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Probing Single-Molecule Protein Spontaneous Folding−Unfolding Conformational Fluctuation Dynamics: The Multiple-State and Multiple-Pathway Energy Landscape Zijian Wang and H. Peter Lu* Center for Photochemical Sciences, Department of Chemistry, Bowling Green State University, Bowling Green, Ohio 43403, United States S Supporting Information *

ABSTRACT: Protein conformational dynamics often plays a critical role in protein functions. We have characterized the spontaneous folding−unfolding conformational fluctuation dynamics of calmodulin (CaM) at thermodynamic equilibrium conditions by using single-molecule fluorescence resonance energy transfer (FRET) spectroscopy. We have identified multiple folding transition pathways and characterized the underlying energy landscape of the single-molecule protein conformational fluctuation trajectories, using a model analysis based on the detailed balance rate process principle. Our results suggest that the folding dynamics of CaM molecules involves a complex multiple-pathway multiple-state energy landscape, rather than an energy landscape of two-state dynamical process. Our probing single-molecule FRET fluctuation experiments demonstrate a new approach of studying spontaneous protein folding−unfolding conformational dynamics at the equilibrium that features recording long time single-molecule conformational fluctuation trajectories.

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INTRODUCTION Single-molecule fluorescence spectroscopy is powerful for studying complex biological processes, such as enzymatic reactions,1−10 protein folding dynamics,11 protein−protein and protein−DNA interactions,12,13 and cellular dynamic processes.14,15 Single-molecule fluorescence resonance energy transfer (smFRET) spectroscopy probes conformational changes of dye labeled individual biomolecules with a sensitivity down to about 1 to 8 nm spatial range and submillisecond temporal resolution,16−19 being a powerful approach to analyzing protein conformational dynamics. Over the past 50 years or so, there have been intensive studies on protein folding mechanism and dynamics.20−23 A widely accepted perspective suggests that a protein folding process is essentially a navigation on a funnel-shaped energy landscape toward a global energy minimum.24−26 This multidimensional energy landscape is typically rugged and complex involving multiple folding routes and metastable states. smFRET is powerful enough to selectively reveal the folded and unfolded protein conformational subpopulations from otherwise hidden in ensemble-averaged overall population distribution.18,21,22,27−29 Using smFRET, there are significant advances in analyzing the protein unfolded state dynamics,27,29−32 and protein folding dynamics.33−37 Generally, a twostate model with two distinct folded and unfolded conformations is sufficient to describe the protein folding dynamic processes. However, for multiple-domain proteins with more than 100 amino acids, multiple-state protein folding dynamics is © XXXX American Chemical Society

often involved beyond the typical two-state folding mechanism.38 CaM, a 148-residue protein responsible for intracellular calcium-sensing, plays a crucial role in a number of biological processes including cell signaling, muscle contraction, and energy metabolism.39,40 CaM has two globular domains, and the conformational dynamics of the domains has been under extensive study.41,44 Traditional methods, such as nuclear magnetic resonance and X-ray crystallography, have provided detailed insights into the mechanisms and dynamics of CaM conformational changes.40,42,43 However, the complexity of the protein dynamics, especially the conformational fluctuations involved in CaM biological functions, is still not fully characterized by using ensemble-averaged dynamic measurements or by static structural analyses alone. Single-molecule spectroscopy is capable of dissecting protein conformational fluctuations under physiological conditions in real time.41,44 In this paper, we report our work on probing the folding− unfolding conformational fluctuation dynamics of CaM under denatured conditions of denaturant guanidinium chloride (GdmCl) by using smFRET spectroscopy. We have identified the critical concentration of GdmCl at which single-molecule CaM undergoes spontaneous folding−unfolding conformational fluctuation with about equal probability of dwelling on Received: January 23, 2015 Revised: March 26, 2015

A

DOI: 10.1021/acs.jpcb.5b00735 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B the folded and unfolded conformational states. Thermodynamically, this titration midpoint condition is ideal for studying equilibrium spontaneous fluctuations without any external driving force.49,50 Recording single-molecule conformational fluctuation trajectories and analyzing equilibrium fluctuation dynamics, we are able to identify the nature of the CaM folding dynamics involving multiple pathways and multiple states. Using a dynamic model analysis, we have further identified the distribution of the folding transition pathways and the energetic features of the folding energy landscape of CaM.

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MATERIALS AND METHODS 1. Sample Preparation and Characterization. The CaM is mutated with cysteine residues on N-terminal domain at residue 34 and C-terminal domain at residue 110, and a FRET dye pair Cy3/Cy5 as donor−acceptor (D−A) is covalently tethered onto the protein on mutated cysteine residue 34 at the N-terminal domain and on mutated cysteine residue 110 at the C-terminal domain via thiolation reactions.46 In our experiment, the change of FRET efficiency (EFRET) ranges from 0.6 to 0.2 corresponding to the D−A distance change of about 5.0 to 6.7 nm in single-molecule protein conformation, and the Forster radius R0 of the Cy3-Cy5 pair is ∼5.4 nm at which EFRET of Cy3-Cy5 is 0.5. The samples for single-molecule conformational folding−unfolding dynamics measurements were prepared with a 1% agarose gel containing 99% of buffer solution (Type VII, Sigma). Single CaM molecules can rotate freely to perform biological functions and chemicals such as electrolytes and denaturant GdmCl can diffuse without interruption.12,46,48 We prepared samples of CaM in different concentrations of denaturant GdmCl in the mixture of 1 nM CaM, 1.25 μL, and oxygen scavenger with Trolox solution to obtain a 10 μL mixture of protein and denaturant solvent GdmCl solution. The Trolox solution was prepared by dissolving about 1 mM 6-hydroxy-2,5,7,8-tetramethylchroman-2-carboxylic acid (Trolox) in 1 mg/mL glucose oxidase, 0.8% D-glucose, and 0.04 mg/mL catalase in order to protect fluorescent dye from photobleaching or blinking due to the triplet state oxygen quenching as well as other photophysical processes.46 We then heated the 10 μL 1% agarose gel just above its gel-transition temperature (26 °C) and quickly mix the above protein solution with denaturant solution GdmCl and the gel between two clean cover glasses to form a sandwiched sample. All solutions were prepared with 4-(2hydroxyethyl)-1-piperazineethanesulfonic acid (HEPES buffer) at pH 7.4 with 1 mM EGTA.44 To probe the conformational dynamics of single-molecule protein at different concentrations of denaturant solvent, we carried out concentration-dependent experiments with different ratios of mixture in the sample (Supporting Information).46 2. Single-Molecule Imaging and FRET Measurements. We used the single-molecule photon-stamping spectroscopic approach to record smFRET D−A trajectories of CaM at different concentrations of denaturant solvent GdmCl (Figure 1A). Using this approach, we were able to record the emission photon time trajectories from both donor and acceptor with specific detection time for each detected photon. The experimental setup is an inverted confocal microscope (Axiovert 200, Zeiss) that uses a laser (532 nm, CW) as the light source for excitation.46,47 The laser beam focuses through a 100× oil immersion objective lens (1.3 NA, 100×, Zeiss) onto the upper surface of the coverslip after the excitation light is reflected up by a dichroic beam splitter (z532rdc, Chroma

Figure 1. (A) Single-molecule fluorescent experimental setup. It is an inverted confocal microscope (Axiovert 200, Zeiss) which uses a laser (532 nm CW) as the excitation light source.41,46 Fluorescence photons from donor and acceptor are both directed onto avalanche photodiodes to acquire emission images and FRET intensity trajectories. (B) Image obtained from Confocal Microscope of single CaM molecule. The left-hand side is the image obtained from the Cy3 donor channel and the right-hand side is the image obtained from the Cy5 acceptor channel. The bright spots are single molecules with diffraction limited (∼300 nm in diameter) images. Each image is obtained by laser focus raster-scanning and collecting fluorescence of Cy3-Cy5 D−A labeled single-molecule CaM.

Technology). To obtain confocal microscopy image, we used an x−y closed-loop piezo position scanning stage for raster-scan of the sample sandwich (Figure 1A). The fluorescence was collected through the same objective, and the FRET photon signal was split by a dichroic beam splitter (640dcxr, Chroma Technology) into two different wavelengths, 570 nm for the donor channel and 670 nm for the acceptor channel, with two Si avalanche photodiode single photon counting modules (SPCM-AQR-16, PerkinElmer Optoelectronics). A detailed description of the experimental setup has been reported previously.41,46 3. Statistical Analyses of Single-Molecule Intensity Trajectories. To analyze the smFRET signal fluctuation trajectories in order to obtain the CaM folding−unfolding conformational dynamics, we have applied a newly developed 2D regional correlation mapping analysis to analyze our singlemolecule photon-stamping trajectories.45,46 The detailed discussion and description of the analysis algorithm approach have been reported in our previous publications.45,46 Briefly, the 2D regional correlation mapping analysis calculates a twodimensional cross-correlation amplitude distribution and acts as a guide to find the anti-correlated D−A intensity portion of a selected single-molecule FRET signal fluctuation trajectory. In such an analysis, each of the smFRET D−A intensity trajectories are scanned with different starting and ending times, and the cross-correlation function amplitudes of each time segment with distinct starting and ending times are calculated. A color bar is used to indicate the amplitude of cross-correlation in order to locate the anti-correlated portion along a selected single-molecule trajectory with negative values of cross-correlation function amplitudes. B

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Figure 2. (A) Typical D−A signals and EFRET trajectories. Green and red lines indicate the donor and acceptor channel, respectively. (B) Singlemolecule EFRET trajectory calculated following formula 4. (C) Histogram distribution from EFRET trajectory gives the average EFRET of a certain singlemolecule. (D−F) Distribution of EFRET of different samples at different concentration of GdmCl; we carried out concentration dependent experiments of GdmCl at 0, 1, and 2 M. The decrease in mean EFRET values characterizes the unfolding of CaM molecules at higher concentrations of denaturant solvent.

intensity drifts as a result of local environment fluctuations.1,8,9,45 Our two-dimensional cross-correlation amplitude distribution analysis enables us to distinguish true anticorrelated D−A FRET signal fluctuation segments resulting from measured protein conformational motion of smFRET trajectories for further dynamics analyses from shot noises or averaged intensity drifts as a result of local environment fluctuations.45,46 We apply autocorrelation function and cross-correlation function calculations45,46 to analyze our single-molecule photon-stamping trajectories after the identification of each anti-correlated single-molecule time segments. The crosscorrelation and autocorrelation functions are defined as

Ccross(τ , tstart: tstop) =

∫t

tstop

IA(t )ID(t − τ ) dt

start

tstop

=

∑ IA(t )ID(t − τ) tstart

(1)

where IA and ID are the two-band photon count intensities signal of donor and acceptor, and tstart and tstop give the scanning window width. The cross-correlation functions are calculated with different tstart and tstop along a pair of smFRET trajectories {IA(t)} and {ID(t)}. Applying this method, we are able to identify the time segments along a specific two-band photon counting intensities signal of donor and acceptor trajectories with a significant anti-correlated cross-correlation indicated by a negative value of cross-correlation amplitudes. We use a cold color spot to indicate such negative values of cross-correlation amplitudes on a 2D regional correlation map. Typically, due to real experimental complexity, smFRET D−A intensity trajectories are dominated by shot noises or averaged

Ccross(t ) = = C

⟨ΔIA(0)ΔID(t )⟩ ⟨ΔIA(0)ΔID(0)⟩ ⟨(IA(0) − ⟨IA ⟩)(ID(t ) − ⟨ID⟩)⟩ ⟨(IA(0) − ⟨IA ⟩)(ID(0) − ⟨ID⟩)⟩

(2)

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Figure 3. (A) Part of single-molecule intensity−time trajectories of the donor and acceptor from the long trajectories shown in Figure 2A. (B) Autocorrelation functions of the donor (green) and the acceptor (red) calculated from single-molecule fluorescence intensity trajectory shown in (A). Time interval at which we calculated the autocorrelation functions and cross-correlation function is indicated by the blue frame. The fitted single exponentials (green and red) yield fluctuation rate kdonor = 2.2 ± 0.7 s−1 and kacceptor = 3.1 ± 0.6 s−1. (C) Part of calculated EFRET trajectory by using formula 4. (D) Cross-correlation function from the single-molecule intensity−time trajectories of the donor and acceptor shown in (A). The fitted single exponentials (black) yield fluctuation rate kcross = 2.0 ± 0.2 s−1 which is similar to the fluctuation rates captured by autocorrelation functions of the donor and the acceptor. (E) Distribution of EFRET measured in (C) yields a mean EFRET = 0.42 ± 0.04. (F) Result of analysis of the singlemolecule donor−acceptor fluorescence trajectories shown in (A). The cold color shows that the D−A intensity fluctuation is anti-correlated, whereas the warm color shows that the D−A is correlated.

Cauto(t ) = =

spot features due to single CaM molecules confined in agarose gel. Single CaM molecules embedded in agarose gel can rotate freely to perform biological functions and chemicals such as electrolytes and denaturant GdmCl can diffuse without interruption.46,48 From each specific single-molecule CaM imaging spot, we are able to obtain continuous D−A fluorescence intensity trajectories (Figure 2A). Typically, we gather a number of single-molecule fluorescence intensity trajectories at different concentrations of denaturant GdmCl, and we calculate the EFRET of each of the single-molecule fluorescence intensity trajectories measured using eq 4 (Figure 2B).

⟨ΔIA(0)ΔIA(t )⟩ 2

⟨ΔIA(0) ⟩ ⟨(IA(0) − ⟨IA ⟩)(IA(t ) − ⟨IA ⟩)⟩ ⟨(IA(0) − ⟨IA ⟩)2 ⟩

(3)

where IA(t) and ID(t) represent acceptor and donor intensities, and ⟨IA⟩ and ⟨ID⟩ are the means of the intensity trajectories, respectively. Ccross(t) and Cauto(t) are cross-correlation and autocorrelation functions.

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RESULTS AND DISCUSSION Single-Molecule FRET Trajectories Monitored Unfolding of Single CaM Molecules. Figure 1B shows typical images from our smFRET imaging microscopy using an inverted confocal microscope. By raster-scanning the sample, a 20 μm × 20 μm sample image yields bright spots with 300 nm resolution of single molecules. We attribute that to the imaging

E FRET(t ) =

IA(t ) IA(t ) + ID(t ) ×

ϕA × ηA ϕD × ηD

(4)

where ϕA and ϕD are the emission quantum yields of acceptor and donor dyes, respectively, and ηA and ηD are the acceptor D

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Figure 4. (A) Contour plot of EFRET vs fitted autocorrelation function correlation rate. We see as the proteins become gradually unfolded, the autocorrelation fluctuation rate of the protein dynamics becomes slower by a factor of 100. We construct this contour plot by calculating the autocorrelation function at a specific time interval along a trajectory and compute the EFRET at this time interval (Figure 3A,C,E). These two parameters, autocorrelation function fluctuation rate and EFRET, correspond to one point on the contour plot, serving as a characteristic value to represent a local fluctuation dynamic behavior. (B) Distribution of fluctuation rate of correlation functions at various concentrations of denaturant solvent. The autocorrelation fluctuation rates are 18 ± 10 s−1, 9 ± 11 s−1, and 3 ± 3 s−1 at 0 M GdmCl (blue), 1 M GdmCl (black), and 2 M GdmCl (red), respectively.

The EFRET distribution measured under the condition of no GdmCl, at which CaM molecules are in folded native states, has a mean of 0.55 ± 0.07 that corresponds to 5.2 ± 0.2 nm D−A distance, a folded conformation state; whereas the distribution of EFRET measured under 2 M GdmCl has a mean of 0.36 ± 0.17 that corresponds to 6.1 ± 0.8 nm D−A distance, an unfolded conformation state. On average, a decrease in EFRET corresponds to a distance change between the D−A Cy3-Cy5 dye pair which further indicates a conformational distance change along the FRET coordinate. Therefore, under the unfolding effect of denaturant GdmCl, we measured the distance change of the D−A Cy3-Cy5 dye pair to be 0.9 ± 1.0 nm for the folded and unfolded states of CaM. In our experiments, single-molecule CaM is embedded in 1% agarose gel in which a single CaM molecule can rotate freely to perform its biological functions, and chemicals, such as electrolyte and denaturant GdmCl, can diffuse without interruption.46,48 2. Autocorrelation Analyses of CaM Folding−Unfolding Conformational Fluctuation Dynamics. We analyze the fluctuation dynamics of our single-molecule FRET trajectories by calculating the autocorrelation functions of selected segments using our 2D regional correlation mapping analysis.45,46 By applying the 2D regional correlation mapping analysis, we identify specific anti-correlated segments along a trajectory and zoom into that segment to study detailed dynamic fluctuation information by calculating autocorrelation and cross-correlation functions. Fitting the autocorrelation functions with single exponential functions, we are able to characterize the rate of protein conformational fluctuation (Figure 3).45,46 The correlation rate of protein conformational fluctuation measured by single-molecule D−A fluorescence intensity fluctuation trajectories gives a broad distribution due to the local environment heterogeneity, thermal fluctuations, dynamic disorder, and static disorder (Figure 4).8−10 At low concentration of GdmCl, the autocorrelation function analysis of single-molecule trajectories of CaM molecules shows a typical autocorrelation rate that corresponds to the millisecond time scale, the characteristic time scale of native state protein motions.8−10,45,46 Nevertheless, Figure 4B shows significantly distinct distributions of autocorrelation rates between different individual CaM molecules under different concentration of

and donor detection efficiencies, respectively. Here the correction factor (ϕA × ηA)/(ϕD × ηD) is ∼1 in our experiment conditions. The histogram distribution from EFRET trajectory gives the average EFRET of single molecule (Figure 2C). To identify the specific condition that facilitates a measurable probability of CaM staying in either folded or unfolded state, we have characterized the folding and unfolding state distributions by single-molecule fluorescence intensity trajectory measurements of EFRET under different denaturant conditions (Figure 2D−F). The decrease in mean EFRET values of the single-molecule CaM EFRET distributions is due to the increased unfolding probability of the single CaM molecules under increased denaturant GdmCl concentrations from 0 to 2 M. The EFRET distribution consists of both the folded subpopulation with EFRET 0.6 and unfolded subpopulation with EFRET 0.2. Single-molecule FRET spectroscopy allows folded and unfolded molecules to be distinguished on the basis of the significant distance dependence of energy transfer between smFRET donor and acceptor dyes. More importantly, single-molecule subpopulations of partially folded with equal amount of time in folded and unfolded conformational states can be thoroughly examined by studying detailed fluctuation dynamics of each single-molecule fluorescence intensity trajectory. We locate and record singe-molecule trajectories from individual molecules undergoing spontaneous folding− unfolding conformational fluctuation under the condition providing roughly 50%/50% folding−unfolding conformational state probability, i.e., the titration midpoint. At the titration midpoint, approximately half of the population of CaM is in the folded conformational state, whereas the other half of the population is in the unfolded conformational state. Our ensemble-averaged titration experiment of Cy3-Cy5 D−A labeled CaM molecules in solution with different concentrations of GdmCl yields a 2 M concentration of denaturant GdmCl as the titration midpoint where the folded and unfolded conformational states of CaM molecules are equally populated (Supporting Information). In terms of single-molecule experiments, such a titration midpoint condition is ideal for studying spontaneous folding−unfolding conformational fluctuation dynamics, since individual molecules undergo spontaneous conformational fluctuations without an external driving force. E

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GdmCl increases, resulting in denaturing of single-molecule CaM which turns single-molecule CaM into random coils. From autocorrelation fluctuation conformational dynamic analyses, the mean autocorrelation fluctuation rates are 18 ± 10 s−1, 9 ± 11 s−1, and 3 ± 3 s−1 at 0 M GdmCl, 1 M GdmCl, and 2 M GdmCl, respectively. We attribute the significantly different fluctuation dynamics to the CaM folding−unfolding conformational states and fluctuation dynamics under the different denature GdmCl conditions, which create heterogeneous local environments. In 2 M concentration of GdmCl, CaM molecules denature into random coils without ordered secondary and tertiary structures, and the conformational dynamics probed by autocorrelation function reveals slow fluctuation dynamics at the rate of 3 ± 3 s−1 which is a result of loss of regulated protein motion on the millisecond time scale, as is expected for native state protein. The distribution of fluctuation rates also becomes narrower as the single-molecule CaM unfolds in 2 M denaturant GdmCl. The narrower distribution of conformational fluctuation rates is related to protein conformational space sampling speed which is an essential part of various recognition processes involved in biological function.8−10,19,41 Under the conditions with or without the denaturant GdmCl, the conformational fluctuation rate can be calculated from the fitted exponential function of autocorrelation function, and essentially the same fluctuation rate of autocorrelation functions of the donor and the acceptor in both autocorrelations and cross-correlation indicates that the fluctuations are from the same origin, the single-molecule CaM folding− unfolding conformational dynamics FRET (Figure 3B,D). We have analyzed about 130 different FRET trajectories of individual molecules under conditions of folded and unfolded states to obtain the distribution of the single-molecule CaM conformational fluctuation rates and the correlated FRET efficiency (Figure 4A). The distributions in Figure 4B show a significant change in the distribution broadness under different conditions of folded and unfolded single-molecule CaM: The distribution is narrower for the unfolded states and broader for the folded states. Such changes in the conformational fluctuation rates of single-molecule CaM folded and unfolded states clearly reveal how conformational fluctuation rates correlated with EFRET. Higher EFRET associates more folded protein with faster conformational fluctuation rates, and lower EFRET associates more unfolded protein with slower conformational fluctuation rates (Figures 2 and 4). To further understand the folding−unfolding conformational fluctuation dynamics probed by autocorrelation function analyses of single-molecule D−A fluorescence intensity trajectories measured at the folding−unfolding titration equilibrium conditions, we use a kinetic model to account for the single-molecule protein CaM folding−unfolding conformational fluctuation dynamics.9,49,50 Conventional kinetics experiments measure the relaxation of concentration of a large ensemble of molecules after a perturbation (such as fast mixing or a temperature jump). In contrast, single-molecule experiments measure the probability at specific states of an individual molecule conformational fluctuation, and the dynamic information can be extracted by measuring single-molecule spontaneous fluctuation dynamics at equilibrium, specified by the Onsager’s regression hypothesis and the fluctuation dissipation theorem.50−52 The fluctuation dissipation theorem dictates that the relaxation of macroscopic nonequilibrium disturbances is governed by the same laws as the regression of

denaturant GdmCl. The CaM molecules, in native states, show more significant autocorrelation rates on the millisecond time scale. However, the significant correlation rates diminish for unfolded CaM single-molecules, which is due to the unfolding of CaM molecules turning to random coils under the denaturant GdmCl condition; and the correlation fluctuation rates of single-molecule protein regulated conformational motion become slower. The two-dimensional regional crosscorrelation analyses help us to identify time segments along each single-molecule intensity trajectory with significant anticorrelations (Figure 3F). We zoom in to study each of the anticorrelated time segments by calculating the cross-correlation and autocorrelation, respectively. The fluctuation dynamics of correlation function analyses yields fluctuation rates of conformational dynamics of single protein molecules.9,10 Since the fitted exponential function of the autocorrelation function has a similar fluctuation rate of donor and acceptor channels, which indicates that autocorrelation comes from the same source, the single-molecule protein conformational fluctuations (Figure 3B,D). The fitted exponential function of the cross-correlation function yielding a similar fluctuation rate further confirms the anti-correlated cross-correlation function between the donor and acceptor fluctuation intensity trajectories with essentially the same correlation rate within the error bar (Figure 3D), a typical anti-correlation FRET D−A trajectory fluctuation dynamics showing a protein conformational motion measured by anticorrelated smFRET D−A intensity trajectories. Such protein conformational motion captured by smFRET gives anticorrelated two-band D−A fluorescence intensity trajectories (Figure 3A). Both autocorrelation functions from the donor and acceptor signal trajectories and the anti-correlated cross correlation function between the donor and acceptor signal fluctuation trajectories have essentially the same fluctuation rate, which strongly indicates that the fluctuations are dominated by the protein folding−unfolding conformational fluctuation probed by the smFRET D−A signal fluctuation trajectory measurement (Figure 3). We focus on the autocorrelations from the donor channel only, and we plot the fitted exponential autocorrelation function rates at various concentrations of denaturants GdmCl in which the singlemolecule protein CaM undergoes different conformational fluctuation dynamics as a result of different local environments. In the time interval at which the autocorrelation functions are calculated (Figure 3A, 3C blue frame), we also compute the EFRET (Figure 3A,C,E), and these two parameters, autocorrelation function fluctuation rate and EFRET, serve as characteristic values to represent local fluctuation dynamics. The twodimensional contour plot of EFRET vs the fitted autocorrelation function correlation rate is shown in Figure 4A. As the single CaM molecules get unfolded, not only does the EFRET decreases, but the fluctuation rate of the autocorrelation function also decreases. Quantitatively, the fitted exponential functions obtained from fitting the single-molecule FRET intensity trajectory donor autocorrelation functions give the fluctuation rates from different single-molecule trajectories and distinct individual molecules at different concentrations of denaturant GdmCl. Single molecules experience different local environments and undergo distinct conformational fluctuation dynamics probed by autocorrelation function analyses. Figure 4B, the distribution of these various autocorrelation fluctuation rates, shows a significant shift of the distribution as the concentration of F

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Figure 5. (A) Donor intensity trajectory obtained from our single-molecule confocal microscopy with 1 ms binning time. The black dotted line indicates the threshold criterion separating the on- and off-time. (Inset) Zoomed-in trajectory portion with marks illustrates the reading of the ontimes and off-times. (B) Histogram of such a trajectory yields two peaks separating the “on” and “off” times which in our experiment correspond to folding and unfolding waiting time. Subsequently, distributions of such “on” and “off” can be constructed. (C) The on-time distribution is a gamma shaped distribution representing a multistep dynamic scheme. It is different in shape from both the exponential distribution resulting from the twostate dynamic scheme and the Gaussian distribution. Simulated data (red line) from a multistep Markovian dynamic process which facilitates a onedimensional random walk type of conformation diffusional process.

the conformational fluctuations of single proteins under spontaneous detailed balanced folding−unfolding conformational fluctuations. By probing autocorrelation function fluctuation rates from single-molecule D−A intensity fluctuation dynamic trajectories, we are able to directly probe and monitor protein conformational fluctuation dynamics rates. 3. Nonexponential Distribution of Folding Waiting Time Indicates Multiple Folding Intermediates. Subtle conformational dynamic signatures are straightforwardly analyzed by the distribution of on-times and off-times of mFRET D−A intensity fluctuation trajectories.8,9,19,53−59 The on-time and off-time are the “waiting time” for the CaM folding and unfolding conformational dynamics, respectively. Such ontime and off-time or “waiting time” correspond to the dwelling time of protein folding−unfolding conformational states (Figure 5A, inset). The nature of single-molecule CaM folding dynamic processes lies in the unique feature of such “waiting time” distributions. To characterize such detailed dynamic behavior, we further bin our data in the smFRET intensity trajectories to 1 ms bin (Figure 5A). The histogram of donor intensity trajectory yields a distinct two-peaked occurrence distribution (Figure 5B). The peak corresponding to the high photon counts is the unfolded state of single-molecule CaM, while the low photon count peak represents the folded state of single-molecule CaM. To obtain a folding waiting time distribution, we set up a threshold, at which the waiting times of the folding and unfolding states are separately identified and read out. For the trajectory shown in Figure 5, we choose the value 10 as the threshold value, and the subsequent folding waiting time distribution is shown in Figure 5C.9 Those very short unfolding events that last less than three binning times of 3 ms total cannot be reliably differentiated

spontaneous microscopic fluctuations in an equilibrium system.50−52 For a two-state spontaneous fluctuation dynamic model, the autocorrelation function directly probes the fluctuation dynamic process under detailed balance by reflecting the summation of forward and backward reaction rate kf + kb as the fluctuation rate of fitted exponential function of autocorrelation. We generalize this argument by expressing a multistep fluctuation dynamic process by separately modeling the forward and backward reaction rates kf and kb by dividing the whole dynamic process into forward and backward half reactions.49−52 For a three-state kinetic scheme, the mean first passage time of the rate process, or reaction, is calculated by the flux method under detailed balance rate processes.49 We express the observed forward and backward reaction rates by a polynomial of all reaction rates involved. k1

k2

k −1

k −2

U HooI I1 HooI F

where k1 and k−1 are the forward and backward reaction rates of step-one reaction, and k2 and k−2 are the forward and backward reaction rates of step-two reaction. In such a case, by assuming k−2 to be small, we derive the expression of the mean first passage time of the forward reaction: ⟨t⟩ = (k−1 + k2)/k1k2 + 1/ k2. By further assuming k−1 equals zero, we get the mean first passage time of the forward half reaction ⟨t⟩ = 1/k1 + 1/k2. The reciprocal of ⟨t⟩ gives us the observed forward reaction rate kf = (k1k2)/(k1 + k2). By the same argument, the observed backward reaction rate is kb = (k−1k−2)/(k−1 + k−2). For a two-state dynamic scheme, the autocorrelation function is C(t) = e−(kf + kb)t.9 For a generalized three-state (or easily generalized n-state) dynamic scheme, the observed kf and kb is a polynomial of the rate constant of each step.49−52 It is clear that the autocorrelation function analysis is a capable approach to probe G

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mean value of the original nonexponential folding waiting time distribution is 13.2 ± 9.0 ms (Figure 5C, distribution). The simulation data yields mean value 13.0 ± 7.2 ms (Figure 5C, fitting line) which reproduces the original distribution well. The simulated distribution involves two Poisson rate processes which indicate a two-step three-state dynamic process (Figure 5C).

from the measurement of photon counting shot noise and are not counted as unfolding events.19,59 Noticeably, the folding waiting time distribution is nonexponential resulting from a non-two-state folding− unfolding dynamic model.8,9,19 The distribution is also distinct from a Gaussian distribution and has a broad coverage of time scales.9 The broad folding waiting time distribution, reflecting the heterogeneity, is typically associated with complex protein conformational dynamics involving multiple states and multiple pathways. This distinction clearly rules out the possibility of two-state folding−unfolding dynamics for single-molecule CaM. The gamma distribution shaped folding waiting time distribution indicates a more complex dynamic mechanism involving a convolution of multiple Poisson processes of the folding−unfolding conformational fluctuations associated with multiple intermediates and multiple steps, which suggests a multiple-state and multiple-pathway funnel-shaped folding energy landscape involving transition states, metastable states, and misfolded states.59,69 Detailed spontaneous conformational fluctuation dynamics measurements yield such distinct statistics of folding waiting time distribution which, from a dynamic perspective, is likely associated with a multistep Markovian dynamic process.59 Our detailed analysis of the folding waiting time distribution confirms a multiple folding pathway with multiple intermediates. Compared to a random search for the folded native state, our single-molecule spectroscopic analysis reveals that the CaM folding process is much more complex with multiple folding pathways running parallel with one another. Opposite to the one-dimensional reaction coordinate which is considered in most two-state chemical reaction modeling of protein folding, the single-molecule CaM folding process involves a folding network without a rate-limiting step.59 4. Model Analyses of Conformational Dynamics and Energy Landscape of Single-Molecule CaM Folding. We attribute the nonexponential folding waiting time distribution to the existence of multiple folding intermediates.9,19,59,60 To further characterize this multiple intermediate state dynamics, we exploit a one-dimensional random walk model, which has been used successfully in the modeling of a multistep singlemolecule enzymatic reaction.19,59 The basic approach is as follows: without prior knowledge of the shape of the energy landscape, we assume a uniform rate k to each step of the overall rate process involved in single-molecule CaM folding− unfolding conformational dynamics. Each step of the onedimensional random walk model is considered to be a Poisson process modeling the single-molecule protein folding intermediate conversion process. The convolution of several Poisson processes with uniform rate k gives a gamma function shaped distribution (eqs 5−7) to model the nonexponential non-Gaussian shaped folding waiting time distribution of single-molecule CaM folding−unfolding conformational dynamics. This distribution reproduces the mean and standard deviation of the original folding waiting time distribution of single-molecule CaM folding−unfolding conformational dynamics. The number of Poisson process steps modeling the single-molecule protein folding intermediate conversion process involved in this convolution calculation gives an estimation of how many Poisson rate processes are present in the overall rate process of single-molecule CaM folding− unfolding conformational dynamics. This number is the lower bound estimation of the number of intermediates involved in the dynamics process of protein folding. Quantitatively, the

P(t ) = A[exp( −t /τ )]

(5)

where P(t) is the probability distribution of the Poisson rate process step times, τ is the averaged Poisson rate process step time, and A is the distribution weight constant. The Poisson rate process step time is the duration between two adjacent states, and it is different from the formation time of the intermediate states or folded state of single-molecule CaM folding−unfolding conformational dynamics. In our model the folding waiting time distribution is the convolution distribution of Poisson rate process step times. To calculate the convolution of function f(t) and g(t), the integration equation is (f *g )(t ) =

∫0

t

f (v )g (t − v ) d v

(6)

Based on eq 6, consecutive intermediate steps involved in a two-state dynamic model of folding−unfolding conformational fluctuation are expressed as a convolution of two consecutive exponential waiting-time distributions. The general probability function involving an arbitrary number of folding intermediates is deduced to be P(Tn) = An

t n − 1[exp(−t /τ )] (n − 1)!

(7)

where n (1, 2, 3, ..., N) is the index of the intermediate steps; τ is the mean formation time of a folding intermediate through a single-step process; and A is the normalizing factor of this probability distribution. To further specify the shape of the energy landscape, we carry out a more detailed dynamic analysis by calculating the conformational diffusion coefficients of single-molecule CaM folding−unfolding conformational dynamics. Via dynamic modeling, the folding waiting time distribution gives the conformational diffusion coefficients.9 Briefly, we model the folding−unfolding conformational dynamics of single-molecule CaM as a classical particle one-dimensional multiple-step random walk in the presence of a force field, and the position distribution density function can be calculated by the Master equation.19 Following derivation of our previous work, the conformational diffusion coefficient of the single-molecule CaM folding−unfolding dynamic process is D=

( ⟨Δtunfold 2⟩ ⟨XN (t )⟩)2 2⟨tunfold⟩3

(8)

where D is the conformational diffusion coefficient. The mean unfolding time, ⟨tunfold⟩, and the standard deviation of the unfolding time distribution, (⟨Δtunfold2⟩)1/2, are directly measured in our experiment. The total drifting distance of the folding−unfolding conformational motion, ⟨XN(t)⟩, is associated with the folding−unfolding conformational distance change. From EFRET distribution data, we already obtain the mean conformational drift distance XN(t) to be 0.9 ± 1.0 nm. Using eq 8,19 we calculate the diffusion coefficient D of this two-step dynamic process to be 1.4 × 10−13 cm2/s. The H

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Figure 6. (A) We postulate a schematic representation of the protein folding process along one specific pathway based on our single-molecule spectroscopy measurements. The nuclear coordinate Q is a projected one-dimensional coordinate of a three-dimensional funnel shaped energy landscape. We estimate the roughness of the potential energy surface and the free energy barrier height based on dynamic modeling. The roughness of the potential surface 3.8 ± 0.8kT is likely to be entropic traps which are the thermodynamic features of multiple folding intermediates observed in the folding−unfolding dynamic process of CaM. The 11kT obtained from Kramer’s theory of barrier crossing is likely to be the empirical overall free energy barrier height. (B) Distribution we constructed with the number of steps involved in each of the dynamic process vs the conformational diffusion coefficient. The vertical axis is occurrence. (C) We extract the distribution of folding pathways of single CaM molecules by measurements of spontaneous conformational fluctuations at folding−unfolding thermodynamic titration equilibrium. Different single-molecule CaM protein folding pathways are labeled with different colors with distinct probabilities. The two-step folding dynamic process involving one intermediate labeled with red has a probability of 23% among all single-molecule protein folding−unfolding conformational fluctuation dynamic analyses. Different folding dynamic process involving multiple intermediate labeled with black, purple, and blue have a probability of 17%, 33%, and 21%. (D) Conceptual representation of the single-molecule CaM protein folding−unfolding conformational dynamic process measured by our single-molecule FRET. In general, a multiple-state folding dynamic process involving more than one intermediate is observed.

τlimit of 148 amino acid CaM is 3.0 μs. Since the CaM polypeptide chain is more like a heteropolymer than a homopolymer, there are additional factors to be taken into account while describing protein collapse. The estimation of the theoretical upper bound of single-molecule CaM folding time τlimit based on the homopolymer collapse theory is essentially accurate.67 We note that the folding time τ = ⟨r2⟩/ (3D) with D as the diffusion coefficient and ⟨r2⟩ = 6Rg2 = 0.7 N1.1 nm2 is the mean square end-to-end distance of a polymer.65 We estimated Dlimit to be 1.9 × 10−7 cm2/s as the theoretical upper bound of single-molecule CaM conformational diffusion.65 The roughness of the free energy barrier involved in the folding−unfolding dynamic process is determined by D = Dlimit exp(−β2ε2), ε is a measurement of the roughness of the single-molecule CaM protein folding free energy barrier.68,70 For CaM folding−unfolding conformational diffusion, the diffusion coefficient D is 1.4 × 10−13 cm2/s from our experimental measurement discussed above, which gives the roughness of the single-molecule CaM protein folding free energy barrier ε of 3.8 ± 0.8kT; k is the Boltzmann constant and T is the temperature. Compared with the hydrogen bonding (∼2−12kT),81,82 this value of free energy barrier roughness is on the scale of hydrogen bonding interaction energy in a protein folding process associating with breaking and forming of a number of hydrogen bonds. On the other

diffusion coefficient is directly related to the shape of the underlying energy landscape, since it reflects the roughness of the potential energy surface of single-molecule CaM folding− unfolding conformational dynamic process. Theoretically, we estimate the size of single-molecule CaM in the presence of denaturant GdmCl. Although there are different models involving complex intrachain molecular interactions, we choose the Gaussian chain model because it is the most simplified model and catches the essential properties of an unfolded protein.61−64 The unfolded radius of gyration Rg of the single-molecule CaM protein in GdmCl, Rg = 0.345N0.55 nm is 5.4 nm, which is consistent with the experimental value of 6.1 ± 0.8 nm. N is the number of monomers in the polypeptide chain. The scaling relation of Rg is strongly supported by smallangle X-ray scattering.62 The folding speed limits of single domain proteins are provided experimentally,65 and the singlemolecule single domain protein folding time is wellcharacterized under the Gaussian chain assumption.62,65 The vast majority of measurements yield the formation time of the loop of less than 0.1 μs, and the α-helix formation time is approximately 0.5 μs. The formation time of the β hairpin is greater than 0.5 μs.65 Since linear scaling theory holds for a small degree of polymerization, using the linear length scaling suggested by the homopolymer collapse theory,66 the theoretical upper bound of single-molecule CaM folding time I

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folding process associates with breaking and forming of a number of hydrogen bonds which comprise the nature of “principle of minimum frustration”: every foldable protein minimizes frustration in its amino acid sequence, the cornerstone of the funnel-shaped protein folding energy landscape.24,25 The roughness of the free energy barrier, 3.8 ± 0.8kT, on the same scale as hydrogen bonding, is the key to understanding molecular interactions, recognitions, and associations, since such local fluctuations are likely to reflect the conformational flexibility of CaM, which is a mechanistic characterization of intra- and intermolecular interaction processes such as single-molecule protein folding conformational dynamic processes. The roughness 3.8 ± 0.8kT as a result of our bunched folding waiting time distribution is an averaged value of roughness along a specific protein folding pathway. This 3.8 ± 0.8kT roughness, which is on the same scale with hydrogen bonding, possibly results from hydrogen bonding dynamics and fluctuations in the protein folding process.81,82 The multiple-pathway and multiple-state dynamic scheme associated with 11kT folding free energy barrier and 3.8 ± 0.8kT roughness of the free energy barrier is a rough folding energy landscape on which a single-molecule CaM conformational diffusion process occurs. To study the heterogeneity of the CaM folding process, we have analyzed 52 single-molecule protein folding−unfolding conformational fluctuation trajectories to acquire a distribution of the conformational diffusion coefficient versus the number of steps involving Poisson rate processes (Figure 6B). For each of the single-molecule FRET folding−unfolding signal fluctuation trajectories, we simulate the number of steps involved in the single-molecule CaM folding−unfolding conformational dynamic process, and we also calculate the single-molecule CaM folding−unfolding conformational diffusion coefficient. A twodimensional distribution is constructed with the conformational diffusion coefficient vs the number of steps (Figure 6B). This distribution is clearly a manifestation of a multiple-pathway multiple-state energy landscape with distinct folding pathways on which single-molecule CaM navigate. Moreover, these various folding pathways have a distribution with distinct probabilities. We count the total number of occurrences of each single-molecule CaM folding−unfolding fluctuation dynamics with the same number of steps involving Poisson rate processes, and calculate the ratio of each protein folding pathway among all the single-molecule folding−unfolding fluctuation dynamics trajectories we measure. Our results suggest that there are at least four different folding pathways of CaM molecule folding process (Figure 6B,C) involving multiple intermediate states have different probabilities ranged from 17% to 33%, based on all single-molecule protein folding−unfolding conformational fluctuation trajectories analyzed (Figure 6C,D). Furthermore, we have identified that the different protein folding pathways involve different protein folding conformational diffusion coefficients, a significant indication of multiple-pathway and multiple-states protein folding energy.24,25 The comparable energy scale of the roughness 3.8 ± 0.8kT and hydrogen bonding is a further manifestation of a protein folding energy landscape with many local minima corresponding to multiple folding intermediates (Figure 6D). Presumably, the folding intermediates are likely to have partially folded domains of single-molecule CaM.71 Our single-molecule CaM folding−unfolding conformational dynamics experiment recording single-molecule conformational fluctuation trajectories under thermodynamic equilibrium

hand, the height of the free energy barrier is estimated by using Kramer’s barrier-crossing theory.65 We first estimate the actual folding time of CaM by using τ = ⟨r2⟩/(3D) to be 0.89s in which (⟨r2⟩)0.5 is the experimental value of 6.1 ± 0.8 nm deduced from the EFRET and D is the experimental value measured by the time bunching effect of the single-molecule CaM folding process. In our experiment the single-molecule CaM folding−unfolding conformational diffusion coefficient D is 1.4 × 10−13 cm2/s. We use Kramer’s theory to estimate the empirical free energy barrier of the single-molecule protein folding process. Kramer’s theory assumes that the dynamic process of protein folding can be described as a onedimensional diffusion along a reaction coordinate, and the minimum and maximum of the free energy surface are parabolic.65 From Kramer’s theory τfolding = 2πτlimit exp(ΔG /kT )

(9)

where τfolding is the folding time of single-molecule CaM measured in our experiment, τlimit is the theoretical upper bound of single-molecule CaM folding time, and ΔG is the free energy barrier, k is the Boltzmann’s constant, and T is the temperature. The height of the free energy barrier ΔG is estimated to be 11kT, which is consistent with previous optical tweezers measurement for CaM.69 Our measurement yields a free energy barrier slightly lower than the optical tweezers measurements; this difference is likely due to the distinct structure involved in the measurement assays using different denaturation methods resulting in slightly different energetic features of the energy landscape.71 Nevertheless, our analysis reveals that, instead of a two-state dynamic scheme, a more complex and higher dimensional dynamic process exists on a multiple-pathway multiple-state energy landscape. By recording detailed fluctuation dynamics at thermodynamic equilibrium under the conditions of 2 M GdmCl, we characterize such unique dynamic features via conformational diffusional modeling of the single-molecule CaM folding−unfolding conformational dynamic process. Multiple-pathway and multiple-state dynamic scheme associated with the 11kT folding free energy barrier and 3.8 ± 0.8kT roughness of the free energy barrier are the most intriguing features of the single-molecule CaM folding− unfolding conformational dynamic process observed in our experiments. The single-molecule CaM protein folding process is a conformational diffusion process on a complex funnelshaped energy landscape involving multiple-pathway and multiple-state dynamics as predicted theoretically and proven experimentally in our work. The 3.8 ± 0.8kT roughness of the free energy barrier is likely to be an entropic trap which is the thermodynamic feature of multiple folding intermediates observed in the folding−unfolding dynamic process of CaM (Figure 6A). In Figure 6 the nuclear coordinate Q is a projected one-dimensional coordinate of a three-dimensional potential energy surface. The nature of such a funnel-shaped energy landscape of the single-molecule CaM folding−unfolding conformational dynamic process is entropic rather than enthalpic due to the comparable energy scale of 3.8 ± 0.8kT roughness of the single-molecule CaM folding potential energy surface and 11kT the free energy barrier height.72 An energy landscape is entropic, and it is dominated by local minima which are captured by our single-molecule CaM folding− unfolding conformational fluctuation dynamic process “waiting time” distribution. If the energy scale of hydrogen bonding is taken into consideration, the single-molecule CaM protein J

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The Journal of Physical Chemistry B conditions is an effective approach to study single-molecule protein folding conformational dynamics, combined by using detailed balance dynamic model analysis. More importantly, by recording equilibrium fluctuation dynamics at the protein folding−unfolding denaturing titration midpoint, we are able to quantitatively identify the single-molecule CaM folding− unfolding conformational diffusion coefficients that are crucial for characterizing the underlying energy landscape of CaM folding. Nevertheless, protein folding−unfolding conformational fluctuations involve multiple dimensional nuclear coordinate motions; for example, the conformational changes may involve cis−trans isomerization or intradomain and interdomain rotations and motions, and it should be desirable to have a single-molecule conformational dynamics measurement sensitive to most of the multiple dimensional motions with higher time resolution.73−76 Technically, this is still a big challenge as the motion detection requires different temporal resolution for different types of motions, as well as different spectroscopic parameter sensitivities: such as anisotropy, polarization, FRET donor−acceptor lifetime, and FRET donor−acceptor photon counting rates. In this work, we only probed the FRET donor−acceptor photon intensity fluctuations, as a typical and conventional single-molecule FRET approach. In the future, we will probe the complex conformational dynamics by using our recently developed multiple spectroscopic parameter simultaneous detection approach for further comprehension and analysis of the protein folding− unfolding conformational fluctuation dynamics.77−80 Moreover, since the single-pair FRET typically probes the fluctuations along a one-dimensional FRET coordinate, which is essentially the projection of the real three-dimensional protein conformational motion, a multiple-site labeling FRET imaging and spectroscopic measurement as demonstrated in the multicolor FRET may provide additional detailed information on the three-dimensional protein conformational motion dynamics.83−85

provide the primary energetic features of CaM folding− unfolding conformational dynamics. The comparable energy scales of roughness and free energy barrier suggest that the multiple intermediate states serve as entropic traps on the folding energy landscape. More interestingly, our dynamic model gives an estimation of the number of folding intermediates along each of the folding pathways. Overall, the approach of our dynamics measurements under equilibrium dynamic is demonstrated to be effective and powerful for exploring the folding funnel energy landscape and to verify the theoretical model of the multiple-pathway and multiple-state folding energy landscape based on the fluctuation dissipation theorem. Presumably, a fluctuating folding energy landscape with multiple-state multiple-pathway is likely to be more energetically efficient and kinetically effective for a consecutive protein folding dynamical process in living cells.

CONCLUSIONS Using more sensitive single-molecule spectroscopic measurements, we have achieved the specification of the underlying protein folding pathways by monitoring the protein folding− unfolding dynamics at equilibrium. We have characterized the multiple-pathway multiple-state folding dynamics and energy landscape of single CaM molecules under conditions of various denaturant GdmCl by using single-molecule FRET spectroscopy measurements and correlated model analysis. We utilized the protein folding−unfolding at the denaturant titration midpoint condition in our single-molecule CaM folding− unfolding experiments to probe single-molecule CaM spontaneous folding−unfolding conformational fluctuations, identifying the correlated conformational fluctuation rates and singlemolecule EFRET distributions. The folded state of singlemolecule CaM corresponds to a high EFRET and a fast fluctuation rate on the millisecond time scale, whereas the unfolded state of single-molecule CaM corresponds to a low EFRET and a slow fluctuation rate on the second time scale. We characterized detailed single-molecule CaM folding−unfolding fluctuation dynamics by analyzing the folding waiting time distribution that indicates the existence of multiple intermediate states. Furthermore, we have identified the specific folding routes with distinct probability distributions. The free energy barriers and roughness of the free energy barriers calculated from the dynamic model based on our experimental results

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ASSOCIATED CONTENT

S Supporting Information *

Additional information about ensemble titration measurements of protein folding−unfolding and single-molecule imaging sample preparations. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +1 419 3721840. Fax: +1 419 3721840. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by NIH/NIGMS and Ohio Eminent Scholar endowment.

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