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On the precision of experimentally determined protein folding rates and -values

¹ Department of Chemistry and Biochemistry, and Interdepartmental Program in Biomolecular Science and Engineering, University of California, Santa Barbara, Santa Barbara, California 93106, USA
² Biochemistry and Cell Biology Department and Chemistry Department, Rice University, Houston, Texas 77251, USA
³ Department of Molecular and Cell Biology, University of California, Berkeley, Berkeley, California 94720, USA
⁴ Department of Biochemistry and Molecular Biology, Institute for Biophysical Dynamics, University of Chicago, Chicago, Illinois 60637, USA
⁵ Department of Biostatistics, Bloomberg School of Public Health, Johns Hopkins University, Baltimore, Maryland 21205, USA

Reprint requests to: Ingo Ruczinski, Department of Biostatistics, Bloomberg School of Public Health, Johns Hopkins University, Baltimore, MD 21205, USA; e-mail: ingo{at}jhu.edu; fax: (410) 955-0958.

(RECEIVED October 3, 2005; FINAL REVISION November 14, 2005; ACCEPTED November 14, 2005)

	Abstract

TOP Abstract Introduction Results Discussion Materials and methods References

 
-Values, a relatively direct probe of transition-state structure, are an important benchmark in both experimental and theoretical studies of protein folding. Recently, however, significant controversy has emerged regarding the reliability with which -values can be determined experimentally: Because is a ratio of differences between experimental observables it is extremely sensitive to errors in those observations when the differences are small. Here we address this issue directly by performing blind, replicate measurements in three laboratories. By monitoring within- and between-laboratory variability, we have determined the precision with which folding rates and -values are measured using generally accepted laboratory practices and under conditions typical of our laboratories. We find that, unless the change in free energy associated with the probing mutation is quite large, the precision of -values is relatively poor when determined using rates extrapolated to the absence of denaturant. In contrast, when we employ rates estimated at nonzero denaturant concentrations or assume that the slopes of the chevron arms (m_f and m_u) are invariant upon mutation, the precision of our estimates of is significantly improved. Nevertheless, the reproducibility we thus obtain still compares poorly with the confidence intervals typically reported in the literature. This discrepancy appears to arise due to differences in how precision is calculated, the dependence of precision on the number of data points employed in defining a chevron, and interlaboratory sources of variability that may have been largely ignored in the prior literature.

Keywords: -values; protein folding; stopped-flow mixing; FynSH3 domain

Abbreviations: GuHCl, guanidine hydrochloride

Article and publication are at http://www.proteinscience.org/cgi/doi/10.1110/ps.051870506.

	Introduction

TOP Abstract Introduction Results Discussion Materials and methods References

 
Since its introduction some 15 years ago (Garvey and Matthews 1989; Goldenberg et al. 1989; Matouschek et al. 1989), -value analysis has been applied with varying levels of completeness to more than two dozen proteins and has become the benchmark experimental method for characterizing folding transition states (Daggett and Fersht 2003). Recently, however, significant controversy has emerged over the precision that can be assigned to measures of this important experimental parameter (Sanchez and Kiefhaber 2003; Fersht and Sato 2004; Garcia-Mira et al. 2004; Settanni et al. 2005).

The controversy regarding precision stems from the following arguments: When G_U is plotted against RTln(k_f) (= G) for multiple mutations at a given position, the data cluster closely about a single line with a slope equal to the weighted of all of the mutations (Mok et al. 2001; Northey et al. 2002). Under these circumstances, however, the slopes of the lines connecting individual mutants, which correspond to the -values associated with specific substitutions, scatter about the slope of the best-fit line. Sanchez and Kiefhaber (2003) believe that this scatter reflects experimental error rather than real, context-dependent changes in . Based on this assumption they conclude that the significant variations observed when G_U is <7 kJ/mol indicate, in turn, that the -value reliability falls off rapidly below this cutoff. There appears, however, to be little direct evidence that experimental error dominates the observed scatter (Garvey and Matthews 1989). Indeed, it has been argued that the observed variations are dominated instead by real, mutation-specific changes in the folding mechanism (Fersht and Sato 2004).

In this paper we describe the results of a more direct test of the claimed relationship between -value reliability and G_U and also explore the relative merits of the various methods employed in the literature for calculating from experimental kinetic data. We have performed this study by employing blind, triplicate measurements of the folding of multiple mutants of the FynSH3 domain. We have used these measurements to determine the precision with which folding rates and are measured in our laboratories. The results of this study provide insights into the sources of variability that affect the precision of estimates under typical laboratory conditions using generally accepted laboratory practices.

	Results

TOP Abstract Introduction Results Discussion Materials and methods References

 
We have performed independent, triplicate measurements of the folding kinetics of the wild-type and seven-point mutations (at two sites) of the FynSH3 domain, a small, well-characterized two-state protein (Plaxco et al. 1998; Northey et al. 2002). To do so, the relevant proteins were expressed and purified in one laboratory and provided to the other two laboratories. Each laboratory was then assigned the task of collecting chevron curves for each protein under previously defined conditions (Maxwell et al. 2005) using the methods traditionally employed in that laboratory. No further guidance or instruction was provided, and thus the results presented here represent fully independent measurements. Of note, the methods employed in this study do not appear to differ in any significant, reported detail from the large majority of previously described protocols.

Cross-wise comparison of the 24 data sets we have obtained allows us to estimate the precision with which folding kinetics are typically determined in our laboratories. Moreover, given that a unique links any two pairs of folding and unfolding rates, irrespective of whether the -value so obtained is readily interpreted (Fersht and Sato 2004), these 24 data sets define 16 single- and 12 double-mutant -values per laboratory. We have used these to define the relationship between -value precision and G_U.

Chevron curve precision
We have defined experimental chevron curves for eight FynSH3 sequences (Fig. 1) using standard stopped-flow techniques. In order to judge whether the precision of these measurements is comparable to that of typical literature reports, we have evaluated the root-mean-squared errors in our chevron plots relative to those of 28 previously reported chevron curves (Maxwell et al. 2005) collected in 16 different laboratories (Fig. 2). In doing so we find that the mean, the median, and the maximum root-mean-squared residuals for our 24 individual chevron curves are smaller than the mean, the median, and the maximum errors in this large, diverse data set. It thus appears that the precision of our measurements compares favorably with typical literature values, suggesting that the magnitude of the experimental errors in our experiments may reflect what is typically encountered and reported on in the literature.

View larger version (25K): [in this window] [in a new window]   Figure 1. Chevron plots for the wild-type FynSH3 domain and the seven point mutants studied here. All three chevron fits are illustrated in each column, but the data points produced by each laboratory (white, gray, and black) are presented in separate columns for clarity. These data provide an indication of the precision with which folding and unfolding rates are measured in our laboratories under typical experimental conditions using generally accepted laboratory practices.

View larger version (24K): [in this window] [in a new window]   Figure 2. The precision of the kinetic measurements described here compares favorably with many previously reported in the literature. For example, the mean, the median, and the maximum root mean squared residuals associated with the 24 chevron curves we have determined (triplicate measurements of each of eight sequence, upper panel) are less than those of a set of 28 previously reported chevron curves determined in 16 different laboratories (lower panel) (Maxwell et al. 2005). Indicated is the precision with which the FynSH3 wild-type folding rate is defined by the previously reported data.

View larger version (25K): [in this window] [in a new window]   Figure 3. A comparison of the three sets of measurements described here. No systematic errors are observed in the scatter in the kinetic parameters ln(kf), ln(ku) (both extrapolations to zero denaturant), mf, or mu (the slopes of the folding and the unfolding chevron arms, respectively); no one research group systematically over- or underestimated any of these parameters. The error bars represent 95% confidence intervals for fits of the chevron data. The symbol scheme is as described in Fig. 1.

-Value precision
-Values can be defined from chevron data via the equation

(1)

where the prime mark (') denotes the relevant parameters for the mutant protein. At least three methods of estimating the relevant folding and unfolding rates for use in this equation have been reported in the literature. Perhaps the most straightforward and most commonly employed of these is to calculate from folding and unfolding rates extrapolated to zero denaturant conditions (we term this the "extrapolation method"). An advantage of the extrapolation method is that it makes no a priori assumptions about whether mutation can affect the slope of a chevron, nor does it require the perhaps arbitrary selection of nonzero denaturant concentrations at which to estimate rates. Conversely, however, the approach often relies on long extrapolations between the actual, experimental observations (necessarily collected at nonzero denaturant concentrations) and the estimated rates employed in the final determination. These extrapolations tend to degrade the precision with which rates, and thus , are determined. In order to avoid this potential difficulty, many groups have defined using folding and unfolding rates estimated at nonzero denaturant concentrations (we have arbitrarily picked 1 M and 5 M for k_f and k_u, respectively) or by assuming that m_f and m_u are fixed to a common value for all mutants (we term these approaches "nonzero" and "fixed-m", respectively). We have calculated -values using all three approaches and find that, when G_U is high, the nonzero and fixed-m methods produce more precisely defined -values than the extrapolation method. However, while we find that the -values of these mutations as derived using each of the three methods are closely similar, small but statistically significant deviations are observed (data not shown). This is not surprising given the differing assumptions that underlie the three methods. Moreover, given these differing assumptions, the fixed-m and nonzero approaches cannot be regarded as better approaches per se.

The relationship between -value precision and folding free energy changes
The precision of all three approaches for measuring depends significantly on the extent to which the probing mutation affects G_U. For example, using the extrapolation method (Figs. 4, 5, left) we find that the standard deviations (across three independent measurements) of the majority of our estimates rise >0.2 if G_U falls < ~7.5 kJ/mol (~1.8 kcal/mol). (We note, too, that the width of the 95% confidence intervals associated with this will be several times greater than this standard deviation.) Above the 7.5 kJ/mol cutoff, however, replicate measurements of are much more consistent. In contrast, the range over which reasonably precise -values can be determined is noticeably improved using the nonzero and fixed-m method (Figs. 4, 5, middle and right).

View larger version (14K): [in this window] [in a new window]   Figure 4. Shown here are the average Gu values and average -values obtained from independent measurements across three laboratories. The horizontal and the vertical bars present the standard deviations of the respective measurements. As indicated by the rapid increase in the size of the error bars on the left-hand sides of these plots, we find that the precision with which we can measure is reasonable when the estimated GU is high but becomes quite poor at lower GU. At larger GU the precision of estimates of is significantly improved when we employ the nonzero and fixed-m analysis methods, an observation that also holds for estimates of GU. Several data points are simply indicated with the symbol "x" at the top of the plot for clarity.

View larger version (12K): [in this window] [in a new window]   Figure 5. Shown are the observed experimental standard deviations of the 28 -values reported here as a function of the mean observed GU for each of the three analysis methods we have employed. To illustrate the differences in -value variability that result from the three different estimation approaches, a line using a scatter-plot smoother was added. For both the nonzero and the fixed-m approach, the standard deviation of the -values among the three laboratories drops below the arbitrarily chosen line at 0.2 at a value of GU of ~5 kJ/mol, while for the approach using extrapolation to zero denaturant, this value is ~7.5 kJ/ mol. The "x" symbols on the upper axis denote estimates that lie above = 2.

Systematic errors in analysis
The inability to measure precisely when G_U is low appears to hold across each of the three laboratories participating in this study. For example, pair-wise comparisons among the laboratories produce no evidence that any one group is systematically over- or underestimating , and none of the laboratories’s -values are systematically farther from the three-laboratory mean (e.g., Fig. 6; data not shown). An additional check for systematic variation is provided by data collected by Davidson and coworkers at the University of Toronto (UT) (Northey et al. 2002), who have previously characterized the folding kinetics of several of the mutants employed in this study. Using their data we have calculated "nonzero" -values for 10 of the 28 single- and double-mutant substitutions characterized here (Table 2). Despite the differing experimental conditions employed in the two studies, the individual data sets produced by our laboratories are well correlated with those calculated from Davidson’s data when G_U is large. Nevertheless, almost all of the estimates produced by the three laboratories differ significantly from the corresponding UT-derived values when G_U is lower (Fig. 6).

View larger version (13K): [in this window] [in a new window]   Figure 6. When GU is large, -values derived independently in each of our groups closely approach those calculated using previously reported data from an independent laboratory (UT) (Northey et al. 2002). In contrast, rather large deviations are observed when GU is smaller. The crossed bars indicate the mean of the values reported here. Otherwise, the symbol scheme is as described above (Fig. 1).

View this table: [in this window] [in a new window]   Table 2. Pairwise values

Potential sources of error in analysis

View larger version (32K): [in this window] [in a new window]   Figure 7. The precision of a single estimate of (i.e., based on kinetic data collected by a single laboratory in a single experiment) is approximately proportional to the square root of the number of kinetic observations employed to define the relevant chevron curves. For substitutions producing both large (12.5 kJ/mol, left) and small (2.6 kJ/mol, right) GU we used the fitted chevron curves and the estimated experimental errors to carry out a simulation study. Ten thousand new chevron curves were generated from each data set by picking a fixed number of equally spaced points on the original chevron curves and adding Gaussian noise with standard deviation equal to the observed experimental error. When plotted as histograms, the resulting 10,000 estimated -values illustrate the dependency of precision on both GU and the number of data points employed.

	Discussion

TOP Abstract Introduction Results Discussion Materials and methods References

 
We find that, using the practices and conditions generally employed in our laboratories we can estimate ln(k_f) and ln(k_u) with a precision of a few percent. , however, is a ratio of the differences between these experimental observables and thus is extremely sensitive to errors when the difference is small. Because of this, and despite the precision in ln(k_f) that we achieve, the standard deviation of our blind, triplicate -value measurements based on these extrapolations is poor unless G_U is rather large. In contrast, our ability to measure reliably is significantly improved when we employ folding and unfolding rates estimated at nonzero denaturant concentrations or when we adopt the assumption that m_f and m_u are fixed. We note, however, that the latter approaches reflect a trade-off. The improved precision is matched by a requirement to select somewhat arbitrary denaturant concentrations or on the potentially mechanistically unjustified assumptions that m_f and m_u are invariant upon mutation.

It appears universally accepted that estimates of become uselessly imprecise as G_U becomes arbitrarily small. The results reported here nevertheless appear to violate conventional wisdom, which typically places the reliability cutoff at 0.8–2.9 kJ/mol (0.2–0.7 kcal/mol) (e.g., Riddle et al. 1999; Hamill et at. 2000; Friel et al. 2003; Fersht and Sato 2004; Garcia-Mira et al. 2004; Settanni et al. 2005). Several potential sources for this discrepancy are apparent. First, because detailed description of the methods employed to estimate confidence intervals on are almost universally lacking in the prior literature, it is difficult to directly compare previous claimed levels of precision with those reported here. Furthermore, even assuming that proper error propagation has been employed in the literature (Zarrine-Afsar and Davidson 2004), such methods typically assume independent errors in ln(k_f) and ln(k_u) (Sanchez and Kiefhaber 2003). As described above, however, this assumption does not hold for our data sets, suggesting that the literature estimates of standard errors may, by ignoring this potentially important effect, underestimate the true experimental errors. Second, the precision of -values derived using data collected within a single laboratory depends not only on the magnitude of G_U, but also on the precision with which individual rates are measured, the position of the inflection of the chevron curve (see, e.g., V55G in Fig. 1), and the quantity of data employed to define a chevron. Also, as some previous reports employed larger data sets, more suitable mutations, and/or more stable proteins than those employed here, it is reasonable to assume that some prior studies have achieved better precision (for single, intralaboratory -value estimates) than that reported here. We also note, however, that even the most reproducible single-laboratory measurement can miss important interlaboratory effects, a source of additional variability that appears to have been ignored in prior estimates of precision.

Our results cannot be taken as proof of the impossibility of accurately determining whenever G_U falls below some arbitrary cutoff, and we do not mean to imply that any single cutoff will hold universally across all studies and for all applications. It is clear, for example, that the appropriate cutoff will depend at least in part on the degree of precision required for a given study and on both inter- and intralaboratory sources variability (such as the number of data points employed) that, at least in principle, can be controlled. The cutoff also will depend on the degree to which the kinetic m-values change upon substitution (with each analysis method having a different dependence on this effect) and the denaturant dependence of , if any. It is thus our hope that, instead of being considered a "one-size-fits-all" benchmark for -value analysis, our work will encourage the field to address the critical issue of -value precision with more rigor and completeness than has historically been the norm.

Last, the results reported here should provide some guidance for the comparison of simulation and experiment. Contemporary theoretical and computational methods have achieved a level of sophistication that allows them to predict -value patterns (Daggett et al. 1998; Alm et al. 2002; Clementi et al. 2003; Daggett and Fersht 2003; Ejtehadi et al. 2004; Garbuzynskiy et al. 2004; Marianayagam and Jackson 2004; Settanni et al. 2005). But even if a simulation is arbitrarily accurate, the correlation between predicted and observed -values will ultimately be limited by the error inherent in the experimental measurements. For this reason, the observation that is poorly defined for mutations that do not significantly alter G_U suggests that greater emphasis should be placed on the prediction of -values associated with large G_U (albeit keeping in mind the important caveats raised by Fersht and Sato [2004], who note that the nonconservative mutations required to generate large G_U may produce difficult-to-interpret values because the mutations affect multiple side-chain interactions simultaneously). This, in turn, suggests that confidence-weighted fits of predicted versus observed -values might be a more appropriate test of theoretical results than simple, unweighted correlations. More generally, the results reported here also suggest that experimental -values must be employed with care when used to validate simulation results or test theoretical models of folding.

	Materials and methods

TOP Abstract Introduction Results Discussion Materials and methods References

 
Wild-type FynSH3 was expressed and purified as previously described (de los Rios and Plaxco 2005). Four mutations at position 55 and three at position 28 (see Table 1) were generated using standard protocols and confirmed by DNA sequencing. The protein was expressed in BL21 cells, purified via nickel affinity chromatography, dialyzed, lyophilized, and used without cleavage of the His-tag. All eight constructs were expressed and purified in one laboratory and shipped to the two collaborative laboratories. All experiments were conducted in 50 mM phosphate (pH 7), 25°C (Maxwell et al. 2005). Of note, the stability of wild-type FynSH3 is effectively independent of pH over the range from 6 to 9 (data not shown).

Kinetic and thermodynamic measurements
Characterization of the eight proteins was performed in replicate, with each of three laboratories performing one replicate experiment. The replicates were conducted blind; no data were shared until after the relevant experiments were completed, and—save for defining consensus temperature, buffer, and pH—no discussion regarding methods was conducted. The values of composite parameters (parameters derived from two or more kinetic or equilibrium measurements, such as G_U and ) were determined independently from each laboratory’s data before being averaged.

The "white group" monitored folding using a pneumatically driven Applied Photophysics SX18MV stopped-flow fluorometer in pressure-hold mode. The protein was equilibrated in either 6 M GuHCl or in buffer for 1–2 h before measurements. Refolding/unfolding were initiated by dilution of these solutions into buffer with the appropriate concentration of GuHCl (solutions made volumetrically) at 25°C and monitored via fluorescence, with an excitation of 280 nm and emission detected >310 nm via a cutoff filter. At each GuHCl concentration, data from four to six experiments were fitted to single exponentials using the supplied software and the fitted rates averaged.

The "gray group" performed kinetic measurements of unfolding and folding using a -Star pneumatically drive stopped-flow reaction analyzer (Applied Photophysics) in the fluorescence mode. Refolding was initiated by 1:10 dilutions of the protein in 5.98–6.37 M GuHCl into buffer with appropriate concentrations of GuHCl. GuHCl solutions were made volumetrically by diluting commercial, 8 M stock (Sigma). Folding and unfolding were monitored via fluorescence, with excitation and emission at 280 nm and 340 nm, respectively. Data were collected between 0 and 5 sec in oversampling and pressure-hold modes. For each condition, a minimum of 6–10 kinetic traces were averaged and fit to a monophasic decay equation using a nonlinear algorithm supplied. A final protein concentration of 10 µM was used both in folding and unfolding experiments.

The "black group" measured folding and unfolding rates using a Biologic SFM 4 stopped-flow device coupled to a Fluoromax 3 fluorometer. Excitation was from 270–290 nm and emission was recorded from 330–350 nm. All samples were temperature-equilibrated for 10 min each time the syringes were reloaded. A three-syringe, two-mixer setup was employed, where syringe 1 contained buffer at 0 M GuHCl for refolding and 8 M GuHCl for unfolding; syringe 2 contained buffer at the concentration midpoint for denaturation; and syringe 3 contained either native or unfolded protein. All GuHCl concentrations were determined from the index of refraction of the solutions (Nozaki 1972). Reactions were initiated by diluting the protein 10-fold into various concentrations of GuHCl, which were determined by adjusting the relative volumes delivered by syringes 1 and 2. Five curves were recorded for each denaturant concentration and averaged. The resulting traces were fit to a single exponential decay using the program SigmaPlot.

Comparison with prior literature estimates of experimental chevron precision
In order to compare the precision of our rate data with the experimental precision typically obtained in the field, we calculated root-mean-squared errors, as residuals of ln(k_obs), for the 24 individual chevron plots we have obtained (one per lab per sequence) (Fig. 2) with those from previously published chevron data reported in Maxwell et al. (2005). These were defined as the square root of the mean residual squared error, obtained by fitting our data and the previously published data to chevron curves. Maxwell et al. describes the folding kinetics of 30 apparently two-state protein domains characterized in 18 different laboratories under experimental conditions to similar or identical those employed here. Omitted from our analysis were the proteins EC298 and U1A; only six T-jump data points are reported for the former and the latter exhibits significantly curved chevron arms.

Statistical analysis
Average estimates and standard deviations for blind triplicate measurements of folding and unfolding rates (extrapolated to zero denaturant) are reported (Table 1). Estimates and standard deviations are also reported for -values independently measured in each laboratory (i.e., using only that laboratory’s estimated folding and unfolding rates) (Tables 2,3).

View this table: [in this window] [in a new window]   Table 3. Change in folding free energies and the standard deviations of triplicate

	Footnotes

 
⁶ Present address: Department of Pharmacology and Toxicology, University of Texas Medical Branch, Galveston, TX 77555, USA.

	Acknowledgments

 
We thank Alan Fersht and Alan Davidson for extensive and very helpful critical comments regarding this paper. This work was supported by NIH funding via grants GM62868-01A2 (to K.W.P.), GM59663 (to P.W.-S.), and GM50945 (to S.M.), and by grant C-1588 from the Robert A. Welch Foundation (to P.W.-S.). I.R. was supported by a faculty innovation award from Johns Hopkins University.

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