Much progress has been made on the problem of assigning free energy values to substructures. Although considerable theoretical work has been done, the most useful free energy data have been extrapolated from experiments on particular kinds of RNA. Much of the most important work has been carried out by Tinoco and Uhlenbeck [Tinoco, Uhlenbeck and Levine, 1971; Tinoco et al., 1973].
A reasonable first attempt at solving the 2˚RNA problem would probably incorporate a detailed physical model of molecular structure.
A mathematician might define a ball and stick model (balls for nucleotides, sticks for bonds) of an RNA molecule of length N, with 2N – 4 variable angles and (N – 1)(N – 2)/2 potential energy functions for all pairwise hydrogen bond interactions. But the number of possible conformations is then an exponential function of the degrees of freedom. Such a model would prove computationally intractable for even the smallest RNA molecules.
Fortunately, Tinoco and others have simplified the problem, arguing that only the existence or nonexistence of particular hydrogen bonds matters; they have also provided empirical evidence that this simpler model has predictive power. Methods for relating free energy values to the size, shape, and base composition of secondary substructures, sometimes known as the “Tinoco Rules”, can be viewed as a means of abstracting away from much of the complex thermodynamics of hydrogen bonding, Van der Waals forces, rotation of covalent bonds, and steric hindrance.
The Tinoco free energy data may be found in Tinoco, Uhlenbeck and Levine, 1971; Tinoco et al., 1973]. Summarized below are the most important general ideas. It is important to qualify these ideas by noting that the E(c) free energy estimates for cycles are only estimates. The values cannot be determined with great accuracy, but they serve as useful, if sometimes crude, approximations of physical reality.
The most stable secondary structures, those having the lowest free energy, are long chains of stacked pairs. That is, a stem is the only kind of cycle which contributes negative free energy to the structure. The particular free energy value for a given stacked pair depends upon the two bases that are bonding, as well as a local context, i.e., the base composition of the closest stacked pairs to its upper right and/or lower left in the matrix.
Loops and bulges raise the free energy roughly in proportion to their size, that is, the number of elements that are left unpaired between the two elements that are paired. Beyond a certain size, loop and bulge energies seem to grow proportionally to the log of the unpaired length. Thus, a certain minimum number of stacked pairs is required to support a loop or bulge interior to the stacked pairs