RNA Folding and Combinatory Landscapes
Walter Fontana, Peter F. Stadler, Erich G. Bornberg-Bauer, Thomas
Griesmacher, Ivo L. Hofacker, Manfred Tacker, Pedro Tarazona, Edward D.
Weinberger, and Peter Schuster
In this paper we view the folding of polynucleotide (RNA) sequences as a
map that assigns to each sequence a minimum free energy pattern of base
pairings, known as secondary structure. Considering only the free energy
leads to an energy landscape over the sequence space. Taking into
account structure generates a less visualizable non-scalar ``landscape'',
where a sequence space is mapped into a space of discrete ``shapes.'' We
investigate the statistical features of both types of landscapes by
computing autocorrelation functions, as well as distributions of energy
and structure distances, as a function of distance in sequence space.
RNA folding is characterized by very short structure correlation lengths
compared to the diameter of the sequence space. The correlation lengths
depend strongly on the size and the pairing rules of the underlying
nucleotide alphabet. Our data suggest that almost every minimum free
energy structure is found within a small neighborhood of any random
The interest in such landscape results from the fact that they govern
natural and artificial processes of optimization by mutation and
selection. Simple statistical model landscapes, like Kauffman's $n-k$
model, are often used as a proxy for understanding realistic landscapes,
like those induced by RNA folding. We make a detailed comparison between
the energy landscapes derived from RNA folding and those obtained from
the $n-k$ model. We derive autocorrelation functions for several
variants of the $n-k$ model, and briefly summarize work on its fine
structure. The comparison leads to an estimate for $k = 7$ to $8$,
independent of $n$, where $n$ is the chain length. While the scaling
behaviors agree, the fine structure is considerably different in the two
cases. The reason is seen to be the extremely high frequency of neutral
neighbors, that is: neighbors with identical energy (and structure), in
the RNA case.
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