RNA-like polymer model: Exact calculation on the Bethe lattice

We consider a lattice polymer model (random walk), in which the walk is allowed to visit lattice bonds at most twice. Such a model might have some relevance to describe statistical properties of RNA molecules. In order to mimic base pairing, we assign an attractive energy term to each doubly visited bond, and a further contribution to each pair of consecutive doubly visited bonds. The latter term is expected to mimic the stacking effect, whereas no effect of sequence, that is, of chemical specificity, is taken into account. The phase diagram is worked out exactly on a Bethe lattice, in a grand-canonical formulation. In the single molecule limit, the system undergoes two different phase transitions upon decreasing temperature: a $\Theta$-like collapse from a swollen “coil” state to a “molten” state, with a low fraction of doubly visited bonds, and subsequently to a “paired” state, with empty or doubly visited bonds only. The stacking effect drives the latter transition from second to first order.

Figure 1

Sketch of a Bethe lattice with $k=2$.

Figure 2

Chemical potential-temperature ($\mu ∕\beta$ vs $1∕\beta$) phase diagram for the zero stacking case ($\gamma =0$, upper graph) and a nonzero stacking case ($\gamma ∕\beta =0.2$, lower graph). Solid lines denote first order transitions; dashed lines denote second order transitions. The ordinary dense phase is denoted by I, whereas the fully paired dense phase is denoted by II. The zero density phase is left blank. The insets display the regions enclosed in the small rectangles.

Figure 3

Fraction of paired segments as a function of temperature ($\varphi$ vs $1∕\beta$) in the single-molecule limit, for different values of the stacking ratio $\gamma ∕\beta$. Dashed lines denote discontinuities; a thin solid line connects transition values. The inset displays the region enclosed in the small rectangle.

Figure 4

Entropy per segment as a function of temperature ($s∕\rho$ vs $1∕\beta$) in the single-molecule limit, for different values of the stacking ratio $\gamma ∕\beta$. Dashed lines denote discontinuities; thin solid lines connect transition values. The inset displays the region enclosed in the small rectangle. A dash-dotted line indicates the entropy value $\left(\ln k\right)∕2$.

Figure 5

Average stacking length as a function of temperature ($\lambda$ vs $1∕\beta$) in the single-molecule limit, for different values of the stacking ratio $\gamma ∕\beta$. Dashed lines denote discontinuities; thin solid lines connect transition values; a dash-dotted line, determined by Eqs. (18, 21), connects $\Theta$-point values.

Figure 6

Transition temperatures as a function of the stacking ratio ($1∕\beta$ vs $\gamma ∕\beta$) in the single-molecule limit. Solid lines denote first order transitions; dashed lines denote second order transitions. The inset displays the region enclosed in the small rectangle.