Worked solutionOne step at a time
Count the shapes the way Levinthal did
Give each of ubiquitin's 76 residues three backbone positions. The choices multiply. Here every residue is counted; counting only the 75 links between residues gives 375 ≈ 6 × 1035, the same story.
Run the brute-force clock
The fastest anything happens in a protein is a bond vibration, about 1013 per second. Try one shape per vibration.
Compare with the measurement
Mammalian ubiquitin refolds with a rate constant of about 1,800 s-1. The time constant is the inverse.
Answer: B, a downhill funnel
A random search is impossible, so folding is not a random search. Each native contact that forms lowers the free energy, so it tends to stay, and it makes the neighbouring contacts easier. The energy landscape is shaped like a funnel: from almost any unfolded start, downhill steps lead to the same native fold. The chain only explores a vanishing fraction of the 1036 shapes.
What the computation showedThe numbers behind the reel
The reel reads ubiquitin's real structure (PDB 1UBQ, 76 residues, mean Cα–Cα distance 3.81 Å) and computes everything in Python.
- 376 is computed as an exact integer: 1,824,800,363,140,073,127,359,051,977,856,583,921, about 1.8 × 1036 (the 1.9 × 1036 often quoted is a rounding up).
- At 1013 shapes per second, the exhaustive search takes 5.8 × 1015 years (365.25-day years), or 419,000 times the age of the universe (13.8 billion years). In the reel's 6-second clock it tries about 6 × 1013 shapes: 3 × 10-21 % of them.
- The unfolded chains in the reel are self-avoiding random walks with 3.8 Å steps (no two Cα atoms closer than 4 Å), superposed on the native trace. Their radius of gyration is 16 to 32 Å; the folded protein's is 11.5 Å.
- The folding time: Vallée-Bélisle, Turcotte & Michnick, "raf RBD and ubiquitin proteins share similar folds, folding rates and mechanisms despite having unrelated amino acid sequences", Biochemistry 43:8447–8458 (2004), PMID 15222756, which reports refolding at 1,800 s-1 for mammalian ubiquitin. Krantz & Sosnick (Biochemistry 39:11696, 2000) measured rates up to about 1,000 s-1, the same millisecond scale.
- Assumptions: three states per residue, all 76 residues counted, one shape per 10-13 s. Change any of them and the answer moves by a few powers of ten, still far past the age of the universe.
Wrong answersWhy each one is tempting
| Pick | What went wrong |
|---|---|
| A · tries faster | 1013 per second is already the physical limit, and it is 1026 times too slow. |
| C · helper proteins | Ubiquitin folds alone, in a test tube, in about half a millisecond. Chaperones prevent clumping; they do not do the search. |
| D · only one shape | Unfolded ubiquitin does take many shapes; the paradox is how it leaves them so fast. |
Follow-ups an interviewer asks nextTry these before opening them
If the landscape is a funnel, why do some proteins fold in seconds instead of microseconds?
The funnel is rough. Wrong contacts form traps, prolines must switch between cis and trans, and long-range contacts (strands far apart in sequence) take longer to find. Folding rates correlate with contact order: proteins whose contacts are mostly local fold fastest.
How would you design a sequence that folds fast?
Make the native state much lower in energy than any alternative (a deep funnel) and make wrong pairings unfavourable (negative design). Designed proteins such as Top7 are often more stable than natural ones and fold in milliseconds.
Is Levinthal's count even the right number?
It overcounts: many of those shapes clash, so they are impossible. Remove clashes and you still have a number far too large to search, so the paradox survives any reasonable count.
Why can AlphaFold predict the structure without simulating the folding?
It learns the mapping from sequence (and evolutionary covariation) to the bottom of the funnel, not the path. Because the native state is a deep, unique minimum, the answer is a property of the sequence, which is what makes prediction possible at all.
