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Nature CommunicationsGeneticsPeer-reviewed journal article

The simplicity of protein sequence-function relationships

Yeonwoo ParkiD, Brian P. H. MetzgeriD, Joseph W. ThorntoniD

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Abstract

How complicated is the genetic architecture of proteins - the set of causal effects by which sequence determines function? High-order epistatic interactions among residues are thought to be pervasive, making a proteins function difficult to predict or understand from its sequence. Most studies, however, used methods that overestimate epistasis, because they analyze genetic architecture relative to a designated reference sequence - causing measurement noise and small local idiosyncrasies to propagate into pervasive high-order interactions - or have not effectively accounted for global nonlinearity in the sequence-function relationship. Here we present a new reference-free method that jointly estimates global nonlinearity and specific epistatic interactions across a proteins entire genotype-phenotype map. This method yields a maximally efficient explanation of a proteins genetic architecture and is more robust than existing methods to measurement noise, partial sampling, and model misspecification. We reanalyze 20 combinatorial mutagenesis experiments from a diverse set of proteins and find that additive and pairwise effects, along with a simple nonlinearity to account for limited dynamic range, explain a median of 96% of total variance in measured phenotypes (and >92% in every case). Only a tiny fraction of genotypes are strongly affected by third- or higher-order epistasis. Genetic architecture is also sparse: the number of terms required to explain the vast majority of variance is smaller than the number of genotypes by many orders of magnitude. The sequence-function relationship in most proteins is therefore far simpler than previously thought, opening the way for new and tractable approaches to characterize it.

In-depth summary

Claude Opus 5.5 · xhigh effort · 8 Nov 2026 · from the full text of version 2 (7 Feb 2024) · CC BY 4.0

Reference-free analysis with a sigmoid link shows additive plus pairwise effects explain a median 96% of phenotypic variance across 20 combinatorial protein mutagenesis datasets.

Reported extents of protein epistasis conflict, partly because reference-based analyses propagate noise and wild-type idiosyncrasies into spurious high-order terms and many analyses ignore global nonlinearity. The work introduces a reference-free, multi-state framework jointly fit with a sigmoid model of nonspecific epistasis, giving interpretable, variance-partitionable terms that are more robust than reference-based, Fourier, and background-averaged formalisms.

Key results

  1. ShownAdditive and pairwise effects with a sigmoid nonlinearity explain nearly all variance in every dataset. Strong higher-order epistasis affects very few genotypes, while reference-based analysis (RBA) of the same data wrongly implies pervasive high-order terms.

    Across 20 datasets, first-order median out-of-sample R^2 0.91 (max 0.97), second-order median 0.96 (min 0.92), third-order gain median 0.02 (max 0.04). Second-order residuals >20% of phenotype range: no genotypes in six datasets, 0.02 to 2% elsewhere. RBA fit up to triple mutants: median R^2 <0.2 for higher-order mutants at every order. · Simplicity of protein sequence-function…; Figure legends

  2. ShownModeling global nonlinearity with a two-parameter sigmoid greatly improves low-order fit and shrinks the variance attributed to epistasis. Gains track the fraction of genotypes at phenotype bounds, implicating bounding as the main source of nonspecific epistasis.

    Median out-of-sample R^2 without vs with sigmoid link: first-order 0.59 vs 0.92 (epistatic variance cut by a factor of 5), second-order 0.87 vs 0.96, third-order 0.95 vs 0.98. In CR9114-B, where 99.9% of genotypes sit at the lower bound, first-order variance explained rises from 1% to 97%. · The primary cause of nonspecific epistasis is…; Figure legends

  3. ShownProtein genetic architecture is sparse: a handful of reference-free terms reach 90% predictive accuracy (T90), and T90 grows very slowly as the total number of genotypes increases.

    T90 is 6 to 44 terms (out of up to 160,000 possible) for every dataset except GB1, whose sites were picked to be epistasis-rich; the fraction of terms needed falls almost linearly with genotype count, and the T90 trend projects about 10,000 terms for two states at 100 sites (about 10^30 genotypes). · Sparsity of protein sequence-function…; Figure legends

  4. ShownDespite this sparsity, L1-regularized fits to random genotype samples need large samples to reach R^2 0.9 (N90). The requirement rises steeply with phenotype bounding but depends only weakly on architecture complexity (T90).

    N90 (sample size for out-of-sample R^2 0.9), excluding three small datasets: 0.2 to 25% of genotypes, median 5%; weakly related to T90. CR9114-B (65,536 genotypes): 10 additive effects give 90% of variance but >16,000 genotypes are needed; CH65-MA90, same size with almost no bounding, needs 99. · Inferring genetic architecture by sparse…; Figure legends

  5. ShownIn simulations, reference-free estimates hold up under measurement noise, missing genotypes, and model truncation. Reference-based estimates accumulate error with epistatic order, and Fourier and background-averaged models lose accuracy when many states per site are inferred from incomplete data.

    Simulated 3-state, 5-site space: RFA effect errors below single-measurement noise, RBA errors larger and rising with order. Missing-genotype tests (200 trials): at four states all methods similar until sampling falls below 1%; at 16 states RFA beats BA markedly and FA moderately. Third-order-only architecture: first- and second-order RFA explain zero variance. · Robustness to measurement noise, missing…; Figure legends

Methods, data and reproducibility

Reference-free analysis (RFA) defines each state or combination effect as the deviation of its subset-mean phenotype from lower-order expectations, fit jointly with a two-parameter sigmoid link by L1-regularized least squares. RFA was compared with Fourier and background-averaged analyses in simulations, and with reference-based analysis in simulations and on 20 datasets using cross-validated R^2.

  • Combinatorially complete mutagenesis libraries of antibodies, enzymes, GFP, transcription factors, viral and toxin-antitoxin proteins; 3 to 16 variable sites, 2 to 20 states per site · multiple; species per dataset not stated in text · n = 20 datasets (32 to 160,000 possible genotypes each)
  • CR9114-B (99.9% of genotypes at lower bound, 81 above it) and CH65-MA90 (virtually no bounding), equal-size datasets contrasted for sampling efficiency · n = 65,536 genotypes each
  • CR9114-H3: binding of antibody variants (all ancestral/derived combinations at 16 affinity-maturation sites) to influenza hemagglutinin · n = 16 variable sites
  • avGFP fluorescence for all combinations of two amino acids at 13 sites; mutated sites mapped onto the crystal structure · n = 13 variable sites · PDB 3e5w
  • ParD3 interface library (all 20 amino acids at 3 sites) assayed separately for binding cognate ParE3 and noncognate ParE2 · n = 20^3 genotypes per ligand
  • Protein G GB1 domain fitness library, unreplicated; variants with fewer than 100 pre-competition reads removed · n = 68.6% of 160,000 possible genotypes
  • Edited reanalyses: methyl-parathion hydrolase (Ni^2+ cofactor only), DHFR antibiotic IC75, H3N2 hemagglutinin (Bei89 background), ParB binding to parS · influenza A (H3N2) for hemagglutinin; others not stated · n = DHFR: 48 of 96 variants (protein variants only)
  • Simulated genetic architectures (e.g., 3-state, 5-site space with standard-normal noise; third-order-only architectures under a sigmoid link) · n = 200 trials per missing-genotype setting
Methods & tools
Reference-free analysis (RFA) scripts and tutorials (github.com/whatdoidohaha/RFA), Two-parameter sigmoid link for nonspecific epistasis, L1-regularized least-squares regression with cross-validation, R package lbfgs (numerical optimization), Reference-based analysis (RBA) with fitted link function, Fourier analysis (simplex encoding / graph Fourier transform), Background-averaged analysis, multi-state recursive matrix extension (custom R script)
Software & environment
R with the lbfgs package for numerical optimization, plus custom R scripts for multi-state background averaging; no software versions reported.
Reproducibility gaps
Table 1, which lists the 20 reanalyzed datasets, has only its caption in the provided text; no dataset accessions, data URLs, software versions, or compute details are stated.
Limitations (6)
  • Each library was assayed for a single phenotype, so architectures that govern specificity among multiple functions could still be more complex. · stated by the authors
  • Libraries varied few sites and likely kept the overall fold fixed (or unfolded it), so conformation-dependent high-order epistasis that specifies folds may go undetected without larger libraries spanning distinct conformations. · stated by the authors
  • Genotypes with large second-order residuals are only candidates for strong high-order epistasis because erratic measurement noise cannot be ruled out; variance left by third-order models also mixes noise with inadequacy of the sigmoid link. · stated by the authors
  • Random sparse sampling is inefficient (excluding three small datasets, N90 is 0.2 to 25% of genotypes, median 5%), so large spaces need a non-random design, possibly single- and double-mutant scans across diverse homologs, still to be developed. · stated by the authors
  • Two large libraries were filtered by pre-competition read count (coverage reduced to 42.2% and 68.6%). Read-count filtering is not random, whereas the unbiasedness of regression-estimated RFA terms assumes randomly sampled genotypes. · noted in this summary
  • The robustness advantage over Fourier and background-averaged analyses comes from simulated architectures with standard-normal effects. The text does not compare those formalisms on the empirical datasets, which have real noise structure and phenotype bounding. · noted in this summary

Read this if you analyze combinatorial mutagenesis or protein genotype-phenotype data and need an interpretable, noise-robust way to separate specific epistasis from global nonlinearity.

Citations
70
FWCI
17.52
Field percentile
top 1%

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