Observation of strange kinetics in protein folding

Abstract

Highly nonexponential folding kinetics in aqueous solution have been observed during temperature jump-induced refolding of two proteins, yeast phosphoglycerate kinase and a ubiquitin mutant. The observations are most easily interpreted in terms of downhill folding, which posits a heterogeneous ensemble of structures en route to the folded state. The data are also reconciled with exponential kinetics measured under different experimental conditions and with titration experiments indicating cooperative folding.

Figure 1

(A) Protein energy landscape; the radial coordinate correlates with compactness (configurational entropy S_C); the angular coordinate symbolizes another >10³ coordinates. More compact states generally have lower contact energy E_C. The native ensemble F lies near the global minimum; there may be other deep minima I. (B) Thermally averaging the landscape over all but two reaction coordinates removes uninteresting energy spikes (e.g., because of steric hindrance) and leads to a two-dimensional free energy plot. The denatured state D is now a local minimum because of its large entropy. The arrows indicate an intermediate, two-state, and downhill pathway; not all of these necessarily appear simultaneously at constant T. Whether I is an obligatory intermediate or a trap that slows folding depends on its location, relative saddle point heights, etc. (C) Further reduction to one-dimensional pathways. The protein may switch pathways when conditions in B are changed, shifting the location and free energy of minima, maxima, and saddles. Some “nonspecial” states are shown in addition to transition states and intermediates. At weak native bias (e.g., [GuHCl] ≠ 0) a two-state or intermediate scenario results: only D, I, and F are significantly populated during folding (indicated by maximum population histograms). At strong native bias (S_C nearly compensated by E_C during folding), a type 0 scenario results. The rate is dictated by downhill folding, and I and ^† join the ranks of the “nonspecial” states, many of which are now populated. Strange kinetics (14) are not caused by thermodynamic barriers, but rather by unproductive diffusive motions “perpendicular” to the one-dimensional reaction coordinate.

Figure 2

Representative steady-state cold denaturation measurements. (A) Fluorescence of yeast phosphoglycerate kinase excited at 280 nm; λ_max redshifts by over 30 nm on cooling, as evidenced by the second component of a singular value decomposition of the temperature-dependent data. The tryptophan and tyrosine residues become solvent exposed. (B) ¹H NMR of Ub*G acquired in D₂O with residual water presaturation. The upfield Ile-61 peak is caused by contact with the core tryptophan 45; it disappears in the cold denatured state, indicating core solvation. Both the F and D components of the His-68 peak are resolved; the F peak rapidly decreases in intensity and shifts downfield at lower T. The chemical shift dispersion of the sample decreases toward a random coil.

Figure 3

PGK at pH 2 has a rapid fluorescence response attributed to relaxation of the unfolded chain (t₁ in Eq. 1 at 0.1 μs). Similar <20 μs phases are also observed preceding the folding kinetics of PGK and Ub*G at higher pH. (Inset) Tryptophan has only an “instantaneous” (<20 ns) temperature-dependent fluorescence change (t₁ at −25 μs).

Figure 4

(A) After a T-jump, time-resolved fluorescence transients change shape from f_t1 to f_t2 because of folding between t₁ and t₂. If only D and F are significantly populated during folding, f_t1, f_t2 and f_t are linear combinations of the denatured fluorescence signal f_D and native fluorescence signal f_F; otherwise, other populations contribute to f_t1 and f_t2. (B) Two-state model. [D]_t1 and [F]_t1 relax exponentially in time to [D]_t2 and [F]_t2 without significantly populating “^†”. f(t) is a weighted average of f_F and f_D and hence of f_t1 and f_t2. χ₁(t) therefore decreases exponentially, no matter at which time f_t1 and f_t2 are picked. The same holds true for a three-state reaction with two time scales τ₁ and τ₂, if τ₂ ≫ τ₁ and f₂ is picked at t ≪ τ₂. (C) Sequential intermediates. f_I is not necessarily well represented by a linear combination of f_t1 and f_t2. If it is, χ₁ is a linear combination of exponentials with uncorrelated lifetimes and amplitudes, some of which may be negative (e.g., if f_I ≈ f_D ≠ f_F). If it is not, χ² for the fit to Eq. 1 is greater at intermediate times than at t₁ and t₂. (D) Downhill scenario. A heterogeneous downhill folding ensemble is populated during folding because of the small transition state barrier ^†. If the subensembles {s} with different fluorescence f_s interconvert slowly, the rate laws of individual s add separately (Eq. 2), yielding a smooth nonexponential χ₁(t). The nonexponentiality can be homogeneous or inhomogeneous depending on the interconversion rates among and within subensembles (11, 12, 19).

Figure 5

Folding kinetics of PGK and Ub*G. (A) PGK folds by a slow single exponential at 5°C, a faster stretched exponential at 19°C (Table 1) (t₁≈ 8 μs, t₂ ≈ 2 ms in χ₁ fit). (Inset) χ² for the fit to Eq. 1 is approximately constant, ruling out intermediates with a fluorescence signature not represented by a linear combination of the endpoints. (B) Ub*G folds by a slow single exponential at 2°C, a stretched exponential at 8°C. (Inset) The full 2°C χ₁ trace: an ≈20 μs phase attributed to D → D′ relaxation in the unfolded well precedes slow exponential folding (τ ≈ 5 ms) over a barrier. The fast initial phase has been subtracted from the 2°C trace in the main figure (t₁ ≈ 100, 3 and 1 μs in χ₁ fit for 2°C, 8°C, and Inset, respectively).

Figure 6

Free energy as a function of T and reaction coordinate (e.g., protein compactness). At T₀, the cold denatured state D_C is most stable; at T₄, the heat denatured state. At T₂ the native state is most stable and at T₁ or T₃ the equilibrium constant is approximately 1. F is separated from D_C and D_H by barriers that always lead to cooperative temperature titrations (thick line shows average value of q during titration). A sudden small T-jump from T₀ to T₁ leaves q constant and the protein in the D well, followed by type 1 activated exponential kinetics (left Inset plot). A larger jump from T₀ to T₂ leaves the protein closer to the transition state or on a purely downhill type 0 surface if the free energy bias from D to F is too steep (middle Inset plot); the protein folds nonexponentially downhill. A jump from T₀ to T₃ would presumably result in activated kinetics again because of the onset of heat denaturation.