No calibrated quantitative accounting exists of how much of functional protein sequence space life on Earth has actually explored. The source paper (Dryden, Thomson & White 2008) addressed this question only with deliberately liberal envelope estimates — between 4×10214\times10^{21} and 4×10434\times10^{43} distinct amino-acid sequences sampled over roughly 4 billion years, set against a reduced functional space of roughly 101010^{10}–103010^{30} sequences — and concluded it 'entirely feasible' that functional protein sequence space was fully explored, leaving only the fold-to-function relationship explicitly open. Later literature sharply contests that conclusion rather than confirming it: the sole comprehensive direct measurement of the fraction of random sequences that are functional (ATP binding, about 1 in 101110^{11} for 80-residue segments), computational estimates spanning 10910^{9}–1012210^{122} functional sequences per protein, and a 2026 geometric analysis of extant sequence distributions reporting explored fractions of order 10−6410^{-64} per protein family with common descent as the dominant restriction. The problem is to derive a transparent, empirically calibrated mapping from biological population, genome, and temporal parameters to the number of amino-acid sequences life has actually sampled — treating phylogeny, selection, and epistasis explicitly rather than assuming independent sampling — to determine the size and structure of functional sequence space, and to compare the two so as to decide the explored fraction with quantified uncertainty at both the per-protein-family and the aggregate all-of-life scales, stating where each of the 'fully explored' and 'astronomically under-explored' positions holds.

Background

Protein sequence space is the set of all possible amino-acid sequences of a given length LL; because proteins are built from 20 standard amino acids, it contains 20L20^L sequences, so a 100-residue protein has about 1013010^{130} possible sequences. The concept was introduced by John Maynard Smith in 1970 through his 'protein space' analogy for stepwise adaptive evolution. Only a minute subset of these strings folds into a stable three-dimensional structure and performs a useful biological function; that subset is the functional sequence space, and its size relative to the whole space is often expressed as fff_f, the fraction of random sequences of a given length that are functional for a chosen task. Assigning a fitness (reproductive contribution) to every sequence turns the space into a fitness landscape, and epistasis — the dependence of a mutation's effect on which amino acids are present at other positions — makes the functional subset form irregular connected networks rather than a uniform block.

Two questions have hung over this picture since the 1970s. First, is functional sequence space really as vast as 2010020^{100}? Second, how much of it has life actually sampled during roughly 4 billion years (Gyr) of evolution? If the answer to the second question is 'essentially none', rerunning evolution would yield entirely different proteins — the evolutionary contingency position crystallized in Stephen Jay Gould's 'replay the tape of life' thought experiment (discussed in Dryden et al. 2008 and Louis 2016); if 'essentially all', protein evolution would be largely repeatable (convergence).

Dryden, Thomson and White (2008) answered both with deliberately liberal back-of-envelope estimates. They argued that 2010020^{100} is a 'straw man': protein structure tolerates reduced amino-acid 'alphabets' (for example classifying residues only as hydrophobic or polar), and many functional domains are only about 33–50 residues long, which shrinks functional space to roughly 101010^{10}–103010^{30} sequences. They then bounded the number of distinct sequences life could have tested, using the census of about 103010^{30} prokaryotic cells on Earth (Whitman et al. 1998), about 10410^{4} genes per genome, about 10910^{9} bacterial species, one substitution per genome per generation, and generation times from one year upward: an extreme upper limit of 4×10434\times10^{43} and a conservative lower limit of 4×10214\times10^{21} distinct sequences explored since the origin of life. Because these limits bracket the reduced functional space, they concluded it is 'entirely feasible' that functional protein sequence space has been fully explored, so molecular evolution shows contingency-free convergence at the sequence level. The relation they explicitly left open is fold-to-function: only about a few thousand discrete protein folds exist (a fold is a characteristic three-dimensional packing pattern; the TIM barrel, named after triosephosphate isomerase, is a famous one), functional space may exceed folding space by orders of magnitude, and it is unclear whether new functions require new folds.

The load-bearing quantities in that argument remain essentially unmeasured. Direct empirical determination of fff_f has one comprehensive data point: Keefe and Szostak (2001) found ATP-binding proteins at a rate of about one in 101110^{11} random 80-residue sequences. Axe (2004) estimated that random sequences adopting functional enzyme folds are vanishingly rarer, and computational estimates of the number of functional sequences for individual proteins span from 10910^{9} to 1012210^{122} (as compiled in Miller 2024 and Isakova et al. 2026). On the exploration side, Isakova et al. (2026) measured, across thousands of protein families, the dimensionality and volume of sequence space occupied by extant homologs (sequences in living organisms related by common ancestry) and found it minuscule — for Gammaproteobacteria on the order of 10−6410^{-64} of a hypothetical functional volume derived from the ratio of non-synonymous to synonymous substitution rates (dN/dSdN/dS, a standard signal of how strongly selection constrains a protein), with descent of each family from a single common ancestor (the 'point of origin effect') identified as by far the largest restriction. Maynard Smith's question 'What fraction of functional sequences has been explored by evolution?' was recently described as having 'hardly progressed' in fifty years (Isakova et al. 2026). The literature therefore contains conclusions ranging from 'functional space is fully explorable' to 'the explored fraction is astronomically small', with no calibrated reconciliation.

Problem Statement

Derive a transparent, empirically calibrated quantitative accounting of how much of functional protein sequence space life on Earth has explored. The accounting has three linked components serving this single objective:

(1) A mapping from measurable biological parameters — organismal census and effective population sizes, genome sizes and gene inventories, mutation and substitution rates, generation times, and the roughly 4 Gyr temporal history of life — to the number of distinct amino-acid sequences actually sampled by evolution, with phylogenetic correlation (descent of protein families from common ancestors), selection, and epistasis treated explicitly rather than assuming independent sampling across individuals or lineages.

(2) A determination of the size and structure of functional sequence space: for at least three named function classes, both the fraction fff_f of random sequences that are functional as a function of sequence length and a quantitative characterisation of the connectivity of the functional set that tests whether the epistasis-implied connected-network picture holds in each case.

(3) A comparison of (1) and (2) yielding a defensible estimate of the fraction of functional sequence space that evolution has explored, reported both per protein family and in aggregate over the reduced functional space of all life, with uncertainty quantification sufficient to state over which of these scopes each position in the literature holds and to reconcile them — from 'essentially all functional space has been explored' (Dryden et al. 2008) to 'the explored fraction is of order 10−6410^{-64} per family because common descent restricts exploration' (Isakova et al. 2026) — and an identification of which biological parameters dominate the answer.

The answer must state its assumptions and the direction of bias of each, and be calibrated against at least one independent empirical anchor (for example random-sequence library measurements of fff_f, observed sequence diversity in protein databases, or deep mutational scanning data — high-throughput assays that measure the effects of very many sequence variants simultaneously).

References

  1. David T. F. Dryden, Andrew R. Thomson, John H. White (2008), 'How much of protein sequence space has been explored by life on Earth?', Journal of the Royal Society Interface 5:953–956. DOI: 10.1098/rsif.2008.0085. https://doi.org/10.1098/rsif.2008.0085
  2. John Maynard Smith (1970), 'Natural Selection and the Concept of a Protein Space', Nature 225:563–564. DOI: 10.1038/225563a0. https://doi.org/10.1038/225563a0
  3. William B. Whitman, David C. Coleman, William J. Wiebe (1998), 'Prokaryotes: The unseen majority', Proceedings of the National Academy of Sciences 95:6578–6583. DOI: 10.1073/pnas.95.12.6578. https://doi.org/10.1073/pnas.95.12.6578
  4. Anthony D. Keefe, Jack W. Szostak (2001), 'Functional proteins from a random-sequence library', Nature 410:715–718. DOI: 10.1038/35070613. https://doi.org/10.1038/35070613
  5. Douglas D. Axe (2004), 'Estimating the Prevalence of Protein Sequences Adopting Functional Enzyme Folds', Journal of Molecular Biology 341:1295–1315. DOI: 10.1016/j.jmb.2004.06.058. https://doi.org/10.1016/j.jmb.2004.06.058
  6. Tom C. B. McLeish (2015), 'Are there ergodic limits to evolution? Ergodic exploration of genome space and convergence', Interface Focus 5:20150041. DOI: 10.1098/rsfs.2015.0041. https://doi.org/10.1098/rsfs.2015.0041
  7. Ard A. Louis (2016), 'Contingency, convergence and hyper-astronomical numbers in biological evolution', Studies in History and Philosophy of Science Part C: Studies in History and Philosophy of Biological and Biomedical Sciences 58:107–116. DOI: 10.1016/j.shpsc.2015.12.014. https://doi.org/10.1016/j.shpsc.2015.12.014
  8. Jorge A. Vila (2020), 'About the Protein Space Vastness', The Protein Journal 39:472–475. DOI: 10.1007/s10930-020-09939-4. https://doi.org/10.1007/s10930-020-09939-4
  9. C. Brandon Ogbunugafor (2020), 'A Reflection on 50 Years of John Maynard Smith's "Protein Space"', Genetics 214:749–754. DOI: 10.1534/genetics.119.302764. https://doi.org/10.1534/genetics.119.302764
  10. Brian J. Miller (2024), 'A percolation theory analysis of continuous functional paths in protein sequence space affirms previous insights on the optimization of proteins for adaptability', PLOS ONE 19:e0314929. DOI: 10.1371/journal.pone.0314929. https://doi.org/10.1371/journal.pone.0314929
  11. Lada H. Isakova, Elizaveta Streltsova, Olga O. Bochkareva, Peter K. Vlasov, Fyodor A. Kondrashov (2026), 'Descent from a common ancestor restricts exploration of protein sequence space', Proceedings of the National Academy of Sciences 123(14):e2532018123. DOI: 10.1073/pnas.2532018123. https://doi.org/10.1073/pnas.2532018123