**8. Results and discussion for the modeling based on double helix initiation parameters**

The free energy for a duplex sequence of *N* bases in the NN approximation can be calculated as the pairwise sum, using the initiation free energy, as a function of the 10 parameters for free energy from Eqs. 12 and 43 as follows:

A Review on the Thermodynamics of Denaturation Transition of DNA Duplex Oligomers in the Context... http://dx.doi.org/10.5772/62574 205

$$
\left\langle \Delta \mathcal{G}\_{\text{init}} \right\rangle + \sum\_{i=1}^{N-1} \Delta \mathcal{G} \left( b\_i b\_{i+1} \right) + \Delta \mathcal{G}\_{\text{sym}} \tag{51}
$$

Simultaneous least-mean-square-deviation fit of this model to the 108 sequence data set compiled by Allawi and SantaLucia [2] gave the values for the free energy parameters, as listed in Table 3 [17].

where Δ*G*(*ww*) is the mean free energy of a stack of two weak base pairs, *χww* is its composition, and so on. Using Eq. 12, we obtain the following values for the mean dimer free energies:

> .

*G ww S V M G ss S V M G ws G sw S M*

D =- - - - + - - +

D =+ + D =- + D =D = -

*z zz z zz*

( )( )

*z ww ss zz ww ss ws sw*

 c

Equation 42 can be used to predict the free energy of any duplex oligomer if we know the values of all the polymeric irreducible parameters for free energy plus the free energy changes associated to the formation of the first base pair. Now, we can return to the set of 108 sequences compiled by Allawi and SantaLucia to obtain the set of eight polymeric irreducible parameters together with these two additional parameters. This will be done in the following section.

**Figure 3.** The formation of the first WC base pair. (a) Strands *S* and *S'* are sufficiently distant one from the other. All the bases in both the chains are in the single strand state. (b) It occurs an approximation between strands *S* and *S'.* However, all the bases are still in the single strand state. (c) It is formed the first base pair, namely the base pair *bk* / *bk*

**8. Results and discussion for the modeling based on double helix initiation**

The free energy for a duplex sequence of *N* bases in the NN approximation can be calculated as the pairwise sum, using the initiation free energy, as a function of the 10 parameters for free

'

*zz*

 cc

D +D (50)

 c (49)

'

. After that, the double helix propagates in both the

,

( ) ( ) () ()

> c

() ()

*G SV M G AT GCG* c

 c° °

+ / /

*w s*

through the establishment of H bonds between the bases *bk* and *bk*

directions extending to the ends of the chain [17].

energy from Eqs. 12 and 43 as follows:

**parameters**

Inserting Eq. 49 into Eq. 48, we can obtain:

204 Nucleic Acids - From Basic Aspects to Laboratory Tools

nuc

c


**Table 3.** Irreducible Parameters for Free Energy at Standard Conditions (37°C and 1 M Salt and DNA)

Given the root-mean-square deviation per dimer, as defined in Eq. 22, the parameters for free energy in Table 3 are those that minimize *χ*. The value obtained for *χ* was 0.14 kcal/mol per dimer [17], which coincides precisely with that obtained for the 12 parameter models using values reported by SantaLucia for the free energies of the 10 duplex dimers [2, 4–6]. Thus, how it happened for the modeling by end effects, from the overall data ensemble quality, there would not be practical reason to prefer a model with a greater number of parameters. The mean values and the errors of the parameters for free energy, as listed in Table 3, were estimated by Guerra in the following way [17]: he selected 1000 sets of 70 sequences chosen randomly, and then he calculated the mean and the deviation for the parameters obtained from each set. Some immediate conclusions can be done with respect to the data contained in Table 3. First, the intrinsic errors of the free energies related to the formation of the first base pairings are only a little larger than the errors of the other irreducible parameters for free energy. Second, considering only the bar of errors, the free energy changes for the initiation of a double chain through the formation of an A/T or C/G base pair are essentially similar. Thus, if it is correct the hypothesis of that the duplex formation can be initiated by the formation of a base pair at any site along the double helix with equal probability (independently of the local composition), then, the initiation free energy is essentially independent on the local composi‐ tion. Finally, once we have obtained the initiation free energy parameters, as listed in Table 3, we are ready for to estimate the nucleation free energy of any duplex oligomer, using Eq. 50. Equation 50 establishes that observable nucleation free energies depend on the mean global composition of the DNA double strand and vary within a range that goes from

$$
\Delta G\_{\text{mac}}^{\text{poly}\,A\cdot T} = 1.38 - 0.58 + 0.04 + 1.7 = 2.54 \frac{\text{kcal}}{\text{mol}}\%
$$

for a poly *A*⋅*T* homopolymer to

$$\Delta G\_{\text{max}}^{\text{poly }C.G} = 1.38 + 0.58 + 0.04 + 1.8 = 3.80 \frac{\text{kcal}}{\text{mol}} \text{J}$$

for a poly *C*⋅*G* homopolymer. Observe that the difference between these values for nucleation free energies, which is ∼1.3 kcal/mol, is greater than the bar of errors estimated for nucleation free energies, which is ∼0.7 kcal/mol. On the another hand, the results obtained above, for the poly *A*⋅*T* and poly *C*⋅*G* homopolymers, are in total discordance with results obtained previ‐ ously using the modeling by end effects [16], as was to be expected. In fact, heteropolymers must have one value for the nucleation free energy that must be inside such interval, and it must depend on their composition. Finally, the mean observable nucleation free energy is Δ*G*nuc mean =(Δ*G*nuc poly *A*⋅*<sup>T</sup>* <sup>+</sup>Δ*G*nuc poly *C*⋅*G*) / 2=3.17 kcal/ mol. This value is only a little lower than that obtained by Manyanga et al. [19].

Comparing the results obtained for the eight polymeric irreducible parameters for free energy, as listed in Table 3 of this section, with results obtained recently using the end effects [16], and contained in Tables 1 and 2 of the Section 6 of this review, we can conclude that the irreducible parameters are not essentially affected with the alteration in the modeling. In another words, if we substitute one modeling by another, the end effects, which, obviously, do not depend essentially on the compositions of the two terminal base pairs, are substituted by the initiation free energies, which do not depend essentially on the global composition of the chain. Therefore, dimer free energies, which depend only on the irreducible parameters for free energy, also are not essentially affected.

Free energy changes associated to the formation of the second base pair are given by the following equation:

$$\left\langle \Delta G \left( b\_{k \pm 1} / b\_{k \pm 1}^{\cdot} \right) \right\rangle = \left\langle \Delta G \left( b\_{k} / b\_{k}^{\cdot} \right) \right\rangle + \left\langle \Delta G \left( O \Phi\_{k} \right) \right\rangle \tag{52}$$

depending if the second base pair formed is located at the *k*+1th site or at the *k−*1th site of the chain. Using Eq. 43 for Δ*G*(*bk* / *bk* ' ) = Δ*G*init , Eq. 46 for Δ*G*(*Obk* ) , and also the approximations given by Eq. 49, we obtain the following:

A Review on the Thermodynamics of Denaturation Transition of DNA Duplex Oligomers in the Context... http://dx.doi.org/10.5772/62574 207

$$\left\langle \Delta G \left( A \,\right/T \right) \right\rangle\_{\text{baso pairaig}} = \left( 0.7 \pm 0.3 \right) \text{ kcal/mol}$$

and

tion. Finally, once we have obtained the initiation free energy parameters, as listed in Table 3, we are ready for to estimate the nucleation free energy of any duplex oligomer, using Eq. 50. Equation 50 establishes that observable nucleation free energies depend on the mean global

kcal 1.38 0.58 0.04 1.7 2.54 , mol

kcal 1.38 0.58 0.04 1.8 3.80 , mol

poly *C*⋅*G*) / 2=3.17 kcal/ mol. This value is only a little lower than that

<sup>11</sup> / / *G b b G b b G Ob k k* ± ± *k k <sup>k</sup>* D =D +D (52)

) = Δ*G*init , Eq. 46 for Δ*G*(*Obk* ) , and also the approximations

composition of the DNA double strand and vary within a range that goes from

*A T G* <sup>×</sup> D = - + +=

*C G G* <sup>×</sup> D = + + +=

for a poly *C*⋅*G* homopolymer. Observe that the difference between these values for nucleation free energies, which is ∼1.3 kcal/mol, is greater than the bar of errors estimated for nucleation free energies, which is ∼0.7 kcal/mol. On the another hand, the results obtained above, for the poly *A*⋅*T* and poly *C*⋅*G* homopolymers, are in total discordance with results obtained previ‐ ously using the modeling by end effects [16], as was to be expected. In fact, heteropolymers must have one value for the nucleation free energy that must be inside such interval, and it must depend on their composition. Finally, the mean observable nucleation free energy is

Comparing the results obtained for the eight polymeric irreducible parameters for free energy, as listed in Table 3 of this section, with results obtained recently using the end effects [16], and contained in Tables 1 and 2 of the Section 6 of this review, we can conclude that the irreducible parameters are not essentially affected with the alteration in the modeling. In another words, if we substitute one modeling by another, the end effects, which, obviously, do not depend essentially on the compositions of the two terminal base pairs, are substituted by the initiation free energies, which do not depend essentially on the global composition of the chain. Therefore, dimer free energies, which depend only on the irreducible parameters for free

Free energy changes associated to the formation of the second base pair are given by the

( ) ( ) ( ) ' '

'

depending if the second base pair formed is located at the *k*+1th site or at the *k−*1th site of the

poly nuc

poly nuc

for a poly *A*⋅*T* homopolymer to

206 Nucleic Acids - From Basic Aspects to Laboratory Tools

Δ*G*nuc

mean =(Δ*G*nuc

following equation:

poly *A*⋅*<sup>T</sup>* <sup>+</sup>Δ*G*nuc

energy, also are not essentially affected.

chain. Using Eq. 43 for Δ*G*(*bk* / *bk*

given by Eq. 49, we obtain the following:

obtained by Manyanga et al. [19].

$$\left\langle \Lambda \mathcal{G} \left( \mathbf{C} \,/ \, \mathbf{G} \right) \right\rangle\_{\text{basa pairing}} = \left( 0.1 \pm 0.3 \right) \text{kcal} \,/\,\text{mol}.$$

The values listed above are just the base pairing contributions for the dimer free energies, which were encountered experimentally by the Frank-Kamenetskii Group [18]. Yakovchuk et al. obtained for the A/T and C/G base pairings, the base pairing free energies of 0.57 kcal/mol and −0.11 kcal/mol, respectively [18]. Therefore, we have obtained values that agree reasonably well with those obtained by the Frank-Kamenetskii Group. In addition, the values for the base pairing free energies are reasonably well defined because their ranges of allowable values have only an unique common intercept.

### **9. Conclusions**

A geometrical representation of four-nucleotide sets as a tetrahedron (Eq. 3 and Fig. 2) allows for the association of the three most distinctive molecular group classifications with corre‐ sponding orthogonal cubic axis. Physical properties of nucleotide sequences may be calculated with an optimal set of tensor coefficients (Eq. 4), assuming projections within this tetrahedral representation. The coefficients are expressed in hierarchical differential form, so lower levels of approximation are explicitly embodied in the description. This includes an ensemble mean expectation from scalar coefficient *S* alone and a global composition approximation, as expressed through *V*-component contributions. The symmetrical set is shown to provide a frame for the analysis of DNA duplex free energy fully compatible with experimental data. Such a symmetrical set of coefficients allows for the translation among different decomposition frames. It also gives a proper irreducible representation for dimer properties (Eqs. 8 and 12). It solves an old indeterminacy of dimer sets by establishing self-consistency relations among the dimer coefficients (Eqs. 14 and 15).

Using the modeling based on end effects, for predicting correctly physical properties of duplex oligomers, we saw that end contributions are fit with large errors to experimental data, as compared to the fits of other NN or dimer contributions. Besides, we could not distinguish between the weak and the strong terminal base pairs. However, using both the sets constituted by two- or four-ending parameters, one calculates free energies for DNA oligomers at least as well as standard models, considering a larger set of parameters do [2, 4–6].

The modeling based on the double helix initiation parameters substitutes the end effects by the initiation parameters. The free energy changes associated to the formation of the first base pair, in the duplex formation, are fit to experimental data with errors only slightly larger than those for the NN or dimer contributions. Furthermore, we obtained that the values for the first base pairing free energies are essentially similar (because the difference between them had a value smaller than the estimated bar of errors). Thus, this could indicate an invariance of the initiation free energy with respect to the composition of the chain. Nucleation free energy, however, depends on the composition, and it can be calculated from the initiation free energy by using Eq. 34. What supports this statement is the fact of that the difference between its maximal and minimal values is larger than the error bars. The model based on the double helix initiation parameters is constructed by using the simplifying hypothesis, which establishes that the nucleation can occur at any site of the chain with equal probability, independently of the local composition. An important result, which becomes such hypothesis quite reasonable, is the fact of that the base pairing contributions for the dimer free energies seem to agree well with values experimentally obtained by the Frank-Kamenetskii Group. Finally, this modeling uses a set of 10 parameters, which is constituted by the eight polymeric irreducible parameters already known plus two parameters related to two possible base pairings (the initiation free energy parameters). With this set, one calculates free energies for DNA oligomers at least as well as standard models considering a larger set of parameters do.
