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Ultrafast dynamics of protein collapse from singlemolecule photon statistics

Communicated by William A. Eaton, National Institutes of Health, Bethesda, MD, December 13, 2006 (received for review November 1, 2006)
Abstract
We use the statistics of photon emission from single molecules to probe the ultrafast dynamics of an unfolded protein via Förster resonance energy transfer. Global reconfiguration of the chain occurs on a time scale of ≈50 ns and slows down concomitant with chain collapse under folding conditions. These diffusive dynamics provide a missing link between the phenomenological chemical kinetics commonly used in protein folding and a physical description in terms of quantitative free energy surfaces. The experiments demonstrate the potential of singlemolecule methods in accessing the biologically important nanosecond time scales even in heterogeneous populations.
The discovery of proteins that fold rapidly in the absence of intermediates (1) has substantially advanced our mechanistic understanding of protein folding. The simplicity of their folding behavior allows thermodynamic and kinetic analyses that have led, e.g., to the characterization of transition states for folding (2) and the prediction of folding rates from native structure (3–5). However, because of their limited experimental accessibility, energetic and dynamic differences between the unfolded states of different proteins have usually been ignored. Similarly, the application of rigorous theories of protein folding in terms of statistical mechanics (6–8) has been hampered by our lack of understanding of structure and dynamics in the unfolded state ensemble. The role of unfolded states in determining protein folding mechanisms is thus largely unknown. Although considerable information about the overall dimensions and residual structure of unfolded proteins has been obtained from methods such as smallangle xray scattering and NMR, their dynamics in the submicrosecond range have largely eluded experimental determination. The importance of these time scales has become particularly obvious through the identification of proteins that fold in a few microseconds (9). In this regime, the free energy barrier to folding is assumed to be extremely low or even absent, and diffusive chain dynamics become the dominate factor in folding kinetics. Here, we determine these dynamics for an unfolded protein and investigate how they are affected by the collapse of the unfolded chain under nearphysiological conditions.
An ideal way to probe the dynamics of the heterogeneous ensemble of unfolded protein conformations is singlemolecule spectroscopy (10). The absence of averaging over many molecules allows spontaneous intramolecular distance fluctuations to be observed at equilibrium, without the need for perturbations to synchronize the ensemble. Förster resonance energy transfer (FRET) between two chromophores attached to the polypeptide chain has been suggested as an approach for investigating its submicrosecond dynamics (11, 12), but has eluded experimental implementation. Here, we use this method to directly probe the unfolded state dynamics of the cold shock protein (Csp) from Thermotoga maritima, a small, 7.5kDa βbarrel protein that exhibits twostate thermodynamics and kinetics (13–17). The protein was labeled terminally with a green fluorescent donor and a red fluorescent acceptor dye via amino and carboxylterminal cysteine residues, and freely diffusing molecules were observed in confocal singlemolecule experiments (Fig. 1). During the transit of the protein through the observation volume, its donor chromophore is excited by the laser beam. Depending on the distance r to the acceptor, energy transfer results with a rate that determines the relative probabilities of photon emission from donor and acceptor. Correspondingly, distance dynamics within the protein can be measured by fluctuations in the transfer efficiency and thus in the fluorescence emission of the chromophores (Fig. 1).
Results
Measurement of Unfolded State Dynamics.
In our experiments, we combine the separation of folded and unfolded subpopulations by singlemolecule spectroscopy with the high time resolution available from the photon statistics of a FRETcoupled dye pair. First, a transfer efficiency histogram is created from the photon bursts of individual molecules diffusing through the focus (15, 18). Fig. 2A shows an example at a concentration of the denaturant guanidinium chloride (GdmCl) of 1.4 M, where three subpopulations are resolved: folded protein molecules with a transfer efficiency E close to 1, unfolded molecules with E ≈ 0.5, and molecules lacking an active acceptor chromophore with E ≈ 0 (15, 19). A second synchronized counting card continuously records the time intervals τ between consecutive detected photons with picosecond time resolution. Dead times of detectors and counting electronics are avoided by using a Hanbury Brown and Twiss detection scheme (20–22). Under our conditions, normalized histograms of the resulting interphoton times are essentially equivalent to the donor or acceptor intensity autocorrelation functions g_{ii}(τ)(i = A, D) (23), which report directly on the time scales of fluctuations in fluorescence intensity (see Materials and Methods). We can obtain g_{ii}(τ) for each of the subpopulations shown in Fig. 2A by using only the interphoton times from molecules assigned to a single subpopulation.
The resulting intensity autocorrelation functions of donor [g_{DD}(τ)] and acceptor [g_{AA}(τ)] emission for every subpopulation are shown in Fig. 2 B–G. In all cases, g_{ii}(τ) exhibits a drop in amplitude at τ = 0. This photon antibunching is characteristic of individual quantum systems that cannot emit two photons simultaneously; it decays within a few nanoseconds (22, 24). More interestingly, pronounced photon bunching, i.e., an additional component in g_{ii}(τ) with positive amplitude, is observed in the 50ns range. The decay of this bunching signal in the unfolded subpopulation (E ≈ 0.5) is described by similar time constants for donor and acceptor autocorrelation functions (Fig. 2 C and F). The behavior of g_{ii}(τ) agrees with theoretical predictions for a flexible polypeptide chain (11, 12): if, for example, a donor photon is emitted at τ = 0, the chain ends are likely to be far apart at that instant, corresponding to a low rate of energy transfer, k_{F}. A very short time later, the ends will still be far apart, and the likelihood of emitting another donor photon will still be increased. However, at times much greater than the reconfiguration time τ_{r} of the chain, the molecule will have lost the “memory” of its initial configuration at τ = 0, and the probability of donor emission will be determined by the average transfer efficiency. In other words, we expect an increased autocorrelation of the emission intensity around τ = 0 that decays approximately on the time scale of chain reconfiguration.^{¶}
The acceptor autocorrelation function g_{AA}(τ) of the native subpopulation also exhibits a small, but significant, relative bunching signal (Fig. 2G). This signal may in part be caused by the imperfect separation of subpopulations (Fig. 2A), but the slower relaxation compared with the unfolded state could indicate an additional contribution from interactions of the chromophores with the specific environment presented by the native structure. To exclude an influence of native molecules on the determination of unfolded state dynamics, we restrict our measurements and analysis to the donor signal, where only unfolded molecules contribute to bunching (Fig. 2 B–D). The optimal signaltonoise ratio is obtained at a protein concentration of 500 pM, where a separation of subpopulations is no longer possible, but the determination of the correlation times is most accurate. An example for such a measurement is shown in Fig. 3B; the pronounced photon bunching yields an intensity autocorrelation time of 44 ± 3 ns. Under identical conditions, this signal is absent in stiff (19) polyproline peptides with the same average transfer efficiency as unfolded Csp (Fig. 2D), confirming that photon bunching is indeed caused by chain dynamics in the unfolded protein. The bunching signal is also absent in Csp labeled only with a donor chromophore (data not shown), excluding dye–protein interactions and rotational dynamics of the entire chain as origins of the bunching signal we observe. An additional slower component with a relaxation time of several microseconds was observed in all data sets, as expected from the tripletstate lifetimes of the chromophores (25). Fluorophore reorientation occurs on the time scale of ≈300 ps (19); resulting fluctuations in the energy transfer rate would thus affect the correlation functions only on time scales much shorter than the range relevant for the dynamics investigated here.
Theory and Analysis of Photon Statistics.
To analyze our measurements in terms of protein dynamics, we describe the relative motion of the chain ends as a diffusive process on the potential of mean force that corresponds to the endtoend distance distribution of the unfolded protein. By combining these diffusive chain dynamics with the distancedependent stochastic photon emission from the coupled dye pair (Fig. 1), the complete photon statistics of the system can be obtained (12). To avoid timeconsuming simulations, we use a recently developed theory (26) that allows us to calculate the intensity correlation functions of donor and acceptor emission numerically. Briefly, if we assume for the unfolded protein the endtoend distance distribution function of a Gaussian chain [which has recently been shown to be a good approximation for Csp (27)], p_{eq}(r) = 4πr^{2}(2π〈r^{2}〉/3)^{−3/2} exp(−3r^{2}/2〈r^{2}〉) (see Fig. 5), protein dynamics can be combined with the photophysics of FRET in the rate matrix: where D is the relative diffusion coefficient of the chain ends, K_{0}(r) describes the distancedependent kinetics of interconversion between the four electronic states illustrated in Fig. 1, and I is the 4 × 4identity matrix. The time dependence of all electronic and conformational transitions in the system is then described by the rate equation dp/dt = Kp, where p is the vector of the populations of the four electronic states in Fig. 1. By discretizing the diffusion operator, the problem is reduced to matrix algebra, and the intensity correlation functions can be calculated to high accuracy with numerical methods (for details, see Materials and Methods and SI Text). All parameters needed to define the model in terms of our system are known: the photophysical rate constants of the FRET process (Fig. 1) were measured independently, and p_{eq}(r) is defined uniquely by the mean square endtoend distance 〈r^{2}〉, which can be calculated from the average transfer efficiency of the unfolded subpopulation (15, 27) (see SI Text). We can thus determine the only remaining parameter, the effective endtoend diffusion coefficient D, by adjusting it such that the calculated intensity autocorrelation function fits the experimental result. Finally, the chain reconfiguration time τ_{r} (the decay time of the endtoend distance autocorrelation function) for a Gaussian chain is obtained from τ_{r} ≃ 〈r^{2}〉/6D (see SI Text).
In summary, we can thus determine, from the measured intensity correlation function, the dynamics of an unfolded protein in terms of the diffusion coefficient D and the corresponding reconfiguration time τ_{r}. Examples for calculated intensity correlation functions both with and without chain dynamics are shown in Fig. 3 A and C, respectively. The excellent agreement between the functional forms of experimental and calculated correlation functions (Fig. 3 A and B) suggests that our single reaction coordinate is a reasonable approximation for the dynamics of unfolded Csp. It is worth stressing that what we observe here are largescale chain dynamics along this reaction coordinate, whereas the wide range of subnanosecond dynamics known to occur in polypeptides (28, 29), such as dihedral angle rotations, essentially enter into the diffusion coefficient D (or the energetic roughness of the free energy surface; see The Free Energy Surface of Collapse and Implications for Protein Folding).
Effect of Collapse on Unfolded State Dynamics.
The unfolded state of Csp has been found to collapse in response to decreasing GdmCl concentrations (15, 17, 27, 30). This collapse precedes the folding reaction in kinetic experiments (17, 30), and singlemolecule FRET can be used to quantify the resulting change in chain dimensions under equilibrium conditions by virtue of the separation of folded and unfolded subpopulations (15, 27). Fig. 4A shows the corresponding decrease in the rms endtoend distance 〈r^{2}〉^{1/2} determined from the mean transfer efficiencies of the unfolded state (27). For an ideal chain, compaction is expected to lead to faster relaxation of intramolecular distances because of the reduced size of the accessible conformational space (Fig. 4B). This behavior has indeed been observed for some unstructured peptides (31) (however, also see ref. 52), but what happens in the case of a real protein? To answer this question, we measured the change in the decay time of the donor intensity autocorrelation function τ_{DD} in response to decreasing GdmCl concentrations (Fig. 4B). With the above analysis, we find a decrease in the viscositycorrected diffusion coefficient D_{η} from ≈0.5 nm^{2}/ns at 8 M GdmCl to ≈0.1 nm^{2}/ns at low GdmCl concentrations [corrected for change in solvent viscosity η_{s} with increasing denaturant concentration according to D_{η} = Dη_{s}^{−1} 1 mPa s(31–35)]. The corresponding chain reconfiguration time τ_{r} increases from ≈20 ns at 8 M GdmCl to ≈65 ns under nearnative conditions (Fig. 4B). In conclusion, we find very fast global chain reconfiguration in the unfolded state of Csp, but, concomitant with chain collapse at low denaturant concentrations, we observe a deceleration of these dynamics. This finding is opposite to the behavior expected for an ideal chain with invariant D_{η} (Fig. 4B), suggesting that interactions within the polypeptide chain, or “internal friction” (36), significantly affect unfolded state dynamics upon chain collapse, similar to what has recently been found for unstructured GlySerrepeat peptides (52).
The Free Energy Surface of Collapse and Implications for Protein Folding.
With the shape of the endtoend distance distribution p_{eq} and the effective endtoend diffusion coefficient D_{η} as a function of the denaturant concentration, we have obtained the two key parameters for describing the collapse of Csp in terms of a quantitative free energy surface. The change in D_{η} upon collapse can be expressed in terms of an effective energetic “roughness”, i.e., a distribution of small energy barriers caused by intramolecular interactions that slow down diffusion along the reaction coordinate. Possible physical origins of the dependence of D_{η} on denaturant concentration, as captured by the roughness, are changes in the packing and interaction strength of the polypeptide backbone and side chains in the unfolded state. These interactions may also contribute to the increase in the βstructure content recently observed upon collapse of unfolded Csp (27). Assuming a random amplitude with a Gaussian distribution independent of r, the rms roughness ε is given by ε = k_{B}T
Discussion
The free energy surface in the absence of denaturant (Fig. 5) summarizes the global structure and dynamics of unfolded Csp molecules under folding conditions. The rate of folding is determined both by the effective free energy barrier ΔG ^{‡} separating folded and unfolded states, and the “attempt frequency” for crossing the barrier from the unfolded state. In generalized transition state expressions for the folding time τ_{f} of type τ_{f} = τ_{0} exp(ΔG ^{‡}/k_{B}T), the reconfiguration time τ_{r} of the chain within the unfolded state potential is directly related to the inverse attempt frequency, or the preexponential factor τ_{0}. In a simplified Kramers description (39–41), where the curvature of the free energy surface and the effective diffusion coefficient are assumed to be similar in the unfolded well and on the top of the barrier, the preexponential factor is given by τ_{0} ≈ 2πτ_{r}. In the absence of a barrier, τ_{0} remains as an approximation to the “speed limit” (42), i.e., the minimum time scale of folding. With our value of 65 ns for τ_{r} of unfolded Csp under native conditions, largescale chain diffusion thus sets a lower limit of τ_{0} ≈ 0.4 μs to the folding time for a protein the size of Csp^{‖}, remarkably similar to estimates based on the length scaling of folding rates (9, 43). It remains to be clarified whether the decrease in the intramolecular diffusion coefficient we observe upon collapse (Fig. 4) continues toward the transition state region, as suggested from theory (44–47). The rapid folding observed for some proteins [folding times ranging from 0.7 μs for a variant of villin headpiece (48) to several microseconds (9)] suggests that this effect might not be large. This issue could be addressed with a combination of singlemolecule experiments of the type described here, and laser temperaturejump experiments, which have been used to determine the molecular time scale of ≈2 μs for the relaxation from the transition state region to the unfolded state in variants of the fivehelix bundle protein λ_{6–85} (49, 50).
Note, however, that the preexponential factor for Csp folding as approximated above is virtually invariant compared with the nine orders of magnitude decrease in its folding time τ_{f} from 8 to 0 M GdmCl (13). The large change in τ_{f} is thus almost entirely caused by a change in ΔG ^{‡}, supporting the assumption of a linear dependence of ΔG ^{‡} on denaturant concentration frequently made in the analysis of protein folding kinetics (2, 13).
Comparison to Peptide Dynamics.
Short unstructured peptides have been investigated intensively as model systems for the chain dynamics of real proteins, using ensemble lifetime fluorometry (51, 52) and quenching methods (32–35, 53, 69, 70). Extrapolating the rates of contact formation observed in quenching experiments of peptides to the length of the Csp chain yields values between ≈0.5 × 10^{6} s^{−1} and 5 × 10^{6} s^{−1}. For a direct comparison to our results, we estimate the endtoend contact formation rate for unfolded Csp assuming a Gaussian chain as k_{c} = 4πDa/(2π〈r^{2}〉/3)^{3/2} (54), where a is the contact distance. With our values for D and 〈r^{2}〉 (Fig. 4), and a = 0.4 nm (34, 52), we calculate contact rates of (0.8 ± 0.1) × 10^{6} s^{−1}, within the extrapolated values from peptide dynamics. Similarly, the endtoend diffusion coefficients between 0.04 and 0.2 nm^{2}/ns, obtained for peptides under a variety of denaturant concentrations (34, 52), are close to the values we find for unfolded Csp**. The similarity of the dynamic time scales observed in unfolded Csp and unstructured peptides, and the single exponential decay of our measured intensity correlation functions (Fig. 3C), indicate the absence of specific interactions and corresponding large energy barriers (55), even in collapsed unfolded Csp under native conditions.
Rate of Collapse.
Finally, we point out that, according to Onsager's regression hypothesis, or, more generally, the fluctuation dissipation theorem, small fluctuations decay on the average in exactly the same way as macroscopic deviations from equilibrium (56). In other words, the time correlation functions of the spontaneous endtoend distance fluctuations we observe in single molecules (Figs. 2 and 3) decay with the same time constants as the macroscopic signal would in ensemble perturbation experiments, such as rapid changes in solution conditions or laserinduced temperature jumps. The reconfiguration times measured in our experiments are thus equivalent to the collapse time of unfolded Csp. The similarity in time scale to the dynamics of unstructured peptides shows that the collapse of Csp is a purely diffusive, “downhill” process. Conversely, previous observations of denatured state dynamics in the microsecond range (57, 58) thus probably involve the crossing of substantial barriers.
Our measurement of the reconfiguration time of ≈50 ns is in good agreement with the relaxation time expected for an ideal chain (59–61) and theoretical estimates of the collapse times for proteins (43, 62). Additional evidence for the connection between reconfiguration dynamics and the collapse time comes from laserinduced temperature jump experiments on the 40residue protein BBL; the collapse of its aciddenatured state occurs on a time scale of ≈60 ns (63), remarkably similar to the reconfiguration time of Csp. Future experiments will have to address in more detail issues such as the dependence of collapse times on chain length, temperature, and other parameters (43). For BBL, the enthalpy change involved in collapse evidently is large enough to allow temperatureinduced perturbations of chain compactness. For a closely related Csp, laserinduced temperature jump experiments have not resulted in an observable signal (64), raising the possibility that its collapse does not involve a sufficient enthalpy change. Other perturbation methods, such as pressure jump and even the fastest mixing methods, are currently limited to time scales in the tens of microsecond range and above. Spontaneous fluctuations of individual molecules at equilibrium, however, are a universal property of sufficiently thermalized systems. Singlemolecule photon statistics thus provide a versatile approach for investigating the dynamics of macromolecules on the biologically important nanosecond time scales.
Materials and Methods
Samples and Instrumentation.
Labeled Csp and polyproline peptides were produced as described (15, 19, 27). All experiments were performed in 50 mM sodium phosphate buffer adjusted to pH 7, containing 0.001% Tween 20 to prevent surface adhesion of the polypeptides. Highpurity GdmCl solutions (Pierce, Rockford, IL) were used for denaturation experiments. Singlemolecule fluorescence was observed by using a PicoQuant MicroTime 200 confocal microscope equipped with a continuous wave solidstate diodepumped laser (Sapphire 488200; Coherent, Santa Clara, CA) operating at 488 nm (average radiant power at the sample: 100 μW), a 1.2 NA, ×60 microscope objective (UplanApo ×60/1.20W; Olympus, Melville, NY), and a 100μm confocal pinhole. A dichroic mirror (585DCXR; Chroma, Rockingham, VT) separated donor and acceptor fluorescence. Subsequently, each fluorescence component was divided randomly by a 50/50 beam splitter between a pair of two avalanche photodiodes (APDs; Optoelectronics SPCMAQR15; PerkinElmer, Wellesley, MA). Additional interference filters [HQ525/50 (Chroma) and 525AF45 (Omega Optical, Brattleboro, VT) for the donor APDs, 600HQLP (Chroma) and HQ640/100 (Chroma) for the acceptor APDs] completed spectral separation of the sample fluorescence and served to suppress the mutual detection of APD breakdown flashes in the infrared (65). Each of the two APD pairs was connected to a photon counter: the donor detectors to a PicoHarp 300 and the acceptor detectors to a TimeHarp 200 (both PicoQuant, Berlin, Germany). In their histogram mode, which we used for the measurements at 500 pM Csp and Pro20, the two input channels of these cards operate as start and stop channels. The measured time intervals Δt between start and stop were histogrammed in 256ps (PicoHarp) or 304ps (TimeHarp) bins. To avoid crosstalk between the two channels at short time intervals and simplify data analysis, an electronic time delay Δt_{0} was imposed onto the stop channel (SI Fig. 6). Mean transfer efficiency data for calculating rms endtoend distances (Fig. 4A) were taken from ref. 27. The dependence of the Förster radius R_{0} on denaturant concentration was determined by measuring changes in spectral overlap, donor quantum yield, and the refractive index of the solvent and was found to be dominated by the change in refractive index. For more details on data acquisition and analysis and the measurement of subpopulationspecific correlation functions (Fig. 2), see SI Text.
Analysis of Interphoton Time Distributions.
The histograms shown in Fig. 3 B and C (integration time 10 h) represent interphoton time distributions φ̃_{ii}(Δt) (i = A, D), which are, in the limit of very low mean photon detection rates τ_{i}^{−1}, proportional to the intensity autocorrelation function g_{ii}(τ) with the lag time τ = Δt − Δt_{0}. At higher count rates, shorter interphoton times are detected more frequently than longer ones. To correct for this pileup effect, we fitted the data to: where g_{ii}(τ) was approximated by assuming separation of time scales. A is an overall amplitude, exp(−Δt/τ_{i}) accounts for the pileup effect (compare Eq. 8), g_{AB}, g_{CD}, and g_{T} correspond to the contributions of photon antibunching, chain dynamics, and tripletstate dynamics, respectively, to the overall autocorrelation function g_{ii}(τ). For each sample, we determined the tripletstate correlation time τ_{T} from independent conventional fluorescence correlation spectroscopy measurements. Typical values were in the range of 1–4 μs. All remaining parameters of φ̃_{ii}(Δt) were determined by leastsquare fitting to the histogram data. The values obtained for τ_{i}^{−1} (≈10^{5} s^{−1}) are in good agreement with the mean photon count rates during a photon burst, i.e., during the passage of a molecule through the observation volume. The pileupcorrected correlation data shown in Fig. 3 were obtained by dividing each histogram value by A exp(−Δt/τ_{i}).
Calculation of Intensity Correlation Functions and Interphoton Time Distributions.
The intensity autocorrelation functions of fluorescence from donor and acceptor are defined as: where n_{i}(τ) is the fluorescence count rate (photon counts per unit time), which fluctuates because of the stochastic processes of photophysics and protein dynamics. The correlation functions are normalized such that g_{ii} (τ) → 1 as τ → 170). The distribution of the times between donor or between acceptor photons are denoted by φ∞. The distribution of the times between donor or between acceptor photons are denoted by φ_{DD}(τ) and φ_{AA}(τ) and are normalized according to ∫_{0}^{∞}φ_{ii}(τ)dτ = 1, i = A, D.
Recently, a general theory of these quantities has been developed (26). In this theory, the rate matrix K describes all electronic and conformational transitions in the system, and the matrix elements of the offdiagonal matrices V_{A} and V_{D} specify those transitions that result in the emission of acceptor or donor photons. Let p_{ss} be the vector of normalized steadystate probabilities obtained by solving Kp_{ss} = 0 (1^{⊤} p_{ss} = Σ_{i} p_{ss} (i) = 1, where 1 is the unit vector). The mean time between detected acceptor (donor) photons, τ_{i}, and the mean fluorescence count rate 〈n_{i}〉 (i = A, D) are then given by: (see equation 2.16 of ref. 26). The intensity correlations are: (see equation 2.18 of ref. 26). The interphoton time distributions are: (see equation 2.25 of ref. 26). The interphoton time distributions can be related to the corresponding intensity correlation functions if the mean time between photons τ_{i} is much longer (in our case on the microsecond time scale) than the relaxation time of the correlation function (in our case on the nanosecond time scale). Then, using the separation of time scales, the interphoton time distribution can be approximated as: We confirmed numerically that this approximation holds very well under the conditions of our experiments. Thus, the intensity correlation function can be obtained by multiplying the interphoton time distribution by an exponential (pileup effect correction; compare Eq. 2).^{††} Details of the theoretical model used for calculating g_{ii}(τ) and the procedures for comparing measured and calculated correlation functions are given in SI Text.
Acknowledgments
We thank Attila Szabo for guidance on theoretical issues, and William Eaton, Peter Hamm, Christian Hübner, Gerhard Hummer, and Rudolf Rigler for discussion and helpful comments on the manuscript. This work was supported by the Swiss National Science Foundation and the Human Frontier Science Program. I.V.G. was supported by the Intramural Research Program of the National Institutes of Health, National Institute of Diabetes and Digestive and Kidney Diseases.
Note Added in Proof.
Two papers (71, 72) published since submission of this manuscript are closely related to the present work.
Footnotes
 ↵^{§}To whom correspondence should be addressed. Email: schuler{at}bioc.unizh.ch

Author contributions: D.N. and B.S. designed research; D.N., I.V.G., A.H., and B.S. performed research; D.N., I.V.G., and B.S. contributed new reagents/analytic tools; D.N. and B.S. analyzed data; and D.N., I.V.G., and B.S. wrote the paper.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/cgi/content/full/0611093104/DC1.

↵¶ Because of the nonlinear dependence of the transfer rate k_{F} on distance, τ_{DD}, τ_{AA}, and τ_{r} are not strictly identical, but they can be related accurately using our analysis [see Theory and Analysis of Photon Statistics and supporting information (SI) Text]. In our range of distances and endtoend diffusion coefficients (Fig. 4), the three time constants differ by <15%.

↵‖ The folding time of 12 ms for labeled CspTm (15) results in a free energy barrier for folding of 10 kT at 0 M GdmCl (Fig. 5).

↵** The somewhat lower intramolecular diffusion coefficients compared to Csp reported recently for GlySerrepeat peptides (52) may be caused by the strong intramolecular hydrogen bonding that has been suggested to occur within these very compact peptides.

↵†† Note that the experimentally determined φ̃_{ii} of Eq. 2 and the calculated φ_{ii} of Eq. 8 are related according to φ̃_{ii}(Δt)=Ae^{−Δt0/τi}τ_{i}^{−1}φ_{ii}(Δt−Δt_{0}).
Abbreviations
 FRET,
 Förster resonance energy transfer;
 Csp,
 cold shock protein;
 GdmCl,
 guanidinium chloride;
 APD,
 avalanche photodiode.
 Received November 1, 2006.
 © 2007 by The National Academy of Sciences of the USA
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 Water pulls the strings in hydrophobic polymer collapse
 Configurationdependent diffusion can shift the kinetic transition state and barrier height of protein folding
 Dynamics of equilibrium structural fluctuations of apomyoglobin measured by fluorescence correlation spectroscopy
 Development of a technique for the investigation of folding dynamics of single proteins for extended time periods