**6. Results and discussion for the modeling based on the end effects**

From now on, the thermodynamical property **E** will be, for us, the free energy of the duplex formation. According to the model based on the end effects, the free energy of a duplex sequence of *N* bases in the NN approximation could be calculated as the pairwise sum including end effects as a function of 12 irreducible parameters from Eqs. 12, 18, and, 19, as follows:

$$
\Delta G\_{\tau} = \Delta G \left( E b\_{1} \right) + \sum\_{i=1}^{N-1} \Delta G \left( b\_{i} b\_{i+1} \right) + \Delta G \left( b\_{N} E \right) + \Delta G\_{\text{sym}} \tag{21}
$$

where Δ*G*sym =0.43 kcal/ mol is a symmetric correction term applicable to self-complementary duplexes.

Simultaneous least-mean-square-deviation fit of this model to the 108 sequence data compiled by Allawi and SantaLucia [12] gave the values for the free energies, which are listed in Table 1. Guerra and Licinio [16] calculated irreducible parameters for the thermodynamic properties of free energy, entropy, and enthalpy but, in Table 1, only the irreducible parameters for free energy are shown. In Table 1, Δ*G*(*ET* ) is the contribution for the free energy of the sequence from the T/A end base pair, and so on. Here, we prefer to use the irreducible parameters Δ*G*(*EA*), Δ*G*(*ET* ), Δ*G*(*EC*), and Δ*G*(*EG*) in the place of the parameters *A*, *B*, *C*, and *D* as defined in Eq. 18 or 19. In fact, there is no loss of generality in the use of the firsts once that they are linear combinations of the seconds.

For comparison, we performed another calculation, supposing that the contributions from the ends do not depend on the orientation of the end base pairs, that is, an A/T end pair would contribute in the same way as a T/A end pair, as it is usually found in the literature [2–6]. As a result, we obtained Table 2.

base pair would not produce the same effect as one T/A end base pair. Therefore, at least in theory, it would be necessary to discriminate four-end pairings, which are listed in the

,

Finally, we can conclude that the four possible end base pairs in Eq. 20 can be expanded in terms of four parameters, namely *A*, *B*, *C*, and *D*. Consequently, for a duplex oligomer, the additional four parameters related to the ends should be added to the eight polymeric parameters already known, producing a total of 12 irreducible parameters, in the light of the

.

*E EA EG EC <sup>T</sup>* (20)

(21)

, ,

,,, *EA ET EC EG*

**6. Results and discussion for the modeling based on the end effects**

From now on, the thermodynamical property **E** will be, for us, the free energy of the duplex formation. According to the model based on the end effects, the free energy of a duplex sequence of *N* bases in the NN approximation could be calculated as the pairwise sum including end effects as a function of 12 irreducible parameters from Eqs. 12, 18, and, 19, as

() ( ) ( ) <sup>1</sup>

+

where Δ*G*sym =0.43 kcal/ mol is a symmetric correction term applicable to self-complementary

Simultaneous least-mean-square-deviation fit of this model to the 108 sequence data compiled by Allawi and SantaLucia [12] gave the values for the free energies, which are listed in Table 1. Guerra and Licinio [16] calculated irreducible parameters for the thermodynamic properties of free energy, entropy, and enthalpy but, in Table 1, only the irreducible parameters for free energy are shown. In Table 1, Δ*G*(*ET* ) is the contribution for the free energy of the sequence from the T/A end base pair, and so on. Here, we prefer to use the irreducible parameters Δ*G*(*EA*), Δ*G*(*ET* ), Δ*G*(*EC*), and Δ*G*(*EG*) in the place of the parameters *A*, *B*, *C*, and *D* as defined in Eq. 18 or 19. In fact, there is no loss of generality in the use of the firsts once that they are

For comparison, we performed another calculation, supposing that the contributions from the ends do not depend on the orientation of the end base pairs, that is, an A/T end pair would

*G G Eb G b b G b E G*

1 *<sup>T</sup> i i <sup>N</sup> i*

= D =D + D +D +D å *N*


1 1 sym

following:

follows:

duplexes.

modeling based on the end effects.

194 Nucleic Acids - From Basic Aspects to Laboratory Tools

linear combinations of the seconds.


**Table 1.** Irreducible Parameters for Free Energy at Standard Conditions (37 ℃ and 1 M Salt and DNA)


**Table 2.** Irreducible Parameters for Free Energy at Standard Conditions (37 ℃ and 1 M Salt and DNA)

Considering the values obtained for the irreducible parameters for free energy presented in Tables 1 and 2, some observations must be carried out:


$$\chi = \sqrt{\sum\_{i=1}^{\text{loss}} \left[ \Delta G\_{\text{eq}} \left( i \right) - \Delta G\_{\text{thor.}} \left( i \right) \right]^2 / \sum\_{i=1}^{\text{loss}} \mathcal{N} \left( i \right)} \tag{22}$$

the free energy irreducible parameters in Tables 1 and 2 are such that they minimize *χ.* The quantity*χ* defines a global minimal deviation, between the theoretical values calculated from the irreducible parameter set for the free energies of the 108 sequences and the experimental values. In Eq. 22, Δ*G*exp(*i*) is the experimental value for the free energy for the *i*th sequence,

<sup>Δ</sup>*G*theor. (*i*) is its corresponding theoretical value, and ∑ *i*=1 108 *N* (*i*) is the total number of duplex

dimers for the ensemble of 108 sequences. The value obtained for*χ* considering 10 (or 12) parameters is precisely the same, namely 0.14 kcal/mol per dimer [16], which also coincides with the 12-parameter model using values reported by SantaLucia for the free energies for the 10 duplex dimers [3–6]. This means that, considering only the overall data ensemble quality, there is no practical reason to prefer a model with a greater number of parameters.


As shown, end contributions are fit with large errors to experimental data, as compared to the fits of other NN or dimer contributions. Besides A/T from T/A as well as C/G from G/C, ending contributions could not be respectively differentiated. More than that, we could not distinguish between the weak and the strong terminal base pairs. However, using both the sets, one can calculate free energies for DNA oligomers at least as well as standard models considering a larger set of parameters do [3–6]. Guerra and Licinio [16] also extended their analysis and obtained equivalent sets of irreducible parameters for enthalpy and entropy. By simultane‐ ously minimizing the deviations from melting temperatures and entropies of the chains, they obtained the most precise set, which is capable of predicting melting temperatures for DNA chains with a standard deviation of 2.2°C for sequence against a deviation of 2.5°C for previous parameters found in the literature [3–6].

In the light of our finding, the formulation based on the use of end effects, according to the NN approach, proves to be naive, even heuristic. The extra parameters (up to now, the end parameters), which must be summed to the eight (polymeric) irreducible parameters for predicting thermodynamical properties of duplex oligomers, seem not to depend on the composition of the terminal base pairs. From now, we will invoke a new hypothesis, which will be detailed later in this review. With base on this hypothesis, we will conclude that, in the light of the NN model, 10 is the number of parameters expand the free energy of any DNA oligomers: eight (polymeric) irreducible parameters for free energy, already described, plus two parameters related to the initiation of the double helix (related to two possible base pairings).
