Case study A · Investigate the molecules

Protein folding and sequence space

A protein is a chain of amino acids that, for most enzymes, must fold into one particular shape to work. Two different 'searches' are often confused here: the physical search a chain makes for its folded shape, and the historical search for a sequence that folds at all.

1 · Fold a chain

Edit the amino-acid sequence (4 to 18 residues). Each residue counts as hydrophobic (H, filled blue: A C F I L M V W) or polar (P, open). The Lab tries every possible shape on a square grid and keeps the one with the most H–H contacts: the 2D HP model of Dill (1985) and Lau & Dill (1989).

Its lowest-energy shape

Shapes searched (all of them)…
Lowest energy…
Shapes sharing it…
Gap to next level—

2 · How many sequences of length 12 fold uniquely?

In the HP model only a minority of sequences have a single lowest-energy shape, and a few shapes are the ground state of far more sequences than others. Li et al. (1996) called this designability, and found that the most designable structures are also the most protein-like and the most stable against mutation.

3 · Two searches that are often confused

The folding search is the physical search a chain makes for its own shape. Levinthal pointed out that trying every shape at random would take absurdly long, yet real proteins fold in microseconds to seconds. Folding is therefore not random: it runs downhill on an energy landscape (Zwanzig et al. 1992; Dill & Chan 1997).

3100 = 5.2 × 10⁴⁷ shapes; trying them all at random would take 1.6 × 10²⁷ years.

This number argues against random folding, which nobody proposes. It says nothing about how a sequence that folds came to exist. That is the second search.

4 · The sequence search, and functional constraints

The sequence search is the historical question of how a sequence that folds and works came to exist. The answer hangs on one number, the share of sequences that work. Published estimates differ enormously, partly because they measure different kinds of function.

Possible sequences: 20150 = 1.4 × 10¹⁹⁵

Share of sequences that work

Axe 2004: upper estimate for a specific function by any domain-sized fold, from β-lactamase experiments.

10³⁰ is about the number of prokaryotic cells on Earth (Whitman et al. 1998). That number counts cells, not new sequences tried.

Expected working sequences found: 10⁻⁴⁷ · chance of at least one: 1 in 10⁴⁷

Caveats. Each estimate was measured for a particular length and function, so applying it to another length is an extrapolation. Random sampling is the worst case: evolution mostly modifies sequences that already work, which this calculator does not model. The Probability Simulator shows the difference.

Methodology and limitations

How this works, and what it leaves out

Method

  • Folding model. The two-dimensional HP lattice model (Dill 1985; Lau & Dill 1989). Residues A, C, F, I, L, M, V and W count as hydrophobic (one common convention); the rest as polar. Each non-bonded H–H contact on the grid scores −1.
  • Exact enumeration. Every self-avoiding conformation is generated. Rotations and reflections are removed by fixing the first bond and the direction of the first turn, so each distinct shape is counted once. A sequence "folds uniquely" when exactly one shape has the lowest energy; the energy gap to the next level is a simple stand-in for stability.
  • Survey. For up to 12 residues every one of the 2L H/P patterns is tested against every shape; for 13–18 residues, 400 seeded random patterns are tested.
  • Calculators. The folding-search calculator multiplies shapes per residue; the sequence-search calculator uses log-space arithmetic for 20L and 1 − (1 − f)n.

Limitations

  • Real proteins are three-dimensional, have 20 chemically different side chains, solvent, and folding kinetics. The HP model captures only the hydrophobic-core effect, and designability results depend on the lattice used (Li et al. 1996 used a 3D cube).
  • Folding uniquely is not the same as functioning: an enzyme also needs an active site. Some functional proteins never fold (Dyson & Wright 2005).
  • The functional fractions are measurements for particular proteins, lengths and functions. Applying them elsewhere is extrapolation, and the Keefe & Szostak figure shown is a raw ratio computed here, not the authors' own estimate.
  • The Levinthal numbers (shapes per residue, sampling rate) are illustrative values for stating the paradox, not measurements.
  • Random sampling of sequences is the worst case. Most evolutionary proposals start from sequences that already work; see the Probability Simulator for how much that changes.
Evidence and debate

Weighing the evidence

Addresses: The origin of biological informationBiological evolution

01 What is observed?

  • EstablishedMany small proteins refold on their own after being unfolded: the amino-acid sequence carries the information for the structure. (Anfinsen 1973)
  • EstablishedProteins fold in microseconds to seconds, far faster than a random trial of every conformation would allow (Levinthal's paradox). Folding is a biased descent down a funnel-shaped energy landscape, not a random search. (Levinthal 1969; Zwanzig et al. 1992; Dill & Chan 1997)
  • EstablishedSequences are constrained unevenly: in λ repressor, buried positions tolerated few substitutions while most surface positions tolerated many. (Reidhaar-Olson & Sauer 1990)
  • EstablishedFrom a library of 6 × 10¹² random 80-residue proteins, in vitro selection recovered four unrelated families that bind ATP. (Keefe & Szostak 2001)
  • Active researchWorking from a weakened β-lactamase, Axe estimated that about 1 in 10⁶⁴ sequences consistent with that fold's hydropathic pattern form a working domain, and that the prevalence of sequences performing a specific function by any domain-sized fold may be as low as 1 in 10⁷⁷. (Axe 2004)
  • Active researchA co-evolution model of ten natural protein families found that the number of sequences able to fold to a given structure is enormous (more than Avogadro's number even for small domains), but that as a fraction of all sequences it is tiny and falls steeply with length. (Tian & Best 2017)
  • EstablishedMost proteins are only marginally stable, many mutations destabilise them further, and extra stability makes a protein more tolerant of new mutations. (Taverna & Goldstein 2002; Bloom et al. 2006; Tokuriki & Tawfik 2009)
  • EstablishedStructures can now be predicted accurately from sequence for most single-domain proteins, and some functional proteins have no fixed fold at all. (Jumper et al. 2021; Dyson & Wright 2005)

02 What does evolutionary theory propose?

03 What are the unresolved questions and criticisms?

  • Active researchHow rare are functional folds? Published figures differ by more than 60 orders of magnitude, largely because they measure different things: binding any ATP, catalysing a reaction, adopting a particular fold. Which definition matters for a given evolutionary step is itself disputed. (Keefe & Szostak 2001; Axe 2004; Tian & Best 2017)
  • Active researchAxe, and Gauger & Axe, argue that functional folds are too rare, and too isolated from each other, to be reached by mutation and selection. Mainstream researchers have generally not accepted the extrapolation from one enzyme to protein space in general. (Axe 2004; Gauger & Axe 2011)
  • EstablishedAn evolution experiment that began with a random 139-residue polypeptide climbed easily to the middle of its fitness landscape, but the landscape above that was rugged; the authors concluded that 'an enormous range of sequence diversity' would be needed to climb further. (Hayashi et al. 2006)
  • EstablishedInteractions between mutations (epistasis) close off most mutational paths: of 120 possible orders for five β-lactamase mutations, 102 were inaccessible to selection. (Weinreich et al. 2006)
  • SpeculativeThe first proteins, before any existed to duplicate, belong to the origin-of-life question, and no tested account of them exists. (Koonin & Novozhilov 2009)

04 What evidence would help distinguish competing explanations?

  • Measurements of the functional fraction for several folds and lengths, all using the same definition of function. This would show whether the gap between estimates is real or a matter of definitions.
  • Laboratory evolution of a genuinely new fold, not just a new activity in an existing one, from an unrelated or random sequence.
  • Resurrected ancestral sequences tested for activity at each step of a proposed path between two different folds.
  • De novo genes in natural populations shown to encode stable, folded proteins with a selectable function.

Established Measured or observed repeatedly; not in serious dispute.Active research An open research question, or a claim that specialists still dispute.Speculative A proposed explanation that has not yet been tested directly.

Your conclusions

Before you move on

Your notes stay in this browser. Nothing is sent anywhere unless you choose to include them in a link you share.

References

  1. Anfinsen CB (1973). Principles that govern the folding of protein chains. Science 181:223–230. doi:10.1126/science.181.4096.223
  2. Axe DD (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
  3. Bloom JD, Labthavikul ST, Otey CR, Arnold FH (2006). Protein stability promotes evolvability. PNAS 103:5869–5874. doi:10.1073/pnas.0510098103
  4. Carvunis AR, Rolland T, Wapinski I, et al. (2012). Proto-genes and de novo gene birth. Nature 487:370–374. doi:10.1038/nature11184
  5. Dill KA, Chan HS (1997). From Levinthal to pathways to funnels. Nature Structural Biology 4:10–19. doi:10.1038/nsb0197-10
  6. Dill KA (1985). Theory for the folding and stability of globular proteins. Biochemistry 24:1501–1509. doi:10.1021/bi00327a032
  7. Dyson HJ, Wright PE (2005). Intrinsically unstructured proteins and their functions. Nature Reviews Molecular Cell Biology 6:197–208. doi:10.1038/nrm1589
  8. Gauger AK, Axe DD (2011). The evolutionary accessibility of new enzyme functions: a case study from the biotin pathway. BIO-Complexity 2011(1). doi:10.5048/bio-c.2011.1 BIO-Complexity was founded by intelligent-design researchers.
  9. Hayashi Y, Aita T, Toyota H, Husimi Y, Urabe I, Yomo T (2006). Experimental rugged fitness landscape in protein sequence space. PLoS ONE 1:e96. doi:10.1371/journal.pone.0000096
  10. Jumper J, Evans R, Pritzel A, et al. (2021). Highly accurate protein structure prediction with AlphaFold. Nature 596:583–589. doi:10.1038/s41586-021-03819-2
  11. Keefe AD, Szostak JW (2001). Functional proteins from a random-sequence library. Nature 410:715–718. doi:10.1038/35070613
  12. Knopp M, Andersson DI (2018). No beneficial fitness effects of random peptides. Nature Ecology & Evolution 2:1046–1047 (comment on Neme et al.). doi:10.1038/s41559-018-0585-4
  13. Knopp M, Gudmundsdottir JS, Nilsson T, et al. (2019). De novo emergence of peptides that confer antibiotic resistance. mBio 10:e00837-19. doi:10.1128/mbio.00837-19
  14. Koonin EV, Novozhilov AS (2009). Origin and evolution of the genetic code: the universal enigma. IUBMB Life 61:99–111. doi:10.1002/iub.146
  15. Lau KF, Dill KA (1989). A lattice statistical mechanics model of the conformational and sequence spaces of proteins. Macromolecules 22:3986–3997. doi:10.1021/ma00200a030
  16. Levinthal C (1969). How to fold graciously. In Debrunner P, Tsibris JCM, Münck E (eds) Mössbauer Spectroscopy in Biological Systems, University of Illinois Press, pp. 22–24. Conference proceedings, not a peer-reviewed journal article.
  17. Li H, Helling R, Tang C, Wingreen N (1996). Emergence of preferred structures in a simple model of protein folding. Science 273:666–669. doi:10.1126/science.273.5275.666
  18. Neme R, Amador C, Yildirim B, McConnell E, Tautz D (2017). Random sequences are an abundant source of bioactive RNAs or peptides. Nature Ecology & Evolution 1:0127. doi:10.1038/s41559-017-0127
  19. Reidhaar-Olson JF, Sauer RT (1990). Functionally acceptable substitutions in two α-helical regions of λ repressor. Proteins 7:306–316. doi:10.1002/prot.340070403
  20. Tautz D, Domazet-Lošo T (2011). The evolutionary origin of orphan genes. Nature Reviews Genetics 12:692–702. doi:10.1038/nrg3053
  21. Taverna DM, Goldstein RA (2002). Why are proteins marginally stable?. Proteins 46:105–109. doi:10.1002/prot.10016
  22. Tian P, Best RB (2017). How many protein sequences fold to a given structure? A coevolutionary analysis. Biophysical Journal 113:1719–1730. doi:10.1016/j.bpj.2017.08.039
  23. Tokuriki N, Tawfik DS (2009). Stability effects of mutations and protein evolvability. Current Opinion in Structural Biology 19:596–604. doi:10.1016/j.sbi.2009.08.003
  24. Tokuriki N, Tawfik DS (2009). Protein dynamism and evolvability. Science 324:203–207. doi:10.1126/science.1169375
  25. Weinreich DM, Delaney NF, DePristo MA, Hartl DL (2006). Darwinian evolution can follow only very few mutational paths to fitter proteins. Science 312:111–114. doi:10.1126/science.1123539
  26. Weisman CM, Eddy SR (2017). Gene evolution: getting something from nothing. Current Biology 27:R661–R663. doi:10.1016/j.cub.2017.05.056
  27. Whitman WB, Coleman DC, Wiebe WJ (1998). Prokaryotes: the unseen majority. PNAS 95:6578–6583. doi:10.1073/pnas.95.12.6578
  28. Zwanzig R, Szabo A, Bagchi B (1992). Levinthal's paradox. PNAS 89:20–22. doi:10.1073/pnas.89.1.20