💻 bioinformatics

PKProbDesign: RNA inverse folding including pseudoknots by optimizing thermodynamic folding probability

PKProbDesign is a novel sampling-based framework that addresses the challenge of RNA inverse folding for pseudoknotted structures by directly optimizing thermodynamic folding probabilities through scaffold decomposition and conditional ensemble evaluation, outperforming existing methods like DesiRNA and MODENA on benchmark targets.

Original authors: Otagaki, T., Iwakiri, J., terai, g., Asai, K., Sato, K.

Published 2026-07-11

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Otagaki, T., Iwakiri, J., terai, g., Asai, K., Sato, K.

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine you're trying to design a secret code (an RNA sequence) that, when you whisper it to the universe, folds itself into a very specific, intricate origami shape (a target structure). This is the game of "RNA inverse folding." Usually, the shapes we want to make are simple, nested boxes. But the most interesting, functional shapes in nature often have "pseudoknots"—twists and turns where the paper crosses over itself, like a pretzel. These are notoriously hard to design because standard folding rules break down when things cross.

Enter PKProbDesign, a new tool created by Takumi Otagaki and his team. Think of it as a master architect who doesn't just guess if a blueprint looks right; instead, they calculate the exact odds that a specific design will actually snap into place.

The Old Way vs. The New Way

For a long time, designers played a game of "Minimum Free Energy" (MFE). It's like asking, "If I build this, is this the single most stable shape it could possibly be?" If the answer is yes, they called it a win.

But the paper argues that this is a trap. Just because a shape is the most stable doesn't mean it's the only shape the RNA will try to make. It's like building a house where the roof is the strongest part, but the walls keep collapsing into a pile of bricks. The RNA might spend most of its time as a pile of bricks, even if the roof is technically the "best" spot.

The authors explicitly reject relying solely on whether the predicted structure matches the target. Instead, they suggest we should care about folding probability: "How likely is it that this RNA will actually be in the shape we want, out of all the wiggly possibilities it could be?"

How PKProbDesign Works: The Scaffold and the Extension

Designing a pretzel-shaped RNA is tough. So, PKProbDesign breaks the problem into two manageable pieces, like building a house with a pre-fab frame and then adding the tricky roof.

The Scaffold (The Frame): First, the tool picks out a simple, non-crossing part of the target structure. It calculates how likely the RNA is to form this basic frame.
The Extension (The Roof): Then, it looks at the remaining "crossing" parts (the pseudoknots) and asks, "Given that the frame is already built, how likely is the RNA to fold the rest of the way to complete the pretzel?"

By combining these two probabilities, the tool creates a "pseudo-joint" score. It's not just asking, "Does it look like the target?" It's asking, "What are the thermodynamic odds that this sequence will actually become the target?"

The Big Test: 354 Pretzels

The team put PKProbDesign to the test against three other famous design methods: DesiRNA, MODENA, and antaRNA. They used a database called PseudoBase++, which contains 354 real-world RNA targets with pseudoknots (specifically, a class called "density-2," which covers nearly all the targets in their dataset).

Here is what the numbers say:

PKProbDesign came out on top for 221 of the 354 targets.
DesiRNA won on 117 targets.
MODENA won on only 16 targets.

The authors suggest that while DesiRNA is great at building the "scaffold" (the frame), PKProbDesign is better at figuring out the "extension" (the tricky crossing parts). When you look at the combined score, PKProbDesign suggests it is the most reliable method for getting the highest folding probability.

The "Hotspot" Twist

The paper also tested how well these designs looked when predicted by standard software. They used three different "prediction protocols," including one that uses "hotspots" (areas where the RNA is likely to start folding) to guide the search.

In a specific case study involving a target called PKB00340, PKProbDesign produced a sequence that not only had a high folding probability but also looked very close to the target when predicted using the hotspot-guided method. However, the authors are careful to note that a high probability score doesn't guarantee the predicted structure will match perfectly every time. Sometimes a high-probability design still looks a bit different in the simulation, and sometimes a moderate-probability one looks perfect. They suggest we should view the probability score as the primary measure of a design's "promise," while structure prediction is a secondary check.

What This Means (and What It Doesn't)

The paper suggests that we can successfully design complex, knotted RNA shapes by focusing on thermodynamic probabilities rather than just guessing if a structure matches.

However, there are limits. The method is currently restricted to "density-2" structures (a specific type of knot). It doesn't cover every single possible RNA shape in nature yet. Also, the results are based on computer simulations and mathematical models; the authors note that real-world biological validation (like testing in a lab) is still needed to prove these designs work in living cells.

In short, PKProbDesign is a powerful new compass for navigating the messy, knotted world of RNA design, showing us that if we want the RNA to fold correctly, we need to care about the odds of it happening, not just the possibility.

Technical Summary of PKProbDesign

Problem Statement
RNA inverse folding involves designing RNA sequences that fold into specified target secondary structures. While essential for functional RNA engineering and synthetic biology, this task is particularly challenging for targets containing pseudoknots. Standard thermodynamic folding models assume nested structures, which pseudoknots violate, rendering exact folding prediction computationally infeasible for general cases (NP-hard). Existing inverse-folding methods for pseudoknotted targets (e.g., antaRNA, DesiRNA, MODENA) typically rely on multi-objective search strategies that optimize for agreement with a predicted structure or use structure-predictor-based objectives. A significant gap remains in directly optimizing the thermodynamic folding probability of a specified pseudoknotted target within a pseudoknot-aware ensemble. Such an approach requires an evaluator capable of assigning target-specific folding probabilities to serve as a direct optimization signal.

Methodology
The authors present PKProbDesign, a sampling-based inverse-folding framework designed specifically for density-2 pseudoknotted targets. This class of structures, defined by HFold and CParty, allows crossing interactions to be organized into "bands" such that no sequence position is spanned by more than two bands.

The methodology proceeds through three main stages:

Target Decomposition:
Each density-2 target structure $R$ is decomposed into two disjoint, pseudoknot-free components: a scaffold ( $G$ ) and an extension component ( $G'$ ), such that $R = G \cup G'$ . This is achieved via a greedy algorithm that sorts base pairs by increasing span and assigns non-crossing pairs to the scaffold until the remaining pairs form the extension.
Probabilistic Objective Formulation:
Instead of optimizing for minimum free energy (MFE) structure agreement, PKProbDesign optimizes a pseudo-joint folding probability objective. For a sequence $S$ , the score is defined as:
$\tilde{P}_{joint}(S; R, G) = P(G | S) \cdot P(G' | G, S)$
- $P(G | S)$ : The folding probability of the scaffold, evaluated using LinearPartition.
- $P(G' | G, S)$ : The conditional folding probability of the extension component given the fixed scaffold. This is evaluated using a CParty-consistent approach, which computes the partition function over extension structures conditioned on the scaffold, ensuring consistency with the hierarchical folding hypothesis.
  The optimization target is the expected negative log of this pseudo-joint probability: $J(\theta) = E_{S \sim p_\theta}[-\log \tilde{P}_{joint}(S; R, G)]$ .
Sampling-Based Optimization:
Following the approach of SamplingDesign, the method optimizes a probability distribution $p_\theta(S)$ over the valid sequence space rather than a single sequence.
- Representation: Unpaired positions are modeled as 4-way categorical variables ({A, C, G, U}), while base pairs are modeled as 6-way joint categorical variables over canonical pairs ({AU, UA, GC, CG, GU, UG}). This ensures sampled sequences inherently satisfy pairing constraints.
- Gradient Estimation: The gradient of the expected score is approximated using Monte Carlo sampling. At each iteration, $K$ sequences are sampled from the current distribution, scored, and used to estimate the gradient $\nabla_\theta J(\theta)$ .
- Update: Distribution parameters are updated using the Adam optimizer. The final output is the highest-scoring sequence sampled from the optimized distribution.

Key Contributions

Direct Probability Optimization for Pseudoknots: PKProbDesign is the first framework to directly optimize a thermodynamic folding-probability objective for pseudoknotted targets, moving beyond structure-prediction agreement.
Conditional Ensemble Evaluation: It introduces a scoring mechanism that combines scaffold folding probability with a CParty-consistent conditional probability for the extension component, enabling probability-based design within the density-2 class.
Integration of HFold/CParty Semantics: The method leverages the hierarchical folding semantics of HFold and the partition function capabilities of CParty to evaluate sequences in a pseudoknot-aware ensemble.

Results
The method was evaluated on a benchmark of 354 density-2 targets from PseudoBase++ (sequence length $\le$ 100 nt).

Folding Probability Performance: PKProbDesign achieved the highest pseudo-joint folding probability on 221 out of 354 targets. In comparison, DesiRNA achieved the highest probability on 117 targets, and MODENA on 16 targets.
Component Analysis: While DesiRNA showed stronger performance on the scaffold term ( $P(G|S)$ ), PKProbDesign demonstrated superior performance on the conditional extension term ( $P(G'|G,S)$ ) and the combined score. A paired Wilcoxon signed-rank test confirmed PKProbDesign's overall superiority over DesiRNA ( $p = 1.96 \times 10^{-2}$ ).
Structure Prediction: When evaluated using normalized base-pair Hamming distance under three different prediction protocols (Viterbi on target scaffold, Two-stage MFE, and HotKnot hotspot-guided Viterbi), the ranking of methods varied. Notably, MODENA performed strongly under the hotspot-guided protocol, suggesting that high folding probability does not always guarantee the best structure-prediction agreement under all protocols, and vice versa.

Significance and Claims
The paper claims that PKProbDesign demonstrates that pseudoknot inverse folding can be effectively formulated around target folding probabilities rather than relying solely on structure-prediction agreement. By decomposing targets into scaffolds and extensions and utilizing conditional-ensemble evaluation, the method provides a practical, probability-based framework for designing sequences for density-2 pseudoknotted targets.

The authors emphasize that their objective serves as a primary metric for candidate promise, while structure prediction should be treated as a secondary, complementary criterion. They acknowledge limitations, including the use of a greedy decomposition (approximating the exact joint probability) and the restriction to the density-2 class, noting that broader structural classes may be necessary for more complex RNAs. The work suggests that separating folding-probability evaluation from structure-prediction behavior offers a clearer framework for future studies in pseudoknot inverse folding.