H if if if

Fig. 2.6 Four cloverleaf RNA structures used tures in natural tRNAPhe with stack lengths in the application of inverse folding to study 6/4/5/5. (b)-(d) Successively longer stacks,

RNA shape space. The structures differ in the 6/5/5/5, 6/5/6/6, and 6/6/6/6, respectively lengths of stacks and hairpin loops. (a) Struc-

Herein nbp(S) is the number of base pairs in structure S and p is the number of allowed base pairs of the alphabet. In the natural alphabet, AUGC, we have p = 6 and k = 4, and hence Rcov(S) =0.625 nbp(S). Evolutionary studies on aptamers and ribozymes essentially confirmed the concept of shape space covering in any sufficiently large part of sequence space (Huang and Szostak, 2003; Hughes et al., 2004). The distribution of RNA sequences belonging to a neutral network is not, of course, completely random and deviations from the predictions of the random graph model can be derived from details of the secondary structure (Reidys et al., 1997; Schuster and Stadler, 2004).

In order to illustrate the influence of structure on the frequency of occurrence and the degree of neutrality we consider in Table 2.1 the example of the four cloverleaf structures shown in Fig. 2.6. The frequency of occurrence is approximated by the probability of finding a sequence for the structure by means ofinverse folding (pinv). The degree of neutrality, l, has been computed by consecutive summation over the one-error neighborhoods of individual sequences calculated by inTable 2.1 Degrees of neutrality and probabilities to find a sequence through a successful inverse folding trial are shown for four cloverleaf structures (Fig. 2.6) in five different nucleo-tide alphabets

Alphabet Structure a Structure b Structure c Structure d

l

Pinv

l

Pinv

l

Pinv

l

Pinv

AU

-

-

-

-

-

-

0.073

± 0.032

0.051

AUG

-

-

0.217

± 0.063

0.003

0.0207

± 0.055

0.026

0.201

± 0.056

0.374

AUGC

0.275

± 0.064

0.794

0.279

± 0.063

0.884

0.289 ±

0.062

0.934

0.313

± 0.058

0.982

UGC

0.263

± 0.071

0.584

0.257

± 0.070

0.628

0.251 ±

0.068

0.697

0.250

± 0.064

0.818

GC

0.052

± 0.033

0.067

0.057

± 0.034

0.086

0.060 ±

0.033

0.087

0.068

± 0.034

0.127

verse folding until convergence was obtained. The four cloverleaf structures differ by the lengths of individual stacks. Since A=U base pairs are too weak to form stable short stacks, structures a, b, and c are rarely formed by sequences from AU and AUG alphabets. The degree of neutrality, one the other hand, is strongly influenced by the cardinality of the alphabet (k). Although sequences forming clo-verleaves with structure d are 2.5 times more likely to be found in the GC than in the AU alphabet, the degree of neutrality is approximately the same.

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