Factorization Machine with Quadratic-Optimization Annealing for RNA Inverse Folding and Evaluation of Binary-Integer Encoding and Nucleotide Assignment

Abstract: The RNA inverse folding problem aims to identify nucleotide sequences that preferentially adopt a given target secondary structure. While various heuristic and machine learning-based approaches have been proposed, many require a large number of sequence evaluations, which limits their applicability when experimental validation is costly. We propose a method to solve the problem using a factorization machine with quadratic-optimization annealing (FMQA). FMQA is a discrete black-box optimization method reported to obtain high-quality solutions with a limited number of evaluations. Applying FMQA to the problem requires converting nucleotides into binary variables. However, the influence of integer-to-nucleotide assignments and binary-integer encoding on the performance of FMQA has not been thoroughly investigated, even though such choices determine the structure of the surrogate model and the search landscape, and thus can directly affect solution quality. Therefore, this study aims both to establish a novel FMQA framework for RNA inverse folding and to analyze the effects of these assignments and encoding methods. We evaluated all 24 possible assignments of the four nucleotides to the ordered integers (0-3), in combination with four binary-integer encoding methods. Our results demonstrated that one-hot and domain-wall encodings outperform binary and unary encodings in terms of the normalized ensemble defect value. In domain-wall encoding, nucleotides assigned to the boundary integers (0 and 3) appeared with higher frequency. In the RNA inverse folding problem, assigning guanine and cytosine to these boundary integers promoted their enrichment in stem regions, which led to more thermodynamically stable secondary structures than those obtained with one-hot encoding.

• The paper demonstrates that FMQA with one-hot and domain-wall encodings generally outperforms binary and unary representations for RNA inverse folding, while using fewer evaluations than random search, TPE, and genetic algorithms.
• The paper finds that domain-wall nucleotide assignments materially affect outcomes because asymmetric Hamming distances bias nucleotide frequencies, with boundary placement of G and C promoting favorable GC-rich stem formation.
• The paper establishes FMQA as a promising evaluation-efficient method for short, pseudoknot-free RNA targets, but notes that results remain limited by small benchmarks, computational objectives, and the absence of experimental validation.

Kikuchi and Tanaka apply factorization machine with quadratic-optimization annealing (FMQA) to the RNA inverse folding problem and, in doing so, conduct what is, to their knowledge, the first systematic analysis of how binary-integer encoding choices and integer-to-nucleotide assignments affect FMQA performance on categorical variables. The work is notable less for its absolute solution quality than for its methodological finding: in FMQA, representation choices that are mathematically equivalent at the integer level can produce materially different search behavior and thermodynamic outcomes.

Background and motivation

The RNA inverse folding problem seeks a nucleotide sequence whose minimum free energy (MFE) structure matches a target secondary structure. The problem is NP-hard under both general and simplified energy models (2602.16643). Existing approaches—stochastic local search, genetic algorithms, constraint programming, Monte Carlo tree search, reinforcement learning, generative models—achieve high success rates but typically require many objective evaluations or large training corpora. The authors motivate FMQA as a discrete black-box optimization (BBO) framework suited to settings where each evaluation is expensive, e.g., wet-lab validation.

FMQA trains a factorization machine (FM) surrogate,

$$\mathcal{H}_{\mathrm{FM}}(\{\bm{x}\}) = \omega_0 + \sum_i \omega_i x_i + \sum_{i<j} \langle \bm{v}_i, \bm{v}_j \rangle x_i x_j,$$

and minimizes it directly with an Ising machine by exploiting the algebraic equivalence between the FM's interaction terms and a QUBO matrix: diagonal entries absorb $\omega_i$ (since $x_i^2 = x_i$), and off-diagonal entries are inner products of latent vectors. This bypasses manual QUBO construction, which is otherwise a prerequisite for Ising-machine solvers.

Objective function and problem formulation

The black-box function is the normalized ensemble defect (NED), defined as the Boltzmann-weighted expected number of nucleotides whose pairing status differs from the target, divided by sequence length $L$. NED was chosen over MFE-based structure distance because prior work (Dirks et al.; Ward et al.) shows it correlates better with sequences that uniquely adopt the target structure. Folding predictions use ViennaRNA v2.7.2 under the Turner nearest-neighbor model with --dangles=2 at 37 °C; pseudoknots are excluded.

Nucleotides (categorical) are mapped to integers 0–3 via one of 24 permutations, then to binary variables via one-hot ($4L$ bits), domain-wall ($3L$), binary ($2L$), or unary ($3L$) encoding. One-hot and domain-wall encodings require penalty terms enforcing the exactly-one-active-variable and single-domain-wall constraints respectively ($\mu = 2$, with no infeasible solutions observed).

Main results

Encoding comparison. Across all 24 assignments × 4 encodings × 10 runs on the 26-nt Eterna100 structure stickshift, one-hot and domain-wall encodings consistently achieved lower NED than binary, with unary worst (some conditions yielding success rate 0.0). One-hot performance was assignment-invariant; domain-wall was not. The authors attribute this ordering to surrogate expressivity rather than Ising-machine optimization quality: binary encoding compresses four states into two bits, limiting the FM's capacity for nonlinear categorical interactions, while unary's many-to-one mapping introduces representational degeneracy the FM must implicitly resolve. This contrasts with earlier Ising-machine studies where binary/unary encodings sometimes outperformed one-hot—supporting the interpretation that in FMQA, the bottleneck is surrogate modeling, not annealer search.

Assignment effects in domain-wall encoding. Four assignments—(A,G,C,U), (A,C,G,U), (U,G,C,A), (U,C,G,A)—yielded high NED and low success rates; the complementary set placing G or C at integer 0 yielded the best results. Nucleotide-frequency analysis explains this mechanistically: domain-wall encoding exhibits asymmetric Hamming distances between integer representations, and boundary integers 0 and 3 have only one feasible neighbor versus two for middle integers, biasing transitions away from them. Consequently, nucleotides assigned to 0 and 3 appear more frequently in solutions. Assigning G and C to boundary integers therefore enriches G-C pairs in stem regions—the most favorable stacks in the Turner model—producing lower MFE structures than one-hot encoding achieves. This is the paper's central practical insight: domain-wall encoding converts an arbitrary labeling choice into a controllable thermodynamic bias, whereas one-hot's uniform pairwise Hamming distance of two makes it assignment-insensitive.

Evaluation-efficiency comparison. Against random search, TPE Bayesian optimization, and a GA (all sharing the same 10-point initial dataset and 1000-evaluation budget), FMQA reached lower best-NED with fewer evaluations. This is the key claim relevant to the paper's motivating scenario of experimentally costly evaluation.

Multi-structure evaluation. On eight short Eterna100 targets (12–36 nt) using the (G,A,U,C) assignment, domain-wall encoding generally outperformed one-hot in both NED and MFE. Success rates were high for G-C Placement, Simple Hairpin, stickshift, and Corner bulge training—including a bulge-containing motif often considered difficult—but near zero for Shortie 4 and Small and Easy 6, both featuring very short stems flanked by unpaired nucleotides, and degraded for the 36-nt structures, reflecting both local thermodynamic instability and the $4^L$ growth of the search space.

Limitations and open questions

The authors are explicit about several constraints. All experiments used computationally evaluated NED rather than experimental measurements, so the claimed sample efficiency advantage in wet-lab contexts remains inferred rather than demonstrated. Only one small benchmark family (Eterna100, 12–36 nt, pseudoknot-free) was tested, with 10 runs per condition; no direct comparison against established inverse-folding tools (RNAinverse, INFO-RNA, MODENA, SentrnaEtc.) was possible because their objective functions differ. GC content was not constrained, making free-energy minimization the de facto driver of success—an assumption that may not hold for design objectives requiring controlled composition. Whether the boundary-integer bias of domain-wall encoding transfers beneficially to longer sequences or other categorical BBO problems is left open, as is systematic benchmarking of FMQA against the established inverse-folding literature.

Conclusion

The paper establishes FMQA as a viable, evaluation-efficient approach to RNA inverse folding on short pseudoknot-free targets and demonstrates that encoding choice and integer-to-nucleotide assignment are consequential design parameters, not implementation details. The identification of a Hamming-distance-driven frequency bias in domain-wall encoding—and its exploitation to enrich G-C stems—offers a concrete guideline for applying FMQA to categorical optimization problems more broadly.