Funnel-like protein energy landscapes emerge from functional evolution under thermal fluctuations

The paper investigates whether funnel-like protein energy landscapes, characteristic of foldable proteins, can emerge solely from selection for biological function, without any direct selection for foldability. Using a two-dimensional lattice protein model with 20 residues and four amino-acid types (two hydrophobic, two polar), the authors define protein fitness as the equilibrium probability that a prescribed local structure forming an active site is realized at environmental temperature T. This probability is used as the evolutionary fitness, and amino-acid sequences spanning the entire fitness range are sampled by multicanonical Monte Carlo methods. The energy and free-energy landscapes of these sequences are then constructed from the equilibrium conformational ensemble.

The key findings are:

1. Temperature-dependent emergence of funnel-like landscapes: Only within an appropriate range of environmental temperatures (e.g., T = 1.0) do high-fitness sequences spontaneously acquire funnel-like energy landscapes, despite foldability never being included in the fitness definition. At T = 1.0, the energy spectrum of the highest-fitness sequence shows that the energy decreases systematically as the number of native contacts increases, demonstrating a funnel-like energy landscape toward the native conformation. This is reflected in a free-energy landscape with two distinct minima corresponding to denatured and native-state ensembles, separated by a free-energy barrier, characteristic of two-state folding.

2. Limited native conformations at intermediate temperature: At T = 1.0, among the 1000 highest-fitness sequences (forming 500 chain-reversal pairs), only four distinct native-structure classes are observed. The paper states: Despite the large number of sequences, only a few distinct native conformations were observed... Thus, high fitness is achieved only by a limited set of native conformations. This suggests functional selection strongly restricts the accessible fold space.

3. Glass-like landscapes at low temperature: At T = 0.1, high-fitness sequences (with fitness around f ≃ 0.9) exhibit rugged, glass-like energy landscapes. The paper notes: "the energy levels are confined to a much narrower energy range and exhibit no systematic decrease with increasing number of native contacts. Instead, the spectrum is characteristic of a glass-like energy landscape, in which structurally similar conformations are not energetically organized toward the native state. The free-energy landscape is highly rugged and lacks the well-defined two-state structure observed at T = 1.0." At this low temperature, essentially every high-fitness sequence adopts its own native conformation (187 distinct conformations among 187 independent sequence patterns), in contrast to the four classes at T = 1.0.

4. Genotypic entropy differences: At T = 1.0, the genotypic entropy decreases markedly with increasing fitness, indicating high-fitness sequences become progressively rarer. At T = 0.1, the entropy remains nearly constant over the entire fitness range, showing high-fitness sequences remain abundant. At T = 1.4, the entropy terminates at f ≃ 0.4, indicating highly functional proteins cannot be realized under such strong thermal fluctuations.

5. Mechanistic explanation: The origin of this temperature dependence is explained as follows: "At finite temperatures, high fitness requires that many thermally populated conformations preserve the active-site geometry rather than only the ground state. Functional selection therefore acts indirectly on the organization of the entire low-energy spectrum. At low temperatures, thermal excitation out of the ground state is strongly suppressed, so high fitness can be achieved as long as the ground-state conformation contains the prescribed active-site geometry. Functional selection therefore no longer requires global energetic organization, allowing rugged, glass-like energy landscapes to persist."

6. Consistency with explicit evolutionary simulations: Conventional evolutionary simulations at T = 1.0 produced representative high-fitness sequences exhibiting funnel-like energy spectra and two-state free-energy landscapes, consistent with the multicanonical sampling results. Two of 40 independent runs reached the globally optimal high-fitness sequences identified by multicanonical sampling.

The paper concludes that funnel-like energy landscapes need not be regarded as an independent target of evolution. Instead, they may emerge naturally as a thermodynamic consequence of maintaining protein function under thermal fluctuations. This provides a unified evolutionary framework linking functional selection, Gö's consistency principle, minimal frustration, and the emergence of funnel-like energy landscapes. The authors also suggest implications for protein design, proposing that stabilizing only the functional motif under thermal fluctuations may be sufficient, with the surrounding scaffold and overall fold emerging as thermodynamic consequences.

Improvement 1: Physics-Aware Evolutionary Fitness Models

The AI system can incorporate temperature-dependent fitness functions (e.g., equilibrium probability of active-site geometry at a given T) instead of static fitness scores. This enables generative models for protein design to explicitly optimize for thermal robustness, producing sequences that maintain function across environmental fluctuations—not just at a single ground state.

Improvement 2: Constraint-Based Fold Space Exploration

The AI can learn to predict the limited set of native conformations that satisfy functional selection (e.g., only 4 classes at T=1.0). This allows the system to prune vast conformational search spaces, accelerating de novo protein design by focusing on functionally viable folds and avoiding energetically disallowed scaffolds.

Improvement 3: Temperature-Sensitive Landscape Generation

The AI can generate energy landscapes conditioned on environmental temperature, distinguishing between funnel-like (T=1.0), glass-like (T=0.1), and non-functional (T=1.4) regimes. This enables the system to design proteins with tunable folding cooperativity—e.g., creating two-state folders for stable enzymes or rugged landscapes for intrinsically disordered proteins under specific thermal conditions.

Improvement 4: Entropy-Aware Sequence Sampling

By modeling genotypic entropy as a function of fitness (e.g., decreasing entropy at T=1.0 vs. constant at T=0.1), the AI can predict sequence rarity and evolutionary accessibility. This improves directed evolution algorithms by identifying high-fitness sequences that are statistically reachable under mutation-selection dynamics, avoiding unrealistic targets.

Improvement 5: Minimal-Frustration Induction Without Explicit Foldability Loss

The AI can adopt the principle that funnel-like landscapes emerge from functional selection alone. This allows the system to design proteins by stabilizing only the active-site motif under thermal fluctuations, letting the surrounding scaffold self-organize—reducing design complexity and computational cost while maintaining foldability as an emergent property.

Improved AI System Capabilities:

• Thermo-adaptive protein designer: Generates sequences with specified active-site geometry that remain functional across a user-defined temperature range, automatically yielding funnel-like or rugged landscapes as needed.

• Fold-space classifier: Predicts whether a given functional constraint will yield a limited set of native conformations, enabling rapid screening of designable folds.

• Evolutionary simulator with physical realism: Simulates protein evolution under explicit thermal selection, predicting sequence entropy, fitness landscapes, and the emergence of two-state folding without ad hoc foldability penalties.

• Design of minimal scaffolds: Proposes minimal protein constructs (e.g., only the functional motif stabilized) and predicts the thermodynamic consequences for the rest of the fold, reducing experimental trial-and-error.

Proteins perform biological functions by folding into specific native structures that are stabilized by funnel-like energy landscapes shaped through evolution. We investigate whether such foldable proteins can emerge solely from selection for function, without any direct selection for foldability. Using a two-dimensional lattice protein model with four amino-acid types, we define protein function as the equilibrium probability that a prescribed local structure forming an active site is realized at environmental temperature T, and use this probability as the evolutionary fitness. Amino-acid sequences spanning the entire fitness range are sampled by multicanonical Monte Carlo method and their energy and free-energy landscapes are constructed from the equilibrium conformational ensemble. We find that only within an appropriate range of environmental temperatures do high-fitness sequences spontaneously acquire funnel-like energy landscapes, despite foldability never being included in the fitness definition. High-fitness sequences are furthermore supported by only a limited number of native conformations. At low temperatures, in contrast, high-fitness sequences exhibit rugged, glass-like energy landscapes and much greater structural diversity. These results suggest that foldable proteins need not be direct targets of evolution, but can emerge as a thermodynamic consequence of maintaining function under thermal fluctuations, whereby local functional constraints organize the global fold and give rise to funnel-like energy landscapes.