The λ-repressor N-terminal domain fragment λ6-85 and its stabilized variants are canonical microsecond fast-folding proteins used to test where a chain sits between cooperative (two-state, barrier-limited) and downhill (barrier-free) folding. Hydrogen-exchange (HX) and ψ/φ analysis of the fast variant λYA (λ6-85 with G46A/G48A/Q33Y) showed that its major folding phase is cooperative, with the observed HX isotherms lying closest to the highest barrier modeled (~5 RT, implying a significant barrier), and a transition-state ensemble (TSE) containing helices α1, α3, and α4, folding in ~5 μs. However, four landscape features of λYA remain undetermined: (1) the origin and structural nature of a minor slow refolding phase (~2–10 s⁻¹, 5–20% amplitude, nearly denaturant-independent, persisting beyond 6 M urea), proposed but not proven to reflect a trapped intermediate with nonnative β structure; (2) whether a small subpopulation of molecules resides near the TSE on an incipient downhill (barrier-free) landscape; (3) the precise composition of the TSE, in particular whether it contains partial structure of helix α2 or the 3₁₀ helix, and the sequence- and method-sensitive folding order of helices α2 and α3; and (4) the magnitude and molecular origin of the kinetic prefactor (attempt frequency) in kf=kattempte−ΔG‡/RTk_f = k_{\text{attempt}} e^{-\Delta G^{\ddagger}/RT}, currently bounded only as faster than (5 μs)⁻¹.

Background

λ repressor is a bacterial transcription regulator whose N-terminal domain, λ6-85, is a five-helix bundle (α1–α5) that folds in microseconds and has served for three decades as the reference experimental system for fast, nearly barrier-free protein folding. Folding mechanisms are commonly described as diffusion on a free-energy landscape: in conventional two-state folding, native (N) and denatured (D) ensembles are separated by a free-energy barrier of many kBTk_B T with a transition-state ensemble (TSE, the set of structures at the barrier top) crossed on every productive folding event, whereas in downhill folding the barrier is comparable to kBTk_B T or absent, so the molecule slides continuously to the native state. Because observed folding rates combine a barrier height with a prefactor (attempt frequency), kf=kattempte−ΔG‡/RTk_f = k_{\text{attempt}} e^{-\Delta G^{\ddagger}/RT} (Kramers-type rate expression), a small apparent barrier cannot be distinguished from a fast prefactor without knowing the prefactor independently; the reciprocal prefactor is related to the 'folding speed limit' estimated at roughly L/100L/100 μs for a chain of LL residues.

The variant λYA carries the helix-promoting mutations G46A/G48A on top of λ6-85 plus Q33Y, and folds in ~5 μs with an extrapolated folding rate of about 2×10⁵ s⁻¹. Gruebele and co-workers proposed from nanosecond temperature-jump experiments (in which a laser pulse heats the sample within nanoseconds and the relaxation back to equilibrium is followed optically) that such fast, stabilized variants fold in an incipient or fully downhill manner under highly stabilizing (low-denaturant, low-temperature) conditions; Prigozhin and Gruebele additionally reported that λ6-85 has a minor slow refolding phase that they attributed to a trapped intermediate containing nonnative β structure, an idea supported by some atomistic simulations (Bowman–Voelz–Pande; Liu–Schulten) but complicated by later joint experiment–simulation analysis showing that millisecond-scale relaxation seen in simulations can be a property of local structural probes rather than global folding. Earlier Ala→Gly φ-value mapping by Oas and co-workers had established that the λ6-85 TSE contains at least helices α1 and α4. In φ-value analysis, the change in folding activation energy produced by a conservative mutation relative to its change in overall stability reports how native-like the mutated site is in the TSE; φ ≈ 1 means fully formed in the TSE, φ ≈ 0 means unformed.

Yu, Baxa, Gagnon, Freed, and Sosnick (PNAS 2016) attacked the downhill question for λYA with NMR-detected hydrogen exchange (HX), which measures the equilibrium opening of individual backbone amide hydrogen bonds: the rate at which each amide exchanges with solvent deuterons yields a per-residue stability, so that protection in many amides reflects one cooperative global unfolding event rather than independent local fluctuations. Under native-like, highly stabilizing conditions (0–4 M urea, sarcosine), where downhill folding is most probable, they found at least one hydrogen bond on each of helices α1, α3, and α4 breaks in a common global event, and combined HX with their ψ-analysis (introducing engineered bidentate-His metal-binding sites whose divalent-ion binding perturbs stability and activation energy, giving a ψ value analogous to a φ value but without mutating the structure) and mutational φ analysis. They concluded that the major phase is cooperative with a significant barrier — the observed HX isotherms lie closest to the highest barrier modeled in their analysis (5 RT), while the experimental data alone cannot identify the barrier height — that the TSE contains α1, α3, and α4, that folding remains barrier-limited at ~5 μs, and that the prefactor must be faster than (5 μs)⁻¹. Yet their own data leave the landscape incompletely determined, in the specific ways stated in the problem. Related progress exists for other proteins (WW-domain prefactors, transition-path times including λ repressor, non-native entanglement misfolds in simulation), but these leave the λYA-specific questions open.

Problem Statement

For the λ-repressor fast-folding variant λYA (λ6-85 with G46A/G48A/Q33Y, folding time ~5 μs, stability ~7.4 kcal·mol⁻¹ at 298 K), determine the remaining features of its folding energy landscape that the 2016 hydrogen-exchange study left unresolved, namely:

  1. The origin and structural nature of the minor slow refolding phase (rate ~2–10 s⁻¹, amplitude 5–20%, persisting beyond 6 M urea with little or no denaturant dependence, and absent in the biHis and Ala/Gly variants used for TSE mapping): is it, as the source allows, a subpopulation folding along a parallel pathway that accumulates due to a misfolding event — e.g., the trapped intermediate with nonnative β structure proposed by Prigozhin and Gruebele, or a rearrangement/misplacement of helix α2 or α5?
  2. Whether any small fraction of λYA molecules resides near the TSE on an incipient downhill (barrier-free) landscape under native-like conditions, as allowed (but not required) by the HX data, or whether the barrier-limited cooperative picture indicated by the HX data (isotherms closest to the highest modeled barrier, ~5 RT) applies to the whole ensemble.
  3. The precise composition of the transition-state ensemble: does it contain partial structure of helix α2 or the 3₁₀ helix (a tight helical form with i→i+3 backbone hydrogen bonds, distinct from the i→i+4 hydrogen bonding of an α-helix)? The φ value at position 38, φ_A38G = 0.4 ± 0.1, is ambiguous — it could reflect either fractional helix formation or the loss of backbone entropy as the chain becomes constrained upon folding of the flanking helices α1 and α3. HX in the source study detected five α3 amides (F51 and N52 exchanging globally; A48, A49, and G53 with nonglobal processes), yet the actual folding order of helices α2 and α3 remains open: at least computationally (the TerItFix sequential-stabilization algorithm of Adhikari et al., ref. 10), the predicted order switches with small protocol, energetic, and sequence differences.
  4. The magnitude and molecular origin of the kinetic prefactor (attempt frequency) kattemptk_{\text{attempt}} in kf=kattempte−ΔG‡/RTk_f = k_{\text{attempt}} e^{-\Delta G^{\ddagger}/RT}, currently bounded only from below as faster than (5 μs)−1(5\,\mu s)^{-1}.

These four items correspond to separate caveats and limitations acknowledged by the 2016 study rather than a single joint question, and each bears on where λYA sits on the continuum between cooperative two-state and downhill folding: the barrier/prefactor decomposition, the homogeneity of the ensemble with respect to the barrier, and the TSE composition are the quantities that decide whether microsecond folding in this canonical fast folder is genuinely barrier-limited. The cooperative, barrier-limited character of the major folding phase, with a TSE containing helices α1, α3, and α4, is an established result of the source study and is not itself in question.

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