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Robust Biomolecular Complex Design Across Protein Conformational Landscapes

Qingyuan Zeng, Zongqi Xu, Anglin Liu, Ziqi Gong, Pengxiang Cai, Zixin Guan, Yunan Chen, Sen Gao, Min Zhou, Jintai Chen

arXiv:2609.33726v2cs.AIq-bio.QM

TL;DR

Structure-based design often optimizes against one target conformation, although a candidate may lose contacts or develop clashes in another natural state. FlexEvo adapts candidates from a single observed conformation through geometry-constrained, bi-objective evolutionary search without retraining the source model. Across nine binder categories, it reduces category-balanced mean relative performance degradation from 47.8% to 4.4%.

  • Problem

    Structure-based design typically optimizes candidates against a single target conformation, although proteins populate ensembles and candidates may lose favorable interactions or develop steric clashes in other states.

    Source: §1 · p. 1

  • Method

    FlexEvo adapts candidates from one observed conformation using geometry-constrained bi-objective evolutionary search that balances input-state interaction preservation with robustness to geometric perturbations.

    Source: §1 · pp. 2–3

  • Results

    47.8% to 4.4%: FlexEvo reduces category-balanced mean relative performance degradation across nine binder categories, with 1.4–3.1 minutes of adaptation per sample.

    Source: §1 · p. 3

  • Takeaways & Limitations

    Experiments support constrained inference-time adaptation of existing candidates as a route to improving scores on natural target conformations unseen during adaptation while maintaining input-state performance on Vina and DockQ.

    Source: §5 · p. 10

    This computational study does not establish biological safety or clinical efficacy; biomedical use requires further validation.

    Source: ETHICS STATEMENT · p. 10

Abstract

from arXiv · show

Proteins populate conformational ensembles, yet structure-based biomolecular design typically optimizes candidates against a single target conformation. Consequently, a candidate that fits one state can lose favorable interactions or develop steric clashes when the target adopts another. We introduce FlexEvo, a model-agnostic evolutionary framework that adapts candidates once at inference time from a single target conformation to improve compatibility with alternative natural conformations unseen during adaptation, without retraining the source model or requiring a conformational ensemble. FlexEvo casts cross-state adaptation as geometry-constrained bi-objective optimization, balancing preservation of input-state interactions against robustness to plausible conformational perturbations. To limit the search space and reduce invalid structural edits, geometry-derived FlexBoxes define protected anchor regions, adaptable regions for local exploration, and forbidden regions for clash avoidance. A unified all-atom representation supports topology-preserving adaptation across diverse binder categories, while Pareto selection preserves nondominated candidates across the two objectives. We evaluate FlexEvo across multiple generation baselines and nine representative binder categories spanning diverse molecular sizes and structural topologies. FlexEvo reduces the category-balanced mean relative performance degradation from 47.8% to 4.4%, while adding only 1.4--3.1 minutes of adaptation per sample. These results establish single-state inference-time adaptation as a practical route toward robust biomolecular complex design across protein conformational landscapes.

1 INTRODUCTION

Structure-based design commonly optimizes candidates against one target conformation, although proteins occupy conformational ensembles and candidates may lose contacts or develop clashes in other states. FlexEvo adapts candidates from one observed state using geometry-constrained, bi-objective search, and evaluations report reduced performance degradation across nine binder categories.

Motivation

  • FlexEvo asks whether a candidate adapted once from a single observed conformation can remain compatible with multiple natural conformations unseen during optimization.The question matters because the target state encountered after design is often unknown, and a candidate optimized for one state may lose favorable contacts or develop steric clashes in another.Source: §1 · pp. 1–2
  • Accommodating target changes and coordinated binder edits is challenging because atomic variables grow with binder size and unrestricted edits can produce invalid geometries.Target changes can range from side-chain and loop rearrangements to backbone and interface remodeling; binder adaptation may require coordinated pose, torsion, backbone, and side-chain changes.Source: §1 · p. 2

Contributions

  • FlexEvo combines a unified all-atom representation with binder-specific structural variation and repair, and maps local packing, contact geometry, and excluded volume to protected, adaptable, and forbidden FlexBoxes.Structural feedback updates spatial constraints, candidate moves use target geometry and bond topology, and Pareto selection retains nondominated preservation–robustness trade-offs.Source: §1 · pp. 2–3
  • FlexEvo formulates adaptation as geometry-constrained bi-objective optimization, balancing input-state interaction preservation against robustness to reference-derived geometric perturbations.Adaptation requires neither source-model retraining nor access to alternative natural target conformations.Source: §1 · p. 3
  • 47.8% to 4.4%: FlexEvo reduces category-balanced mean relative performance degradation across nine binder categories, with 1.4–3.1 minutes of adaptation per sample.Evaluations use multiple generation baselines and span diverse molecular sizes, structural topologies, and interaction geometries.Source: §1 · p. 3

2 RELATED WORK

Prior work includes methods that condition design on a single target structure and methods that model flexibility or use evolutionary multi-objective search. The related work distinguishes those approaches from adapting one candidate for compatibility with unseen natural states.

2.1 SINGLE-STATE BIOMOLECULAR COMPLEX PREDICTION AND DESIGN

  • Single-state predictors and generators condition candidates on one target realization, leaving conformational variation outside the design objective.The section names small-molecule generators, peptide and protein-binder design methods, and general-purpose complex predictors as examples.Source: §2.1 · p. 3

2.2 FLEXIBILITY-AWARE BIOMOLECULAR COMPLEX MODELING

  • Flexibility-aware methods model receptor–binder changes or optimize over supplied ensembles, but do not resolve compatibility with unknown post-design states beyond conformations available beforehand.FlexEvo is described as adapting one candidate from one observed conformation for compatibility with unseen natural states.Source: §2.2 · p. 3

2.3 EVOLUTIONARY OPTIMIZATION, GRAPH LEARNING, AND ROBUSTNESS

  • The section situates multi-objective evolutionary search alongside applications in graph learning, robustness, and cross-modality optimization.Examples include explanations for fair graph neural networks, multiview subgraph representations, and transferability or robustness studies.Source: §2.3 · p. 4

3 METHODOLOGY

FlexEvo adapts candidates from one observed target conformation by evolving residue-level spatial constraints and applying constrained candidate editing and repair. Its objectives balance interaction preservation with robustness to synthetic geometric perturbations, while structural feedback and Pareto selection guide the outer search.

3.1 PROBLEM SETUP AND OVERVIEW

  • FlexEvo adapts an initial candidate pool from one observed target conformation, reserving alternative natural conformations for evaluation.The search optimizes admissible residue-level constraint configurations, each of which determines a candidate pool through constrained editing and repair.Source: §3.1 · p. 4

3.2 EVOLVABLE FLEXBOX PRIOR

  • FlexBoxes encode residue coverage, editing roles, and scales, assigning stable regions to anchor proximity, flexible regions to local exploration, and forbidden regions to occupancy avoidance.The initial candidate pool is screened once for basic geometric defects and redundancy, then shared across genome evaluations.Source: §3.2 · pp. 4–5, Figure 2 · p. 5

3.3 GENOME-CONDITIONED ADAPTATION AND SINGLE-STATE ROBUSTNESS

  • FlexEditor applies geometry-guided candidate edits and repair under fixed FlexBoxes, with the reported molecular experiments using graph-preserving translations, rotations, and annotated bond-axis torsions.Repair and screening precede survivor selection, so candidates failing geometric checks cannot compensate with favorable scores.Source: §3.3 · p. 5
  • Synthetic probes perturb target regions locally while candidate coordinates remain fixed, assessing contact retention and clash response without re-optimizing for each probe.These probes are constructed solely from the observed conformation; alternative natural conformations are reserved for evaluation.Source: §3.3 · p. 6
  • Genome objectives aggregate leading-candidate and pool-average interaction-preservation and robustness scores, with a parameter balancing maximum and mean and an optional constraint-assignment penalty.The two objective-wise maxima need not correspond to the same candidate.Source: §3.3 · p. 6

3.4 FEEDBACK-DRIVEN GENOME EVOLUTION

  • Genome offspring arise through pruning, fusion, or perturbation of spatial constraints; roles change locally, while scale updates may be nonlocal.The operators alter the prior rather than removing target atoms.Source: §3.4 · p. 6
  • Structural feedback uses constraint usage and proximity to activate uncovered flexible regions or retire low-signal assignments, then evaluates offspring from the shared initial pool.Nondominated sorting and objective-space crowding retain trade-offs, while spatially similar priors are deferred to discourage redundant configurations.Source: §3.4 · p. 7

4 EXPERIMENTS

The benchmark evaluates adaptation from a single Reference conformation across nine binder categories and tests selected candidates on alternative natural conformations. Results show stronger gains under Divergent conditions, alongside metric-specific trade-offs and analyses of initialization, search behavior, and runtime.

4.1 EXPERIMENTAL SETUP

  • The benchmark uses nine binder categories and multiple generation baselines, adapting with Reference and evaluating the same selected candidate on Reference, Matched, and Divergent states.Matched preserves the Reference regulatory state, while Divergent involves a propeptide-associated state change; source models remain frozen and adaptation uses only Reference and the initial candidate pool.Source: §4.1 · pp. 7–9

4.2 MAIN RESULTS

  • Across all 27 source–category combinations, primary-metric gains were larger under Divergent than Reference; DynamicFlow small-molecule Vina improved from −4.85 to −7.80 kcal/mol.The Divergent improvement was obtained using Reference-derived geometric surrogates rather than Vina or other external evaluators.Source: Table 1 · p. 8, §4.2 · p. 9
  • Reference-state gains coexist with secondary-metric trade-offs: BoltzGen linear-peptide DockQ rises, while LRMSD below 2 Å falls from 42.2% to 40.2%.The reported structural measures capture complementary information.Source: §4.2 · p. 9
  • Residues in the highest quartiles of packing sparsity or distance from the chain center have greater normalized RMSF, motivating initial editing-freedom allocation before search refinement.The analysis averages residue RMSF over three molecular-dynamics simulations for each of 200 ATLAS proteins.Source: §4.2 · p. 9
  • Later search populations shift toward lower plotted f1 and f2 values while retaining different objective trade-offs, consistent with Pareto-based survival.These snapshots reflect genome-dependent perturbation probes; held-out natural-conformation compatibility is assessed separately.Source: §4.2 · p. 9
  • In displayed runs, high-affinity fraction reaches 60.0% by about two minutes and changes little near the end, while log P continues to increase.On-target pAE decreases overall and off-target pAE increases overall across displayed intervals of approximately 2–3 minutes, with local reversals in some traces.Source: Figure 4 · p. 8, §4.2 · p. 9
  • One ligand example progresses from Vina scores of −7.900 to −10.491 to −12.666 kcal/mol across early, intermediate, and late poses.The snapshots show candidate-geometry changes within the binding pocket alongside progressively more favorable docking scores.Source: §4.2 · pp. 9–10

4.3 FURTHER ANALYSES

  • Among six optimization strategies, FLEXEVO ranks first on reported HV-AUC and B-pocket top-10 scores, and beats all three component-removal variants on Divergent Vina or DockQ across nine categories.Runtime comparisons report an additional 1.4–3.1 minutes of adaptation per sample.Source: §4.3 · p. 10

5 CONCLUSION

FLEXEVO adapts candidates from a single observed target conformation without retraining the source model, balancing input-state interaction preservation with robustness to local geometric perturbations. The paper reports improved Vina and DockQ scores on unseen natural conformations while maintaining input-state performance on those metrics.

5 CONCLUSION

  • FLEXEVO adapts existing binder candidates once at inference time from a single observed conformation, using an outer evolutionary search and decoded FlexBoxes to guide structural editing.Its bi-objective search balances input-state interaction preservation with robustness to local geometric perturbations, using the latter as a proxy for cross-state compatibility.Source: §5 · p. 10

APPENDIX OVERVIEW

  • The supplement separates implementation details, evaluation protocols, supplementary results, and mathematical analysis, including finite-probe bounds and transfer enclosures under explicit target-proximity conditions.The empirical benchmark evaluates frozen representatives on held-out Matched and Divergent conformations after adaptation from Reference.Source: APPENDIX OVERVIEW · p. 16

A.1 OBJECTS, INFORMATION FLOW, AND SCOPE

  • Each genome specifies residue support, editing roles, and scales; its decoded FlexBoxes guide adaptation of the shared screened pool, while parent-adapted structures provide feedback.Genome evaluations restart from the same initial pool and inner seed, so each proposed spatial prior is compared through a common adaptation procedure.Source: §A.1 · p. 16, A ADDITIONAL DETAILS OF FLEXEVO · p. 16
  • Candidate records retain atom-level coordinates and source correspondence, with supplied covalent connectivity and polymer sequence annotations stored separately rather than inferred from spatial proximity.Edited children receive new identifiers while retaining correspondence between coordinate indices and source atoms.Source: §A.1 · p. 16

C. Physicochemical properties and diversity

  • For small molecules and nonpeptidic macrocycles, the tables report physicochemical summaries under Reference, binding scores by target state, and frozen-pool diversity once under Reference.The table captions identify QED, SA, Lipinski, logP, and PAINS summaries, with directional metric outcomes indicated relative to the base model.Source: Table 2 · p. 17, Table 3 · p. 18
  • Topology-preserving adaptation retains the input candidate’s atom correspondence and supplied covalent connectivity during coordinate editing.The input molecular identity is retained; the appendix separately specifies permitted motions, structural invariants, and molecular-validity checks.Source: C. Physicochemical properties and… · p. 17

A.2 TARGET ENCODING, SCREENING, AND INITIALIZATION

This section outlines FlexEvo’s single-conformation adaptation workflow, covering target and candidate encoding and screening, geometry-preserving edits, perturbation-based robustness scoring, and genome selection and evolution. It also describes the reference execution schedule.

A.2 TARGET ENCODING, SCREENING, AND INITIALIZATION

  • Fixed target descriptors and heuristic role priors initialize FlexBoxes, while shared screening filters invalid, low-support, forbidden-region, and redundant candidates.Descriptors and frames remain fixed during search; stable assignments are geometric anchoring priors, not fitted mobility predictions or identified binding-energy hot spots. Screening retains at most 64 candidates and uses fingerprint and geometric similarity when available.Source: §A.2 · pp. 18–19

A.3 CANDIDATE EDITING AND STRUCTURAL SAFEGUARDS

  • Graph-preserving adaptation uses whole-binder rigid motions and annotated bond-axis torsions, then gates proposals on coordinate, support, shape, bond-list, and covalent-geometry checks.Rigid motions preserve internal geometry and handedness; torsions are restricted to eligible bonds whose removal separates the graph, excluding ring edges. Receptor-overlap repair uses up to two whole-binder displacements before acceptance.Source: §A.3 · pp. 20–22

A.4 SINGLE-STATE PERTURBATION SCORING

  • Single-state robustness scoring evaluates fixed candidate coordinates against the reference target and six perturbed targets, combining interaction preservation, probe robustness, and spatial penalties in two maximized objectives.Probe directions are atom-dependent while displacement strength is assigned by residue; if no flexible residue is present, non-anchor residues supply a fallback perturbation.Source: §A.4 · pp. 25, 28

A.5 GENOME VARIATION, FEEDBACK, AND SURVIVAL

  • Genome evolution generates pruning, fusion, and perturbation variants, then uses surviving edit histories and adapted-coordinate proximity to guide feedback, activation, and retirement.Fusion transfers donor configurations and scales within an 8 Å neighborhood; perturbation changes or removes roles locally and mutates scales separately.Source: §A.4 · p. 27, §A.5 · pp. 28–29
  • Genome selection traverses nondominated fronts by rank and crowding, while diversity rules defer outer genomes within residue-role distance 2 and inner candidates within 1 Å separation.Repair clips scales and attempts to restore missing stable and flexible roles when eligible residues exist; the 0.25R term limits retirement rather than imposing a global minimum support.Source: §A.5 · p. 29

A.6 REFERENCE CONFIGURATION AND EXECUTION SCHEDULE

  • The default run schedules 72 genome evaluations (8 + 4 × 8 × 2), alternating offspring generation, feedback, fresh inner adaptation, and survivor selection after initialization.The count is a search-budget count, not a sample size or replication count; returned records retain selected genomes, adapted coordinates, edit histories, and candidate and genome scores.Source: §A.6 · p. 29

B EXPERIMENTAL PROTOCOL AND REPRODUCIBILITY

  • The single-conformation protocol covers candidate selection, evaluation, and reporting, with coordinate-level search and reference settings specified in Appendix A and Table 13.The execution order begins by computing fixed target descriptors, constructing the initial prior, and screening the source pool.Source: §A.6 · p. 29, B EXPERIMENTAL PROTOCOL AND… · p. 30, Table 13 · p. 30

B.1 EVALUATION DESIGN

The evaluation compares FlexEvo with source-native candidates across nine binder categories and three natural target states, with alternative states held out during adaptation. It also separates adaptive-search gains from source-pool reranking and includes secondary structural and reference-similarity analyses.

Sample size and secondary analyses

  • The benchmark counts 37,299 target–category memberships across nine categories, with each target counted once per evaluated category.Paired observations are aggregated within targets, which receive equal weight; secondary analyses include 200-protein geometry–mobility comparisons and sampled reference-similarity analyses.Source: §B.1 · pp. 30, 32

Scope and candidate sources

  • The benchmark spans nine binder categories and assigns generation sources by modality, comparing each original output with its adapted counterpart.Ligand sources are DynamicFlow, FlexSBDD, and YuelDesign; peptide categories use BoltzGen, Chai-1, and Protenix, as do the other biomolecular categories.Source: §B.1 · p. 31

Baseline execution and source pools

  • The reference protocol requests up to 64 source candidates per case, model, and seed, retaining native order and recording shortfalls without duplication.Imported predictions require provenance for model revision, checkpoint, inference settings, coordinates, and Reference-frame alignment.Source: §B.1 · p. 31

Final representative selection

  • Across all 27 source–category combinations, FLEXEVO improved the primary evaluation metric under the held-out Divergent conformation.Adaptation used only Reference, with source-model parameters fixed and no state-specific redesign during evaluation.Source: §B.1 · p. 31

Pipeline and source-selection comparisons

  • Base, Rerank-only, and FLEXEVO share a source-generation run; reranking versus adaptation isolates the additional benefit of adaptive search over fixed-G0 selection.The full workflow also includes screening and proxy-based final selection; representative selection uses Reference-derived information.Source: §B.1 · pp. 31–32

Natural target-state conditions and alignment

  • Reference is the source and adaptation structure, while Matched and Divergent are alternative natural states reserved for evaluation.Receptors are aligned to the Reference frame and evaluated around fixed candidate coordinates without redocking, minimization, or state-specific adaptation.Source: §B.1 · p. 32

Synthetic probes and fixed candidates

  • Search robustness uses six synthetic local perturbations plus unperturbed Reference, distinct from the held-out natural states used for evaluation.Candidate coordinates remain fixed across all seven probes.Source: §B.1 · p. 32

B.2 EVALUATION METRICS

The evaluation combines fixed-pose ligand scores, native-complex comparisons, confidence-derived interaction metrics, and molecular-property summaries. It distinguishes candidate validity and evaluator success from endpoint scores and retains evaluator and preparation provenance.

Structural export and evaluation records

  • Validity outcomes, evaluator success, and numerical scores are reported separately, with structure exports and prepared evaluator inputs tracked by distinct checksums and provenance.Recorded adaptation trajectories and computational costs are reported separately from the scoring endpoints.Source: §B.2 · pp. 33–34

Fixed-pose ligand scoring

  • Vina, Vinardo, and AutoDock4 score frozen ligand poses without docking or optimization, using the paired native ligand and candidate on the same receptor preparation and grid.Lower Vina values are preferred; the high-affinity indicator requires a candidate Vina score strictly below the paired native ligand score under matching conditions.Source: §B.2 · p. 33

Molecular and set-level properties

  • QED, SA, Lipinski descriptors, logP, and PAINS screening summarize molecular properties, while diversity characterizes the frozen candidate pool separately from pose metrics.Intrinsic descriptors remain unchanged across receptor states when the molecular graph and settings are fixed; diversity is reported under Reference only.Source: §B.2 · p. 33

Native-complex comparison

  • DockQ evaluates mapped native interfaces, while LRMSD below 2 Å defines pose success; higher DockQ and lower LRMSD are favorable.MM/GBSA and HADDOCK are complementary terms for which lower values are preferred in the supplied tables.Source: §B.2 · p. 33

Confidence-derived interaction metrics

  • The symmetric interchain pAE mean summarizes confidence-derived interaction metrics, with lower on-target and higher off-target pAE preferred; these are not affinity measurements.Confidence must correspond to the evaluated edited candidate, predictor version, and input checksum.Source: §B.2 · p. 34

Primary endpoints for the three-arm comparison

  • The three-arm comparison uses Vina for small molecules and nonpeptidic macrocycles, and DockQ for the other seven binder categories.Raw-score directions are retained, while effect calculations use −Vina or DockQ so positive differences indicate improvement.Source: §B.2 · p. 34

B.3 REPRODUCIBILITY

The reproducibility protocol defines matched comparisons, target-level aggregation, coverage accounting, and runtime boundaries. The supplement also describes source-pool coverage and the scope of its supplementary analyses.

B.3 REPRODUCIBILITY

  • The 72-genome optimizer budget is not a count of distinct genomes or independent replicates, and resetting the inner seed does not estimate run-to-run variability.Source: §B.3 · pp. 34–35
  • Three repeats use seeds 31, 47, and 71; observations are averaged within cases and then targets, with equal target weighting.Source: §B.3 · p. 34
  • Paired contrasts match case, model, seed, and state, then bootstrap target families 10,000 times for 95% intervals; fewer than two eligible families yields no interval.Source: §B.3 · pp. 34–35
  • Coverage records distinguish measured values, executed failures, unsupported metrics, and unrun requests; continuous paired contrasts use complete pairs, while success-rate denominators retain executed failures.Source: §B.3 · p. 35
  • Runtime reports source generation plus adaptation as total time, separates external evaluation, and does not treat imported source predictions as native inference time.Source: §B.3 · p. 36

C SUPPLEMENTARY RESULTS AND ANALYSES; C.1 STRUCTURE-POOL COVERAGE AND BIOLOGICAL CONTEXT

C.2 DETAILED RESULTS ACROSS NINE BINDER CATEGORIES

FlexEvo improves divergent-state primary metrics across all evaluated source–category combinations, with lower category-balanced degradation and gains beyond reranking. Joint plots compare performance across two target conditions for diverse binder categories.

C.2 DETAILED RESULTS ACROSS NINE BINDER CATEGORIES

  • The source pool includes 19,715 small-molecule, 9,858 peptide, and 8,726 protein-binder PDB entries; the small-molecule target annotations cover 1,774 mapped targets.Source: Figure 5 · p. 37, Figure 6 · p. 38
  • FlexEvo achieves the best model-averaged scores over rerank-only in all three states for both metric groups, with divergent-state gains of 0.948 kcal/mol Vina and 0.4801 DockQ.Source: §C.2 · pp. 37–38, Table 14 · p. 37, Table 15 · p. 38
  • 47.8% to 4.4%: FlexEvo reduces category-balanced mean relative degradation across nine categories and 27 source–category combinations.Source: §C.2 · pp. 36, 39

C.3 JOINT CROSS-CONFORMATION COMPARISONS

C.4 ALTERNATIVE OPTIMIZERS AND COMPONENT ABLATIONS

FlexEvo leads the reported optimizer comparison on both displayed metrics, and the full method outperforms its component-removal variants on divergent-state primary metrics across all nine categories.

C.4 ALTERNATIVE OPTIMIZERS AND COMPONENT ABLATIONS

  • The full method achieves the best divergent-state primary metric across all nine categories, though small-molecule Reference Vina is slightly more favorable without FlexBox.Source: §C.4 · p. 43
  • 8.482 versus 0.959: FlexEvo’s HV-AUC is approximately 8.84-fold the Best-of-N value, and its B-pocket top-10 score is −31.409 versus −32.497.Source: Figure 10 · p. 43, §C.4 · p. 43, Table 16 · p. 43

C.5 RUNTIME AND STATE-WISE SUMMARY PROFILES

FlexEvo adds a short adaptation stage across nine workflows, while state-wise profiles and time-resolved examples describe performance across receptor conditions and during optimization.

C.5 RUNTIME AND STATE-WISE SUMMARY PROFILES

  • 1.393–3.141 minutes: FlexEvo’s added adaptation time per sample yields total generation-plus-adaptation times of 6.861–12.966 minutes across nine workflows.Source: §C.5 · p. 44, Figure 11 · p. 44
  • In the illustrated run, the high-affinity fraction reaches 60.0% at about two minutes; the ligand example’s Vina improves from −7.900 to −12.666 kcal/mol.Source: §C.5 · p. 44
  • State-wise profiles compare Reference, Matched, and Divergent conditions; FlexEvo-optimized small-molecule workflows retain 81–83% performance under Divergent conditions, with 4–5-point decreases from Reference.Source: §C.5 · p. 44, Figure 12 · p. 45
  • For protein binders, displayed performance declines from Reference to Divergent by 4 points for BoltzGen and Chai-1 and 5 points for Protenix, with smaller declines than the original methods.Source: Figure 14 · p. 46

C.6 SIMILARITY TO A SAMPLED REFERENCE LIBRARY

The appendix compares generated molecules with sampled references, illustrates binder structures and physicochemical profiles, and presents multim metric diagnostics and adaptation trajectories across binder categories. These materials describe comparison setups and selected trends, but the supplied passages do not report the underlying similarity values or diagnostic percentages.

C.6 SIMILARITY TO A SAMPLED REFERENCE LIBRARY

  • Table 17 compares 10 candidates per category with 47 sampled public references, reporting maximum and median similarity across 470 pairwise comparisons.Figures 15–17 show category-specific heatmaps whose cells encode Tanimoto similarity from 0 to 1; lower similarity indicates limited resemblance within the sampled library.Source: Table 17 · p. 46, §C.6 · p. 46, Figure 15 · p. 47

C.7 STRUCTURAL EXAMPLES AND PHYSICOCHEMICAL CONTEXT

  • Figures 18–20 illustrate representative structures across binder categories, while Figure 21 profiles seven descriptors for 116 reference structures in two illustrative sets of 58.For six descriptors with guideline lines, pooled medians and upper quartiles lie below the corresponding bounds, though some molecules exceed them; median sp3 fraction is 0.45 (IQR 0.31–0.60).Source: §C.7 · p. 48, Figure 18 · p. 49, Figure 19 · p. 50

C.8 MULTIMETRIC PROFILES

  • Figures 22–24 compare base models with FlexEvo variants across reference, matched, and divergent conformations using modality-specific, min–max-normalized metric profiles.Normalization bounds are shared across the three conformation columns within each modality but differ between modalities; interpretation follows each metric axis.Source: §C.8 · p. 52, Figure 22 · p. 53, Figure 23 · p. 54

C.9 ON-TARGET AND OFF-TARGET SCORE DIAGNOSTICS

  • Figures 25–27 identify candidates with on-target pAE below 5 and minimum off-target pAE above 10, with annotated percentages giving the fraction meeting both criteria.The diagnostics summarize the on/off-target contrast within the evaluated panel.Source: §C.9 · p. 56, Figure 25 · p. 56, Figure 26 · p. 57

C.10 OPTIMIZATION TRAJECTORIES ACROSS BINDER CATEGORIES

  • Across antibody-fragment, oligonucleotide, and long-nucleic-acid categories, trajectories show higher DockQ and LRMSD success rates, lower LRMSD and on-target pAE, and increased diversity over optimization.The curves are plotted against cumulative adaptation time and show local fluctuations and category-dependent convergence profiles.Source: Figure 30 · p. 62

D DETERMINISTIC SCORE BOUNDS AND CONDITIONAL ROBUSTNESS OF FLEXEVO

This analysis gives deterministic score bounds for fixed candidate poses and characterizes the geometry and support of the spatial prior. Transfer to unseen natural target geometries is conditional on explicit proximity assumptions, rather than guaranteed by the fixed-pose bounds alone.

D DETERMINISTIC SCORE BOUNDS AND CONDITIONAL ROBUSTNESS OF FLEXEVO

  • Fixed-pose scores use one candidate across synthetic probes from a single observed target; held-out natural conformations are reserved for evaluation, and transfer bounds require explicit target-proximity conditions.The objectives do not take held-out conformations as inputs, and the bounds do not transfer the full genome fitness or a newly run adaptation procedure.Source: D DETERMINISTIC SCORE BOUNDS AND… · p. 63

D.1 GEOMETRY OF THE SPATIAL PRIOR

  • FlexBoxes are nonempty, compact, convex regions with volume Lr^3 and diameter √3 Lr; boxes with a common center and frame nest according to bℓrzr.The volume bound controls the edit prior and ignores overlaps, so it is generally loose.Source: §D.1 · p. 63

D.2 FIXED-POSE SCORES AND FINITE-PROBE BOUNDS

  • For a fixed candidate across K probes, contact scores satisfy 0 ≤ qmin ≤ qrob ≤ q̄ ≤ qmax ≤ 1, while clash scores satisfy 0 ≤ χm ≤ χmax ≤ 1.The contact profile supplies a lower bound for every probe; its bound involving qrob and Δq is sharp over arbitrary tuples in [0, 1]^K, not necessarily realizable by the specified probes or a molecular structure.Source: §D.2 · p. 64
  • For K = 7 and ρ = 0.60, the aggregate contact bound gives qmin ≥ qrob − (12/35)Δq for the fixed candidate.The bound connects the robust summary and contact variation to worst-probe contact retention.Source: §D.2 · p. 64

D.3 COMPUTABLE BOUNDS UNDER TARGET PERTURBATIONS

The appendix bounds score changes under target perturbations and transfers those bounds to natural targets only when explicit neighborhood conditions hold. It also gives objective-range bounds under stated assumptions.

D.3 COMPUTABLE BOUNDS UNDER TARGET PERTURBATIONS

  • Hausdorff-bounded target changes place each score within a computable interval, whose endpoints are exact on a product relaxation but may not be attainable by one target geometry.The same enclosure contains the reference score; it does not imply that every intermediate score is realizable.Source: §D.3 · pp. 65–66
  • A centered deviation bound scales with the fraction of distance intervals crossing kernel thresholds; zero sensitive fraction gives exact score invariance.The centered bound can be tighter than the full envelope width, while the envelope can also be tighter in other cases.Source: §D.3 · pp. 66–67
  • Neither contact nor clash scores has a finite global Lipschitz constant with respect to target Hausdorff distance.Arbitrarily small target shifts can cross score-kernel thresholds and produce finite score changes.Source: §D.3 · p. 67

D.4 TRANSFER FROM PROBE NEIGHBORHOODS

  • Bounds transfer to a target in any declared probe neighborhood, and intersecting enclosures tightens them when the same target is known to lie in multiple neighborhoods.An unspecified single covering probe supports only the union bound; natural-target coverage requires an additional hypothesis or independently justified variation model.Source: §D.4 · pp. 68–69

D.5 OBJECTIVE RANGES

  • With bounded candidate coordinates and edit-history records, the preservation and robustness objectives have finite componentwise ranges, but their endpoints need not be jointly attained by one candidate.The penalty bound depends on coordinate count, history length, and forbidden-residue count.Source: §D.5 · pp. 69–70

D.6 SELECTION: WHAT IS AND IS NOT PRESERVED

The selector uses nondominated fronts, crowding, and residue-role distance, but it does not guarantee preservation of every nondominated record or every dominator. The appendix also states when target-transfer score control applies and how structural safeguards are enforced.

D.6 SELECTION: WHAT IS AND IS NOT PRESERVED

  • The selector orders records by nondominated front and crowding, then applies residue-role distance and backfill; it does not guarantee retention of every nondominated record.Accepted records are separated by role distance at least three before backfill, but that separation can be lost during backfill.Source: §D.6 · pp. 70–71

D.6 SELECTION: WHAT IS AND WHAT IS NOT PRESERVED

  • At capacity one, the sole survivor, when present, lies on the first front.The selector returns a subset of the input of size min{capacity, input size}.Source: §D.6 · p. 70

D.6 SELECTION: WHAT IS AND IS NOT PRESERVED

  • For capacity N≥2, there are scored inputs at target sizes R≥3(N−1)+2 where selection retains a dominated record and excludes its dominator.The stated target-size condition is sufficient, not necessary; this is an operator-level counterexample, not a claim of molecular realizability.Source: §D.6 · p. 71

D.7 ESTABLISHED PROPERTIES AND THEIR ROLE IN ADAPTATION

  • The established score enclosures control target transfer only under stated Hausdorff-neighborhood conditions; molecular safeguards separately check graph, bond-length, and category-specific structural consistency.Permitted transformations preserve the supplied molecular graph and covalent bond lengths, with an acceptance gate checking coordinate and bond-length consistency before survivor selection.Source: §D.7 · p. 72
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