Direct Observation of Parallel Folding Pathways Revealed Using a Symmetric Repeat Protein System

Abstract

Although progress has been made to determine the native fold of a polypeptide from its primary structure, the diversity of pathways that connect the unfolded and folded states has not been adequately explored. Theoretical and computational studies predict that proteins fold through parallel pathways on funneled energy landscapes, although experimental detection of pathway diversity has been challenging. Here, we exploit the high translational symmetry and the direct length variation afforded by linear repeat proteins to directly detect folding through parallel pathways. By comparing folding rates of consensus ankyrin repeat proteins (CARPs), we find a clear increase in folding rates with increasing size and repeat number, although the size of the transition states (estimated from denaturant sensitivity) remains unchanged. The increase in folding rate with chain length, as opposed to a decrease expected from typical models for globular proteins, is a clear demonstration of parallel pathways. This conclusion is not dependent on extensive curve-fitting or structural perturbation of protein structure. By globally fitting a simple parallel-Ising pathway model, we have directly measured nucleation and propagation rates in protein folding, and have quantified the fluxes along each path, providing a detailed energy landscape for folding. This finding of parallel pathways differs from results from kinetic studies of repeat-proteins composed of sequence-variable repeats, where modest repeat-to-repeat energy variation coalesces folding into a single, dominant channel. Thus, for globular proteins, which have much higher variation in local structure and topology, parallel pathways are expected to be the exception rather than the rule.

Introduction

Protein folding studies are aimed at understanding the principles that dictate the sequence of events connecting an unfolded polypeptide ensemble to the native state. Experimental studies have focused on intermediates and transition states; however, due to the low resolution of experimental techniques, results are typically abstracted onto one-dimensional coordinates, which are interpreted using a single pathway (1). In contrast, theory and simulation suggest a complex network of parallel pathways along a funneled energy landscape, challenging the single-pathway view (2,3).

Although a few experimental studies have suggested parallel pathways (4–9), these experiments often involve destabilizing the dominant pathway by chemical (4,5,7,8) or physical perturbation (4–6), to attempt to shift the flux to an alternative pathway. In such studies, it is difficult to rule out the possibility that shifts in pathway result from the perturbations themselves, which significantly alter the local distribution of stability and chain topology (1,10). Thus, these studies do not stringently test whether folding occurs on multiple distinct pathways for a given sequence under a single set of conditions—a key prediction of a highly funneled folding scenario.

The linear symmetry of repeat proteins can be exploited to probe pathway diversity without changing sequence, topology, or stability. Repeat proteins are composed of the same secondary structures and packing interactions that constitute globular proteins, but their folds are maintained exclusively by nearest-neighbor interactions between adjacent repeats. The regular symmetry of repeat proteins has enabled quantification of local equilibrium stability and coupling of individual repeats (11–14).

Studies using natural repeat proteins, which show sequence variation among repeats, have an uneven stability distribution (15), and fold along a major pathway (16,17) involving the most stable repeats (18). Studies on designed repeat proteins with identical, consensus repeats eliminate sequence variation, and thus, variation in local stability (11,19,20). With flat energy landscapes, consensus repeat proteins may be expected to fold through multiple transition states of equal size. However, kinetic studies of consensus repeat proteins have so far failed to identify a key feature expected of parallel pathways, namely that the folding rate should increase linearly with repeat number (14,21). Thus, the existence of parallel pathways for consensus repeat proteins remains an open question. Answering this question is critical, because if these symmetric repeat proteins do not fold through parallel pathways, parallel pathways in globular proteins seem highly unlikely.

To shed light on pathway diversity, we performed kinetic refolding studies on a consensus ankyrin repeat protein (CARP) array with high structure, sequence, and energy symmetry. We see clear evidence for increases in rate constant with repeat number, consistent with a mechanism that includes parallel pathways. By analyzing the kinetic data globally with a kinetic Ising model, we are able to uniquely determine rate constants for nucleation and propagation for a protein-folding reaction.

Materials and Methods

Cloning, protein expression, and purification

CARP(Ala) and (Pro) gene arrays containing N- and C-terminal tryptophan residues (see Fig. 1B) were assembled as described in Aksel et al. (20). Protein expression and purification was carried out essentially as described (20).

Figure 1.

Structure and stability of consensus Ankyrin repeat proteins (CARPs) of different lengths and cap sequences. (A) Schematic and (B) ribbon diagram (2L6B) of constructs used in this study; N, R, and C repeats are in green, blue, and red, respectively. In panel B, polar and charged residues unique to N and C repeats are in space-filling representation. (Orange) Prolines; (magenta) asparagines (substituted to tryptophans). (C) Two-state fitting (solid lines) to GdnHCl unfolding transitions for CARP(Ala), and (D) fitted free energy changes (Eqs. 1 and 2) and (E) m-values. Error bars indicate 95% confidence intervals determined by bootstrapping. To see this figure in color, go online.

Analytical ultracentrifugation

Sedimentation velocity experiments of NR₂ (Pro) were performed on a model No. XLI Analytical Ultracentrifuge (Beckman Coulter, Fullerton, CA) equipped with interference optics, using meniscus-matching cells at 50,000 rpm, 20°C, as described in Aksel et al. (20). NR₂ (Pro) was dialyzed overnight three times against 150 mM NaCl, 25 mM Tris-Cl, pH 8.0. Analysis of g(S^∗) distributions and model-fitting to dc/dt difference data were performed using the software SEDANAL (22).

Circular-dichroism spectroscopy and equilibrium unfolding transitions

Far- and near-UV spectra of CARPs were measured on an Aviv Model 400 circular-dichroism (CD) spectrometer model (Aviv Associates, Lakewood, NJ) as described in Aksel et al. (20). Samples were in 150 mM NaCl, 25 mM Tris-Cl, pH 8.0, 20°C. Automated GdnHCl titrations were performed with an ML500 titrator (Hamilton, Reno, NV) as described in Aksel et al. (20). To quantify the unfolding transitions of CARP species with a simple two-state approximation (Fig. 1), a linear free energy relationship (23,24)

$$\text{Δ}{G}^{°}=\text{Δ}{G}_{{\text{H}}_2\text{O}}^{°}+m\left[\text{GdnHCl}\right]$$ (1)

was used separately for each construct. The folding free energy in the absence of denaturant, ΔG°_H20, and the denaturant dependence, m, were obtained by fitting unfolding transitions using

$$Y_{\text{obs}}\left(\left[\text{GdnHCl}\right]\right)=f\left(A_f+B_f\left[\text{GdnHCl}\right]\right)+\left(1-f\right)\left(A_u+B_u\left[\text{GdnHCl}\right]\right),$$ (2)

where f is the fraction of native protein in a two-state scheme, and depends on the folding free energy as

$$f=\frac{e^{-\text{Δ}{G}^{°}/RT}}{1+e^{-\text{Δ}{G}^{°}/RT}}.$$ (3)

Ising model analysis of equilibrium GdnHCl denaturation

Equilibrium intrinsic folding and interfacial energies (ΔG°_i, ΔG°_i,i+1) and m-values (m_i, m_i,i+1) were determined by globally fitting a heteropolymer Ising model to CARP equilibrium unfolding transitions (20,25), using an in-house program ISINGBUL. In the heteropolymer Ising model, there are 2ⁿ microstates in the ensemble (where n is number of repeats), each with a free energy (relative to the fully unfolded state) of

$$\text{Δ}{G}_i^{°}\left(\delta \right)=\sum _{i=1}^n{\delta }_i\text{Δ}{G}_i^{°}+\sum _{i=1}^{n-1}{\delta }_i{\delta }_{i,i+1}\text{Δ}{G}_{i,i+1}^{°},$$ (4)

where δ_i is unity if the ith repeat is folded, and is zero otherwise. The partition function Q for the ensemble can be calculated from the matrix equation

$$Q=\left[\begin{array}{cc}0& 1\end{array}\right]\left(\prod _{j=1}^n{\mathbf{W}}_j\right)\left[\begin{array}{c}1\\ 1\end{array}\right]=\left[\begin{array}{cc}0& 1\end{array}\right]\left(\prod _{j=1}^n{\left[\begin{array}{ll}{e}^{-\left(\text{Δ}{G}_i^{°}+\text{Δ}{G}_{i,i+1}^{°}\right)/RT}\hfill & 1\hfill \\ e^{-\text{Δ}{G}_i^{°}/RT}\hfill & 1\hfill \end{array}\right]}_j\right)\left[\begin{array}{c}1\\ 1\end{array}\right].$$ (5)

The CD and fluorescence signals of each CARP at different GdnHCl concentrations can be expressed from manipulation of the partition function as

$$Y_{\text{Ising}}\left(\text{GdnHCl}\right)=F\left(A_f+B_f\left[\text{GdnHCl}\right]\right)+\left(1-F\right)\left(A_u+B_u\left[\text{GdnHCl}\right]\right).$$ (6)

F is the fraction of folded repeats, and is obtained from differentiation of Q (25). The parameters A_f, B_f and A_u, B_u are folded and unfolded baseline denaturant signals and denaturant sensitivities, respectively. By globally fitting Eq. 6 to GdnHCl denaturation equilibrium data of a library of CARP constructs with a different number of repeats and cap structures, ΔG°_i, ΔG°_i,i+1, m_i, and m_i,i+1, were obtained. In our previous analysis, m_i,i+1 was determined to be close to half of m_i (20). Hence, in this study, we applied the constraint m_i,i+1 = 0.5 m_i to all kinetic and equilibrium fitting.

Stopped-flow fluorescence kinetics

Fluorescence-detected stopped-flow kinetic measurements were performed on an SX-18MV-R stopped-flow fluorometer (Applied Photophysics, Leatherhead, UK). Proteins were diluted 11-fold to concentrations of 2–6 μM. At least 10 fluorescence traces were collected at each condition. Intrinsic tryptophan fluorescence, after excitation at 280 nm, was detected using a 320-nm cutoff filter. Diluents at different GdnHCl concentrations were prepared automatically by combining different volumes of high and low GdnHCl solutions via an in-house program controlling an ML500 titrator (Hamilton). Before analysis, kinetic traces at each GdnHCl concentration were averaged.

Model-free kinetic analysis of individual decay curves

To gain general insight into the complexity of the kinetic refolding and unfolding data, we initially fit fluorescence decay data using a simple sum of exponentials. Amplitudes and rate constants of different phases at each GdnHCl concentration were determined by fitting the minimum number of decaying exponentials using

$$Y_{\text{obs}}\left(t\right)=Y_{\infty }+{\sum _i\text{Δ}{Y}_{i}e}^{-k_{i}t},$$ (7)

where Y_obs(t) is the fluorescence measured at time t, Y_∞ is the equilibrium fluorescence, and ΔY_i and k_i are the amplitude and the rate constant of the ith kinetic phase, respectively. In addition to providing numerical values for observed rates and amplitudes, this analysis provided a starting point for mechanistic models used in subsequent global analysis.

Global analysis of kinetic progress curves with specific models

Although the model-free analysis above provides a quick and fairly unbiased way to quantify the folding and unfolding kinetics for a given construct and GdnHCl concentration, analyzing the data with a mechanistic model requires a second level of fitting, which is dependent on the assumptions (number of exponentials) used in the first round (Eq. 7). To avoid the subjectivity of this two-stage fitting approach, we directly fitted kinetic models to raw fluorescence time-courses. To constrain model parameters, we globally fitted to data at all GdnHCl concentrations and all constructs of different length and cap structure. Equilibrium unfolding transitions for each construct were also included in the global fit, to better define the amplitudes and denaturant dependences of the kinetic phases (see F, in Eqs. 8, 9, and 14 below).

To equally weight each time course in the global analysis, folding and unfolding fluorescence traces were normalized via the following transformations, respectively, where

$$Y_{\text{norm}}^f\left(t\right)=F\left(1-\frac{\left(Y_{\text{obs}}\left(t\right)-Y_{\infty }\right)}{\sum Y_i}\right),$$ (8)

$$Y_{\text{norm}}^u\left(t\right)=1-\left(1-F\right)\left(1-\frac{\left(Y_{\text{obs}}\left(t\right)-Y_{\infty }\right)}{\sum Y_i}\right).$$ (9)

F is the equilibrium fraction of folded repeats determined by the Ising analysis. As a result of this normalization (which assumes there is no burst phase within the dead time of the stopped flow experiments—see Fig. S1 in the Supporting Material), folding and unfolding curves have an amplitude of ±1 away from the transition, and ±F in the transition region.

To relate the fluorescence signal to a kinetic model for CARP folding a master equation was built in a matrix representation,

$$\frac{d\mathbf{x}\left(\mathbf{t}\right)}{dt}=\mathbf{M}\mathbf{x}\left(\mathbf{t}\right),$$ (10)

where x(t) is the time-dependent concentration of all species (in vector form), and M is a rate coefficient matrix whose structure is determined by the kinetic model used (sequential multistate, parallel Ising, see below). Equation 10 is solved by eigenvalue decomposition of M,

$$\mathbf{x}\left(t\right)=\mathbf{V}{e}^{\mathbf{D}t}\mathbf{c},$$ (11)

where V is a matrix of column eigenvectors, and D is a diagonal matrix where each diagonal element is an eigenvalue. The vector c is obtained by evaluating x(t) at time zero:

$$\mathbf{x}\left(\mathbf{0}\right)=\mathbf{V}\mathbf{c}.$$ (12)

The fitted fluorescence signal evolution is obtained from the expression

$$Y_M\left(t\right)={\mathbf{s}}^{\mathbf{T}}\mathbf{V}{e}^{\mathbf{D}t}\mathbf{c},$$ (13)

where s is a column vector containing the fluorescence signal of each species.

CD-detected equilibrium GdnHCl titration measurements were simultaneously fitted by using the stationary eigenvector (corresponding to the zero eigenvalue) to calculate F,

$$F=\frac{{\mathbf{c}}_{\infty }{\mathbf{r}}^{\mathbf{T}}{\mathbf{V}}_{\infty }}{n},$$ (14)

where c_∞ is the constant that corresponds to the stationary eigenvector, r is a column vector with the number of folded repeats for each species, and n is the number of repeats.

The difference in fluorescence signals from the tryptophans in the N and C repeats of NR_xC constructs was represented by introducing a parameter, q, the fraction of the native state fluorescence from the Trp in the N-terminal repeat. For example, for NRC, the following fluorescence signals were assumed:

$$\mathbf{s}=\left[\begin{array}{c}1\\ q\\ 1-q\\ 0\end{array}\right].$$ (15)

Here the elements of the s vector correspond to the species NRC, NRc, nRC, and nRC (upper-case represents a folded repeat).

Fitted parameter values were obtained by globally minimizing

$${\chi }^2={\left(Y_M-Y_{\text{norm}}\right)}^2+{\left(Y_{\text{Ising}}-Y_{CD}\right)}^2$$ (16)

for the kinetic and the equilibrium data for all CARP constructs simultaneously. Y_CD is the CD signal from equilibrium GdnHCl titrations. To capture minor kinetic phases with small amplitudes, the data were normalized iteratively using Eqs. 8 and 9 during minimization, by replacing F by Y_M,∞, which is the equilibrium fluorescence signal estimated by the model. All data were analyzed and plotted using in-house PYTHON scripts called PYKINFIT and ISINGKINFIT (available upon request).

Sequential multistate kinetic model

A detailed graphical representation of the multistate model is shown in Fig. S2. Parallel pathways are not explicit in this model. If parallel pathways contribute to folding kinetics, they would produce variation in the rate constant for the nucleation and propagation steps from construct to construct. There are 82 and 59 free parameters in the model for CARP(Ala) and CARP(Pro), respectively.

Parallel-Ising kinetic model

A detailed graphical representation of the parallel Ising model is shown in Fig. S3. Intrinsic and interfacial parameter definitions were adapted from the previous study (20,25). In that study, we found m_i,i+1 to be very close to half of m_i (20). In the parallel-Ising kinetic model, we fix m_i,i+1 at 0.5 m_i. There are 14 and 28 free parameters in the models for CARP(Ala) and CARP(Pro), respectively, significantly fewer than for the sequential multistate kinetic model.

Results

Ankyrin repeats consist of two antiparallel α-helices, with neighboring repeats connected by a β-hairpin loop (Fig. 1). To maximize symmetry among repeats, we have created a series of consensus-based ankyrin repeats that have identical internal R repeats (blue, Fig. 1, A and B), and that have four polar sequence substitutions at exposed sites in the N- and/or C-terminal repeats (green and red, Fig. 1, A and B) to enhance solubility (14,20). N and C repeats are similarly folded (Fig. 1 B), although they are less stable than internal R repeats (20). Approximating equilibrium unfolding using a two-state mechanism, each repeat decreases the folding free energy (increases stability, Fig. 1, C and D) and increases the m-value (denaturant sensitivity of unfolding free energy, Fig. 1 E) by a constant, additive amount (20).

Kinetics of consensus ankyrin repeat protein folding

To monitor CARP refolding kinetics, we introduced a tryptophan at position 5 of the N and C repeats. Because the consensus sequence has a proline at position 7, which could potentially mask pathway diversity through slow isomerization in the unfolded state, we constructed a series of CARPs in which the prolines are substituted with alanines. We refer to these two series as CARP(Ala) and CARP(Pro). These substitutions do not alter the CARP fold (see Fig. S4) or its monomeric state (see Fig. S5).

Fluorescence-detected refolding of CARP(Ala) constructs shows single-exponential folding kinetics (Fig. 2, A and B, and see Fig. S1 A) with no detectable burst-phase (see Fig. S1 B). Single exponential folding is seen at both low and high GdnHCl concentrations. Notably, within all series, the exponential decay is faster for longer constructs (with more repeats, Fig. 2, A and B, and see Fig. S6). This is true regardless of cap structure (NR_xC, NR_x, and R_xC). This is also true for the major phase of the CARP(Pro) series, although a minor slow phase is seen in the folding of all CARP(Pro) constructs (but none of the CARP(Ala) constructs). This slow phase has a rate constant of ∼0.02 s⁻¹, and is insensitive to GdnHCl, consistent with proline isomerization.

Figure 2.

Fluorescence-monitored refolding and unfolding traces of consensus Ankyrin constructs of differing lengths. (A and B) Refolding of NR_x(Ala) at 1.2 M and 3.2 M GdnHCl, respectively. Lines are from fitted single-exponential decay curves. (C) Unfolding of NR_x(Ala) at 6.5 M GdnHCl. Lines are from model-free fitting with multiple exponentials (Eq. 7). To see this figure in color, go online.

Our rationale for parallel pathway detection is based on a simple relationship between the folding rate and number of pathways, in which the observed folding rate constant is the sum of rate constants through each pathway. If the number of pathways is doubled, the observed rate will also double (compare the model in Fig. 3 C with Fig. 3 A). For CARPs, because internal repeats are energetically equivalent, adding repeats should introduce energetically equivalent pathways, increasing the folding rate. Indeed, the folding rate increases linearly with the number of repeats (Figs. 2 and 4, and see Fig. S6), as expected for parallel pathways.

Figure 3.

Plausible models for NR_x(Ala) folding. The Upper and lower panels show the energy landscapes and chevrons. The middle panel shows kinetic mechanisms, with transition state ensemble (TSE) locations marked by ‡ (for the sequential multistate model, fast steps are marked by ‡′). In the energy landscapes, native state free energy differences from the unfolded state (U) are determined from equilibrium denaturation (see Fig. 1D). (A) Two-state/single pathway/fixed TSE model. Barrier heights are equal regardless of repeat number. Thus, slopes and y intercepts of the refolding arm of the chevron are the same for all constructs. (B) Two-state/single-pathway moving TSE model. The extent of folding in the TSE increases linearly with repeat number, and its energy decreases linearly, but its relative compactness is equal for all constructs (β_T = −m_f/m_eq = 0.5). (C) Two-state/parallel pathways model. The number of pathways increases linearly with repeat number, although the free energy and compactness of the TSE remains unchanged. As a result, y intercept of the refolding arm of the chevron plot increases with repeat length, but its slope remains the same. (D) Sequential multistate model. For an n repeat construct, there are n−1 transitions and n−2 ensembles of intermediates (X_i) in which i tandem repeats are folded (1 < i < n). The distance between states is proportional to the m-value difference (see Fig. S2). To see this figure in color, go online.

Figure 4.

CARPs fold through parallel pathways. (A) Chevron plots of NR_x(Ala), with rate constants for major phases determined from the model-free fitting equation (Eq. 7). (Inset) Expanded scale, highlighting an increase in folding rate with repeat number, without a change in slope. This is evidence of parallel folding pathways. Lines are the result of fitting the sequential multistate model fit to raw fluorescence data. (B) Folding rate constants determined by fitting the sequential multistate model. Error bars are 95% confidence intervals from 100 bootstrap iterations (see Fig. S2, Fig. S8, and Fig. S11). To see this figure in color, go online.

Another mechanism by which folding rate may increase with repeat number is a progressive stabilization of the transition state ensemble by additional repeats (Fig. 3 B). However, expansion of the transition state in longer constructs would increase the m-value of the folding rate (chevron plots, Fig. 3), which we clearly do not see. Rather, kinetic m-values for folding are the same regardless of repeat number (Fig. 4 A), implying that the transition state structure is the same for folding of all CARPs, but the longer constructs have additional transition states available.

Although a simple two-state model with parallel pathways (Fig. 3 C) captures the increase in folding rates with repeat number, a two-state model alone cannot explain the kinetic complexities we see in unfolding. As constructs get longer, the unfolding arms of the chevron plots show downward curvature, and for the longest constructs, curvature is also seen on the refolding arm at high GdnHCl. Moreover, multiple unfolding phases are observed (Fig. 2 C, and see Fig. S7). These complexities are consistent with sequential on-pathway intermediates (sequential multistate model; see Fig. 3 D). Because kinetic m-values for folding are the same for all constructs at low GdnHCl concentrations, we model the rate-limiting (slow) step for all constructs as formation of the same transition state structure (containing two repeats or fewer). Because curvature in the unfolding arm is seen for all constructs longer than two repeats, and because the number of observed unfolding phases increases with each additional repeat starting from three-repeat constructs, we model subsequent (fast) steps as the addition of single repeats (up to four steps, see Fig. S2). We model the relative positions of barriers and intermediate states globally using only four kinetic m-values (two for an initial slow step, and two for subsequent fast steps, with each pair constrained to the equilibrium m-values, see Fig. S2). In contrast, forward and reverse rate constants for slow and fast steps are modeled separately for each construct.

All raw fluorescence folding and unfolding decay curves at different denaturant concentrations are fitted quite well by the sequential multistate model (not shown). Fitted rate constants match very closely to the rate constants determined from model-free exponential fitting at different denaturant concentrations (Figs. 4 A, and see Fig. S8), reproducing the multiple unfolding phases and their nonlinear denaturant dependence (see Fig. S8). Importantly, the fitted rate constants for the rate-limiting folding steps (k_f) of this multistate mechanism increase with repeat number. Within each capping series, this increase is linear with repeat number (Fig. 4 B). As described above, this increase is consistent with a set of parallel pathways that connect the denatured ensemble to the rate-limiting step.

A simple parallel pathway model that captures multistate unfolding

The analysis above shows evidence of parallel pathways in folding, and multiple steps in unfolding. Given the complexity of the sequential multistate model (there are 82 adjustable parameters in the sequential multistate model, because rate constants for each of the 10 CARP(Ala) constructs are treated separately) and its goodness of fit, further expansion of the sequential multistate model to include parameters that capture parallel pathways is not likely to lead to unique parameters. To model both the parallel and sequential multistate aspects of the data without overparameterizing, we developed a model that takes advantage of the symmetry of the CARP constructs. This kinetic Ising model (Fig. 5) is equivalent to a nucleation-propagation model, requiring a single rate constant for nucleation (structuring of two repeats and forming a single interface) and a second rate constant for propagation (addition of a single repeat to a folded array). We assume that n−1 parallel nucleations emanate from U (where n is the number of repeats), generating partly folded states involving two adjacent repeats. From these partially folded states, one or two parallel propagations emanate. As folding progresses, these pathways converge on a fully folded state. By treating nucleation and propagation in terms of intrinsic repeat folding and interfacial interactions, we can constrain the unfolding rate constants for the nucleation and propagation steps (and their denaturant dependences) using the equilibrium unfolding transitions and Ising analysis (Fig. 1, and see Fig. S3, Ising Parameters box).

Figure 5.

A parallel-Ising kinetic model for CARP(Ala) folding kinetics. (A) Folding mechanism of NR₃C(Ala). The first step involves folding two contiguous repeats; subsequent steps involve folding an adjacent repeat. The magnitude of folding flux is indicated by arrow thickness. (B) Three elementary nucleation and propagation steps are sufficient to model all CARPs (see Fig. S2). (C) Energy landscape of NR₃C(Ala), constructed with parameters obtained from the parallel-Ising kinetic model. The ground-states and barriers are colored according to their free energies. The folded region of each ground state is depicted on the energy level. The grayscale intensity of the paths between neighboring states is proportional to the magnitude of the folding flux. To see this figure in color, go online.

By globally fitting this parallel Ising model to fluorescence-monitored folding and unfolding kinetics in different GdnHCl concentrations along with equilibrium CD melts for CARP(Ala) constructs of different lengths (see Fig. S9, A and C), we determined the folding rates of nucleation and propagation for all repeats (Fig. 5 B, and see Table S1 in the Supporting Material), and as a result, fluxes through each parallel folding pathway (Fig. 5, A and C). The parallel-Ising model captures CARP(Ala) folding as well as the sequential multistate model, but with significantly fewer parameters (14 vs. 82 parameters). These 14 fitted parameters are sufficient to reproduce chevron plots for all 10 constructs, including increases in k_f with repeat number (parallel pathways), and multiphasic unfolding with nonlinear GdnHCl dependence (Fig. 6). The complexity in unfolding displayed by the model reflects the fact that, like the more complicated 82 parameter model, the kinetic Ising model includes multistate sequential folding (along each parallel pathway), but it treats parallel steps as kinetically equivalent reactions. The nonlinearity in the unfolding arms of the chevron plots, and for the refolding arms of longer constructs, is consistent with a shift in the transition state from two to three repeats with increasing GdnHCl concentration within each parallel pathway (see Fig. S10).

Figure 6.

The parallel-Ising kinetic model captures the folding and unfolding kinetics of CARP(Ala) constructs of various length and capping structures at different GdnHCl concentrations. (A) Comparison of eigenvalues obtained from the parallel-Ising global fit to fluorescence kinetic traces and CD-detected equilibrium unfolding transitions (lines, see Fig. S9, A and C) with rate constants (points) obtained from the model-free multiexponential fitting. The area of each point marker is proportional to its fitted kinetic amplitude; only eigenvalues and rate constants with relative amplitude higher than 10% are shown. (B) The parallel-Ising model captures the increase in the folding rates with repeat number for all constructs; only major phases are shown here for clarity. To see this figure in color, go online.

The fitted folding rate constants for nucleation, which represent the rate-limiting step(s) in the absence of GdnHCl, are between 80 and 630 s⁻¹, depending on whether nucleation involves an N and an R, two R repeats, or an R and a C repeat (see Table S1). The variation in the nucleation rate constants parallels the variation in the intrinsic stabilities of the three types of repeats: the R-repeat is most stable, and formation of an RR nucleus has the largest value of k_nuc, whereas the C-repeat is least stable, and formation of an RC nucleus has the smallest value of k_nuc. As expected for a nucleation-propagation mechanism, the propagation reaction is significantly faster than the rate-limiting nucleation in the absence of GdnHCl, ranging from 11,000 to nearly 60,000 s⁻¹.

Our observation of parallel folding pathways is at odds with an earlier study by Wetzel et al. (14) of doubly capped consensus ankyrin arrays with 3–5 repeats. The internal repeats of that study have 73% identity to the CARPs studied here, whereas the N- and C-capping repeats have 34 and 20% identity, respectively. One potentially significant difference between the two series is that in our study the conserved prolines at position 7 were substituted with alanines (CARP(Ala)) to avoid any potentially obscuring effects of slow prolyl isomerization. To test whether the independence of folding rate on repeat number seen by Wetzel et al. (14) results from prolyl isomerization, we measured and analyzed folding kinetics of a series of CARPs with prolines (Pro, Fig. 7, and see Fig. S6, Fig. S7, Fig. S11, and Fig. S12). As with the Ala variants, the folding rate constants for the CARP(Pro) series increase with repeat number (Figs. 4 B, and see Fig. S6 and Table S1). An additional proline isomerization phase is clearly observed in refolding of CARP(Pro) constructs ((26), Fig. 7, A and D, and see Table S2), although it is significantly slower than the major folding phase. These findings show that the absence of parallel pathways in the study of Wetzel et al. (14) is not a result of prolyl cis-trans isomerization. Our data suggests that the amplitude and rate of the proline isomerization does not depend on the number of repeats (Fig. 7 D, and see Table S2), which suggests that only one proline in the unfolded state needs to be in the trans isomer for folding to initiate. Isomerization of the remaining prolines may be accelerated in subsequent propagation steps.

Figure 7.

The effect of prolines on CARP folding kinetics. (A) Chevron plots of NR₂ with (▵) and without (▴) prolines. The box highlights the cis-trans proline isomerization phase observed in NR₂(Pro) folding traces. Similar phases are seen for other CARP(Pro) constructs. The area of each point marker is proportional to its fitted kinetic amplitude. (B) Parallel-Ising kinetic reaction net for the folding of NR₂C(Pro). Unlike the reaction net for CARP(Ala), for CARP(Pro) constructs, a cis-trans proline isomerization reaction in the unfolded state precedes the nucleation step. (C) Global fit of the parallel-Ising kinetic model to CARP(Pro) stopped-flow fluorescence data (see Fig. S9, B–D, and Fig. S12). Only the rate constants with relative amplitude higher than 10% are displayed. Lines and symbols are as described in Fig. 6. (D) Relative amplitudes of the proline isomerization phase of NR_x (where x = 1–3) do not depend on the number of prolines. Lines indicate expected folding amplitudes for constructs of different lengths, assuming that all prolines must be trans to fold, and assuming a trans/cis equilibrium constant of 8.9 (26). Our measured amplitudes for the prolyl isomerization phase do not depend on the number of repeats, suggesting that only one proline needs to be in trans to initiate folding. To see this figure in color, go online.

Discussion

Although parallel pathways have been a central tenet of several models of protein folding from both theory and computation (chiefly the folding funnel model (3) and more recently the kinetic hub model (2)), they have been difficult to test for experimentally. Importantly, the system studied here allows chain length to be varied in a structurally conservative way. We find clear evidence for parallel folding pathways by comparing linear proteins composed of different numbers of symmetrical structural elements.

We see a linear increase in the folding rate constant with the number of repeats, a central prediction of introducing multiple pathways for folding as repeats are added. The guanidine dependence of the folding rate constant is the same, regardless of length, as would be expected for parallel pathways through similar transition states (but not for changes in transition state structure; see Figs. 3 and 4). These parallel pathways do not result from heterogeneity in the denatured state (in contrast to larger proteins such as dihydrofolate reductase (27)), but instead result from multiple routes that depart from an equilibrated denatured state ensemble toward distinct barriers. Based on both kinetic modeling (Fig. 5, A and B) and denaturant sensitivity (Fig. 4 A), these barriers appear to involve the intrinsic folding of two adjacent repeats that have yet to develop interfacial stabilization.

In addition to the parallel pathway behavior we observe for both the Ala and Pro series, we see evidence that folding progresses through one or more intermediate states, especially for longer constructs. We see rollover in the unfolding limbs of chevron plots for longer constructs, and also see multiple, faster kinetic phases for unfolding of longer constructs (Fig. 6 A, and see Fig. S12). Although downward curvature in chevron plots can be interpreted as the gradual movement of a single transition state ensemble (28), the additional kinetic phases are not consistent with transition state movement. Instead, these multiple phases imply discrete, barrier-limited transitions between intermediates that populate to a significant level during unfolding (29).

The kinetic Ising model not only captures the increase in folding rate constant with repeat number, it also captures the unfolding rollover and the multiple kinetic phases seen in unfolding. This is because this relatively simple model includes partly folded states between the native and denatured states (Fig. 5, and see Fig. S3). At high GdnHCl concentrations, the fully folded states are more destabilized than the partly folded states (with one or more end repeats unfolded). High GdnHCl also destabilizes the nativelike barriers between these largely folded states more than the two-repeat barrier that limits folding at low GdnHCl, allowing a significant population of partly folded states (with a shallower unfolding GdnHCl dependence) to build up during unfolding (see Fig. S10). This interpretation is consistent with the experimental observation that the fast, minor kinetic unfolding phases for long constructs match the dominant phases of shorter constructs.

Thus, the kinetic Ising model describes both parallel pathways and multistep kinetics. Each pathway is a sequence of kinetic steps, and denaturant modulates the stabilities and lifetimes of different steps in the reaction. Because the model takes advantage of the structural symmetry inherent in CARPs, it requires a relatively small number of parameters (14 in the absence of prolyl isomerization) to globally fit constructs of different lengths and cap structures. Although a set of stepwise multistate models that ignore structural symmetry (Fig. 3, and see Fig. S2) can also be fitted to the each CARP construct (see Fig. S8 and Fig. S11), the number of parameters required is significantly higher (82 in the absence of prolyl isomerization).

The length dependence of CARP folding and unfolding rates

A number of theoretical and comparative studies have examined how the folding rates of globular proteins depend on chain length. Theoretical models predict a decrease in folding rate (log k_f) with increase in chain length (L), ranging from an −L^0.7 dependence (30) to a −L^0.5 dependence (31,32). These include capillarity models for globular proteins (33,34). Comparisons of folding rates of different globular proteins support such a decrease (30,32), and are interpreted to result from an overall decrease in sequence-local contacts, relative to sequence-distant contacts, in native and transition states (35). However, such correlations compare proteins of different fold, sequence, and stability.

The studies here with CARPs allow us to vary chain-length substantially, while holding local topology, structure, sequence, and local stability constant. We find the length dependence of CARP folding rates to differ significantly from rates predicted from theory (31) and measured experimentally for globular proteins (32): CARP folding rates increase with length, whereas rates for globular proteins (both predicted and measured) decrease (Fig. 8 A). We view this not as failure of theory or correlative studies, but as a result of the linearity and structural symmetry of CARPs, which are lacking in globular proteins, and correspondingly, a difference in mechanism (as has been argued in application of capillarity ideas to repeat protein folding (36,37)). For CARPs, the increase in folding rate with length results from an increased number of parallel pathways. In contrast, for globular proteins, the decrease in folding rate likely reflects a single dominant pathway, rather than parallel pathways. The nucleation reaction in the kinetic Ising analysis can be thought of as analogous to a single-pathway event for globular protein folding. Indeed, the rate constant for this two-repeat reaction, k_nuc = 80–600 s⁻¹, is in close agreement with predictions from correlative studies based on contact order (30), taking two folded repeats as the native structure (Fig. 8, B and C).

Figure 8.

Comparison of CARP folding rates with values predicted based on chain length and topology. (A) CARP folding rates increase with chain length, in contrast to the decrease observed for globular proteins (31,32). The black solid line shows the negative square-root dependence obtained by fitting to experimentally determined folding rates (lnk_f = 16.15–1.28L^1/2 (32)). The green line shows the overall folding rate constants expected for NR_x (x ≥ 2) by addition of nucleation rate constants for each available path for folding (i.e., k_f(NR_x) = k^nuc_NR + (x−1) k^nuc_RR). Solid circles show nucleation rates for NR (green), RR (blue), and RC (red). (B and C) Comparison of folding rates estimated by relative contact order (RCO, B) and absolute contact order (ACO, C) with CARP nucleation rates. Solid symbols show experimentally determined folding rate constants (lnk_f) for globular proteins with different secondary structure composition (○, all α; ⋄, all β; +, α/β). Lines show linear fits between RCO (B) and ACO (C) and lnk_f for all globular proteins, regardless of secondary structure. To see this figure in color, go online.

Previous studies of folding kinetics in repeat proteins

Two other studies have been published that examine the folding rates of consensus repeat proteins. Consistent with the findings here, folding rates for a three-repeat α-helical consensus TPR construct are faster than that for a two-repeat construct (38), although a more complete study including longer constructs shows the opposite dependence, with longer constructs folding slower (21). As described above, a series of consensus ankyrin repeats with significantly different capping sequences showed folding rate constants that appear to be independent of length (14). Our studies with CARP(Pro) series show that this insensitivity is not a result of proline isomerization. It seems unlikely that the sequence differences in the cap structures limit the number of pathways in the previous ankyrin repeat study, because these cap structures are typically less stable than internal consensus repeats.

Interestingly, although a Gō-model simulation of ankyrin and TPR folding shows decreases in rates of folding of both proteins with increasing repeat number (39), this simulation is restricted to analysis close to the unfolding transition midpoint. Indeed, chevron analysis of our CARP constructs shows that if rate constants are compared at unfolding midpoints (minima in chevron plots), longer constructs have smaller rate constants (Fig. 6). However, this is a direct effect of the fact that these rate constants are being compared at significantly different GdnHCl values, which results in an exponential change in folding rate. When extrapolated to a common set of conditions, folding rate constants increase with length, regardless of whether the comparison is made in GdnHCl or in water.

Previous studies of parallel folding pathways in globular proteins

Multiple kinetic phases have been observed in both unfolding and refolding of medium, globular-sized proteins for decades (27,40,41). For a number of proteins, this heterogeneity has been shown to result from slow reactions in the denatured state, and in particular, prolyl isomerization (e.g., RNase A, mixed α/β, 124 residues (41,42)). Under stabilizing conditions, this heterogeneity can transfer into partly folded states on route to the native state, creating chemically distinct pathways, as is seen for the α-subunit of Trp synthetase (αβ-barrel, 270 residues (40)). For dihydrofolate reductase (mixed α/β, 159 residues), four parallel channels are seen for both folding and unfolding, but appear to be independent of prolyl isomerization (27,43). Given the slow interconversions of the start- and end-points along these channels (in the D and N states), these parallel pathways are consistent with a model in which distinct denatured-state energy basins converge to a native state hub (or to multiple native states). For hen lysozyme (129 residues), amide hydrogen exchange suggests there may be fast and slow parallel tracks for folding in which the α-helical and β-sheet subdomains fold differentially (44,45).

For globular proteins <100 residues in length, there is less evidence for parallel pathways. Aside from the effects of prolyl isomerization, small globular proteins typically show simple single-exponential folding kinetics. Many of these proteins fold by a kinetically two-state process through what appears to be a single barrier, though as discussed above, single-exponential kinetics does not rule out parallel pathways from a kinetically homogeneous denatured state. One way to identify parallel folding pathways is through upward curvature in chevron plots (4,46). Such behavior has been seen for the unfolding arm of the titin I27 domain, a β-sheet protein of 98 residues (4), and for single-chain monellin, a largely β-sheet protein (47), although the latter protein also displays heterogeneity in the denatured state.

Another method that has been used to test for parallel pathways in small globular proteins is to make multiple substitutions in different protein structural elements and test for kinetic additivity (7). Although comparing point substitutions in different circular permutants suggests that different barriers limit folding when topology is rearranged (8), combining point substitutions in single-domain proteins typically preserves additivity. Interestingly, the most notable exception to this observation is protein L, which possesses approximate twofold symmetry (48). As we report here for CARPs, the symmetry in protein L provides multiple, topologically equivalent routes for folding, although for protein L, high sequence variation is likely to favor one route over the other.

Like protein L, naturally occurring repeat proteins display significant sequence variation from repeat to repeat (15), which results in significant local stability variation (12,13). The observation that a single path dominates for natural (i.e., nonconsensus) repeat proteins (16,18,19,49) suggests that modest repeat-to-repeat energy differences (with a standard deviation of ∼1.4 kcal/mol (12)) are enough to collapse parallel folding into a single channel, which can be shifted to other well-defined locations by shifting stability. Only for very large repeat proteins (426 residues for ankyrin D34; 590 residues for the HEAT-repeat PR65/A) does evidence for multiple pathways become clear (50,51, although these studies are restricted to unfolding at high denaturant concentrations). Therefore, it seems unlikely that a general mechanism for single-domain globular proteins, which lack structural symmetry, will involve more than one major pathway. In our experimentally determined energy landscapes, this manifests as a channel that opens early in folding, defining a single rate-limiting step (52). Thus, single folding pathways in globular proteins should be identifiable from the asymmetries of both stability and local topology (53).