ASYMPTOTICS OF CANONICAL AND SATURATED RNA SECONDARY STRUCTURES
Bibliographic record
Abstract
It is a classical result of Stein and Waterman that the asymptotic number of RNA secondary structures is 1.104366 . n(-3/2) . 2.618034(n). In this paper, we study combinatorial asymptotics for two special subclasses of RNA secondary structures - canonical and saturated structures. Canonical secondary structures are defined to have no lonely (isolated) base pairs. This class of secondary structures was introduced by Bompfünewerer et al., who noted that the run time of Vienna RNA Package is substantially reduced when restricting computations to canonical structures. Here we provide an explanation for the speed-up, by proving that the asymptotic number of canonical RNA secondary structures is 2.1614 . n(-3/2) . 1.96798(n) and that the expected number of base pairs in a canonical secondary structure is 0.31724 . n. The asymptotic number of canonical secondary structures was obtained much earlier by Hofacker, Schuster and Stadler using a different method. Saturated secondary structures have the property that no base pairs can be added without violating the definition of secondary structure (i.e. introducing a pseudoknot or base triple). Here we show that the asymptotic number of saturated structures is 1.07427 . n(-3/2) . 2.35467(n), the asymptotic expected number of base pairs is 0.337361 . n, and the asymptotic number of saturated stem-loop structures is 0.323954 . 1.69562(n), in contrast to the number 2(n - 2) of (arbitrary) stem-loop structures as classically computed by Stein and Waterman. Finally, we apply the work of Drmota to show that the density of states for [all resp. canonical resp. saturated] secondary structures is asymptotically Gaussian. We introduce a stochastic greedy method to sample random saturated structures, called quasi-random saturated structures, and show that the expected number of base pairs is 0.340633 . n.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.020 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".