1440 DOI: 10.1021/la104130n Langmuir 2011, 27(4), 1440–1447Published on Web 11/30/2010

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© 2010 American Chemical Society

Single-Molecule Force-Clamp Spectroscopy: Dwell Time Analysis and

Practical Considerations†

Yi Cao‡ and Hongbin Li*

Department of Chemistry, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada.‡Present address: Department of Physics, Nanjing University, Nanjing, Jiangsu Province, PR China

Received October 13, 2010. Revised Manuscript Received November 8, 2010

Single-molecule force-clamp spectroscopy has become a powerful tool for studying protein folding/unfolding, bondrupture, and enzymatic reactions. Different methods have been developed to analyze force-clamp spectroscopy data onpolyproteins to obtain kinetic parameters characterizing the mechanical unfolding of proteins, which are often modeled as atwo-state process (a Poisson process). However, because of the finite number of domains in polyproteins, the statisticalanalysis of the force-clamp spectroscopy data is different from that of a classical Poisson process, and the equivalency ofdifferent analysismethods remains to be proven. In this article, we show that thesemethods are equivalent and lead to accuratemeasurements of the unfolding rate constant. We also demonstrate that distinct from the constant-pulling-velocityexperiments, in which the unfolding rate extracted from the data is dependent on the number of protein domains in thepolyproteins (theN effect), force-clamp experiments do not show anyN effect. Using a simulated data set, we also highlightedimportant practical considerations that one needs to take into account when using the single-molecule force-clampspectroscopy technique to characterize the unfolding energy landscape of proteins.

Introduction

Atomic force microscopy (AFM)-based single-molecule forcespectroscopy has evolved into a powerful tool for studying thedynamics and the involved weak noncovalent interactions of manyprocesses, including conformational changes of polymers (includ-ing synthetic polymers and biomacromolecules1-5 and DNA6-11),

ligand binding,12-18 chemical/enzymatic reactions,19-24 and proteinfolding andunfolding.25-40Over the past decade, the use ofAFMtoinvestigate the folding and unfolding dynamics of proteins has pro-vided a tremendous amount of new information, opening up newavenues for studying protein folding and unfolding. To identifysingle-molecule stretching events unambiguously, polyprotein ap-proaches are widely used in typical protein folding/unfolding experi-ments41-43 in which the protein of interest is constructed into apolymer consisting of identical tandem repeats of the protein.Stretching suchpolyproteins at a constant velocity gives rise touniquesawtooth-like force-extension curves with each force peak corre-sponding to the unfolding of a protein domain in the polyproteinchain. Such polyprotein approaches allow for single-molecule force

† Part of the Supramolecular Chemistry at Interfaces special issue.*To whom correspondence should be addressed. E-mail: hongbin@chem.

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spectroscopy to be performed in a well-controlled fashion and havemade it possible to interpret data without any ambiguity. However,in such experiments, the force that is applied to the protein changesas a function of the extension of polyproteins in a complexmanner.Such a complex evolution of forcemakes it challenging to derive ananalytical solution for extracting unfolding kinetic parameters ofproteins. Instead, kinetic parameters of mechanical unfolding reac-tions are often obtained by using Monte Carlo simulation44-46 ornumeric fitting.47-49 Moreover, the compliance of the polyproteinchain varies with the number of folded protein domains in thepolyprotein and the linker length, resulting in the dependence of theunfolding force on the number of protein domains in the poly-protein chain, the so-called N effect, in constant-velocity single-molecule force spectroscopy experiments.50

The development of force-clamp and force-ramp spectroscopyhas made it possible to stretch proteins under a predefinedconstant force or a force increasing at a constant rate.31,51 Thesenew methods use a force feedback loop to precisely control theforce being applied to a protein molecule and monitor its lengthchanges over time. Stretching a polyprotein in force-clamp moderesults in a staircase-like length-time profile where each step

corresponds to a mechanical unfolding event of a protein domain(Figure 1). Because all of the domains are independent andexperience the same stretching force, the unfolding traces of apolyprotein can be regarded as the superposition of individualunfolding events, as suggested by Fernandez and co-workers.51,52

Therefore, the unfolding probability or unfolding probabilitydensity can be directly measured, and the unfolding rate constantcan be readily determined at a given stretching force. This led totwo widely used data-analysis methods: length average52,53 andsurvival time.51,54 In these two methods, the number of domainsin the polyproteins is not considered.

Recently, Fernandez and co-workers introduced a dwell-timeanalysis method to test the two-state Markovian process, whichconsiders the total number of domains in the polyproteins and theorder of unfolding events.53 In this method, the dwell time isdefined as the survival time. In a different effort, we proposed apseudo-dwell-time method, where the dwell time is defined as thetime interval between consecutive unfolding events in force-clampexperiments.55 Apparently, these two methods explicitly consid-ered the number of domains in polyproteins, and the resultsshowed that the number of domains in the polyprotein affects thedistribution of each dwell time that is categorized with respect tothe order of unfolding events and the total number of domains,raising the question of whether there is a similarN effect in force-clamp experiments on polyproteins and whether these twomethods are equivalent to the length-average and survival-timemethods. Similar questions also exist for force-ramp experiments.

In this article, we reconcile these issues by showing that thesedifferent data-analysis methods are equivalent and measure thesame unfolding kinetics. We show that the unfolding kinetics ofproteins in force-clamp experiments is not affected by the numberof domains in the polyprotein and there is no N effect in force-clamp or force-ramp experiments. Using simulated data sets, wealso compared the accuracy and versatility of these differentmethods in some representative experimental situations (e.g., fordifferent numbers of protein domains, small data pools, theabsence of some initial unfolding events because of the presenceof nonspecific interactions at the beginning of the traces, and theabsence of late unfolding events because of the detachment ofmolecules from the cantilever or substrate). These practicalconsiderations will help to optimize the experimental conditionsin single-molecule force spectroscopy experiments to characterizethe unfolding energy landscape of proteins.

Materials and Methods

Monte Carlo Simulation. The simulated data set for themechanical unfolding of polyproteins using force-clamp spec-troscopy was generated using a Monte Carlo procedure asdescribed.41 The protein unfolding is modeled as a two-stateprocess, and the unfolding rate constant depends on the stretchingforce following the Bell-Evans model,56,57 R(F) = R0 exp(FΔxu/kBT), whereR(F) is the unfolding rate constant for a force F, R0 isthe spontaneous unfolding rate constant in the absence of astretching force, 4xu is the unfolding distance between the nativestate and the mechanical unfolding transition state, kB is theBoltzmann constant, and T is the temperature. The parameters

Figure 1. Schematics of force-clamp experiments and the force-clamp spectrum (red trace). (i) A polyprotein is picked up from thesolid substrate by an AFM cantilever and held at a constantstretching force. (ii) The unfolding of a protein domain gives riseto the stepwise elongation of the polyprotein molecule. (iii) Thelength of the polyprotein molecule will remain constant after all ofthe domains are unfolded. The length-time profiles of polypro-teins show a characteristic staircaselike appearance, where eachstep corresponds to the mechanical unfolding of an individualdomain in the polyprotein chain. As shown in the schematiclength-time curve,we define the time interval between consecutiveunfolding events as the dwell time (td) and the time between zeroand the unfolding event as the survival time (ts).

(44) Rief, M.; Gautel, M.; Oesterhelt, F.; Fernandez, J. M.; Gaub, H. E. Science1997, 276, 1109.(45) Rief,M.; Gautel,M.; Schemmel, A.; Gaub, H. E.Biophys. J. 1998, 75, 3008.(46) Oberhauser, A. F.; Marszalek, P. E.; Erickson, H. P.; Fernandez, J. M.

Nature 1998, 393, 181.(47) Williams, P.M.; Fowler, S. B.; Best, R. B.; Toca-Herrera, J. L.; Scott, K.A.;

Steward, A.; Clarke, J. Nature 2003, 422, 446.(48) Brockwell, D. J.; Beddard, G. S.; Paci, E.; West, D. K.; Olmsted, P. D.;

Smith, D. A.; Radford, S. E. Biophys. J. 2005, 89, 506.(49) Schlierf, M.; Rief, M. J. Mol. Biol. 2005, 354, 497.(50) Zinober, R. C.; Brockwell, D. J.; Beddard, G. S.; Blake, A. W.; Olmsted,

P. D.; Radford, S. E.; Smith, D. A. Protein Sci. 2002, 11, 2759.(51) Oberhauser, A. F.; Hansma, P. K.; Carrion-Vazquez,M.; Fernandez, J. M.

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used in theMonte Carlo simulations are R0 = 0.0004 s-1,Δxu=0.25 nm, T= 298 K, Δt= 0.0001 s, and F= 140 pN. The totalnumber of domains is 10, and the total number of traces generatedis 2000. This data pool was then used for all of the data analysis inthis article. To evaluate the effect of the total number of domainson the unfolding rate, we used different numbers for the totalnumber of domains and kept the remaining parameters un-changed.

Data Processing and Fitting. All of the data analysis wasdone using custom-written procedures in Igor 5.1 (Wavemetrics).For the length-average method, the unfolding traces were first nor-malized against the total length changes (lengthfinal - lengthinitial).Then thenormalized curveswere averaged toobtain the final length-average traces. Exponential fitting to the length-average traces gaverise to the unfolding rate of the protein. The survival time andpseudo dwell time were calculated using the method specified in theResults section. Then, the histograms of both the survival time andpseudo dwell time were calculated with a bin size of 0.08 s. Thesehistograms were fit to a single-exponential function to obtain theunfolding rate constant. For the force-ramp experiments and con-stant-velocity experiments, the unfolding probability densities werecalculated using a bin size of 5 pN.

Estimation of Fitting Errors.To estimate the fitting errors ofdifferent sample sizes, we first randomly select a desired numberof traces from the data pool with a total of 2000 traces. Then theselected subdata set was analyzed using length-average, survival-time, and pseudo-dwell-time methods to extract the unfoldingrate. This procedurewas repeated50 times toget a set of unfoldingrates with the same data size. The average and standard deviationof the 50 unfolding rates were used to estimate the quality of thefitting.

Results and Discussion

Brief Introduction to Different Analysis Methods. AFM-based single-molecule force-clamp spectroscopy uses a forcefeedback system to maintain the constant force at the presetvalue by varying the length of the proteins being stretched.51 Asshown in Figure 1, the unfolding of a polyprotein in force-clampmode gives rise to a staircaselike unfolding pattern with each stepcorresponding to the unfolding of a protein domain in thepolyprotein. Here we define the time from the beginning (t0) tothe kth unfolding event as the survival time of the kth domain(ts,N,k), where N is the total number of domains being stretched.We define the dwell time (td,N,k) of the kth domain as the timeinterval between the (k - 1)th and the kth unfolding events.

Different methods have been developed to analyze force-clamp data on polyproteins, and we focus on four of them:length-average, survival-time, dwell-time, and pseudo-dwell-time methods.

The length-average method is based on the reasoning thatbecause individual domains in the polyprotein are identical andindependent, the averagednormalized lengthLn(t) is equivalent tothe unfolding probability Ps(t).

51 For two-state processes,

LnðtÞ ¼ PsðtÞ ¼ 1- e-Rt ð1ÞSimilarly, the distribution of the survival time equals the

probability density of unfolding at time t.

fsðtÞ ¼ Re-Rt ð2ÞThe survival-time method fits the distribution of survival time

to eq 2 to extract the unfolding rate at a given stretching force.For the dwell-time analysis proposed by Fernandez et al.,53 the

dwell time of the kth domain is defined as the survival time of thekth domain. It is of note that this definition is different from the

arrival time used in a classical Poisson process.58 The unfoldingprobability of the kth domain to unfold at time t is

Psðt,N, kÞ ¼ Nk

!ðPsðtÞÞkð1-PsðtÞÞN- k

¼ N!

k!ðN- kÞ!ðPsðtÞÞkð1-PsðtÞÞN- k ð3Þ

The probability density of the unfolding of the kth domain isthe probability of having exactly k- 1 domains unfolded at timetmultiplied by the unfolding probability of the kth domain in the[t, t þ dt] time interval, which equals (N - k þ 1)R dt.

fsðt,N, kÞ ¼ Psðt,N, k- 1ÞðN- kþ 1ÞR

¼ Nk- 1

!ðPsðtÞÞk- 1ð1-PsðtÞÞN- kþ1ðN- kþ 1ÞR

¼ N!

ðk- 1Þ!ðN- ðk- 1ÞÞ!ð1- e-RtÞk- 1 e- ðN- kþ1ÞRtðN- kþ 1ÞR

ð4ÞThis equation is normalized from zero to infinity over t. In the

unfolding probability density of the kth domain out of N do-mains,N is explicitly considered in eq 3. This is different from thegamma function used to describe the classical Poisson processes.58

For pseudo-dwell-time analysis, the dwell time of the kth domainis defined as the time interval between the kth and (k þ 1)thdomain,55 which is the same definition as the arrival time in aclassical Poisson process.58 When considering the number ofdomains in the polyprotein, the unfolding probability of thekth domain of a polyprotein with a total of N domains is

Pdðt,N, kÞ ¼ 1- e- ðN- kþ1ÞRt ð5Þand the probability density function is

fdðt,N,KÞ ¼ ðN- kþ 1ÞRe- ðN- kþ1ÞRt ð6ÞThe pseudo dwell time t0 is then defined as t0= (N - k þ 1)t,and eq 6 can be rewritten as

fdðt0,NÞ ¼ Re-Rt0 ð7ÞThe unfolding rate constant can then be obtained by fittingeq 7 to the distribution of the pseudo dwell time.

In all four methods, only the length-average method uses theprobability function to fit the experimental data. The other threemethods extract the unfolding kinetics by fitting the probabilitydensity function to the distribution of time.

As the first attempt to determine whether these methods areequivalent and lead to the correct estimateofunfolding rate constant,we synthesized a data pool of 2000 unfolding traces (10 unfoldingevents in each traces) using a standard Monte Carlo simulation asdescribed in the Materials and Methods section. As shown inFigure S1, all fourmethods can adequately recover the input unfold-ing rate constant of 2.04 s-1 at a stretching force of 140 pN. More-over, the equations can also reproduce the distribution of the experi-mental data, which is a validation of these data-analysis methods.NoNEffect in Force-Clamp Experiments. In all four data-

analysis methods, dwell-time and pseudo-dwell-time methods

(58) Ross, S. M. Stochastic Processes, 2nd ed.; Wiley: New York, 1996.

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Cao and Li Article

explicitly consider the number of domains in the polyprotein. Theprobability density functions (eqs 4 and 6) contain N and aredifferent from the corresponding functions in a classical Poissonprocess. In contrast, length-average and survival-time methodsare based on the assumption that the unfolding traces of apolyprotein can be regarded as the superposition of individualunfolding events, and N is not in the probability and probabilitydistribution functions.

It is of note that the survival time of the kth domain out of Ndomains follows eq 4.53 If the dwell-time and survival-timemethods are equivalent, then summing all the survival times(from 1 to N) for eq 4 should lead to the same unfoldingprobability density function, fs(t)=R exp(-Rt). Here we providea mathematical proof for this equivalency.

fsðt,NÞ ¼XNk¼ 1

fsðt,N, kÞ

¼XNk¼ 1

N

k- 1

!ðPsðtÞÞk- 1ð1-PsðtÞÞN- kþ1ðN- kþ 1ÞR

¼XN- 1

k- 1¼ 0

N

k- 1

!ðPsðtÞÞk- 1ð1-PsðtÞÞN- kþ1ðN- kþ 1ÞR

fsðt,NÞ ¼XNj¼ 0

N

j

!ðPsðtÞÞjð1-PsðtÞÞN- jðN- jÞR

¼XNj¼ 0

Nj

!ðPsðtÞÞjð1-PsðtÞÞN- jNR

-XNj¼ 0

Nj

!ðPsðtÞÞjð1-PsðtÞÞN- j jR

¼ NR-NRPsðtÞXN- 1

j- 1¼ 0

N- 1j- 1

!ðPsðtÞÞj- 1ð1-PsðtÞÞðN- 1Þ- ðj- 1Þ

0@

þ 0Þ ¼ NRð1-PsðtÞÞBecause Ps(t) = 1 - e-Rt, we can write fs(t, N) = NRe-Rt.The above derivation is based on the following equation:

ðaþbÞn ¼XNk¼ 0

Nk

!bkaN- k

Let a = 1 - Ps(t) and b = Ps(t), hence

XNj¼ 0

Nj

!ðPsðtÞÞjð1-PsðtÞÞN- j ¼ 1

XN- 1

j- 1¼ 0

N- 1j- 1

!ðPsðtÞÞj- 1ð1-PsðtÞÞðN- 1Þ- ðj- 1Þ ¼ 1

BecauseR0¥fs(N, t) dt=1, after normalizationwe get fs(N, t)=Re-Rt.

This result clearly indicated that the dwell-time method isequivalent to the survival-time and length-average methods,and the number of domains N in the polyprotein does not affectthe unfolding kinetics. Hence, there is no N effect in force-clampexperiments on polyproteins.

For pseudo-dwell-time analysis, the definition of dwell timeis the same as that in classical Poisson processes.58 However,

because of finiteN, different dwell times no longer follow the sameexponential distribution as for classical Poisson processes. In-stead, by correcting the effect of a finite numberof foldeddomainsin a polyprotein, the pseudo dwell time follows an identicaldistribution and there is no N effect.

The results clearly show that these methods are equivalent. Bychoosing the appropriate experimental parameter for analysis,there is noN effect. However, it is of note that a finite number ofdomains in the polyprotein does make the unfolding of thepolyprotein slightly different from that in classical Poissonprocesses in that both the survival time and dwell time are nowdependent upon N.

On the basis of the proof of equivalency of themethods, we canimmediately draw the conclusion that we can take into account ofall of the complete unfolding trajectories of different numbers ofdomains in the data pool for the analysis using length-average,survival-time and pseudo-dwell-time methods. An analysis of thesimulated data set generated by Monte Carlo simulations con-firmed this point (Figure 2). However, for a dwell-time analysisthat relies on N, only trajectories of the same N can be groupedtogether for analysis. This can significantly limit the size of thedata pool and makes this method impractical in most experi-ments. In the rest of the discussion, we focus on length-average,survival-time, and pseudo-dwell time methods.No N Effect in Force-Ramp Experiments. Different from

force-clamp experiments, which measure the dwell time underconstant force, in force-ramp experiments the force increases

Figure 2. No N effect in force-clamp experiments. The simulateddata (2000 traces) with different values ofN are superimposable oneach other in the three different analysis methods. Therefore, it ispossible to combine complete length-time traces with different Nto increase the data pool without affecting the analysis.

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linearly and the unfolding force at which each individual domainunfolds is measured to extract the unfolding kinetics.51,52,59 If theunfolding trace of the force-ramp experiments can be consideredto be the superposition ofN independent single-molecule unfold-ing events, then the most probable unfolding force should beindependent of the number of domains N. Following the sameapproach for force-clamp experiments, we define the unfoldingprobability density of the kth domain out ofN domains to unfoldat forceF as f(k,N,F). The sumof f(k,N,F) should be equal to theunfolding probability density of a single domain, which is of theform57,60

f ðFÞ ¼ R0

aeFΔxu=kBTe-R0kBT=aΔxuðeFΔxu=kBT - 1Þ ð8Þ

where a is the force-ramp rate and the other symbols are asdefined as before. We also attempted to prove this point math-ematically. However, such a derivation cannot be done analyti-cally because the unfolding probability density function of the kthevent is conditional on that of the (k - 1)th event and f(1, N, F)has a complex expression similar to eq 8.

f ð1,N,FÞ ¼ NR0

aeFΔxu=kBTe-NR0kBT=aΔxuðeFΔxu=kBT - 1Þ ð9Þ

Instead, we used Monte Carlo simulation to show that indeedthe results are consistent with this statement. We generated 2000force-ramp unfolding traces using Monte Carlo simulation andconstructed the unfolding-force distribution for the ith (from 1 toN) domain in the data set. It is clear that the sum of the unfolding-force distribution from the first domain to the Nth domain is thesame as the unfolding-force distribution of a single domainmeasured in force-clamp experiments (Figure 3A), suggestingthat unfolding trajectories measured from force-ramp experi-ments can be considered to be the superposition of N unfoldingtrajectories of a single domain under the same force-ramp con-dition. Therefore, we can readily combine force-ramp traces of adifferent number of domains to construct the unfolding-forcehistogram (the unfolding probability density function). The mostprobable unfolding force at the given ramp rate is independent ofthe number of domains in the polyprotein chain, and there is noN

effect in the force-ramp experiment. This is in sharp contrastto the constant-velocity experiments (Figure 3B). For example,the average unfolding forces decrease from ∼300 pN at N=1 to∼200 pN at N=10 in constant-velocity experiments. Mixing thedata with different N significantly broadens the unfolding-forcedistribution andmay lead to an incorrect estimation of the unfoldingdistance because the width of the unfolding-force distribution isdirectly related to the unfolding distance between the native stateand the mechanical-unfolding transition state.

What is the origin of theN effect? One intuitive thought mightbe that the total number of domains in the polyproteins speedsup the unfolding rate of the kth unfolding events by a factor of(N- k þ 1) because there are (N- k þ 1) domains folded in thepolyprotein. However, the mathematical proof andMonte Carlosimulation showed that this is not the case. As correctly pointedout by Smith and co-workers,50 the unfolding force is directlyrelated to the effective loading rate at the point of rupture. It is thedifference in the loading rate that causes theN effect. In constant-velocity experiments, the force increases with the length of themolecule and the length of the molecule depends on the totalnumber of domains (N) and the number of unfolded domains inthe polyprotein (k). Therefore, the loading rate is a function ofN.A larger value of N results in a smaller average loading rate.However, in force-ramp and force-clamp experiments, the load-ing rates of all of the domains are identical and independent ofN.To extend this point, as long as the force feedback system is used,any waveform of force, linear or nonlinear, can be used in force-clamp experiments and no N effect will be present.

This point is of critical importance in defining the mechanicalstability of a protein. Mechanical stability is often defined as theaverage unfolding force at a given pulling velocity in constant-velocity experiments. However, because the effective loading ratein constant-velocity experiments is affected by many factors suchas the polyprotein construct (with different values of N anddifferent linker lengths) and the cantilever spring constant, themechanical stability of the same protein measured in differentlaboratories is often quite different, making the comparison ofprotein mechanical stability difficult. However, force-ramp spec-troscopy experiments can effectively circumvent this problem,making these the ideal choice in single-molecule force spectros-copy experiments. Hence, we propose that the mechanical stabi-lity of proteins should be reported as the average unfolding forceat a given ramp rate in force-ramp spectroscopy experiments tofacilitate the comparison of results from different laboratories.

Figure 3. (A) The unfolding-force histogrammeasured by the force-ramp technique does not show anN effect. Unfolding-force histogramsof individual unfolding events categorized by their order in the force-rampdata are gray and their sum is black. It is evident that the sumof allof the unfolding-force histograms is identical to the unfolding-force histogram of a single domain (in red). (B) The unfolding-force histogrammeasured in constant-velocity experiments shows a clear N effect. The average unfolding force depends on the number of domains in thepolyprotein. Combining datawith different numbers of domainsmay potentially lead to relatively large errors in determining themechanicalstability andmechanical unfolding kinetic parameters of proteins. The ramp rate inA is 100 pN/s, and the pulling speed in B is 400 nm/s. Therest of the parameters are the same as those reported in the Materials and Method section.

(59) Wang, M. J.; Cao, Y.; Li, H. B. Polymer 2006, 47, 2548.(60) Izrailev, S.; Stepaniants, S.; Balsera, M.; Oono, Y.; Schulten, K. Biophys. J.

1997, 72, 1568.

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Length-Average Method May Be More Suitable for

Extracting the Unfolding Rate from Small Data Pools.

Because of the difficulties in single-molecule experiments, it ischallenging to get an experimental data pool as large as the simu-lated data pools. How much data is sufficient to give an accurateestimation of the unfolding rate constant? Are the fitting errors forall three data-analysis methods similar? To address these issues, weestimated the fitting errors from simulated data pools of differentsizes. We randomly selected the desired number of traces from atotal of 2000 traces and obtained the unfolding rate constant usingthe three different methods. This process was repeated 50 times tocompare the mean and standard deviation of the unfolding rateconstant from different analysis methods. As shown in Figure 4A,the average unfolding rate constants from the 50 trials are aroundthe input number of 2.04 s-1 for all three methods, which againindicates the correctness of these analysismethods and the accuracyof the fittings.We used the standard deviation of the unfolding rateconstant from 50 trials as the indicator for the precision of the fit.The length-average method gave much smaller deviations than theother twomethods (Figure 4B).Less than5%error in theunfoldingrate constantwasobtainedwhenusing around100unfolding eventswhereas for the survival-time and pseudo-dwell-time methodsobtaining less than 10% error in the unfolding rate constantrequired the use of around 400 unfolding events (dotted line). Thisfinding is somehow unexpected but reasonable. Among these threemethods, only the length-average method calculates the unfoldingprobability at different times instead of the probability density. Inthe length-average method, the data at each t is essentially inte-grated from time 0 to t, which increases the effective data pool andreduces the error in fitting.Choice of the Starting Time Does Not Affect the Esti-

mated Unfolding Rate Constant. In single-molecule AFMexperiments, the beginning parts of the force spectroscopy tracesare often “contaminated” by features associated with nonspecificinteractions, making it difficult to determine the starting time, t0,for each trace. To examinewhether the choice of starting timewilllead to the deviation of the estimated unfolding rate constantfrom the actual value, we changed the t0 for all 2000 simulatedtraces to t0

0 = 0.2 s. Such a change makes some unfolding traceshave less than 10 unfolding events because the unfolding events inbetween t0 and t0

0 are missed. Then, these new data were analyzedusing all three methods. Such changes did not have any effect on

the estimated unfolding rate constant for all three methods, asshown in Figure 5. This reflects important features of a Poisson

Figure 4. Comparison of the accuracy and precision of different data-analysis methods. The accuracy of three different data-analysismethodswas estimated by comparing the unfolding rate constants obtained fromdifferent analysismethods and the input number (A). Thereis no significant difference among the threemethods. However, these three different analysis methods have distinct precision, as shown in thecomparison of the standard deviation of the unfolding rate constant (B). The length average is the most precise method with smaller fittingerrors. Therefore, it requires a smaller data set thanothermethods to estimate the unfolding rate constant accurately.A number (100-200) ofunfolding events are sufficient to obtain the unfolding rate constant with less than 5% error.

Figure 5. The choice of starting timedoes not affect the analysis ofunfolding kinetics. When the starting time t0 was arbitrarily offsetto 0.2 s in the original staircase data, the analysis of the unfoldingkinetics using the three different methods led to the correctunfolding rate constant.

1446 DOI: 10.1021/la104130n Langmuir 2011, 27(4), 1440–1447

Article Cao and Li

process: no memory and history-independent. This is importantfor experiments because we can effectively include trajectoriescontaminated by the nonspecific interactions at the beginning ofthe unfolding trajectories. However, such conveniencemay not beapplicable to unfolding processes that are non-Poisson in nature.Missing Initial Fast Unfolding Events Affect the Esti-

mated Unfolding Rate Constant. We may also face a difficultsituation in choosing the starting point of the traces for real

experimental data. Because of the limited response of the force-feedback loop, the first few fast unfolding events may be missed.This leads to the wrong estimation of the starting point for thosetraces, as shown in Figure 6A. In this situation, the wrong t0 doesnot affect all traces but only those with fast initial unfoldingevents.54,55 We used the simulated data to test whether this biasesthe estimated unfolding rate constant.We artificially changed thestarting points of all traces to the position of the first unfoldingevent with a dwell time longer than 0.03 s. Although the forcefeedback for state-of-the-art AFM instruments can be as fast as1-3ms,54 we choose 0.03 s as the threshold just tomake the effectobvious. As shown in Figure 6B-D, because of missing fastunfolding events, the data for all three analysis methods deviatedfrom a single-exponential distribution. Simply applying a single-exponential fit to the data will cause significant bias in the inputunfolding rate constant (red line in Figure 6B and gray lines inFigure 6C,D). However for both survival-time and pseudo-dwell-time methods, if we omit the first few bins of the distribution wecan still get the correct unfolding rate constant from the data (redlines in Figure 6C,D). For force-clamp experiments carried out ata high clamp force, the unfolding of protein domains can be veryfast at the beginning of the traces. Therefore, choosing a suitabledata-analysis method is very important. It is worth noting thatmissing short unfolding events have been reported in many force-clamp experiments.54,55 To obtain unbiased data, the response

Figure 6. Missing fast unfolding events may bias the data analysis.(A) Initial fast unfolding events may be missed if an improper t0 ischosen in the analysis. (B) Length-average, (C) survival-time, and (D)pseudo-dwell-timemethods;black lines represent thedatawithmissingfast unfolding events (t=0.02 s is the cutoff), and red lines and graylines correspond to single-exponential fitting. (C,D) Gray lines are fitsto all data, and red lines omit the first few data points in the fitting.

Figure 7. Missing slow unfolding events due to a short observa-tion timewindowordetachment of the polyproteinmolecule eitherfrom the cantilever or the substrate may bias the obtained unfold-ing rate .

DOI: 10.1021/la104130n 1447Langmuir 2011, 27(4), 1440–1447

Cao and Li Article

speed of the force-feedback loop may determine the limit of theclamped force in force-clamp experiments used to obtain un-biased data.Including Unfinished Traces Significantly Affects the

Estimated Unfolding Rate . In single-molecule AFM experi-ments, another potential error comes from including unfinishedtraces in the data analysis. Thismay occur quite often in low-forceforce-clamp experiments that require a long time for all of thedomains in the polyprotein to unfold.However, if the polyproteinis detached from either the cantilever or the substrate beforeall domains in the polyprotein unfold, then the total number ofdomains in the polyprotein is also unknown. In these scenarios,the total N in the polyprotein is unknown. Because we haveproven that there is no N effect in force-clamp experiments, doesmissing some unfolding events affect the estimated unfoldingrate ? Because the missed events mainly occur at long survivaltimes, from intuition we may expect that including unfinishedtraces may bias the unfolding rate to higher value. We then usedthe simulated data to demonstrate how significant this effectcould be for all three data-analysis methods. We randomlydetached half of the unfolding traces at time t=1.5 s to mimicthe real experimental condition. As shown in Figure 7, includingunfinished traces significantlymade the estimated unfolding rate sfor all three methods faster than the real value. However, amongall three methods, the survival time can still provide a reasonablygood estimation of the unfolding rate with an error of less than10%. It is worth noting that if the number of domains in thepolyproteins is different, then including unfinished traces maybias the data toward a single-exponential model and show anartificial multiple-exponential distribution. This may result in awrong interpretation of the unfolding mechanism of proteins.Therefore, excluding the unfinished traces in the data analysis iscritical for single-molecule force-clamp experiments. Moreover,for the pseudo-dwell-timemethod, if the total number of domainsin the polyprotein being stretched is known (e.g., in a refoldingexperiment), then the unfinished traces can still be used for theanalysis, as we demonstrated previously.55

Conclusions

Force-clamp (including force-ramp) spectroscopy has playedan increasingly important role in the study of protein unfoldingdynamics. The development of different data-analysis methodsgreatly facilitates the extraction of the unfolding kinetics of proteinsfrom force-clamp experiments. Here, by combining mathematicalderivation and Monte Carlo simulation, we demonstrated thatthese commonly used methods of analyzing force-clamp data areequivalent and provide an accurate estimation of the unfolding rateof proteins. Moreover, we showed that the force-clamp and force-ramp experiments do not have anyN effect, making the extractionof unfolding kinetics from them easier than that from constant-pulling-velocity experiments. Then, we showed that the accuracyand precision of thesemethods under some practical conditions canvary because of the different mathematical treatment of the experi-mental data.This comparisonwill help tooptimize the experimentalconditions in single-molecule force spectroscopy experiments con-ducted to characterize theunfolding energy landscape of proteins. Itis of note that the data-analysis methods available for force-clampexperiments are not limited to the ones discussed in this article.Other methods have also been developed.61,62 Further develop-ments of new analysis methods will be of great importance inextending the use of force-clamp techniques to analyze morecomplex unfolding kinetics of proteins that deviate from simpletwo-state unfolding kinetics.

Acknowledgment. This work is supported by the NaturalSciences and Engineering Research Council of Canada, theCanada Foundation of Innovation, and the Canada ResearchChairs Program.

Supporting Information Available: Plots of data from fourdifferent analysis methods. This material is available free ofcharge via the Internet at http://pubs.acs.org.

(61) Bura, E.; Klimov, D. K.; Barsegov, V. Biophys. J. 2007, 93, 1100.(62) Dudko, O. K.; Hummer, G.; Szabo, A. Proc. Natl. Acad. Sci. U.S.A. 2008,

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