Model for Folding and Aggregation in RNA Secondary Structures

We study the statistical mechanics of RNA secondary structures designed to have an attraction between two different types of structures as a model system for heteropolymer aggregation. The competition between the branching entropy of the secondary structure and the energy gained by pairing drives the RNA to undergo a “temperature independent” second order phase transition from a molten to an aggregated phase. The aggregated phase thus obtained has a macroscopically large number of contacts between different RNAs. The partition function scaling exponent for this phase is $\theta \approx 1/2$ and the crossover exponent of the phase transition is $\nu \approx 5/3$. The relevance of these calculations to the aggregation of biological molecules is discussed.

Figure 1

Abstract representations of RNA secondary structures (from Ref. 4). (a) Helix representation (b) Noncrossing arch diagram. Here, the solid line corresponds to the backbone of the RNA. The dashed arches correspond to the base pairs. The absence of pseudoknots implies that the arches never cross. (c) Mountain representation. Here, as we go along the backbone of the RNA from base 1 to $N$ (represented by the base line), we go one step up for the beginning of a pair, one step down for the closing of a pair, and a horizontal step for no pairing. Such a mountain never crosses the baseline and always returns to the baseline at the end.

Figure 2

The behavior of the partition function ${\widehat{Z}}_d(z;q_1=4,q_2=9,q_3)$. For $q_3=q_{3c}=6$, we observe a square root behavior. For $q_3>q_{3c}$, we see an inverse square root behavior. The inset shows the resulting phase diagram.

Figure 3

Scaling plot for the order parameter. Inset shows the order parameter of the phase transition. In both the plots $q_1=4$ and $q_2=9$, hence $q_{3c}=6$.