Dependence of RNA secondary structure on the energy model

We analyze a microscopic RNA model, which includes two widely used models as limiting cases; namely, it contains terms for bond as well as for stacking energies. We numerically investigate possible changes in the qualitative and quantitative behavior while going from one model to the other; in particular, we test whether a transition occurs when continuously moving from one model to the other. For this we calculate various thermodynamic quantities, at both zero temperature and finite temperatures. All calculations can be done efficiently in polynomial time by a dynamic programming algorithm. We do not find a sign for the transition between the models, but the critical exponent $\nu$ of the correlation length, describing the phase transition in all models to an ordered low-temperature phase, seems to depend continuously on the model. Finally, we apply the $\epsilon$-coupling method to study low-energy excitations. The exponent $\theta$ describing the energy scaling of the excitations seems to depend not much on the energy model.

Figure 1

Schematical explanation of Eqs. (5, 6); e.g., white boxes represent $Z$, gray boxes $\widehat{Z}$.

Figure 2

Average stacking size as a function of energy parameter $E_p$ for different system sizes $(T=0)$. At $E_p=-1$—i.e., $E_s=0$ and $E_p=0$—the curves are discontinuous. Missing error bars are of the size of the symbols or smaller and omitted for legibility. Calculated data points are indicated by symbols. Lines are drawn to guide the eye.

Figure 3

Average overlap $q_{\mathcal{E}}$ as a function of the energy parameter $E_p$ for different system sizes at $T=0$. The local minima at $E_p=-0.75,-0.5,-0.25$ are due to the commensurability of $E_p$ and $E_s$ and indicate a broad distribution of the ground states in configuration space. The left inset is an enlargement to show the discontinuity. The right inset is the $q$ distribution for $E_s=E_p=-0.5$ (solid lines) and $E_s=-0.49,E_p=-0.51$ (dashed lines) for sequence lengths $L=512$ (thin lines) and $L=1024$ (thick lines).

Figure 4

The Binder cumulant $B$ of Eq. (9) with $q=q_{\mathcal{E}}$ [see Eq. (10)] at temperature $T=0$. The curves for different system sizes do not cross, and therefore $B$ gives no hint of a phase transition. Missing error bars are of the size of the symbols or smaller and omitted for legibility. Calculated data points are indicated by symbols. Lines are drawn to guide the eye.

Figure 5

The Binder cumulant $B$ of Eq. (9) with $q=q_{\mathcal{E},{\mathcal{E}}^{\prime }}$ [see Eq. (11)] at temperature $T=0$. Missing error bars are of the size of the symbols or smaller and omitted for legibility. Calculated data points are indicated by symbols. Lines are drawn to guide the eye.

Figure 6

Specific heat $c_V$ as a function of temperature for parameters $E_s=-1.0$ and $E_p=0.0$. The curves for different system sizes crosses at $T\approx 0.25$ and $T\approx 0.11$. The inset is an enlargement of the main plot. Missing error bars are of the size of the symbols or smaller and omitted for legibility. Calculated data points are indicated by symbols. Lines are drawn to guide the eye.

Figure 7

The position of the maxima of ${\chi }_{\mathcal{R}}$ for different energy parameter. The curves are fitted to the form $T_c\left(L\right)=T_c+a{L}^{-1∕\nu }$ and gives the critical parameters of Table .

Figure 8

The parameter $G$ [Eq. (14)] as a function of temperature for different energy parameters $(E_s,E_p)=(0,-1)$, $(-0.5,-0.5)$, and $(-1,0)$. For low temperatures the first two cases approach the theoretical expected value $G(T\to 0)=\frac{1}{3}$, while in the third case—the one with the lowest critical temperature—this is not clear.

Figure 9

$\epsilon$-coupling results for $E_s=-0.5$, $E_p=-0.5$. The energy difference $\Delta E\left(\mathcal{S}\right)$ between the original ground-state structure ${\mathcal{S}}_0$ and the ground-state structure ${\mathcal{S}}_{\epsilon }$ of the disturbed model is plotted as a function of the distance between this structures. The inset is a scaled plot of the data with $\epsilon <1$ of the main plot ($\theta =0.24$; see Table ). Missing error bars are of the size of the symbols or smaller and omitted for legibility. Calculated data points are indicated by symbols. Lines are drawn to guide the eye.