RNA-Undesign Framework

• RNA-Undesign is a framework that certifies undesignability by proving no RNA sequence can uniquely achieve a target structure under the Turner model.
• It employs techniques such as rival structure generation, structure decomposition, and motif-level analysis to identify and isolate energetic failures in RNA folding.
• The framework integrates both discrete proof methods and ensemble-based approximations, guiding improvements in RNA design and model refinement.

Searching arXiv for RNA-Undesign papers and related designability work. RNA-Undesign is a framework for certifying when an RNA secondary structure is undesignable under thermodynamic folding models, particularly the Turner nearest-neighbor model, and for localizing the specific motifs or substructures responsible for that failure. In the RNA inverse-folding setting, the target is to find a sequence whose unique minimum-free-energy fold is a prescribed structure. RNA-Undesign addresses the complementary problem: determining when no such sequence exists, either for an entire structure or for a motif under contextual constraints. The framework has developed across at least three linked lines of work: rival-structure generation and structure decomposition for unique-MFE undesignability (Zhou et al., 2023), scalable motif-level identification of minimal undesignable motifs with rotational invariance (Zhou et al., 2024), and an ensemble-based extension that studies probabilistic designability through ensemble approximation and dynamic decomposition (Zhou et al., 14 Feb 2026).

1. Concept and formal setting

RNA-Undesign is centered on undesignability: the condition that no sequence can make a given target structure the unique optimum of the folding model. In the 2023 formulation, an RNA sequence $x$ of length $n$ is a string $x_1 \ldots x_n$ over $\{A,C,G,U\}$, and a pseudoknot-free secondary structure $y$ is a nesting of base pairs $P \subseteq \{(i,j)\mid 1 \le i < j \le n\}$ such that each index appears at most once and no crossings occur (Zhou et al., 2023). Under the Turner model, the total free energy decomposes over loops,

$$\Delta G(x,y)=\sum_{\ell \in L(y)} \Delta G_{\text{loop}}(x \text{ restricted to } \ell \text{ critical positions}, \ell),$$

with loop classes including hairpins, stacks, bulges, internal loops, multiloops, and the external loop (Zhou et al., 2023).

The unique-MFE design criterion requires that a target $y$ satisfy

$\forall y' \ne y,\ \Delta G(x,y) < \Delta G(x,y').$

A structure is undesignable if there is no compatible sequence whose unique MFE fold is the target; equivalently,

$$\forall x \text{ compatible with } y,\ \exists y' \ne y \text{ such that } \Delta G(x,y') \le \Delta G(x,y).$$

This formalization makes undesignability a universal statement over compatible sequences and an existential statement over competing structures (Zhou et al., 2023).

The 2024 extension shifts the same logic to motifs. A motif $n$0 is any contiguous subset of loops in a structure, and motif free energy is defined as

$n$1

A motif is undesignable if no sequence can make it the unique-MFE motif when the rest of the enclosing structure is fixed:

$n$2

This motif-level formalization is intended to explain why larger structures fail by isolating local energetic obstructions (Zhou et al., 2024).

2. Rival structures, rival motifs, and sufficient conditions

The foundational mechanism in RNA-Undesign is the construction of rivals that always tie or beat the target in free energy. In the 2023 framework, Theorem 1 states that if there exists a rival structure $n$3 such that

$n$4

then $n$5 is undesignable (Zhou et al., 2023). Practical checking is reduced to the set of positions where loop energies differ between $n$6 and $n$7, denoted $n$8, so only those “critical positions” need to be enumerated (Zhou et al., 2023).

Theorem 2 generalizes this to a finite rival set $n$9 such that for every compatible sequence, at least one rival is no worse than the target:

$x_1 \ldots x_n$0

where $x_1 \ldots x_n$1 (Zhou et al., 2023). This makes undesignability certifiable even when no single rival dominates globally.

The 2024 motif framework restates the same logic locally. For a single rival motif, if one can show

$x_1 \ldots x_n$2

then $x_1 \ldots x_n$3 is undesignable; more generally, a small set of rivals $x_1 \ldots x_n$4 suffices when for every sequence at least one rival has no higher energy (Zhou et al., 2024). The paper emphasizes that one only needs to check the energy difference over the “differential positions” where the motifs differ (Zhou et al., 2024). This local formulation is significant because it transforms undesignability from a diffuse global property into a concrete energetic comparison between structurally similar alternatives.

A related sufficient condition is structure decomposition. Theorem 3 of the 2023 work states that if a substructure $x_1 \ldots x_n$5 enclosed by a base pair $x_1 \ldots x_n$6 is context-constrained-undesignable under the requirement that $x_1 \ldots x_n$7 be forced, then the full structure $x_1 \ldots x_n$8 is also undesignable (Zhou et al., 2023). This decomposition principle is one of the main reasons the framework scales beyond brute-force enumeration.

3. Algorithmic framework

RNA-Undesign realizes these theorems through a sequence of algorithms that alternate between search and proof. In the 2023 work, Algorithm 1 takes a target $x_1 \ldots x_n$9 and a candidate rival $\{A,C,G,U\}$0, computes the critical positions in the differing loops, enumerates base assignments on those positions subject to compatibility, and evaluates the free-energy difference. If no assignment violates $\{A,C,G,U\}$1, the target is certified undesignable (Zhou et al., 2023). Its reported complexity is $\{A,C,G,U\}$2 where $\{A,C,G,U\}$3 is the number of differing paired positions and $\{A,C,G,U\}$4 the number of differing unpaired positions (Zhou et al., 2023).

Algorithm 2 handles multiple rivals. It maintains a residual design space of sequences compatible with the target, repeatedly samples sequences, folds them to generate MFE competitors, applies the one-rival test when feasible, and intersects the residual design space with the resulting design constraints. If the residual space becomes empty, the structure is declared undesignable (Zhou et al., 2023). The stated complexity is

$\{A,C,G,U\}$5

for at most $\{A,C,G,U\}$6 rivals, at most $\{A,C,G,U\}$7 foldings per rival, and a constraint-enumeration cost $\{A,C,G,U\}$8 (Zhou et al., 2023).

Algorithm 3 applies decomposition over enclosed substructures. For each base pair $\{A,C,G,U\}$9 in the target, it isolates the context-constrained substructure $y$0, runs the multi-rival procedure on that subproblem, and declares the full target undesignable if any substructure is certified (Zhou et al., 2023). Its stated complexity is

$y$1

The 2024 paper presents a motif analogue, “RivalMotifSearch,” with high-level pseudocode that maintains a candidate set of rival motifs and a design space of sequences not yet ruled out. The procedure repeats by either declaring the motif undesignable when the design space consistent with all rivals becomes empty, or by sampling sequences, folding to obtain a competing motif, and adding that rival if it contributes new differential positions (Zhou et al., 2024). The reported complexity is

$y$2

where $y$3 is the maximum allowed rivals, $y$4 is the limit on differential-position size, and $y$5 is the number of sampled sequences per iteration (Zhou et al., 2024).

A plausible implication is that RNA-Undesign is best understood not as a single decision procedure but as a proof system whose certificates are rival structures, rival motifs, or context-constrained substructures.

4. Motif-level analysis and rotational invariance

The 2024 work extends RNA-Undesign from whole-structure certification to the identification of minimal undesignable motifs. Its central motivation is that understanding local structures that contribute to undesignability is crucial for refining RNA folding models and determining the limits of RNA designability (Zhou et al., 2024). The framework defines motifs as contiguous subsets of loops and uses minimality in the sense that no proper sub-motif is already undesignable (Zhou et al., 2024).

A major technical addition is rotational invariance. The paper argues that because Turner-model loop energies do not depend on absolute embedding in the full structure, motifs that differ only by a rotation of loop-pair connections have identical energy behavior (Zhou et al., 2024). To exploit this, it introduces a loop-pair graph representation. In this graph, there are loop nodes of types such as $y$6, $y$7, $y$8, $y$9, and $P \subseteq \{(i,j)\mid 1 \le i < j \le n\}$0, pair nodes of type $P \subseteq \{(i,j)\mid 1 \le i < j \le n\}$1, and a special root node $P \subseteq \{(i,j)\mid 1 \le i < j \le n\}$2 for the external loop; edges connect each loop to each of its boundary pairs, and each edge is weighted by the number of unpaired nucleotides between those pairs along the backbone (Zhou et al., 2024).

The associated recursive graph-isomorphism procedure canonicalizes a motif graph under rotation by choosing a boundary pair node as a new root and recursively reordering neighbor subtrees by rotating the child list. The paper states that this runs in linear time in the number of nodes in the motif graph (Zhou et al., 2024). Two motifs are rotationally isomorphic if their loop-pair graphs can be made identical by canonical rooting at some boundary pair (Zhou et al., 2024).

This machinery enables grouping equivalent motifs, reusing proofs across occurrences, and constructing a database of unique minimal undesignable motifs rather than reporting each occurrence independently. In interpretive terms, rotational invariance converts motif discovery from a purely local search into a reusable structural taxonomy.

5. Empirical findings and benchmark results

The 2023 study applied RNA-Undesign to the 22 Eterna100 puzzles that two leading design programs, NEMO and SAMFEO, failed to solve under ViennaRNA 2.5.1 parameters. Using parameters $P \subseteq \{(i,j)\mid 1 \le i < j \le n\}$3, $P \subseteq \{(i,j)\mid 1 \le i < j \le n\}$4, and $P \subseteq \{(i,j)\mid 1 \le i < j \le n\}$5, it proved 15 of the 22 to be rigorously undesignable within seconds to minutes each (Zhou et al., 2023). Reported examples include puzzle 50 of length 105 solved with Algorithm 1 in $P \subseteq \{(i,j)\mid 1 \le i < j \le n\}$6, puzzle 52 of length 80 solved with Algorithm 1 in $P \subseteq \{(i,j)\mid 1 \le i < j \le n\}$7, puzzle 57 of length 36 solved with Algorithm 3 in $P \subseteq \{(i,j)\mid 1 \le i < j \le n\}$8, puzzle 88 of length 34 solved with Algorithm 2 using 9 rivals in $P \subseteq \{(i,j)\mid 1 \le i < j \le n\}$9, puzzle 96 of length 358 solved with Algorithm 3 in $$\Delta G(x,y)=\sum_{\ell \in L(y)} \Delta G_{\text{loop}}(x \text{ restricted to } \ell \text{ critical positions}, \ell),$$0, and puzzle 99 of length 364 solved with Algorithm 3 using 2 rivals in $$\Delta G(x,y)=\sum_{\ell \in L(y)} \Delta G_{\text{loop}}(x \text{ restricted to } \ell \text{ critical positions}, \ell),$$1 (Zhou et al., 2023). The same study also reports that “Short String 4” is designable, since Algorithm 2 produced a sequence whose MFE fold is uniquely the target, and that five puzzles remained uncertain under the chosen settings (Zhou et al., 2023).

The 2024 study reports broader motif-level statistics. On the Eterna100 benchmark, it identified 18 structures proved undesignable under the unique-MFE criterion, extracted 36 minimal undesignable motif occurrences, and grouped them into 24 rotationally unique motifs (Zhou et al., 2024). On ArchiveII, described as 3,957 structures across 10 families, it found that 663 structures, or 17%, contain at least one undesignable motif; these yielded 961 total minimal undesignable motif occurrences, collapsed to 331 unique motifs within ArchiveII and 355 unique motifs overall when combined with Eterna100 (Zhou et al., 2024).

The 2026 paper shifts from MFE-based criteria to ensemble-based notions of designability. Its abstract states that recent advances in RNA designability had focused primarily on MFE-based criteria while ensemble-based notions remained underexplored. It introduces a theory of ensemble approximation and a probability decomposition framework for bounding folding probabilities of RNA structures in an explainable way, together with a linear-time dynamic programming algorithm that efficiently searches over exponentially many decompositions and identifies the optimal one yielding the tightest probabilistic bound for a given structure (Zhou et al., 14 Feb 2026). Applied to native and artificial RNA structures in the ArchiveII and Eterna100 benchmarks, these methods produced probability bounds much tighter than prior approaches and provided anatomical tools for analyzing RNA structures and understanding sources of design difficulty at the motif level (Zhou et al., 14 Feb 2026).

6. Representative motifs, software resources, and interpretation

The motif paper gives concrete examples of minimal undesignable motifs. One example from Eterna100 puzzle 52 consists of an internal bulge loop closed by pairs $$\Delta G(x,y)=\sum_{\ell \in L(y)} \Delta G_{\text{loop}}(x \text{ restricted to } \ell \text{ critical positions}, \ell),$$2 plus a hairpin loop on $$\Delta G(x,y)=\sum_{\ell \in L(y)} \Delta G_{\text{loop}}(x \text{ restricted to } \ell \text{ critical positions}, \ell),$$3; the rival motif deletes the internal pair $$\Delta G(x,y)=\sum_{\ell \in L(y)} \Delta G_{\text{loop}}(x \text{ restricted to } \ell \text{ critical positions}, \ell),$$4, leaving only the hairpin $$\Delta G(x,y)=\sum_{\ell \in L(y)} \Delta G_{\text{loop}}(x \text{ restricted to } \ell \text{ critical positions}, \ell),$$5, and the argument is that the stacking bonus from the two-pair stack is never sufficient to overcome the penalty of the extra bulge loop, so $$\Delta G(x,y)=\sum_{\ell \in L(y)} \Delta G_{\text{loop}}(x \text{ restricted to } \ell \text{ critical positions}, \ell),$$6 for every assignment of critical nucleotides (Zhou et al., 2024). A second example from Eterna100 puzzle 57 is a motif involving two stacks plus a 3-way multiloop, where no single deletion suffices but a rival set $$\Delta G(x,y)=\sum_{\ell \in L(y)} \Delta G_{\text{loop}}(x \text{ restricted to } \ell \text{ critical positions}, \ell),$$7 covers all sequences under Theorem 4 (Zhou et al., 2024). A third example from a tRNA family in ArchiveII involves the acceptor stem and D-loop as a two-loop motif closed by pairs $$\Delta G(x,y)=\sum_{\ell \in L(y)} \Delta G_{\text{loop}}(x \text{ restricted to } \ell \text{ critical positions}, \ell),$$8, with a rival formed by deleting the inner pair so that the D-loop merges into the external region; this is argued to always lower the hairpin penalty more than any stacking gain (Zhou et al., 2024).

These examples align with a broader interpretation in the 2023 study: many undesignable cases arise from tiny local motifs, including isolated or double base pairs, that act as unavoidable energetic traps; undesignability is often local and context-decomposable rather than solely a consequence of global symmetry; and when multiple rivals are needed, the number of rivals is small and the rivals are structurally similar, suggesting a compact partition of sequence space (Zhou et al., 2023). These are presented as insights from the empirical study rather than universal theorems.

RNA-Undesign is distributed as software and a motif database. The 2024 paper states that the C++ source code is available at https://github.com/shanry/RNA-Undesign and that a web server is available at http://linearfold.org/motifs (Zhou et al., 2024). It further states that all 355 motifs, spanning 5 to 203 nt and cardinalities 2 to 5 loops, are indexed on the server (Zhou et al., 2024). The 2026 abstract likewise lists the same GitHub repository as the location of source code and data for the ensemble-based extension (Zhou et al., 14 Feb 2026).

7. Relation to RNA designability and open directions

RNA-Undesign occupies a complementary position relative to standard inverse-folding methods. Conventional RNA design seeks a sequence that folds into a target, whereas RNA-Undesign seeks proofs that no such sequence exists under the adopted model. The 2023 work frames this as useful for preventing wasted compute on intractable design tasks, serving as a sanity check for heuristic design tools, and illuminating limitations or bias in energy models (Zhou et al., 2023). The 2024 work sharpens that perspective by locating specific local motifs that make otherwise complex structures fail (Zhou et al., 2024).

A common misconception is that undesignability is necessarily a global property of highly symmetric or otherwise visually complex structures. The reported decomposition and motif results point in a different direction: many failures can be attributed to small local substructures, and these can recur across unrelated RNAs and families (Zhou et al., 2023, Zhou et al., 2024). Another possible misconception is that undesignability is only meaningful under an MFE criterion. The 2026 extension suggests that this is incomplete, since ensemble-based notions of designability can also be bounded and anatomized through interpretable decomposition (Zhou et al., 14 Feb 2026).

The current trajectory of the framework therefore has two distinct but connected axes. One axis is proof-oriented and discrete: rival structures, rival motifs, minimal undesignable motifs, and rotational canonicalization (Zhou et al., 2023, Zhou et al., 2024). The other is probabilistic and ensemble-oriented: approximation theory, probability decomposition, and optimal decompositions for tight folding-probability bounds (Zhou et al., 14 Feb 2026). This suggests that RNA-Undesign has evolved from a method for certifying unique-MFE impossibility into a broader program for understanding the limits of RNA designability at both structure and motif scales.