3.3.1 Thermodynamics of DNA duplex formation

As mentioned before, the NN model allows us to compute the thermodynamic properties of DNA duplex formation starting from the enthalpies and entropies of the elementary motifs that constitute all possible Watson-Crick sequences of DNA. This section describes how to calculate these magnitudes.

The sequence of an $ N$-base ssDNA ($ \sigma_i$) is given from the $ 5'$ end to the $ 3'$ one, where $ i$=$ 1$ is the base located at the $ 5'$ end, $ i$=$ N$ is the base located at the $ 3'$ one. $ \sigma_i$ can take four different values ( $ \sigma=\{A,C,G,T\}$). The standard enthalpy of hybridization of such sequence ( $ \Delta H^0$) with its complementary strand can be calculated according to,

$\displaystyle \Delta H^0 = \Delta H_\mathrm{init} + \sum_{i=1}^{N-1} \Delta h^0(\sigma_i,\sigma_{i+1})$ (3.1)

where $ \Delta H_\mathrm{init}$ is the initiation term that depends on the ends of the sequence ($ i=1,N$); $ \Delta h^0$ is the formation energy of Watson-Crick motifs and it depends on the nearest neighbor ( $ \sigma_i,\sigma_{i+1} \in \{AA,AC,\dots,TT\}$); and the sum is extended over all NN base pairs. The notation is usually simplified by assuming that the $ i+1$ base is already known from the sequence. Thus, the arguments of $ \Delta h^0$ can be removed,

$\displaystyle \Delta H^0 = \Delta H_\mathrm{init} + \sum_{i=1}^{N-1} \Delta h^0_i$ (3.2)

where $ \Delta h^0_i$ is the enthalpy of formation of the motif $ (\sigma_i,\sigma_{i+1})$. Similarly, the entropy contribution ( $ \Delta S^0_i$) in the hybridization reaction can be computed according to

$\displaystyle \Delta S^0 = \Delta S_\mathrm{init} + \sum_{i=1}^{N-1} \Delta s^0_i$ (3.3)

where $ \Delta S_\mathrm{init}$ is the initiation term and $ \Delta s^0_i$ is the entropy of formation of the motif $ \sigma_i\sigma_{i+1}$.

The last bases of the sequence do not have a complete stacking interaction because they only have one neighbor. This is why the NN model has to introduce the initiation term. This term always adds a constant contribution, which is sequence independent. Moreover, a variable contribution has to be added depending on the type of bases located at the ends: an extra contribution of $ \Delta h^0_\mathrm{A/T}$ and $ \Delta s^0_\mathrm{A/T}$ must be added if the sequence starts with a $ 5'$-A...-$ 3'$ or $ 5'$-T...-$ 3'$; or ends with a $ 5'$-...A-$ 3'$ or $ 5'$-...T-$ 3'$. If the molecule has both motifs (start and end) the contribution has to be counted twice. Moreover, if the molecule specifically starts with $ 5'$-TA...-$ 3'$ or ends with $ 5'$-...TA-$ 3'$, there is another extra contribution of $ \Delta h^0_\mathrm{TA}$ and $ \Delta s^0_\mathrm{TA}$ to the initiation terms. Table 3.1 summarizes all these cases.

Table 3.1: Initiation terms. The contributions are not mutually exclusive. It means that they are added to the total $ \Delta H^{\circ }$ and $ \Delta S^{\circ }$ for each motif of the oligo. For instance, the sequence $ 5'$-TA...C-$ 3'$ will have the following initiation term for the enthalpy: $ \Delta H_{\textrm {init}}=0.2+\Delta h^0_{\textrm {A/T}}+\Delta h^0_{\textrm {TA}}$. Enthalpies are given in kcal/mol and entropies, in cal/mol$ \cdot $K.
Term Motif $ \Delta H^\circ$ $ \Delta S^\circ$
Constant contribution All 0.2 -5.6
A/T penalty $ 5'$-A...-$ 3'$ 2.2 6.9
$ 5'$-T...-$ 3'$
$ 5'$-...A-$ 3'$
$ 5'$-...T-$ 3'$
TA penalty $ 5'$-TA...-$ 3'$ -0.4 0.5
$ 5'$-...TA-$ 3'$

In general, $ \Delta H^0_i$ and $ \Delta S^0_i$ are considered to be temperature independent, meaning that the change in heat capacity can be neglected ( $ \Delta C_p=0$). This allows to compute the free energy of formation ( $ \Delta G^0$),

$\displaystyle \Delta G^0=\Delta H^0 - T \Delta S^0$ (3.4)

where $ T$ is the temperature at which the free energy is calculated. Following the same scheme, it is also possible to define the free energy of formation ( $ \Delta g^0_i$) of one NN motif according to

$\displaystyle \Delta g^0_i=\Delta h^0_i - T \Delta s^0_i~.$ (3.5)

JM Huguet 2014-02-12