). In general, the folding process for the tetraplex is very slow relative to unfolding. Therefore, the unfolding and folding processes often show different sigmoidal curves (Figure 3.11b) [19]. Due to the hysteresis of unfolding and folding processes, the thermodynamic parameters for intermolecular tetraplex using UV melting curves are not estimated. However, the equilibrium between unfolding and folding tetraplex is adopted, and the thermodynamic parameters for tetraplex formation can be estimated by the following methods.

Figure 3.11 (a) The unfolding process for the intermolecular tetraplex. (b) Unfolding and folding behaviors for intermolecular tetraplex monitored by UV absorption.

For a folding reaction involving intermolecular and intramolecular tetraplexes (Figure 3.12) from same sequences, the general equilibrium can be written as [20]

      The general expression for the equilibrium constant, K, in terms of α and n is

      (3.25)

      Note that this expression for the equilibrium constant for an association reaction among same sequences is not identical to the corresponding expression for different sequences. If one defines the melting temperature, T_m, as the temperature at which α = 0.5, the general expression for K shown above reduces to

(3.26)

This expression allows calculation of K at the T_m for an association reaction of any molecularity among same sequences. One also can derive a general expression for calculating the transition enthalpy for self-complementary associations. To accomplish this, one substitutes Eq. (3.26) into the van't Hoff equation (Eq. 3.27):

Figure 3.12 (a–c) Typical examples for intermolecular and intramolecular tetraplexes described with equation of nA ⇌ A_n. n indicates the number of strands.

(3.27)

3.6 Conclusion

      1 Study the interactions to determine the stability of canonical nucleic acids.

      Stability of canonical nucleic acids depends on the sequences because hydrogen bonding, base stacking, and conformational entropy affect mainly stability of canonical nucleic acids.

      1 Understand the difference in factors determining stability of canonical and non-canonical nucleic acids.

Factors influencing the non-canonical nucleic acids are different from that of canonical duplex. The non-canonical structures are susceptible to a change in solution condition such as pH and cations. The difference can appear as structural properties, especially charge of the nucleic acids and protonation of bases.

      1 Analyze the stability for canonical and non-canonical nucleic acids.

      The thermodynamic parameters for both canonical and non-canonical structures can be estimated using thermal melting curves for the nucleic acid structures. The different equations for melting treatments are required because the unfolding process for the nucleic acids depends on their structures.

References

      1 1 (a) Breslauer, K.J., Frank, R., Blocker, H., and Marky, L.A. (1986). Proc. Natl. Acad. Sci. U. S. A. 83: 3746–3750.(b) Auffinger, P. and Westhof, E. (2002). Biophys. Chem. 95: 203–210.(c) Auffinger, P. and Westhof, E. (2001). Angew. Chem. Int. Ed. Engl. 40: 4648–4650.(d) Anderson, C.F. and Record, M.T. Jr. (1995). Annu. Rev. Phys. Chem. 46: 657–700.

      2 2 (a) Feig, M. and Pettitt, B.M. (1999). Biophys. J. 77: 1769–1781.(b) Feig, M. and Pettitt, B.M. (1998). Biopolymers 48: 199–209.(c) Spink, C.H. and Chaires, J.B. (1999). Biochemistry 38: 496–508.(d) Nordstrom, L.J., Clark, C.A., Andersen, B. et al. (2006). Biochemistry 45: 9604–9614.(e) Record, M.T. Jr. Anderson, C.F., and Lohman, T.M. (1978).

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