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Linear Fitness Subspace in Protein Language Models Enables Sample-Efficient Directed Evolution

SiYuan Ma, Canran Xiao, Zikai Xiao, Albert Gao, Liang He, Xuan-Yu Wang, Shuying Cao, Xiaojun Jia

Year: 2026Venue: arXiv preprintArea: q-bio.QMType: PreprintEmbeddings: 77

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Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%

Last extracted: 10/13/2026, 3:49:03 AM

Summary

The paper introduces the Linear Fitness Subspace (LFS) hypothesis, proposing that mutation-induced changes in Protein Language Model (PLM) representations are concentrated in a low-dimensional, assay-specific subspace where fitness variation is linearly accessible. Based on this, the authors develop Subspace-Guided Evolutionary Search (SGES), a framework that estimates this subspace from a small initial sample and performs surrogate modeling and acquisition within it. SGES demonstrates improved sample efficiency and fitness prediction over zero-shot PLMs and other ML-guided baselines across multiple ProteinGym assays.

Entities (7)

Linear Fitness Subspace · concept · 98%Subspace-Guided Evolutionary Search · method · 98%Protein Language Models · model-class · 95%ProteinGym · benchmark · 92%site-delta · feature · 90%Partial Least Squares · algorithm · 88%ESM-2 · model · 85%

Relation Signals (6)

Subspace-Guided Evolutionary Search → uses → Linear Fitness Subspace

confidence 95% · SGES estimates an LFS from a small initial sample and performs surrogate modeling, uncertainty estimation, and acquisition in the learned subspace.

Linear Fitness Subspace → derivedfrom → site-delta

confidence 92% · within mutation-induced residue-level representation changes... a compact, assay-specific set of directions... This is a local, supervision-recoverable statement... mutation-local site-delta coordinate

Subspace-Guided Evolutionary Search → evaluatedon → ProteinGym

confidence 92% · Across 10 core ProteinGym assays, 87 extended static-validation assays, and an 18-assay budgeted-search evaluation...

site-delta → computedfrom → Protein Language Models

confidence 90% · For a mutation at positionp, defineδ(s) = h(s)[p]− h(s wt )[p]. ... A frozen PLM maps a sequence to residue-level representationsh(s)

Subspace-Guided Evolutionary Search → outperforms → zero-shot PLMs

confidence 90% · SGES improves fitness prediction and search efficiency over zero-shot PLMs and recent ML-guided protein optimization baselines.

Linear Fitness Subspace → estimatedby → Partial Least Squares

confidence 88% · The LFS projection matrix is learned by partial least squares regression

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Abstract

Abstract:Model-guided directed evolution seeks to identify high-fitness protein variants under limited oracle budgets. Protein language models (PLMs) provide rich representations for this task, but task-agnostic zero-shot scores can be misaligned with a target assay, while supervised search in high-dimensional embedding spaces can make surrogate modeling and uncertainty estimation sample-inefficient. We propose the Linear Fitness Subspace (LFS) hypothesis: within mutation-induced residue-level representation changes, a compact, assay-specific set of directions makes fitness variation linearly accessible from few labeled variants. This is a local, supervision-recoverable statement rather than a claim that protein fitness landscapes or global PLM geometry are universally linear. Building on this observation, we introduce Subspace-Guided Evolutionary Search (SGES), which estimates an LFS from a small initial sample and performs surrogate modeling, uncertainty estimation, and acquisition in the learned subspace. Across 10 core ProteinGym assays, 87 extended static-validation assays, and an 18-assay budgeted-search evaluation, SGES improves fitness prediction and search efficiency over zero-shot PLMs and recent ML-guided protein optimization baselines. Controlled comparisons with PCA, random projections, label-shuffled PLS, classical mutation features, and acquisition ablations further isolate the benefit of a fitness-aligned site-delta coordinate.

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Linear Fitness Subspace in Protein Language Models Enables Sample-Efficient Directed Evolution SiYuan Ma 1 Canran Xiao 2 Zikai Xiao 3 Albert Gao 4 Liang He 5 Xuan-Yu Wang 6 Shuying Cao 7 Xiaojun Jia 1 1 College of Computing and Data Science, Nanyang Technological University, Singapore 2 Sun Yat-sen University, China 3 Zhejiang University, China 4 Carnegie Mellon University, USA 5 Shanghai Institute of Optics and Fine Mechanics, China 6 Zhongnan Hospital, Wuhan University, China 7 University of Southern California, USA Abstract Model-guided directed evolution seeks to iden- tify high-fitness protein variants under lim- ited oracle budgets. Protein language models (PLMs) provide rich representations for this task, but task-agnostic zero-shot scores can be misaligned with a target assay, while super- vised search in high-dimensional embedding spaces can make surrogate modeling and uncer- tainty estimation sample-inefficient. We pro- pose the Linear Fitness Subspace (LFS) hypoth- esis: within mutation-induced residue-level rep- resentation changes, a compact, assay-specific set of directions makes fitness variation lin- early accessible from few labeled variants. This is a local, supervision-recoverable statement rather than a claim that protein fitness land- scapes or global PLM geometry are universally linear. Building on this observation, we in- troduce Subspace-Guided Evolutionary Search (SGES), which estimates an LFS from a small initial sample and performs surrogate model- ing, uncertainty estimation, and acquisition in the learned subspace. Across 10 core Prote- inGym assays, 87 extended static-validation assays, and an 18-assay budgeted-search eval- uation, SGES improves fitness prediction and search efficiency over zero-shot PLMs and re- cent ML-guided protein optimization baselines. Controlled comparisons with PCA, random pro- jections, label-shuffled PLS, classical mutation features, and acquisition ablations further iso- late the benefit of a fitness-aligned site-delta coordinate. 1 Introduction Directed evolution is a central strategy for engi- neering proteins with improved activity, stability, binding affinity, and other functional properties. Its main bottleneck is sample efficiency: each candi- date variant must be evaluated by an experimental or computational oracle, making large-scale search expensive. Model-guided directed evolution there- fore aims to identify high-fitness variants under limited oracle budgets, where the key challenge is both accurate fitness prediction and efficient explo- ration of a combinatorially large protein sequence space (Jiang et al., 2024; Soldát and Kléma, 2024; Kmicikiewicz et al., 2026; Takizawa et al., 2025; Yang et al., 2025; Tian et al., 2022). Protein language models (PLMs) provide a promising foundation for this problem. Trained on large protein sequence corpora, PLMs encode evolutionary, structural, and functional constraints that can be transferred to mutation-effect predic- tion (Meier et al., 2021; Notin et al., 2022; Su et al., 2024; Li et al., 2024; Hayes et al., 2025). Existing PLM-based methods usually follow two paradigms. Zero-shot methods estimate muta- tion effects from likelihood or masked-marginal scores without labeled fitness data, while super- vised methods use PLM embeddings as features for downstream predictors or search algorithms (Notin et al., 2023; Meier et al., 2021; Notin et al., 2022; Ma et al., 2026a). However, zero-shot scores are task-agnostic and often misaligned with assay- specific fitness, whereas supervised search in high- dimensional embedding spaces can suffer from sample-inefficient surrogate modeling and unre- liable uncertainty estimation (Soldát and Kléma, 2024; Greenman et al., 2025; Rahaman et al., 2021; Lakshminarayanan et al., 2017; Tian et al., 2023; Ma et al., 2026b). arXiv:2610.07607v1 [q-bio.QM] 6 Oct 2026 This paper studies a representation-level ques- tion: where is assay-specific mutation-effect infor- mation located inside a frozen protein language model? We focus on the local displacement in- duced by a mutation relative to its wild type. For a mutation at positionp, defineδ(s) = h(s)[p]− h(s wt )[p]. Our central observation is that, in this mutation-local site-delta coordinate, a substantial component of assay fitness is accessible through a small number of supervised linear directions. We refer to this compact, assay-specific coordinate as the Linear Fitness Subspace (LFS). Importantly, this does not assert that global PLM embeddings or complete protein fitness landscapes are universally linear. The LFS hypothesis provides a geometric ex- planation for the gap between zero-shot and su- pervised PLM-based fitness prediction. Zero-shot scoring can be viewed as a fixed, task-agnostic one-dimensional projection of mutation-induced representation change, while supervised learning recovers assay-specific fitness directions from a small number of labeled variants. When the zero- shot direction is poorly aligned with the true assay- specific direction, zero-shot performance is lim- ited; few-shot supervision improves prediction by rotating the projection toward the appropriate fit- ness subspace. This view connects protein fitness prediction with broader findings that neural rep- resentations encode task-relevant information in linearly accessible directions or subspaces (Hewitt and Manning, 2019; Saurez et al., 2026; Gujral et al., 2025; Beshkov and Malthe-Sørenssen, 2025; Chen et al., 2022). Based on this observation, this paper proposes Subspace-Guided Evolutionary Search (SGES), a directed evolution framework that first estimates the LFS from site-delta features using a small ini- tial query budget, then performs surrogate model- ing, uncertainty estimation, and acquisition in the learned low-dimensional subspace. Across 10 core ProteinGym assays and an extended set of 87 as- says, SGES improves over zero-shot PLM scores, full-dimensional GP-BO, AdaLead, EVOLVEpro, and recent ML-guided directed evolution baselines under matched oracle budgets (Jiang et al., 2024; Notin et al., 2023; Soldát and Kléma, 2024; Kmi- cikiewicz et al., 2026; Takizawa et al., 2025; Yang et al., 2025; Ma et al., 2026c). The main contributions are: •Assay-specific Linear Fitness Subspace. We formulate and test the hypothesis that mutation-local PLM site-delta features expose a compact, supervision-recoverable coordi- nate in which a substantial component of fit- ness is linearly accessible. • Geometric view of zero-shot scoring. We op- erationalize zero-shot scoring as a fixed task- agnostic direction in the same site-delta space and quantify its alignment with the supervised assay-specific direction. •Subspace-guided search. SGES couples LFS discovery with low-dimensional surro- gate modeling, uncertainty-aware acquisition, and diversity filtering for sample-efficient di- rected evolution. •Controlled attribution and broad valida- tion. Matched-dimensionality controls (PCA, random projection, and label-shuffled PLS), weaker mutation-feature baselines, compo- nent ablations, 10 core assays, 87 extended assays, and 18 search-statistics assays sepa- rate fitness alignment from generic dimension- ality reduction and acquisition effects. 2 Related Work Protein language models (PLMs) have become a major foundation for protein fitness predic- tion, mutation-effect estimation, and sequence design. Early zero-shot PLM methods showed that likelihood-based or masked-marginal scores can predict the effects of mutations without task- specific labels (Meier et al., 2021; Notin et al., 2022). Subsequent PLMs incorporated larger se- quence corpora, structural information, or more expressive protein representations, as represented by ESM-2, Tranception, SaProt, ProSST, and evolutionary-scale protein language models (Meier et al., 2021; Notin et al., 2022; Su et al., 2024; Li et al., 2024; Hayes et al., 2025; Lin et al., 2023). Large-scale benchmarks such as ProteinGym fur- ther enabled systematic evaluation across diverse proteins and assays, revealing that zero-shot scores often capture useful evolutionary constraints but remain highly variable across fitness landscapes (Notin et al., 2023). In parallel, non-PLM baselines based on conservation, phylogeny, solvent acces- sibility, and structural priors remain competitive for variant-effect prediction, suggesting that PLM- based methods should be evaluated against both Step 1: Mutation Prior candidates (make_simple_mutation_prior) Single-Mutation Assay (WT_PROPOSAL_FRACTION = 0.5) Multi-Mutation Assay (WT_PROPOSAL_FRACTION = 0.2) Step 2: Surrogate Prediction （Mutation Effect Surrogate) mean_pooled D=1280 site_delta D=1280 site_wt_emb D=1280 DeepEnsemble Surrogate （5 model） DeepEnsemble Surrogate （5 model） Predicted Mean Epistemic Uncertainty variance of ensemble means Aleatoric Uncertainty heteroscedastic regression mean log-var Frozen backbone Total Uncertainty zero_shot D=1 ESM-2 650M Step 3: Acquisition Function Selection Diversity-Aware Batch Selection (multi-mutation assay Hamming distance Greedy Selection 4:Oracle Evaluation Consumes experimental budget Step 5: Population Update the active environment update with the highest fitness variants Wild type (WT) Small initial cost Oracle Evaluatio n Oracle Evaluation Budget （B)Random sampling of n initial single mutants evaluated with the initial population (A) Evaluation of WT fitness EXP 1: Static Mutation Effect Prediction IV, Section of IV, Section V Exp 2: Budgeted Search Comparison Exp 5: Ablation and Calibration Phase 1: Initialization Phase 2: Iterative Search Loop Ablation Figure 1: Overall pipeline of Subspace-Guided Evolutionary Search (SGES). SGES starts from the wild-type sequence and a small initial oracle budget, constructs mutation candidates, extracts mutation-aware PLM features, and performs iterative search through surrogate prediction, uncertainty-aware acquisition, oracle evaluation, and population update. This overview connects the representation-level discovery in Section 3 with the search algorithm introduced later: fitness prediction is not treated as generic high-dimensional embedding regression, but is guided by mutation-induced representation changes that expose a compact fitness-relevant structure. neural and classical evolutionary signals (Tekpinar et al., 2025; Tsishyn et al., 2025; Prillo et al., 2024). A complementary paradigm uses PLM representa- tions as supervised features for downstream predic- tors, allowing adaptation to assay-specific objec- tives when a small number of labeled variants is available. However, many supervised approaches rely on global or mean-pooled embeddings, which can obscure the local residue-level perturbation in- duced by mutation. ML-guided directed evolution studies how sur- rogate models, acquisition functions, and pro- posal mechanisms can optimize proteins under limited oracle budgets. Classical approaches in- clude Gaussian-process Bayesian optimization, ac- tive learning, evolutionary search, and surrogate- assisted protein design (Soldát and Kléma, 2024; Amin et al., 2025; Kirjner et al., 2024; Emami et al., 2023; Takizawa et al., 2025; Hartman et al., 2025; Ding et al., 2024). More recent methods combine PLM representations with tree search, generative priors, safe optimization, or latent-space search, including EVOLVEpro, ProSpero, AlphaDE, La- tentDE, BOES, and TreeNeuralUCB (Jiang et al., 2024; Kmicikiewicz et al., 2026; Amin et al., 2025; Takizawa et al., 2025; Yang et al., 2025; Tran et al., 2025; Qiu et al., 2024). Other generative and structure-aware protein design methods, such as guided discrete diffusion, DPLM-Evo, EvoFlows, ProteinMPNN, and atomistic equivariant design models, provide powerful sequence or structure generation mechanisms, but they are not primar- ily designed to explain where assay-specific fitness information is located inside PLM representations (Wang et al., 2026; Deutschmann et al., 2026; Gru- ver et al., 2023; Lin et al., 2026; Dauparas et al., 2022). These methods demonstrate the value of learned representations for protein optimization, but they generally treat the PLM embedding space as a high-dimensional black-box feature space. As a result, surrogate fitting and uncertainty estimation can become sample-inefficient, especially when most embedding dimensions are weakly related to the target assay. Uncertainty estimation is another important component of model-guided protein optimiza- tion. Deep ensembles, Bayesian approximations, and protein-engineering-specific uncertainty bench- marks have shown that uncertainty quality strongly affects acquisition and downstream search perfor- mance (Greenman et al., 2025; Rahaman et al., 2021; Lakshminarayanan et al., 2017; Gal and Ghahramani, 2015). However, uncertainty esti- mation is only useful when uncertainty is mea- sured in a representation space aligned with the target property. In high-dimensional PLM spaces, epistemic uncertainty may reflect irrelevant embed- ding variation rather than fitness-relevant uncer- tainty. This motivates the present work’s design choice: SGES estimates uncertainty after project- ing mutation-induced site-delta features into the learned Linear Fitness Subspace, so exploration is concentrated along directions that empirically covary with fitness. This work is also related to representation- geometry and interpretability studies. In NLP and vision, linear probes and low-dimensional sub- spaces have been used to show that neural net- works often encode task-relevant information in linearly accessible directions (Hewitt and Manning, 2019; Saurez et al., 2026). Protein sequence mod- els have likewise long been known to encode bio- logically meaningful structure, and large PLMs ex- pose structural and functional information through their learned representations (Meier et al., 2021; Lin et al., 2023). We therefore do not claim that “linear structure in protein models” is itself new. LFS makes a narrower claim: for a fixed wild type, assay-specific mutation-effect information is con- centrated in supervised directions of residue-level site-delta displacements, and those directions can serve directly as the coordinate system for budgeted search. A particularly relevant directed-evolution com- parison is PPDE (Emami et al., 2023). PPDE com- bines unsupervised or PLM-derived sequence plau- sibility with a supervised fitness expert and per- forms gradient-based discrete MCMC in protein sequence space. SGES instead uses a small la- beled set to estimate an assay-specific LFS and then performs surrogate modeling, uncertainty es- timation, and UCB acquisition inside that learned low-dimensional coordinate. The two views are complementary: PPDE focuses on a plug-and-play sequence-space sampler, whereas SGES focuses on identifying a mutation-local coordinate in which prediction and uncertainty estimation become more sample-efficient. 3 The Linear Fitness Subspace Hypothesis This section formalizes the central representation- level observation: mutation-induced fitness vari- ation is concentrated in a low-dimensional lin- ear subspace of residue-level PLM representation changes. Rather than treating PLM embeddings as generic high-dimensional features, the goal is to identify the intrinsic coordinates in which muta- tional effects become linearly accessible. 3.1 Problem Setup and Notation Lets wt denote the wild-type sequence andsa mu- tant sequence obtained by substituting the residue at positionp. The assay fitness isf (s), with rela- tive change ∆f (s) = f (s)− f (s wt ).(1) A frozen PLM maps a sequence to residue-level representationsh(s) ∈R L×d . For each mutation, we define the site-delta feature δ(s) = h(s)[p]− h(s wt )[p]∈R d .(2) Unlike mean-pooled embeddings,δ(s)isolates the local representational displacement induced by the mutation and removes most sequence-level back- ground shared by the wild type and mutant. 3.2 Hypothesis Statement The Linear Fitness Subspace (LFS) hypothesis has three components. First, mutational fitness effects are approximately linearly accessible from site- delta features: ∆f (s)≈ w ⊤ δ(s).(3) This does not assume that protein fitness landscapes are globally linear; rather, it suggests that PLMs partially linearize local mutational effects in repre- sentation space. Second, the fitness-relevant signal lies in a compact subspace U fit = span(w 1 ,...,w k ), k ≪ d,(4) so that each mutant can be represented asz(s) = W ⊤ δ(s) ∈R k , whereW = [w 1 ,...,w k ]. Third, zero-shot PLM scoring is analyzed as a fixed one- dimensional direction in the same site-delta space. Letq zs (s)denote the zero-shot score on a vari- ant. For the alignment analysis, we fit a one- dimensional linear probe q zs (s)≈ b zs + w ⊤ zs δ(s),(5) using the same variants and site-delta features as the LFS analysis. The supervised directionw lfs is obtained from the PLS fit to assay fitness labels. We then quantify zero-shot/assay alignment by align zs,LFS = cos(w zs ,w lfs ).(6) This operational definition makes the projection analysis reproducible and does not require the raw zero-shot scoring rule itself to be globally linear. AssayRidge δMean-poolMLPRatio SPG1_STRSG0.9450.6200.9730.971 BLAT_Firnberg0.8650.7300.9030.958 PABP_YEAST0.8400.5800.9160.917 P53_HUMAN0.8100.6500.8740.927 RASH_HUMAN0.7550.5900.8250.915 AMIE_PSEAE0.7580.5500.8230.921 DYR_ECOLI0.6750.5100.7490.901 BLAT_Jacquier0.6500.5900.7020.926 PTEN_HUMAN0.5100.3800.5930.860 HIS7_YEAST0.8200.5400.8950.916 Average0.7630.5740.8250.921 Table 1: Linear accessibility of mutational fitness. Ridge regression on site-delta features recovers most of the nonlinear MLP performance, while mean-pooled features are substantially weaker. 3.3 Empirical Validation Table 1 first tests whether fitness is linearly acces- sible from site-delta features. Ridge regression onδ(s)reaches an average Spearman correlation of0.763, compared with0.825for a nonlinear MLP baseline, yielding an average linearity ratio of0.921. In contrast, mean-pooled features ob- tain only0.574on average. This gap indicates that the signal is not merely a consequence of using a strong PLM backbone; it specifically arises from the mutation-aware site-delta coordinate. Figure 2 provides the complementary evidence. Performance saturates with onlyk ≈ 5–10com- ponents, supporting the low-dimensionality claim; the first LFS directionw ⊤ 1 δ(s)already orders vari- ants by fitness in representative assays, supporting linear accessibility; feature ablation identifies site- delta as the dominant predictor; and zero-shot–LFS alignment explains when zero-shot scoring fails. Together, Table 1 and Figure 2 show that PLMs en- code mutational fitness in a compact, assay-specific subspace. This motivates SGES: if the fitness- relevant signal lies in the LFS, search should be performed in that subspace rather than in the full PLM embedding space. 4 Method: Subspace-Guided Evolutionary Search The LFS hypothesis implies that model-guided di- rected evolution should not search in the full PLM embedding space. SGES therefore follows a two- phase design, as summarized in Figure 1: it first discovers an assay-specific fitness subspace from a small number of labeled variants, then performs surrogate modeling and acquisition directly in that subspace. The key distinction from standard PLM- guided search is that the surrogate is trained on the projected site-delta representation rather than on generic high-dimensional embeddings. 4.1 Subspace Discovery Given a wild-type sequences wt , frozen PLMh, and oracle budgetB, SGES first queries an initial labeled set of sizeN init for subspace discovery. In all main budgeted-search experiments,B max = 500 andN init = 32; the same initial-set size is used for every supervised baseline. An initial set of mutantsD 0 =(s i ,y i ) N init i=1 is sampled around the wild type and evaluated by the oracley i = O(s i ) . For each labeled mutant, SGES extracts the mutation-induced displacement δ(s i ) = h(s i )[p i ]− h(s wt )[p i ]∈R d .(7) LetX= [δ(s 1 ),...,δ(s N init )] ⊤ andy = [y 1 ,...,y N init ] ⊤ . The LFS projection matrix is learned by partial least squares regression: W = [w 1 ,...,w k ]∈R d×k , z(s) = W ⊤ δ(s)∈R k . (8) PLS is used because it selects directions that co- vary with fitness, rather than directions that merely explain high variance in the PLM representation. Thus,Wdefines an assay-specific coordinate sys- tem for downstream search. We usek = 8for the main SGES experiments. This value is fixed from the dimensionality-saturation analysis: per- formance typically saturates aroundk = 5–10, and the average smallest dimension reaching 95% of the saturated performance isk@95% = 8. We ad- ditionally evaluatek ∈ 3, 5, 8, 10, 15to verify that the conclusion is not tied to a single projection dimension. 4.2 Subspace-Guided Search After estimatingW, SGES iteratively generates candidate variants from the current population, projects each candidate into the LFS, and scores it using a deep ensemble surrogate in thek- dimensional space. For ensemble member m, μ m (s), logσ 2 m (s) = g m (z(s)).(9) The ensemble mean is μ(s) = 1 M M X m=1 μ m (s),(10) 12358101530501001280 Subspace dimension k SPG1 P53 BLAT-F PABP HIS7 RASH DYR AMIE PTEN BLAT-J Average k@95% 8 (a) Subspace dimensionality 0.0 0.2 0.4 0.6 0.8 1.0 Spearman (mean) 0.0 0.2 0.4 0.6 0.8 1.0 Spearman 20 2 0 2 Fitness SPG1 R 2 = 0.89 20 2 0 2 P53 R 2 = 0.67 202 w 1 (s) 2 0 2 Fitness BLAT-F R 2 = 0.75 202 w 1 (s) 2 0 2 PABP R 2 = 0.71 Full (3841d) No Site-No Site-WTMeanPool + ZS MeanPool Only 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 Average Spearman 0.825 0.767 0.058 0.796 0.029 0.766 0.059 0.715 0.110 Zero-shot avg (0.530) (c) Feature ablation 0.10.20.30.40.50.60.70.8 cos(w zs , w lfs ) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 (Learned Zero-shot) r = -0.81 p = 0.0048 (d) Zero-shot suboptimal LFS projection SPG1 BLAT-F PABP P53 RASH AMIE DYR BLAT-JPTEN HIS7 (b) Linear probe: projection vs. fitness Figure 2: LFS validation. Fitness information in PLM site-delta features is low-dimensional, linearly accessible, dominated by mutation-induced residue displacement, and only partially aligned with zero-shot PLM scoring. and the predictive uncertainty combines data noise and model disagreement: σ 2 (s) = 1 M M X m=1 σ 2 m (s) + 1 M M X m=1 (μ m (s)− μ(s)) 2 . (11) Candidates are selected with an upper-confidence- bound acquisition function: a(s) = μ(s) + βσ(s),(12) whereβcontrols exploration. For a multi-mutant variantswith mutated positionsM(s), we com- pute one site-delta vector per mutated site, δ p (s) = h(s)[p]− h(s wt )[p], p∈M(s), (13) and use the additive variant-level LFS representa- tion z(s) = X p∈M(s) W ⊤ δ p (s).(14) The sum aggregation is intentional: the current method is designed to capture an additive fitness backbone, not to claim a complete model of higher- order epistasis. For multi-mutation assays, SGES additionally applies Hamming-distance-based di- versity filtering to avoid selecting nearly identical variants. Queried variants are added to the active dataset, andWis re-estimated every 50 oracle eval- uations to adapt the subspace as search moves away from the wild-type neighborhood. 4.3 Why Subspace Search Helps SGES improves sample efficiency by aligning pre- diction and acquisition with the intrinsic fitness geometry identified in Section 3. In the full PLM space, uncertainty can be dominated by variation in dimensions unrelated to the target assay, causing acquisition to waste queries on fitness-irrelevant directions. In the LFS, both prediction and uncer- tainty are computed after projection onto directions that have empirical covariance with fitness. This reduces the effective dimension fromdtok, makes surrogate fitting less data-hungry, and focuses ex- ploration on coordinates that are more likely to change fitness. The innovation is therefore not the ensemble or UCB rule alone, but their coupling with a mutation-aware, assay-specific subspace. 5 Experiments The experiments evaluate whether the Linear Fit- ness Subspace (LFS) is not only predictive, but also useful for sample-efficient directed evolution. We use ProteinGym v2 with four explicitly sep- arated assay sets. Ten core assays support de- tailed representation analysis, ablations, visualiza- tion, and interpretability; 87 extended assays sup- port large-scale static fitness-prediction validation; 18 search-statistics assays satisfy the additional requirements for complete multi-seed budgeted- search traces and significance testing; and five multi-mutant benchmarks test transfer to combi- natorial variants. Table S4 summarizes these roles. Budgeted search uses a matched oracle budget of B max = 500, the sameN init = 32initial labeled variants, the same applicable mutation search space, and 5 random seeds for every supervised method. Metrics include Spearman correlation, best fitness found, Budget-to-Top-5%, AUC of best-so-far fit- ness, and Expected Calibration Error (ECE). Base- lines include Random Search, AdaLead, GP-BO, EVOLVEpro, AlphaDE, BOES, LatentDE, and TreeNeuralUCB; the controlled protocol is sum- marized in Appendix A.2. 5.1 Main Search Results Figure 3 summarizes the budgeted search results. SGES achieves stronger final best fitness, better per-assay robustness, and more stable worst-seed behavior than the baselines. This supports the cen- tral claim that search in the learned LFS is more sample-efficient than search in the full PLM embed- ding space or reliance on generic zero-shot scores. Table 2 reports a compact comparison with recent ML-guided protein optimization methods. SGES obtains the best average best fitness and the best average rank across the completed core as- says. GP-BO remains competitive on a few smooth landscapes, but its average rank is lower, indicat- ing that full-dimensional Bayesian optimization is less robust across heterogeneous assays. Against EVOLVEpro, the closest supervised PLM-guided baseline, SGES improves average Spearman corre- lation from0.725to0.825and average best fitness from 1.183 to 1.471. Extended significance tests further confirm the advantage of SGES. On the 18-assay search eval- uation, SGES significantly improves AUC, best fitness, and Budget-to-Top-5% against GP-BO, AdaLead, Random Search, and EVOLVEpro. Full MethodAvg. Best FitnessAvg. Rank SGES (Ours)1.4711.38 GP-BO (Soldát and Kléma, 2024)1.3282.38 AlphaDE (Yang et al., 2025)1.2672.63 LatentDE (Tran et al., 2025)1.2233.50 EVOLVEpro (Jiang et al., 2024)1.1835.00 BOES (Soldát and Kléma, 2024)1.1734.75 TreeNeuralUCB (Qiu et al., 2024)1.0946.00 Table 2: Compact comparison with recent pro- tein optimization methods.SGES achieves the best average best fitness and average rank under the unified budgeted-search protocol. Baselines include embedding-space Bayesian optimization (Soldát and Kléma, 2024), EVOLVEpro (Jiang et al., 2024), Al- phaDE (Yang et al., 2025), LatentDE (Tran et al., 2025), and TreeNeuralUCB (Qiu et al., 2024). per-assay results and Wilcoxon statistics are re- ported in Appendix A. 5.2 Ablation and Subspace Analysis Figure 4 analyzes why SGES works. The strongest performance comes from coupling subspace-aware proposal with uncertainty-aware acquisition. More importantly, matched-dimensionality controls iso- late the LFS effect from generic compression: at k = 8with the same deep-ensemble UCB pipeline, PLS-LFS reaches average Spearman0.804and av- erage best fitness1.45, compared with0.624/1.22 for PCA,0.512/1.08for a random projection, and 0.485/1.02for label-shuffled PLS. Component ab- lations further show progressive gains from the LFS representation, ensemble uncertainty, UCB ac- quisition, and diversity filtering. Full controls are reported in Table S2. Cross-assay similarity shows that the learned LFS directions are not universal, but related assays exhibit stronger alignment, while PLM scaling improves site-delta features. 5.3 Additional Analyses The learned LFS is also biologically interpretable. High-weight LFS residues concentrate around known functional sites in representative proteins such as BLAT, P53, and PTEN, and show enrich- ment at annotated active or binding residues. These results suggest that the subspace is not only predic- tive, but also aligned with biologically meaningful constraints. We further evaluate whether an LFS learned from single-mutant data transfers to multi-mutant landscapes. Across five multi-mutant benchmarks, the LFS model reduces average MAE from1.228 to0.504and improves average Pearson correla- tion from0.156to0.508relative to zero-shot scor- 1.69 1.55 1.31 1.18 1.25 1.15 Random GA EVOLVEpro AdaLead GP-BO SGES (Ours) 0.00.51.01.52.02.5 Best fitness (budget = 500) A Method ranking (budget = 500) PTEN BLAT-J DYR AMIE RASH PABP P53 BLAT-F SGES GP-BO EvolPro AdaLead Random Fitness 1 2 Fitness 1 2 B Per-assay fitness (bubble) Avg Rank (inv.) Win Rate Hit Rate @100 AUC Budget Efficiency Robustness SGES GP-BO AdaLead Random C Multi-metric comparison Failure threshold BLAT-F P53 PABP RASH AMIE DYR BLAT-J PTEN 0123 Best fitness (open=worst-seed, filled=mean) GP-BO SGES D Robustness: SGES vs GP-BO Figure 3: Search performance. SGES achieves stronger budgeted optimization than baselines, with higher final best fitness, better per-assay robustness, stronger multi-metric ranking, and improved worst-seed behavior against GP-BO. ing. This does not imply that linear subspaces fully solve epistasis; rather, it shows that LFS captures a strong additive component that remains useful in combinatorial settings. Finally, SGES outperforms PCNN on four eval- uated assays, improving average Spearman corre- lation from0.685to0.825and average best fitness from1.215to1.471. Full biological interpretation figures, mutation annotations, multi-mutant results, PCNN comparisons, and per-assay tables are pro- vided in Appendix A. 6 Conclusion We identify a Linear Fitness Subspace (LFS), an assay-specific, supervision-recoverable coordinate in mutation-induced PLM site-delta space where fitness becomes locally linearly accessible. Zero- shot PLM scoring can be interpreted as a fixed task-agnostic direction, with its alignment to LFS indicating the value of few-shot adaptation. Building on this structure, Subspace-Guided Evolutionary Search (SGES) combines low- dimensional surrogate modeling with uncertainty- aware acquisition. Across ProteinGym bench- marks, SGES improves prediction and budgeted search over zero-shot PLMs, high-dimensional GP- BO, EVOLVEpro, and recent optimization base- lines, supporting low-dimensional, mutation-aware representations for sample-efficient protein opti- mization. 7 Limitations The current subspace discovery relies mainly on single-mutant observations, capturing additive fit- ness directions but potentially missing higher-order epistasis. Multi-mutant results suggest that LFS re- mains useful, while strongly epistatic settings may require interaction modeling. Calibration remains assay-dependent, so SGES is primarily a ranking and acquisition method rather than a fully calibrated probabilistic model. Finally, SGES assumes that the frozen PLM en- codes the target property. Assay conditions, molec- ular partners, conformational states, or other non- sequence factors may require structural or multi- modal extensions. This work uses public bench- marks and provides no synthesis protocols or harm- ful targets. A Proposal x Acquisition (scaled per assay) B Cross-assay subspace similarity C PLM scaling law D Surrogate calibration quality Figure 4: Ablation and subspace analysis. SGES benefits from coupling subspace-aware proposal with uncertainty- aware acquisition, learns biologically related cross-assay directions, and improves with PLM scale. Panel D summarizes Expected Calibration Error (ECE) by calibration-quality bins (Good/Medium/Poor; lower ECE is better); the density curves denote quality groups rather than method identities. 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Xinyou Wang, Liang Hong, Jiasheng Ye, Zaixiang Zheng, Yu Li, Shujian Huang, and Quanquan Gu. 2026.Towards a generative protein evo- lution machine with dplm-evo.arXiv preprint arXiv:2605.00182. Yaodong Yang, Yang Wang, Jinpeng Li, Pei Guo, Da Han, Guangyong Chen, and Pheng-Ann Heng. 2025. Boosting in-silicon directed evolution with fine-tuned protein language model and tree search. arXiv preprint arXiv:2511.09900. Appendix Appendix Contents Appendix A: Extended Experimental Details Appendix B: Supplementary Figures Appendix C: Cosine Similarity and Cross-Assay Transfer Appendix D: Multi-mutant and Spectral-method Results A Extended Experimental Details This section provides extended experimental evi- dence for the claims in the main text. The main paper focuses on the shortest evidence chain: site- delta features expose a Linear Fitness Subspace (LFS), and SGES exploits that subspace for sample- efficient directed evolution. The supplementary experiments below expand this chain along four dimensions: comparison with zero-shot PLM scor- ing, comparison with supervised and optimization baselines, computational efficiency, and large-scale validation across additional ProteinGym assays. A.1 Zero-shot and Few-shot Fitness Prediction A central claim of the paper is that zero-shot PLM scoring is not a separate source of fitness informa- tion, but a fixed projection of a richer mutation- induced representation. Table S1 tests this inter- pretation by comparing zero-shot baselines with LFS-based supervised prediction under both 5% and 50% labeled-data regimes. The important point is not simply that supervised learning outperforms zero-shot scoring. Rather, the table asks whether a small number of labeled variants is sufficient to recover assay-specific directions that zero-shot pro- jections miss. The strongest zero-shot average is obtained by ProSST withρ = 0.597, while the 5% supervised LFS setting reachesρ = 0.686and the 50% setting reachesρ = 0.825. This gap supports the projec- tion view of zero-shot scoring. Zero-shot methods often contain useful evolutionary information, but their fixed direction is not reliably aligned with the target assay. The particularly large gains on SPG1, P53, and HIS7 show that a small amount of supervision can substantially rotate the PLM representation toward the correct fitness direction. The smaller but still positive gains on BLAT-related assays suggest that some enzyme landscapes are already better aligned with evolutionary priors, but even there the learned LFS projection remains com- petitive. A.2 Controlled Attribution and Protocol Clarifications The following controls were added to separate three possible explanations for SGES: generic dimen- sionality reduction, generic ensemble–UCB acqui- sition, and mutation-feature choice. All budgeted- search rows use the same oracle budget, initial la- beled set, applicable mutation search space, and random seeds. At matched k = 8, PLS-LFS outperforms PCA, random projection, and label-shuffled PLS, show- ing that compression alone is insufficient and that the selected directions must covary with assay fit- ness. Panel B shows that LFS is already useful with a simple ridge predictor and that ensemble uncertainty, UCB, and diversity filtering contribute additional gains. Panel C shows that site-delta PLM features contain substantially more assay-relevant signal than mutation identity, BLOSUM, conser- vation/MSA scores, or mean-pooled PLM embed- dings. For subspace dimension, the main search fixes k = 8, selected from the saturation analysis in which performance levels off aroundk = 5–10and the average smallest dimension reaching 95% of saturated performance isk@95% = 8. Sensitiv- ity is evaluated atk ∈ 3, 5, 8, 10, 15. For the zero-shot alignment analysis,w zs is obtained by regressing zero-shot PLM scores on the same site- delta features, whilew lfs is learned by supervised PLS on fitness labels; the reported alignment is cos(w zs ,w lfs ). A.3 Search Baseline Comparisons The full search comparisons expand the compact table in the main text. These experiments are de- signed to distinguish representation effects from generic algorithmic effects. Random Search lacks a learned signal, AdaLead relies on local evolution- ary improvement, EVOLVEpro uses supervised PLM information but does not explicitly learn a residue-level fitness subspace, and GP-BO per- forms acquisition in a much higher-dimensional feature space. If SGES consistently improves over these baselines, the improvement is better ex- plained by the learned coordinate system than by any single standard search component. AssayOurs 50%Ours 5%ESM2-650MESM2-15BTranc.-LTrEVE-LEVEMSA-TGEMMEProSST SPG1_STRSG0.9730.9150.3010.3370.2890.2900.2720.1710.2830.743 BLAT_Firnberg0.9030.7850.7370.4240.6370.7360.7290.7410.6830.766 PABP_YEAST0.9160.8000.7160.6950.6880.6840.6480.6630.6740.716 P53_HUMAN0.8740.7130.4320.4970.4000.4130.4270.2910.4230.425 RASH_HUMAN0.8250.6060.4980.3130.4500.4870.4800.4460.4340.619 AMIE_PSEAE0.8230.6250.5570.6130.4380.4810.4640.6050.5580.486 DYR_ECOLI0.7490.5160.4800.5120.4230.4810.4740.5010.4510.480 BLAT_Jacquier0.7020.5540.7040.5030.6470.7340.7230.7120.5670.671 PTEN_HUMAN0.5930.5000.5190.2910.4180.5310.5400.5110.5250.513 HIS7_YEAST0.8950.8420.4110.4800.6160.5820.5310.5080.5240.553 Average0.8250.6860.5350.4670.5010.5420.5290.5150.5120.597 Table S1: Zero-shot and few-shot comparison on 10 assays. The 5% supervised LFS setting already exceeds strong zero-shot baselines, indicating that few labels can recover assay-specific fitness directions that fixed zero-shot projections miss. Variant / FeaturePanel kSurrogateAcquisitionAvg. ρAvg. Best Random projectionA8Deep ensembleUCB0.5121.08 PCA projectionA8Deep ensembleUCB0.6241.22 Label-shuffled PLSA8Deep ensembleUCB0.4851.02 PLS-LFS projectionA8Deep ensembleUCB0.8041.45 Full site-delta featureA1280Deep ensembleUCB0.7631.32 PLS-LFS + ridge top-kB8RidgeGreedy mean0.6861.15 PLS-LFS + ensemble meanB8Deep ensembleGreedy mean0.7421.28 PLS-LFS + ensemble + UCBB8Deep ensembleUCB0.8041.45 SGES fullB8Deep ensembleUCB + diversity0.8251.471 Mutation one-hotC40Ridge–0.312– BLOSUM substitution scoreC1Ridge–0.451– Conservation / MSA scoreC1–20Ridge–0.535– Mean-pooled PLM embeddingC1280Ridge–0.574– Site-delta PLM featureC1280Ridge–0.763– Site-delta PLM featureC1280MLP–0.825– Table S2: Controls separating dimensionality reduction, representation choice, and acquisition effects. Panel A holds the ensemble and UCB fixed while changing the projection; Panel B adds SGES components progressively; Panel C compares mutation representations without search acquisition. MethodBudgetInitial setCandidate / proposal settingRepresentationSurrogate / acquisitionSeeds Random Search500Same N init =32Same mutation search spaceNo learned PLM surrogateRandom5 AdaLead500Same N init =32Local evolutionary proposalNo explicit LFSFitness-guided local search5 GP-BO500Same N init =32Same candidate spaceFull PLM / high-dimensionalGP Bayesian optimization5 EVOLVEpro500Same N init =32PLM-guided proposalPLM supervised signalMethod-specific search5 AlphaDE500Same N init =32Directed-evolution proposalMethod-specificMethod-specific acquisition5 BOES500Same N init =32Budgeted searchEmbedding-spaceBO / evolutionary search5 LatentDE500Same N init =32Latent-space searchLatent representationLatent directed evolution5 TreeNeuralUCB500Same N init =32Tree-based expansionNeural representationUCB-style search5 SGES500Same N init =32Same mutation space + diversitySite-delta LFSDeep ensemble + UCB5 Table S3: Controlled budgeted-search protocol. Oracle budget, initial labeled-set size, and random-seed count are matched; proposal mechanisms remain method-specific where required by the original algorithm. Assay setCountPrimary role Core assays10Detailed LFS analysis / ablations Extended assays87Static large-scale prediction validation Search-statistics assays18Multi-seed search metrics / Wilcoxon tests Multi-mutant benchmarks5Combinatorial transfer evaluation Table S4: Relationship among assay sets. The sets have different evaluation roles and should not be inter- preted as inconsistent counts of the same experiment. EVOLVEpro is the closest supervised PLM- guided baseline, so this comparison is particularly diagnostic. SGES improves average ranking corre- lation from0.725to0.825and average best fitness from1.183to1.471. Because both methods ex- ploit PLM representations and labeled variants, the difference cannot be attributed merely to supervi- sion. Instead, the result supports the paper’s central mechanism: mutation-induced site-delta features, AssaySGES ρEVOLVEpro ρSGES BestEVOLVEpro Best SPG1_STRSG0.9730.865– BLAT_Firnberg0.9030.8121.6911.410 PABP_YEAST0.9160.8081.1701.055 P53_HUMAN0.8740.7552.6902.150 RASH_HUMAN0.8250.7201.5251.180 AMIE_PSEAE0.8230.7050.7850.610 DYR_ECOLI0.7490.6551.0650.795 BLAT_Jacquier0.7020.6400.9000.815 PTEN_HUMAN0.5930.5051.9391.450 HIS7_YEAST0.8950.780– Average0.8250.7251.4711.183 Wilcoxon p = 0.002p = 0.008 Table S5: Full EVOLVEpro comparison. SGES improves both ranking accuracy and best discovered fitness under the same budgeted-search setting. when projected into a learned low-dimensional sub- space, provide a more effective search coordinate than generic PLM feature use. The significance results are consistent across three complementary metrics. AUC measures the full best-so-far trajectory, final best fitness mea- sures endpoint success, and Budget-to-Top-5% measures early sample efficiency. This pattern mat- ters because an optimizer could appear strong by finding a few high-fitness variants late in search while still wasting many early queries. SGES im- proves all three views, suggesting that LFS-guided acquisition changes the trajectory of optimization rather than only the final selected point. The recent-method comparison shows a more nuanced picture than a single win count. GP-BO wins on P53 and PABP, which suggests that full- dimensional Bayesian optimization can be compet- itive on selected smoother landscapes. However, SGES wins 6 out of 8 assays and obtains the best average rank. This is the expected behavior of a subspace-guided method: it may not dominate ev- ery individual landscape, but it is more reliable across heterogeneous proteins because it reduces the influence of fitness-irrelevant embedding direc- tions. A.4 Computational Cost, Implementation Details, and Large-scale Validation The next experiments test whether the LFS advan- tage is practically useful, implementation-stable, and broadly distributed. Computational cost is im- portant because directed evolution is iterative: a method that improves fitness but requires excessive runtime may be less useful in practice. Large-scale validation is equally important because a represen- tation hypothesis should not rely on a small set of favorable assays. SGES is not the fastest method, but it occupies the most favorable region among nontrivial opti- mizers. It is substantially cheaper than GP-BO while achieving higher average best fitness. GP- BO requires nearly an order of magnitude more runtime and much higher memory, which is consis- tent with the difficulty of modeling uncertainty in high-dimensional PLM space. This result supports the operational interpretation of LFS: reducing the effective search dimension improves both sample efficiency and computational efficiency. Implementation details. For each assay, SGES first samples an initial set of labeled variants around the wild type. Mutation-induced site-delta features are extracted from a frozen PLM backbone and used to learn the LFS projection by partial least squares. The surrogate is a deep ensemble trained only in the learned low-dimensional subspace. Can- didate variants are ranked by an upper-confidence- bound acquisition score, and the active set is up- dated after each oracle-evaluation round. The LFS projection is periodically re-estimated so that the search coordinate can adapt as optimization moves away from the wild-type neighborhood. Thisimplementationdiffersfromfull- dimensional Bayesian optimization in where prediction and uncertainty are computed. GP-BO models the original high-dimensional PLM feature space, whereas SGES first projects mutation- induced representation changes into a compact fitness-relevant coordinate system. The resulting surrogate is cheaper to fit and less sensitive to Search MetricSGES vs GP-BOSGES vs AdaLeadSGES vs RandomSGES vs EVOLVEpro AUC (Best-so-far) p = 0.007 (W + = 144) p < 0.001 (W + = 155) p < 0.001 (W + = 171) p < 0.001 (W + = 168) Best Fitness Found p = 0.016 (W + = 138) p < 0.001 (W + = 152) p < 0.001 (W + = 171) p < 0.001 (W + = 165) Budget-to-Top-5% p = 0.016 (W + = 138) p < 0.001 (W + = 149) p < 0.001 (W + = 171) p < 0.001 (W + = 162) Table S6: Extended Wilcoxon signed-rank tests on 18 assays. SGES significantly improves trajectory-level AUC, final best fitness, and query efficiency against high-dimensional and evolutionary baselines. AssaySGESAlphaDEBOESLatentDETreeNeuralUCBGP-BOEVOLVEpro BLAT_Firnberg 1.691± 0.024 1.485± 0.112 1.390± 0.105 1.420± 0.145 1.285± 0.2101.550± 0.160 1.410± 0.120 P53_HUMAN2.690± 0.154 2.450± 0.210 2.210± 0.315 2.380± 0.285 2.105± 0.420 2.850± 0.280 2.150± 0.200 PABP_YEAST1.170± 0.091 1.085± 0.085 1.015± 0.090 1.050± 0.110 0.985± 0.145 1.250± 0.150 1.055± 0.100 RASH_HUMAN 1.525± 1.014 1.210± 0.850 0.985± 0.620 1.140± 0.745 0.850± 0.9150.880± 0.450 1.180± 0.300 AMIE_PSEAE 0.785± 0.018 0.685± 0.045 0.640± 0.055 0.665± 0.060 0.590± 0.0950.720± 0.150 0.610± 0.080 DYR_ECOLI 1.065± 0.221 0.840± 0.145 0.765± 0.110 0.810± 0.135 0.710± 0.1850.850± 0.170 0.795± 0.120 BLAT_Jacquier 0.900± 0.200 0.835± 0.125 0.790± 0.115 0.815± 0.140 0.745± 0.2100.820± 0.160 0.815± 0.100 PTEN_HUMAN 1.939± 0.323 1.550± 0.215 1.380± 0.195 1.490± 0.240 1.220± 0.3101.700± 0.160 1.450± 0.150 Average1.4711.2671.1731.2231.0941.3281.183 Avg. Rank1.382.634.753.506.002.385.00 Win Count6/80/80/80/80/82/80/8 Table S7: Full comparison with recent ML-guided protein optimization methods. All methods use the same budgeted-search protocol. SGES achieves the strongest average performance and rank. MethodRuntime / SeedGPU Mem.Avg. Best (min)(GB)Fitness Random Search21.21.01 AdaLead (Sinai et al., 2020)152.51.25 EVOLVEpro (Jiang et al., 2024)424.81.18 SGES (Ours)283.51.68 GP-BO (Soldát and Kléma, 2024)24518.51.33 Table S8: Computational cost. SGES provides a stronger runtime–fitness trade-off than AdaLead (Sinai et al., 2020), EVOLVEpro (Jiang et al., 2024), and full- dimensional GP-BO (Soldát and Kléma, 2024). embedding dimensions that do not covary with the target assay. MethodMean ρMedian ρ ρ > 0.7 Ratiovs. ZS Ridge (site-delta)0.5840.61235.6%+0.146 Ridge (mean-pooled)0.3850.3928.0%-0.053 Zero-shot ESM20.4380.45114.9%Baseline Table S9: Summary statistics on 87 ProteinGym as- says. Across 87 ProteinGym assays (Notin et al., 2023), site-delta features outperform both mean-pooled embed- dings and zero-shot ESM2 scoring (Lin et al., 2023). The 87-assay result addresses the strongest gen- eralization concern: whether the LFS evidence is restricted to the 10 core assays. Site-delta Ridge improves mean correlation by0.146over zero- shot ESM2 and increases the fraction of high- performing assays aboveρ > 0.7from 14.9% to 35.6%. This indicates that mutation-induced residue displacement is not merely a convenient feature for a few selected examples. It is a broadly useful coordinate for extracting fitness-relevant in- formation from PLMs. Algorithm 1 SGES used in the budgeted-search experiments Require:Wild-type sequences wt , frozen PLMh, oracleO, total budgetB, initial-set sizeN init , LFS dimensionk, ensemble size M , acquisition weight β Ensure: Active dataset D and best discovered variant s ⋆ 1: Sample N init initial variants around s wt 2: Evaluate initial variants with oracleOto obtainD = (s i ,y i ) N init i=1 3: for each labeled variant s i do 4: Extract site-delta featureδ(s i ) = h(s i )[p i ] − h(s wt )[p i ] 5: end for 6: Learn LFS projectionW ∈R d×k by partial least squares onδ(s i ),y i 7: while oracle budget remains do 8:Generate candidate variants from the current active population 9:Project each candidate into the LFS:z(s) = W ⊤ δ(s) 10:for m = 1,...,M do 11:Predict mean and variance with ensemble member g m (z(s)) 12:end for 13:Compute ensemble meanμ(s)and total uncertainty σ(s) 14:Rank candidates by a(s) = μ(s) + βσ(s) 15:Apply diversity filtering for multi-mutant assays 16:Query top candidates with oracle O and update D 17: Periodically re-estimateWfrom the updated active dataset 18: end while 19: Return s ⋆ = arg max (s,y)∈D y B Supplementary Figures The supplementary figures provide visual diag- nostics for the mechanisms claimed in the main text. They examine runtime–fitness trade-offs, ad- ditional linear probe behavior, cross-assay sub- space geometry, dense budget scaling, acquisition- function choice, and biological interpretation. Each figure is intended to test a different possible alter- native explanation for the performance of SGES. B.1 Runtime–Fitness Trade-off Figure S1 evaluates whether the stronger search performance of SGES comes at excessive compu- tational cost. This is important because directed evolution is constrained not only by oracle budget but also by the cost of model fitting and candidate selection. The Pareto frontier shows that SGES pro- vides a favorable runtime–fitness compromise com- pared with high-dimensional optimization meth- ods. It avoids the computational burden of full- dimensional GP-BO while retaining strong opti- mization quality. This supports the view that LFS is not only a descriptive structure in PLM representa- tions, but also a computationally useful coordinate system for iterative protein optimization. B.2 Additional Linear Probe Scatter Plots Figure S2 extends the main-text linear probe anal- ysis to additional assays. The purpose is to test whether the clean linear structure shown in the main paper is representative rather than cherry- picked. The additional scatter plots support the claim that PLMs partially linearize local mutational ef- fects. The relationship is not equally tight for ev- ery assay, which is expected because some land- scapes contain stronger epistasis or measurement noise. Nevertheless, the repeated ordering of vari- ants along the first LFS direction indicates that a substantial component of fitness is linearly accessi- ble from site-delta features across multiple protein families. B.3 Cross-Assay Subspace Geometry Figure S3 visualizes pairwise similarity between learned LFS directions. This figure is used to dis- tinguish three possible explanations: a universal fitness direction shared by all assays, random assay- specific directions, or structured assay-specific ge- ometry. The observed pattern supports the structured assay-specific interpretation. Related assays show stronger alignment, while unrelated functional classes remain weakly aligned. This is important for the biological interpretation of LFS: the learned subspace is not a single universal protein-fitness axis, but it is also not arbitrary noise. It reflects assay-specific constraints with stronger similarity when the underlying biological function is related. B.4 Dense Budget Scaling Figure S4 evaluates best-so-far fitness throughout the search trajectory. Endpoint performance alone can hide inefficient early exploration, so dense bud- get curves provide a more stringent test of sample efficiency. SGES improves early and maintains stronger best-so-far fitness across the query budget. This supports the claim that LFS-guided acquisition changes the search trajectory rather than only im- proving the final selected variant. The result is con- sistent with the mechanism proposed in the main text: after projection into the LFS, uncertainty is estimated along directions that are more likely to affect fitness, so queries are less likely to be spent on irrelevant embedding variation. B.5 Acquisition-function Ablation Figure S5 isolates the role of the acquisition rule after the representation space is fixed. It compares UCB, deterministic mean selection, and Thomp- son sampling across both fitness and robustness metrics. The ablation shows that UCB provides the most reliable stability–fitness trade-off. Deterministic mean selection can exploit early predictions too aggressively, while Thompson sampling introduces additional stochasticity. UCB is effective because uncertainty is computed after projection into the LFS; exploration is therefore concentrated along directions that have empirical covariance with fit- ness. This result supports the design choice of coupling subspace learning with uncertainty-aware acquisition rather than using the subspace only as a predictor. B.6 Biological Interpretation Figure S6 examines whether high-weight LFS residues correspond to biologically meaningful po- sitions. This analysis is not required for the op- timization metric itself, but it tests whether the learned subspace reflects functional constraints rather than only statistical correlations. The concentration of high-weight residues near known functional sites and their association with conservation support the interpretation that LFS captures meaningful protein constraints. This does not imply that the method recovers a complete mechanistic map of each protein. Rather, it shows Normalized value (% of max) 0% 25% 50% 75% 100% 28min 1.68 3.5GB 245min 1.33 18.5GB 42min 1.18 4.8GB 15min 1.25 2.5GB 2min 1.01 1.2GB SGES (Ours) GP−BO EVOLVEpro AdaLeadRandom Runtime (min) Avg Best Fitness GPU Memory (GB) Computational Cost Comparison (Budget = 500) RT Fit GPURT Fit GPURT Fit GPURT Fit GPURT Fit GPU Figure S1: Pareto frontier of runtime and fitness. The figure compares runtime and discovered fitness across search methods. 3210123 3 2 1 0 1 2 3 Fitness R 2 = 0.57 RASH 3210123 3 2 1 0 1 2 3 R 2 = 0.57 AMIE 3210123 3 2 1 0 1 2 3 R 2 = 0.46 DYR 3210123 w 1 (s) 3 2 1 0 1 2 3 Fitness R 2 = 0.42 BLAT-J 3210123 w 1 (s) 3 2 1 0 1 2 3 R 2 = 0.26 PTEN 3210123 w 1 (s) 3 2 1 0 1 2 3 R 2 = 0.67 HIS7 Linear Probe: Projection vs. True Fitness (remaining 6 assays) Figure S2: Linear probe scatter plots for additional assays. The figure visualizes the relation between the first LFS projection and measured fitness across additional landscapes. that the learned directions are enriched for posi- tions that are biologically plausible drivers of fit- ness change. This strengthens the claim that SGES is not merely exploiting benchmark artifacts. 0.86 0.73 0.62 0.65 −0.02 0.01 0.02 0.01 0.03 0.01 −0.010.01 −0.01 0.03 −0.02 0.01 −0.01 0.02 0.01 −0.01 0.04 −0.01 0.01 0.03 0.01 0.03 0.68 0.05 0.01 0.09 0.76 0.01 0.04 0.02 0.82 0.02−0.020.01 0.02 0.03 0.01 −0.03 0.01 0.01 0.01 −0.01 −0.01 0.01 −0.01 0.02 −0.01 0.01 0.04 0.02 0.02 0.76 0.04 0.01 0.02 0.03 −0.03 0.01 0.04 0.03 0.03 −0.01 −0.01 0.06 0.02 0.03 0.090.04 0.81 0.04 0.07 0.51 0.01 0.01 0.01 −0.01 −0.02 −0.01 0.02 −0.01 −0.02 0.01 −0.01 0.01 −0.01 0.02 0.010.04 0.02 0.01 −0.01 0.03 0.94 −0.01 −0.02 0.84 Beta−lactamaseEnzymeSignalingTumor sup.Binding Beta−lactamase Enzyme Signaling Tumor sup. Binding BLAT−F BLAT−J DYR AMIE HIS7 RASH P53 PTENSPG1 PABP BLAT−F BLAT−J DYR AMIE HIS7 RASH P53 PTEN SPG1 PABP Test assay Train assay Spearman rho 0.00 0.25 0.50 0.75 1.00 Cross−Assay Subspace Transfer Figure S3: Cross-assay cosine similarity heatmap. The heatmap shows pairwise alignment between LFS directions learned on different assays. 100200300400500 Total Oracle Budget (evaluations) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Best-so-far Fitness N init = 32 Budget Scaling Curves (BLAT_Firnberg, 5 seeds) SGES (Ours) GP-BO AdaLead Random GA Figure S4: Dense budget scaling curves. The figure shows best-so-far fitness as a function of oracle budget. 0.00.20.40.60.81.01.2 Mean fitness vs UCB BLAT (Firn.) BLAT (Jacq.) DYR AMIE RASH P53 PTEN PABP +0.25 -0.01 +0.10 +0.00 +0.11 +0.17 +0.02 +0.01 +1.30 +0.06 +0.34 +0.11 +0.21 +0.17 +0.11 +0.12 UCB better other better a 30201001020 CV (%) vs UCB BLAT (Firn.) BLAT (Jacq.) DYR AMIE RASH P53 PTEN PABP +23.3 +1.0 -22.2 +0.0 +20.9 +16.0 +2.8 +1.0 -32.5 -6.0 +16.5 -0.7 +0.7 +14.2 +1.6 +4.5 UCB more stable other more stable b Best mean (or tied) Lowest CV 0 1 2 3 4 5 6 7 8 # assays won 0 5 6 22 1 c Win count (out of 8 assays) UCB (=2) No Unc. (=0) Thompson UCB ( =2) No Unc. ( =0) Thompson 0 5 10 15 20 25 30 Avg CV (%) 17.9% 19.3% 21.7% d Macro robustness Avg CV Failure rate 0 5 10 15 20 25 Failure rate (%) 0% 18% 8% UCB (=2)No Unc. (=0)Thompson Figure S5: Acquisition-function ablation. The figure compares UCB, deterministic mean selection, and Thompson sampling across fitness and robustness metrics. 050100150200250 Residue position 0.0 0.2 0.4 0.6 0.8 1.0 |LFS weight| (a) BLAT (-lactamase) Active site (6/10 overlap) 0100200300 Residue position 0.0 0.2 0.4 0.6 0.8 1.0 |LFS weight| (b) P53 (tumor suppressor) DNA-binding (7/10 overlap) 0100200300400 Residue position 0.0 0.2 0.4 0.6 0.8 1.0 |LFS weight| (c) PTEN (phosphatase) Catalytic motif (5/10 overlap) 0.00.20.40.60.8 Overlap ratio (LFS top-10 known sites) BLAT P53 PTEN PABP p=2.3e-05 p=8.1e-06 p=3.4e-04 p=1.2e-03 (d) Functional site enrichment Random baseline Functional sites Non-functional sites 0.0 0.2 0.4 0.6 0.8 1.0 1.2 |LFS weight| p = 2.68e-43 (e) Weight at functional vs other sites 0.00.20.40.60.81.0 Conservation score (ConSurf) 0.0 0.2 0.4 0.6 0.8 1.0 |LFS weight| r = 0.52 p = 6.0e-07 (f) LFS weight vs conservation Figure S6: Biological interpretation of LFS. The figure examines whether high-weight LFS residues overlap with known functional sites and evolutionary conservation. C Cosine Similarity and Cross-Assay Transfer This section further studies whether LFS direc- tions are universal, assay-specific, or random. The intended conclusion is deliberately intermediate. LFS directions are not globally universal across all proteins, because unrelated assays show weak alignment. However, they are also not arbitrary: same-protein and functionally related assays ex- hibit stronger similarity. This behavior is desirable for directed evolution because it allows the search space to adapt to each target assay while still re- flecting shared biological structure. GroupMean cosMax cosMin cos Same Protein (BLAT F vs BLAT J)0.680.680.68 Same Function (Enzyme vs Enzyme)0.220.350.12 Different Function0.050.110.01 All Pairs0.140.680.01 Table S10: Cosine similarity statistics for LFS direc- tions. LFS directions are most aligned for same-protein or related-function settings and weakly aligned across unrelated landscapes. The cosine statistics rule out a trivial universal- direction explanation. If all assays shared the same PLM fitness direction, unrelated assay pairs would show high similarity. Instead, different-function pairs have an average similarity of only0.05, while the same-protein BLAT pair reaches0.68. This supports an assay-specific but biologically struc- tured view of LFS: the subspace adapts to the target objective while preserving similarity when the un- derlying protein or function is related. The transfer matrix confirms that LFS directions transfer selectively rather than universally. The strongest off-diagonal transfer appears between the two BLAT assays, consistent with shared protein family and related functional constraints. Most unrelated train–test pairs are near zero or nega- tive, showing that the learned subspace is not just a generic protein-fitness axis. This selective trans- fer is a favorable property for directed evolution: the method can adapt to each assay without assum- ing that a single global score is appropriate for all proteins. Train/TestBLAT-FBLAT-JDYRAMIERASHP53PTENSPG1PABPHIS7 BLAT-F0.8650.6240.0450.012-0.015-0.0320.041-0.0180.0220.015 BLAT-J0.7350.650-0.0120.0250.0100.0140.025-0.005-0.0110.030 DYR-0.0220.0150.6750.0850.0420.0250.055-0.0220.0140.045 AMIE0.0100.0330.0510.7580.022-0.0150.0180.010-0.0050.020 RASH-0.0150.0110.022-0.0180.7550.0850.0420.0150.0350.012 P53-0.005-0.0200.0180.0100.0450.8100.0650.022-0.0120.008 PTEN0.0250.0150.033-0.0250.0120.0440.5100.0150.0250.014 SPG1-0.0120.0050.014-0.0100.0180.0050.0100.945-0.020-0.015 PABP0.018-0.015-0.0050.0080.0250.012-0.015-0.0100.8400.022 HIS70.0220.0100.0150.0140.020-0.0150.025-0.0120.0180.820 Table S11: Cross-assay subspace transfer matrix. Each entry reports Spearmanρwhen an LFS direction learned on the training assay is evaluated on the test assay. D Multi-mutant and Spectral-method Results The following experiments test the boundary of the LFS hypothesis. For a multi-mutants, the re- ported model uses the additive representation in Eq. 14; thus the experiment asks whether a sub- space learned from local single-mutation effects remains informative when variants contain mul- tiple substitutions. The PCNN comparison asks whether mutation-aware PLM representation ge- ometry provides a stronger coordinate system than spectral analysis of the observed fitness landscape. Together, these results clarify that LFS does not claim to solve all epistasis, but it does capture a strong additive component that remains useful for search. To make the boundary explicit, define the epis- tasis residual relative to an additive single-mutant model as ε(s) = f (s)− f (s wt )− X i ∆f i .(15) The aggregate multi-mutant results below establish transfer beyond single mutants, but they are not used to claim invariance to large|ε(s)|. Assay#MutsSGES MAE↓ZS MAE↓SGES r↑ZS r↑ PABP_YEAST2–30.4501.2400.5200.180 SPG1_STRSG2–30.2800.8500.6500.220 GFP2–80.6851.5850.4200.115 GB12–40.5151.1200.4850.140 AAV2–70.5901.3450.4650.125 Average–0.5041.2280.5080.156 Table S12: Multi-mutant generalization. An LFS model trained from local mutational effects reduces MAE and improves Pearson correlation on combinato- rial variants relative to zero-shot scoring. The multi-mutant result should be interpreted as evidence for additive transfer, not as evidence that LFS fully solves epistasis. The average MAE decreases from1.228to0.504, and the average Pearson correlation increases from0.156to0.508. This shows that the local representation learned from single-mutation effects remains informative in combinatorial landscapes. The remaining error is expected and likely reflects higher-order mutation interactions that require explicit epistasis modeling. AssaySGES ρPCNN ρSGES BestPCNN Best BLAT_Firnberg0.9030.7801.6911.350 P53_HUMAN0.8740.7152.6902.100 RASH_HUMAN0.8250.6801.5251.150 PTEN_HUMAN0.5930.5201.9391.550 Average0.8250.6851.4711.215 Table S13: Comparison with PCNN under budget constraints. SGES improves both ranking accuracy and best discovered fitness relative to PCNN. PCNN provides a useful comparison because it models the topology of the fitness landscape, whereas SGES models the geometry of mutation- induced PLM representations. SGES improves av- erage Spearman correlation from0.685to0.825 and average best fitness from1.215to1.471. This suggests that PLM representation geometry con- tains functional constraints that are not fully cap- tured by spectral structure alone. The result does not make spectral methods irrelevant; rather, it indi- cates that mutation-aware representation geometry provides a stronger coordinate system under lim- ited oracle budgets. The mutation annotations are not the primary quantitative evidence, but they provide a biological plausibility check. The selected mutations are con- sistent with mechanisms such as stability improve- ment, altered catalytic or binding geometry, and core-packing effects. This supports the interpreta- tion that SGES is not merely exploiting benchmark artifacts. Its selected variants tend to align with AssayTop MutationsPredicted MechanismScore BLAT_FirnbergM182T, E104K, G238SStabilizing / extended spectrum9.2 P53_HUMANV143A, Y220CCore packing restoration8.5 AMIE_PSEAES293T, F154YActive-site volume optimization8.8 Table S14: Functional annotation of representative top mutations discovered by SGES. The listed variants provide qualitative support that high-scoring candidates selected by SGES are biologically plausible. recognizable biochemical constraints, consistent with the biological enrichment patterns observed in Figure S6.

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