When does a funnel become a trap? A Gömböc-inspired model of protein folding landscapes

Abstract

The protein folding problem remains a fundamental challenge in biophysics. While energy landscape theory resolves Levinthal's paradox through funnel-shaped landscapes, the quantitative effects of landscape ruggedness on folding kinetics are not fully understood. We introduce a computational model inspired by Gömböc, a convex 3D shape with one stable equilibrium and one unstable equilibrium, as an analogy for protein folding landscapes. Using a tunable ruggedness parameter, we simulate “folding” across multiple ruggedness conditions (r = 0 and r = 10⁻⁸ to 10⁻³), spanning five orders of magnitude in the positive-r regime. At minimal ruggedness (10⁻⁸), the model achieves 83% success in reaching native-like states. Increasing the ruggedness to 10⁻⁷–10⁻⁵ reduces the success rate to 25–39%, reflecting landscape trapping as an analog for kinetic trapping in physical folding. Benchmarking against the TPR1 domain via Rosetta and AlphaFold3 reveals native-like coverages of 77% and 90%, respectively, showing qualitative correspondence with the Gömböc model's success rates at low ruggedness. These findings suggest that the Gömböc-inspired model captures essential features of folding landscape quality and may serve as a useful heuristic for comparing foldability across different systems, bridging mathematical topology with protein biophysics.

The protein folding problem remains a fundamental challenge in biophysics.

1. Introduction

The protein folding problem, which involves understanding how a linear amino acid chain consistently folds into its native three-dimensional structure, has captivated biologists, chemists, and physicists for decades. Anfinsen's thermodynamic hypothesis posits that the native structure represents the global free energy minimum under physiological conditions.¹ However, this principle alone cannot explain the remarkable speed and reliability of folding in vivo.² Levinthal's paradox suggested that a random search for a stable configuration in the conformational space would require astronomical timescales, yet many proteins would fold in milliseconds or less.³

The energy landscape theory of protein folding provides a conceptual framework for resolving this paradox. According to this theory, proteins fold on funnel-shaped energy landscapes where the native state sits at the global minimum, and the landscape's topography, particularly its ruggedness, determines folding kinetics.⁴ A perfectly smooth funnel enables rapid, deterministic folding, whereas excessive ruggedness creates kinetic traps that slow folding and increase the probability of misfolding.⁵

However, natural proteins are not perfectly smooth; they exhibit varying degrees of landscape ruggedness resulting from evolutionary pressures, including the need to avoid aggregation, accommodate functional dynamics, and navigate cellular environments.⁶ Understanding how ruggedness quantitatively affects folding outcomes is therefore essential for both basic science and applications in protein design and disease modeling.

The concept of “frustration” in protein folding captures this ruggedness, the extent to which interactions conflict with each other, creating local energy minima that can trap folding intermediates.⁷ Minimally frustrated proteins have landscapes dominated by the native funnel, whereas highly frustrated proteins exhibit multiple competing minima.⁷ Experimental studies have shown that frustration is unevenly distributed in protein structures, with some regions showing higher frustration than others.⁸

In this study, we introduce a computational model inspired by Gömböc, a convex three-dimensional shape with exactly one stable and one unstable equilibrium point, as an analogy for protein folding landscapes.

The Gömböc (pronounced goemboets) was discovered by Hungarian scientists Gábor Domokos and Péter Várkonyi in 2006.⁹ It is the first known convex 3D shape made of homogeneous material, which is monomonostatic, i.e., it has just one stable and one unstable point of equilibrium, a property long thought impossible. It has been proven that such a 2D shape is impossible; however, monomonostatic bodies can exist in higher dimensions. The shapes of 3D monostatic convex homogeneous bodies are somewhat close to a sphere, although the exact shape is elusive and hard to visualize, describe, and identify; the minimal number of faces for monomonostatic polyhedra might be very large.⁹ These unusual properties gave rise to the name Gömböc, a character in Eastern Europe fairy tales (compared with the Slavic “Kolobok”) and an elusive, almost spherical, bread-like being. Gömböc-like shape capsules have also been manufactured and implemented in biomedical research and drug delivery.¹⁰

The mathematical curiosity of the Gömböc provides a powerful analogy for protein folding: the shape always returns to its unique stable position when perturbed, just as a well-evolved protein consistently refolds to its native state. However, the pathway involved depends on how it is perturbed, mirroring the multiple folding pathways observed in proteins.¹¹

The unique stability properties of Gömböc make it an ideal metaphor: like a well-evolved protein, it has a single, well-defined native state but can approach it through multiple pathways. By systematically varying a ruggedness parameter across several orders of magnitude, we map the relationship between landscape topography and folding outcomes, identifying optimal regimes and paradoxical behaviors. Note that Gömböc reaches its stable equilibrium relatively slowly in comparison with the departure from the unstable equilibrium. This process is similar to protein unfolding being approximately two orders of magnitude faster than folding.¹²

Our computational model (Fig. 1A–C) captures this essence while incorporating key biophysical features of real proteins: chain connectivity, excluded volume, local steric constraints, and long-range interactions that can create frustration. By systematically tuning a ruggedness parameter, we explore the continuum from perfectly smooth funnels to extremely rugged landscapes, providing quantitative insights into how landscape topography determines folding outcomes.

Fig. 1. Computational pipeline using the Gömböc-like model for protein folding. The figure outlines the workflow inspired by Gömböc's unique stability properties to model protein energy landscapes. (A) The Gömböc structure: a convex, homogeneous 3D body with exactly one stable and one unstable equilibrium. (B) Native chain mimicking a protein: the 2D closed chain conformation constructed as a Gömböc-like asymmetric contour. (C) 2D free energy landscape: computed contour/scatter representation of the folding funnel toward the native state; tunable ruggedness introduces local minima (traps) that affect the protein energy landscape. The Gömböc 3D model was generated via MATLAB R2020b.

2. Computational methods

2.1. Gömböc-inspired model

The conformational space of proteins is multidimensional. While the original Gömböc is a 3D shape, it can be generalized to higher dimensions via mathematical induction by revolving an n-dimensional Gömböc in the (n + 1)-dimensional space around a line that connects two equilibria. The resulting n + 1 shape is a Gömböc because any point on its surface, other than the original equilibria, belongs to the n-dimensional Gömböc and, therefore, is not an equilibrium point.

Long molecules, such as polymers, are often modeled by random 3D chains in space; however, simplified models are often 2D chain models; the latter tend to capture the most significant characteristics of the former. An ideal chain has an average end-to-end distance proportional to the square root of the number of segments (the scaling exponent of 0.5). A chain without self-intersection is called a chain with excluded volume. These chains have scaling exponents of 3/5 (or 0.588 based on elaborate scaling arguments) in 3D and 3/4 in 2D.¹³

Protein conformations were represented as closed, two-dimensional chains consisting of N = 48 discrete points (vertices) in the plane, yielding a flattened coordinate vector of length 96. This representation abstracts the polypeptide backbone as a continuous, closed chain with fixed segment connectivity but variable bond angles and positions.

We constructed the target (‘native’) conformation as a two-dimensional analog inspired by Gömböc, a convex 3D shape with exactly one stable and one unstable equilibrium. In this analogy the native conformation corresponds to the unique stable equilibrium of the Gömböc, while the unstable equilibrium is represented conceptually by the high-energy, disordered or randomly initialized chain conformations from which optimization begins; these starting states are unstable with respect to the native basin and spontaneously evolve toward it under the energy gradient (when the landscape is sufficiently smooth). Our 2D representation captures the essential property of a unique global energy minimum while remaining computationally tractable, although it does not fully satisfy the mathematical definition of a Gömböc.

Like the 3D Gömböc shape, which is close to a sphere, the equilibrium chain was assumed to be a perturbed ring (Fig. 1B). The native chain was constructed via multiharmonic Fourier synthesis with incommensurate frequencies to ensure aperiodicity.

The 2D closed chain model used in this study is the constructed native conformation (target shape) generated via multiharmonic Fourier synthesis with asymmetry perturbation, representing a Gömböc-like convex contour with a single pronounced global energy minimum. This 2D analog captures the essential property of a unique stable state while enabling computation of the free energy landscape.

A base polar function r(θ) was supplemented with Cartesian polynomial shear to break the radial symmetry, yielding a unique, smooth, asymmetric target shape analogous to a protein's native state. The native shape was parameterized as follows:r(θ) = C₀ + C₁ sin θ + C₂ sin(k₁θ + φ) + C₃ cos(k₂θ),where θ ∈ [0,2π) is uniformly sampled at N points (endpoint excluded). The coefficients C₀ = 1.0, C₁ = 0.20, C₂ = 0.12, C₃ = 0.08, k₁ = 2.8, k₂ = 4.5, and φ = 0.5 were selected empirically to generate a mildly asymmetric, near-circular convex contour that possesses a single pronounced global energy minimum while remaining free of self-intersections and of radial symmetry that would produce degenerate minima. These values yield a shape qualitatively reminiscent of a 2D projection of a Gömböc-like body. Cartesian coordinates were computed viax = r(θ)cos θ, y = r(θ)sin θ

To break the radial symmetry and ensure a unique global energy minimum, we apply a coordinate transformation that shifts the x-coordinate of each point on the basis of its y-coordinate:x ← x + 0.12y² − 0.04y³, y ← y

The quadratic term (0.12y²) introduces a symmetric lateral deformation, whereas the cubic term (0.04y³) breaks up asymmetry. Only the x-coordinate is modified to preserve the overall convexity and prevent self-intersection. The y-coordinate remains unchanged, ensuring that the perturbation is minimal and that the shape remains close to the original ring.

2.1.1. Energy function

The non-dimensional potential energy function E comprises seven additive terms designed to balance the native attraction, geometric constraints, and tunable frustration:E = E_nat + E_conf + E_rep + E_local + E_bond + E_rug + E_frust

The energy function integrates the following terms: the native attraction energy E_nat, which drives folding via a quadratic root-mean-square deviation (RMSD) potential near the native state. Confinement energy (E_conf to prevent expansion; steric repulsion energy E_rep to avoid self-intersection; local geometry energy E_local to maintain approximately 114.6° bond angles; bond restraint energy E_bond to preserve segment lengths; random ruggedness energy E_rug, which introduces local energy variations via stochastic perturbations according to the following equation:where N = 48 (number of points), so 2N = 96 coordinates; x_i parameters are the flattened coordinates of the current conformation; N_i(0,1) are independent standard normal random variables; r is the ruggedness parameter with c = 0.01 and k = 20; and frustration energy E_frust, which penalizes nonnative long-range interactions for residue pairs separated by at least five positions along the chain according to the following equation:where M is the set of long-range residue pairs whose current distance deviates significantly from the native distance, and where d_ij is the Euclidean distance between two points (residues) i and j on the chain with α = 15. Both E_rug and E_frust are scaled by the same tunable ruggedness parameter r. Scaling both terms by a single control parameter simultaneously modulates the amplitude of stochastic local noise and the strength of non-native long-range penalties. This design choice permits systematic exploration of landscape roughness while preserving the relative balance between the two sources of frustration; independent scaling of the two terms would introduce an additional free parameter without altering the qualitative conclusions of the study. This model belongs to the class of structure-based (Gō-like) models,¹⁴ in which the energy function is constructed with explicit knowledge of the target native conformation.

2.1.2. Numerical optimization

The numerical optimization generates folding trajectories via deterministic gradient-based minimization, which employs a two-stage hybrid strategy to balance global exploration and precise local convergence. In Stage I, the conjugate gradient method was applied for 2000 iterations with a gradient tolerance of 10⁻³, which was chosen for its efficiency in navigating the initial high-energy regions of the landscape. Stage II utilized the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm¹⁵ for 6000 iterations with a stricter gradient tolerance of 10⁻⁶, ensuring accurate convergence near local minima. A total of 700 independent folding experiments were conducted across seven ruggedness conditions (n = 100 per condition), with simulations terminating upon convergence (gradient norm < 10⁻⁶) or after reaching the 10 000-step iteration limit. The model is pattern-reproducible but not bit-reproducible: without a frozen noise field, reruns give different individual trajectories but the same ensemble statistics. The trajectories were classified into six outcome categories on the basis of final RMSD values: perfect (≤0.12), success (≤0.20), good (≤0.30), near-native (≤0.40), misfolded (≤0.55), and poor (>0.55). The energy drop was also recorded to differentiate successful native state recovery from optimizer trapping (analogous to kinetic trapping in physical folding):ΔE = E_init − E_finalwhere E_init and E_final are the potential energies at the start and end of the simulation, respectively. Computationally, all simulations were implemented in Python 3.9.12, leveraging NumPy 1.21.5 for numerical arrays and SciPy 1.7.3 for optimization routines, while visualizations were created via Matplotlib. The calculations were performed on a standard workstation (Intel Core i7, 16 GB RAM), with the full ensemble requiring approximately 4 CPU-hours of computational time.

2.2. Rosetta protein folding model

The Rosetta modeling suite (version 2020.03) was used to perform parallel ab initio folding simulations on a well-characterized protein domain. The TPR1 domain of the Hop adaptor protein (PDB ID: 1ELW, chain A) was selected as the test system, a 118-residue domain solved by X-ray diffraction at 1.60 Å resolution.¹⁶ The MPI-parallelized ab initio relax application was employed to generate 2500 independent decoys across 24 physical cores. The key simulation parameters included increased cycle amplification (cycles = 10) to enhance conformational sampling, radius of gyration reweighting (R_g = 0.5 Å) to bias toward compactness, and balanced secondary structure weights (0.5) for the helix and loop. Following the centroid-based sampling stage, all-atom relaxation was performed via the fast relaxation protocol to refine the local geometry. Folding outcomes were assessed by plotting Rosetta energy scores against the Cα RMSD relative to the native structure, and the lowest-energy decoys (top 1.0%, corresponding to 25 structures) were extracted for visual inspection. To characterize the topography of the energy landscape, all 2500 decoys were clustered via the Rosetta cluster application with a 3.0 Å RMSD radius, grouping similar conformations and revealing the density of states across the folding funnel.

2.3. AlphaFold model

The AlphaFold (AF3) server was used to provide a contemporary benchmark for protein structure prediction and to assess the folding landscape of the TPR1 domain via the AF3 model.¹⁷ The generated protein structure was evaluated by measuring the overall confidence score (pTM) and per-residue confidence, which were assessed via the predicted local distance difference test (pLDDT). The predicted aligned error (PAE) plot was used to analyze expected position errors between residues within the same structural domain, with values typically less than 5 Å confirming the integrity of the TPR fold. Finally, the generated structures were aligned against the experimental X-ray crystal structure to calculate RMSD values via PyMOL software.¹⁸

3. Results and discussion

3.1. Gömböc-inspired model reveals frustration-dependent folding behavior

The Gömböc-inspired computational model clearly reveals a transition in folding behavior as landscape ruggedness increases (Table 1). Protein conformations are represented as closed two-dimensional chains with a tunable ruggedness parameter (r) in the energy function. While the true Gömböc is a 3D convex body with a single stable and unstable equilibrium, our 2D analog captures the essential feature of a unique, stable native state with multiple accessible folding pathways, serving as a conceptual rather than geometric analogy.

Table 1. Folding performance of the Gömböc-inspired model across seven ruggedness levels.

Ruggedness	Successa (%)	Behavior	Reference datab
0.0	93	Perfect funnel	AF3 pLDDT ∼90%
10−8	83	Almost perfect	Rosetta ∼77% cov.
10−7	39	Gradient disruption	—
10−6	26	Noise-dominated	—
10−5	25	Noise-dominated	—
10−4	33	Optimizer failure/random baseline	—
10−3	34	Optimizer failure/random baseline	—

^aGömböc model: fraction with RMSD ≤ 0.4 (dimensionless, normalized). The results are not directly comparable to the reference data.

^bReference metrics for qualitative comparison only: pLDDT = AF3 per-residue confidence score averaged over all residues (0–100); coverage = percentage of Cα atoms aligned within 1.0 Å RMSD after structural alignment.

At zero ruggedness (r = 0.0), the landscape acts as a near-perfect funnel, yielding an average energy drop of 129.5 ± 21.7 arbitrary units, convergence in 21 optimization steps, and a 93% success rate in reaching native-like states (RMSD ≤ 0.4). This behavior mirrors minimally frustrated, efficient folding akin to an ideal Gömböc returning to its stable equilibrium (Fig. 2A). A slight increase to r = 10⁻⁸ reduces the energy drop to 123.9 ± 23.9 units, extends the number of steps to 47, and decreases the success rate to 83%, maintaining near-ideal behavior. However, further escalation to r = 10⁻⁷ sharply impairs the outcome: the energy drop decreases to 67.0 ± 20.7 units, the number of steps increases to 63, and success further decreases to 39%, as stochastic energy fluctuations begin to disrupt the gradient-based optimization pathway. At r = 10⁻⁶ and 10⁻⁵, this noise dominates the landscape, yielding modest energy drops (14–17 a.u.) over just 6 to 10 steps, with success rates falling to 25–26% (Fig. 2B). Higher ruggedness (r = 10⁻⁴) yields even smaller energy changes (2.2 ± 5.9 a.u.) and fewer steps (4), with success rates plateauing and showing a slight recovery to 33% at the highest ruggedness. However, the stochastic noise at extreme ruggedness values (10⁻⁴ and 10⁻³) causes the gradient-based optimizer to terminate prematurely; thus, the apparent recovery in the success rate is merely an artifact of the baseline probability that a random initial conformation happens to fall within the near-native RMSD threshold. This algorithmic failure at intermediate ruggedness (r = 10⁻⁷ to 10⁻⁶) functionally mimics the kinetic trapping described by Porebski et al.,¹⁹ who reported that the rugged folding landscapes of functional proteins render them susceptible to misfolding and aggregation.

Fig. 2. Energy-RMSD scatter plots for the Gömböc-inspired model across varying ruggedness levels, with extremes highlighted at r = 0.0 (A) and r = 10–5 (B). Scatter plots show the final potential energy (in arbitrary units) versus the root-mean-square deviation (RMSD, dimensionless and normalized) to the native conformation, derived from 700 energy landscape exploration trajectories (100 per ruggedness condition).

The present Gömböc-inspired model complements our earlier topological bioscaling analysis of Trp-cage folding, in which the chain was shown to pass through fractional-dimensional intermediate states (1 ≤ D ≤ 1.92) whose stability is controlled by interaction energy.²⁰ Together, the tunable ruggedness parameter r and the fractal scaling approach provide a quantitative bridge between monomonostatic topology and the dimensional properties of native state recovery under optimization.

Overall, our results quantitatively demonstrate how even minute increases in landscape ruggedness dramatically degrade folding efficiency, shifting from rapid, high-fidelity convergence to severe landscape trapping and misfolding. This is similar to how Gömböc is fragile and sensitive to a slight change of its shape.⁹

3.2. Rosetta ab initio simulations characterize the TPR1 domain as a minimally frustrated system

Rosetta ab initio folding simulations generated 2500 independent decoys for the TPR1 domain, revealing a characteristic funnel-shaped energy landscape with a well-defined global minimum (Fig. 3A). The TPR1 domain is a critical component of the Hsp70/Hsp90-organizing protein (Hop), where it serves as a scaffold by binding directly to the C-terminal EEVD motif of Hsp70.¹⁶ This interaction facilitates the transfer of client proteins from Hsp70 to Hsp90, coordinating their maturation and folding.²¹ Consistent with this well-folded, minimally frustrated system, the lowest-energy decoys (after outlier removal) achieved scores near −350 REU at Cα RMSD values of 0.7–1.1 Å relative to the experimental structure; the lowest-energy structure in the cleaned set has E = −350.14 REU and RMSD = 0.822 Å. Analysis revealed a clear energetic gap of ∼30–40 REU between native-like conformations (RMSD < 2.0 Å) and misfolded structures, indicating a minimally frustrated folding landscape dominated by a single, deep funnel (Fig. 3A). The top 1.0% of the decoys (n = 25) achieved RMSD values below 1.5 Å, confirming that the TPR1 domain is a well-behaved, efficient folding system.

Fig. 3. Rosetta energy landscape analysis for the TPR1 domain. (A) Scatter plot of the Rosetta energy score versus the Cα RMSD for decoys in the native-like region (RMSD ≤ 2.0 Å). The blue points represent individual decoys; the red stars represent the native structure. (B) Scatter plot of ΔE (energy relative to native) versus RMSD across all cleaned decoys, colored by ΔE magnitude (red, low; yellow, high), revealing an energetic gap separating native-like from misfolded states. (C) Energy landscape profile. 2D profile along the RMSD: the blue line represents the moving average energy, the light blue shading represents ±1 SD, and the red dashed line represents the native energy.

Favorable energetic contributions included attractive van der Waals interactions (E_attr ≈ −730 REU), backbone hydrogen bonding (E_hbond ≈ −86 REU), and favorable electrostatic interactions (E_elec ≈ −230 REU), balanced by steric repulsion (E_rep ≈ 75 REU) and solvation effects (E_sol ≈ 510 REU). The outlier removal (3.8% of decoys) substantially improved the energy-RMSD correlation from r = 0.434 to r = 0.699 (p < 0.001), a 60.9% increase revealing the underlying funnel topology (Fig. S1A–F). The clean dataset (n = 2404) presented RMSD values ranging from 0.526 to 12.472 Å (mean 2.379 ± 2.392 Å) and energies ranging from −350.140 to −245.833 REU (mean: −322.132 ± 14.458 REU). The energy gap between the lowest-energy structure and the second lowest-energy structure (E = −350.131 REU, RMSD = 0.803 Å) was ΔE = 0.009 REU, indicating that multiple distinct conformations near the native state have similar energies (Fig. 3B and C). The energy functions used in the Rosetta protocol are based on Anfinsen's hypothesis, in which the native state corresponds to a unique, low-energy, thermodynamically stable conformation.^22,23 One-way ANOVA across RMSD tertiles revealed highly significant energy differences (F = 752.43, p < 0.001), confirming that the TPR1 domain is a robust benchmark for comparison with the Gömböc-inspired model.

3.3. AF3 validates the TPR1 domain as a minimally frustrated benchmark

The AF3 predictions for the TPR1 domain exhibited exceptionally high confidence, with an overall pTM score of 0.91 and per-residue pLDDT values ≥90 for >95% of the residues, indicating “very high” confidence.

The PAE plot revealed predominantly low expected position errors (most values <5 Å, with a maximum of approximately 14 Å), confirming a single, well-defined structural fold without domain-swapping or alternative conformations. High-confidence AF3 predictions (pLDDT ≥ 90) with a uniform PAE plot are characteristic of well-folded, minimally frustrated systems, as frustrated proteins often exhibit conformational heterogeneity reflected in lower prediction confidence or alternative domain arrangements.²⁴ These results support the classification of the TPR1 domain as a minimally frustrated system suitable for benchmarking the Gömböc-inspired model.

A quantitative comparison of structural prediction accuracy revealed that AF3 substantially outperformed Rosetta in alignment with the experimental reference (Fig. 4A and B). In this analysis, accuracy benchmarking tests confirmed superior protein structure prediction by AlphaFold compared with established physics-based ab initio methods, including Rosetta.²⁵ The best AF3 alignment achieved an RMSD of 0.3 Å with 90% atom coverage, whereas Rosetta's optimal alignment yielded 0.43 Å RMSD at 77% coverage. AF3 also demonstrated greater consistency (RMSD range: 0.30–0.49 Å) than Rosetta did (0.43–0.72 Å). AF3 alignments began at 0.45 Å RMSD versus 1.26 Å for Rosetta, indicating inherently closer proximity to the native conformation.

Fig. 4. 3D pairwise protein structure alignment of the TPR1 domain via Rosetta (A) and AF3 (B) models. For each method, five independent models (n = 5) are shown superimposed on the experimental structure. Blue: experimental X-ray reference structure; gray: predicted models.

Collectively, these results establish AF3 as a more accurate predictor for this target, achieving sub-0.5 Å RMSD values approaching the experimental resolution. These data were close to the 0.5 Å RMSD benchmark reported for the Trp-cage protein folded by AlphaFold2.¹² The residual differences between the predicted models and the experimental X-ray structure (RMSD 0.3–0.7 Å) can be attributed to several factors: (i) the finite resolution of the experimental structure (1.60 Å), (ii) the absence of crystal-packing contacts in the computational models, (iii) differences in the treatment of side-chain rotamers and solvent, and (iv) the fact that both Rosetta and AF3 sample an ensemble of near-native conformations rather than a single crystallographic snapshot.

A direct comparison between Rosetta/AF3 alignment coverage and Gömböc-inspired model success rates reveals a compelling parallel. AF3 achieved 90% coverage at 0.30 Å RMSD, whereas Rosetta achieved 77% coverage at 0.43 Å RMSD, qualitatively corresponding to Gömböc success rates at low ruggedness: 93% (r = 0.0) and 83% (r = 10⁻⁸), respectively. The success rate of the Gömböc-inspired model decreased sharply to 26–39% at higher ruggedness values (r = 10⁻⁷ to 10⁻⁶), which is characteristic of highly frustrated landscapes where folding becomes trapped (analogous to kinetic trapping in physical folding). These findings suggest that the ruggedness parameter captures essential features of folding landscape quality, with qualitative correspondence between model success rates and prediction method performance. However, further validation of diverse protein targets and explicit physical units is needed to establish a model as a quantitative predictor.

4. Conclusion

In this study, we introduced a computational model inspired by Gömböc, a convex shape with a single stable equilibrium, to investigate how landscape ruggedness influences protein folding landscape optimizability. By systematically varying a ruggedness parameter across multiple conditions (r = 0 and r = 10⁻⁸–10⁻³, spanning five orders of magnitude in the positive-r regime), we demonstrated a sharp transition from efficient, funnel-dominated folding to trapped, misfolded states analogous to kinetically trapped intermediates. At minimal ruggedness, the model achieved 93% success in reaching native-like conformations, showing qualitative correspondence with the high-accuracy predictions of AF3 (90% native-like coverage) for the TPR1 domain. In contrast, increasing the ruggedness by as little as a factor of 10⁻⁷ reduced the success rate to 39%. While the metrics differ, this degradation in optimizability is broadly consistent with the lower native-like coverage achieved by Rosetta (77%) compared with AF3 (90%) for the minimally frustrated TPR1 domain.

These findings suggest that the Gömböc-inspired model captures essential features of folding landscape quality. Unlike conventional Gō models that encode only native contacts, the present formulation combines a monomonostatic (single-stable-equilibrium) geometric target with an explicit, tunable E_frust that penalizes non-native long-range distances and a stochastic ruggedness term E_rug that injects local energy noise. The sharp degradation of success rate with increasing r demonstrates that the optimizability of the landscape depends critically on the interplay between the unique stable native basin (the Gömböc-like property) and the strength of competing interactions; when r is negligible, the funnel dominates, and the model recovers the native state with high fidelity, whereas even minute ruggedness introduces kinetic traps analogous to those observed in frustrated proteins. The correspondence between model success rates and the performance of state-of-the-art structure prediction methods, despite differing metrics and dimensionalities, indicates that the ruggedness parameter may serve as a useful heuristic for comparing foldability across different systems. By bridging mathematical topology with protein folding biophysics, this approach offers a scalable platform for exploring the effects of frustration, evolutionary design principles, and the limits of foldability.

Future work will extend the model to three dimensions, incorporate sequence-specific effects, validate diverse protein targets, and explore its applicability to intrinsically disordered proteins and aggregation-prone systems.

Conflicts of interest

The authors declare that they have no conflicts of interest.

Data availability

The data and code supporting the findings of this study are available in the GitHub repository https://github.com/virtualscreenlab/Gomboc.

Supplementary information (SI) is available. See DOI: https://doi.org/10.1039/d6ra07903b.