Sequence-Dependent Folding-Rate Variation Beyond the Size-Based Foldon-Funnel Mechanism for Protein Folding Kinetics

Rollins and Dill's Foldon Funnel Model (2014) explains the roughly nine-order-of-magnitude spread of observed folding rates across 93 proteins — two-state folders, plus multistate folders for which only the slowest folding phase was considered — using only the number of secondary structures (a proxy for chain size) and two universal mesoscale parameters. Its acknowledged central limitation — still unresolved in the literature — is that structurally similar proteins of identical size and topology, such as members of the spectrin-repeat and homeodomain families, fold at rates differing by orders of magnitude, so the residual scatter must be governed by amino-acid sequence rather than size. The open problem is to identify the physical sequence determinants of this intra-family rate variation, incorporate them into a mechanistic kinetics framework that remains consistent with the established size- and topology-based scaling, and establish whether such a general mechanism extends beyond two-state, single-domain, in-vitro folding to proteins with stable on-pathway intermediates and to multidomain or large (over 400 residue) domains.

Background

For many small single-domain proteins, folding in vitro is kinetically two-state: the observable decay of the unfolded population is a single exponential characterized by one folding rate constant $k_f$, with no detectable intermediates. Since Plaxco, Simons, and Baker (1998) showed that $\ln k_f$ correlates with the contact order — the average sequence separation between residues in native contact, a measure of the localness of the native topology — folding-rate theory has been built on structural descriptors. Ivankov et al. (2003) showed that once chain length is accounted for, size dominates this correlation, and Garbuzynskiy, Ivankov, Bogatyreva, and Finkelstein (2013) showed that the maximal (speed-limit) and typical folding rates form a 'golden triangle' envelope in the rate-versus-length plane: chain length and topology bound the accessible folding times over about nine orders of magnitude. These are envelopes and correlational laws, not mechanisms, and they leave large residuals. Rollins and Dill (2014) proposed a mechanistic mesoscale model, the Foldon Funnel Model, in the spirit of Englander's foldon concept: a foldon is a cooperative unit of secondary structure (a helix or strand with its turns), and folding proceeds by the sequential formation and assembly of $N$ such units. The model uses an Ising-like kinetics with two universal parameters: $K_2$, the equilibrium constant for forming an isolated (intrinsically unstable) secondary structure, and $K_3$, the additional stabilization gained when a structure assembles onto already-formed structures. With best-fit values $K_2 = 0.037$ and $K_3 = 1.96$ shared by all proteins, the model reproduces the nine-order-of-magnitude dependence of $k_f$ on $N$, the near-independence of the folding equilibrium constant on size, and predicts a volcano-shaped free-energy landscape (each assembly step is uphill except the last), nested transition states (each partial structure is the rate-limiting bottleneck for the one before it), and single-exponential two-state kinetics precisely because isolated foldons are unstable. The model's explicit limitation is that it 'only treats how folding rates depend on protein size and not otherwise on the protein's amino acid sequence'. Yet structurally matched proteins fold at very different rates: within the spectrin-repeat three-helix bundle family, different members have folding rate constants differing by orders of magnitude and even fold by different mechanisms (Wensley, Gärtner, Choo, Batey, and Clarke, 2009), and the same holds for homeodomains (Gianni, Guydosh, Khan, Caldas, Mayor, White, DeMarco, Daggett, and Fersht, 2003). Both families appear in the Rollins–Dill dataset, but the global two-parameter fit cannot predict within-family rate variation. The model is likewise restricted to two-state, single-domain, in-vitro folding: it does not treat proteins with stable on-pathway intermediates (whose landscapes resemble a descending rather than ascending staircase), multidomain proteins (for which only a crude largest-domain estimate was made), large domains of more than about 400 residues (predicted in the Escherichia coli proteome to fold on timescales slower than cell division absent mitigating factors), or cellular factors such as chaperones and cotranslational folding on the ribosome. Correlational predictors that do use sequence exist — Ivankov and Finkelstein (2004) derived rates from sequence-predicted secondary structure, and Casier and Duhamel (2023) built a 'blob'-based sequence model reaching a correlation of about 0.71 with measured folding times over 145 proteins — but neither provides a mechanism for intra-family scatter, and machine-learning structure-prediction tools such as AlphaFold2 have been shown to expose folding pathways and intermediates (Chang and Pérez, 2025), not folding rates. What is missing is a physical, mechanistic account that connects amino-acid sequence to the mesoscale kinetic parameters and thereby explains — and predicts — the order-of-magnitude rate differences among proteins of identical size and topology, together with the extension of such a mechanism beyond the two-state single-domain regime in which all current mesoscale theories are formulated.

Problem Statement

The problem, as posed by the acknowledged limitations of Rollins and Dill (2014), has one overarching objective — a genuinely general mechanism of protein folding kinetics — with two linked facets. Primary: identify the physical determinants in a protein's amino-acid sequence that produce folding-rate variation among proteins of the same size and topology — including the order-of-magnitude rate differences observed within the spectrin-repeat and homeodomain families — and incorporate those determinants into a mechanistic kinetic framework consistent with the established size- and topology-based scaling (the nine-order-of-magnitude length dependence and the near size-independence of folding equilibrium constants), such that folding rates and their within-family variation can be predicted from sequence rather than fitted per protein. Secondary, as part of the same generality objective: determine whether the resulting mechanism extends beyond two-state, single-domain, in-vitro folding — to proteins with stable on-pathway intermediates and multistate (down-staircase) landscapes (for which the folding rate is conventionally taken, as in the Rollins–Dill data set, to be that of the slowest folding phase), to multidomain proteins with independently or cooperatively folding domains, and to large (over 400 residue) domains whose in-vivo folding may depend on chaperones and cotranslational folding on the ribosome. The mechanism sought must explain the origin of the rate scatter mechanistically rather than merely correlating with it, must predict folding rates and their within-family variation from sequence rather than by per-protein fitting, and must remain consistent with the existing length/topology envelope laws — extending or sharpening them where possible but not contradicting them in the regimes where they hold. The secondary facet concerns whether the mechanism extends to the three extension regimes — multistate folding with stable on-pathway intermediates, multidomain proteins, and large (over 400 residue) domains. Experimental mutation studies within matched families can causally validate the proposed sequence determinants, but the primary facet concerns the sequence-to-rate mechanism itself.

References

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