Detection of Structural Dynamics by FRET: A Photon Distribution

J. Phys. Chem. B 2010, 114, 7983–7995

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Detection of Structural Dynamics by FRET: A Photon Distribution and Fluorescence Lifetime Analysis of Systems with Multiple States Stanislav Kalinin,§,*,† Alessandro Valeri,§,† Matthew Antonik,‡ Suren Felekyan,† and Claus A. M. Seidel*,† Institut fu¨r Physikalische Chemie, Lehrstuhl fu¨r Molekulare Physikalische Chemie, Heinrich-Heine-UniVersita¨t, UniVersita¨tsstraβe 1, Geb 26.32, 40225 Du¨sseldorf, Germany, and Department of Physics & Astronomy, The UniVersity of Kansas, 2061 Malott Hall, 1251 Wescoe Hall DriVe, Lawrence, Kansas 66045-7582 ReceiVed: March 10, 2010; ReVised Manuscript ReceiVed: April 16, 2010

Two complementary methods in confocal single-molecule fluorescence spectroscopy are presented to analyze conformational dynamics by Fo¨rster resonance energy transfer (FRET) measurements considering simulated and experimental data. First, an extension of photon distribution analysis (PDA) is applied to characterize conformational exchange between two or more states via global analysis of the shape of FRET peaks for different time bins. PDA accurately predicts the shape of FRET efficiency histograms in the presence of FRET fluctuations, taking into account shot noise and background contributions. Dynamic-PDA quantitatively recovers FRET efficiencies of the interconverting states and relaxation times of dynamics on the time scale of the diffusion time td (typically milliseconds), with a dynamic range of the method of about (1 order of magnitude with respect to td. Correction procedures are proposed to consider the factors limiting the accuracy of dynamic-PDA, such as brightness variations, shortening of the observation time due to diffusion, and a contribution of multimolecular events. Second, an analysis procedure for multiparameter fluorescence detection is presented, where intensity-derived FRET efficiency is correlated with the fluorescence lifetime of the donor quenched by FRET. If a maximum likelihood estimator is applied to compute a mean fluorescence lifetime of mixed states, one obtains a fluorescence weighted mean lifetime. Thus a mixed state is detected by a characteristic shift of the fluorescence lifetime, which becomes longer than that expected for a single species with the same intensity-derived FRET efficiency. Analysis tools for direct visual inspection of two-dimensional diagrams of FRET efficiency versus donor lifetime are presented for the cases of static and dynamic FRET. Finally these new techniques are compared with fluorescence correlation spectroscopy. 1. Introduction Ever since their introduction two decades ago, single-molecule (sm) fluorescence methods have matured and branched out to address numerous biological questions, which were inaccessible via ensemble measurements. sm-Fluorescence techniques have capabilities of probing the dynamic interactions of nucleic acids and proteins via Fo¨rster resonance energy transfer (FRET).1 Nowadays FRET is a widely used tool in sm-fluorescence experiments for monitoring changes in distances between a donor and acceptor dye attached to molecules.2 A wide range of applications such as unfolding, structural dynamics, catalysis, and complex formation of proteins and nucleic acids is described in recent reviews.3-6 Due to the stochastic nature of photon detection and the relatively small number of photons collected in sm experiments (shot noise), distributions are obtained for all recorded fluorescence parameters even if a well-defined single state is monitored. Moreover, static and dynamic heterogeneities result in an additional broadening of the fluorescence signal distributions, which is the actual interesting information. In the case of smFRET experiments, the signals distributions can be potentially related to the distribution of donor-acceptor * To whom correspondence should be addressed. E-mail: stanislav.kalinin@ uni-duesseldorf.de (S.K.); [email protected] (C.A.M.S.). † Heinrich-Heine-Universita¨t. ‡ The University of Kansas. § These authors contributed equally to this work.

distances, allowing for an accurate study of the structural features of a molecule. In recent years, several methods have been developed to analyze these measured distributions.7-10 Photon distribution analysis (PDA), proximity ratio histogram (PRH) analysis,8 fluorescence intensity distribution analysis (FIDA),10 and related approaches (e.g., photon counting histogram (PCH)11) allow one to resolve static heterogeneities based on the emission properties of different species. In addition, multiparameter fluorescence detection (MFD) allows for a robust assessment of all possible artifacts arising from smFRET and offers unsurpassed capabilities regarding the identification and analysis of individual species present in a population of molecules. The specific photon detection scheme offered by MFD provides great flexibility regarding the data analysis acquired in a single measurement12 such as joint twodimensional (2D) fluorescence parameter histograms including fluorescence lifetimes measured by time-correlated single photon counting (TCSPC), PDA, fluorescence correlation spectroscopy (FCS) and FIDA.13 At present, MFD, PDA, and PRH analysis are applied mainly to investigate static heterogeneities,14,15 although the potential to study FRET dynamics is also discussed.8 Two major works demonstrating the possibility to characterize dynamic processes by analyzing intensity distributions have been recently published by Kask et al16 as well as Gopich and Szabo.17,18 The main method for the analysis of fluorescence fluctuations arising from dynamic heterogeneities is FCS.19-22 FCS has the

10.1021/jp102156t  2010 American Chemical Society Published on Web 05/20/2010

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advantage to have access to dynamic processes with time scales spanning many orders of magnitudes from picoseconds to seconds.12 On the other hand, FCS has only limited potential to reveal the static composition of a mixture and characterize the individual states due to the complex relation between the amplitude of the FCS relaxation terms and the brightness of the FRET species. In the present work we combine PDA with a two-state interconversion model (state 1 T state 2) or more complex models, which we will refer to as dynamic PDA. This extension of PDA theory represents a way to overcome the FCS problems stated via a direct identification of the FRET levels. Our main goal is to present a toolbox with qualitative and quantitative methods that are directly applicable to experimental data of freely diffusing molecules. To demonstrate the characteristic effects of dynamic FRET on fluorescence parameters, we present a more visual analysis procedure for MFD parameter histograms, where intensity-derived FRET efficiency is correlated with the fluorescence lifetime of the donor quenched by FRET. To consider dynamics, we use a formalism that is largely similar to the one used in refs 16 and 18, but the quantitative analysis within the framework of PDA additionally includes several essential steps toward the analysis of real data, which are essential for high-resolution FRET measurements. In particular, we test the validity of all approximations that in dynamic PDA are used to derive the model and show that most of them must be used with great caution. The corresponding systematic errors and stability of the analysis are discussed. A simple approach to distinguish between a dynamic and a static system based on varying time window (TW) length is proposed. Moreover, we address several common experimental issues such as brightness variations and multiple molecule events. Actually, we show that equal brightness of the states involved is generally not required. We demonstrate how dynamic PDA is able to identify the different interconverting states and to retrieve the rate and equilibrium constants. Finally these new techniques are compared with FCS. 2. Theory Usually the FRET efficiency is affected by dye motions, molecular fluctuations, and conformational changes, so that the experimental system exhibits more than a single FRET efficiency value. Depending on the time scale, such fluctuations may have an effect on the shape of FRET-related parameter histograms. In such cases, it is necessary to develop a suitable model describing the distribution of FRET efficiencies, P(E), and incorporate this model into the PDA analysis. Here, a model will be derived corresponding to a two-state system, where a molecule switches between two states, each with a distinct FRET efficiency. For a given donor-acceptor distance RDA, the transfer efficiency is given by E ) R06/(R06 + RDA6) where R0 is the Fo¨rster radius [Å], which accounts for the specific properties of the donor-acceptor pair.23 Thus, possible reasons for distinct FRET efficiencies are changes in donor-acceptor distance or in system properties influencing the Fo¨rster radius R0. The theory presented in the following can describe both cases. However, fluctuations of R0 are usually considered as a source for artifacts in structural studies. As a typical application, we therefore describe the case where a molecule switches with definite rate constants between two conformations, each with a distinct RDA. Consider a molecule that has two states, “s1” and “s2”, between which it can freely switch. Each state is characterized by a brightness Qi, which defines the count rate of detected photons, and a unique FRET efficiency Ei. The molecule is

Kalinin et al. observed for a length of time ∆t, during which it may switch between s1 and s2 multiple times. The molecule spends a cumulative amount of time T1 in state s1. Assuming that Q1 and Q2 remain constant during the TW ∆t (i.e., the case of immobilized or slowly diffusing molecules), the mean number of photons produced during the time T1 is T1Q1. The remaining time T2 ) ∆t - T1 is spent in s2 producing a mean number of photons T2Q2. Under these conditions, the total number of photons expected to be observed is T1Q1 + T2Q2, and the expected effective efficiency is

E)

Q1T1E1 + Q2T2E2 Q1T1 + Q2T2

(1)

The brightnesses Qi are the total count rate (in both donor and acceptor detection channels) at a given excitation intensity, and at a concentration of one molecule per observation volume. Choosing T1 determines T2, so the problem of determining the probability P(E) that the FRET efficiency is in the range E0 < E < E0 + ∆E is transformed into one of determining the probability that the cumulative time the molecule spends in s1 is in the range T0 < T1 < T0 + ∆T. We define that probability here to be P(T1). To calculate P(T1), it is necessary to have a probability density function, p(T1), which accurately describes the behavior of the system. The probability density function for a two state system with rate constants k1 and k2 has been previously derived.16,18,24,25 The rate constants define the average time the molecule remains in state si as (1/ki). The density function p(T1) consists of three terms (eqs 2). The first two terms give the probability of a molecule spending the entirety of the time ∆t in s2 and s1, respectively. The third term is the probability density of the molecule being in s1 for a time T1, where 0 < T1 < ∆t.

k1 exp(-k2∆t) + k1 + k2 k2 δ(∆t - T1) exp(-k1∆t) + k1 + k2

p(T1) ) δ(T1)

[

k2T1 + k1T2 2k1k2 I0(2√k1k2T1T2) + · k1 + k2 k1 + k2

]

√k1k2 I (2 k k T T ) I √ 1 2 1 2 √T1T2

exp(-k1T1 - k2T2) (2a)

In eq 2a, δ is the Dirac δ-function, and I0() and II() denote Bessel functions of order 0 and 1, respectively. For completeness, the derivation of eqs 2 is reproduced in the Supporting Information (SI), section 3. The probability of the molecule having a mean effective efficiency in the range E0 < E < E0 + ∆E is determined by using eq 1 to determine T0 and ∆T from E0 and ∆E, then integrating p(T1) over this range to determine the probability P(T0 < T1 < T0 + ∆T), which is identical to the probability P(E0 < E < E0 + ∆E). Using a sufficiently small interval ∆T leads to the following form of eq 2, which is directly applicable for computations (see also eq 3 below):

P(0) ) P(T1 ) 0) )

k1 exp(-k2∆t) k1 + k2

(2b)

Detection of Structural Dynamics by FRET

P(∆t) ) P(T2 ) 0) )

P(T0 < T1 < T0 + ∆T) ≈

[

k2 exp(-k1∆t) k1 + k2

J. Phys. Chem. B, Vol. 114, No. 23, 2010 7985

(2c)

]

Subsequently, P(E) may be further manipulated to generate a theoretical histogram of any FRET-related parameter as discussed elsewhere.14,26 In this work, P(E) is calculated as a discrete function consisting of n + 1 values. This discrete P(E) is calculated integrating P(T1) over n + 1 intervals from T1 ) 0 to T1 ) ∆t. For the intervals i ) 1 to n, the width of the interval ∆T is ∆t/n, and the median dwell time for the interval is i∆T. The first and last intervals are special cases with only half the dwell time and including the probability of no change in state from eqs 2b and 2c. Defining P(T1(i)) as the probability that the dwell time T1 falls in the ith interval, P(T1(i)) can be calculated from eq 3 as

P(T1(i)) ) P(0 < T1 < ∆T/2) + P(0) for i ) 0 P(i∆T - ∆T/2 < T1 < i∆T + ∆T/2) for i ) 1...n - 1 P(∆t - ∆T/2 < T1 < ∆t) + P(∆t) for i ) n (3)

{

Average effective efficiency values Ei for each interval are calculated from the T1(i) values according to eq 1, yielding the desired discrete distribution P(Ei). Next, for each Ei the probability of observing a certain combination of signal photon counts S in the “green” (G) and “red” (R) detection channels, P(SG, SR), is calculated by using eq 4:7,26

∑

FG+BG)SG;FR+BR)SR

F! p FG(1 - pG)FR FG!FR ! G

(5)

In eq 5, the probability to register a “green” photon pG is determined by FRET efficiency as given by eq 6:

2k1k2 I (2 k k T T ) + k1 + k2 0 √ 1 2 1 2

k2T1 + k1T2 √k1k2 · II(2√k1k2T1T2) exp(-k1T1 - k2T2)∆T k1 + k2 √T1T2 (2d)

P(SG, SR) )

P(FG, FR |F) )

P(F)P(FG, FR |F)P(BG)P(BR)

(4) “Green” and “red” detection channels cover the emission spectra of the donor and the acceptor, respectively. The names “green” and “red” refer to the emission wavelengths of fluorophores such as Alexa488 (donor, green emission), and Cy5, Alexa647, or Atto647N (acceptors, red emission). These dyes are frequently used in smFRET studies because of their high quantum yields and photostability. The measured signal S consists of fluorescence (F) and background (B) contributions (S ) F + B). The intensity distribution of the fluorescence, P(F), is obtained from the total measured signal intensity distribution P(S) by deconvolution assuming that the background signals BG and BR obey Poisson distributions, P(BG) and P(BR), with known mean intensities 〈BG〉 and 〈BR〉.26 P(FG, FR|F) represents the conditional probability of observing a particular combination of green and red fluorescence photons, FG and FR, provided the total number of registered fluorescence photons is F, and can be expressed as a binomial distribution7

(

pG ) 1 + R +

γE (1 - E)

-1

)

with

γ)

gRΦFA gGΦFD(0)

(6) In eq 6, R is the spectral crosstalk from donor signal into the red detection channel of the acceptor. The relative spectral sensitivity γ is determined by the detection efficiencies, gG and gR, of the two detection channels and the fluorescence quantum yields of the donor and the acceptor, ΦFD(0) and ΦFA, respectively. Compared to eq 1, eq 6 describes the probability to detect a green photon rather than the probability of emitting a donor photon, i.e., takes various experimental imperfections into account. For an “ideal system” (γ ) 1 and R ) 0), eq 6 naturally converges to pG ) 1 - E. In case of equal brightnesses (Q1 ) Q2), the overall P(SG, SR) is given by a fraction-weighted sum of contributions of all Ei states, given by eqs 4-6. The procedure used to obtain P(SG, SR) for the case of Q1 * Q2 additionally involves deconvolution of individual fluorescence intensity distributions, as described in ref 14. 3. Material and Methods 3.1. Simulations of MFD. Simulations of smMFD measurements were performed by using the Brownian dynamics approach,10,27-29 as described elsewhere.30 The spatial intensity distribution of the observation volume was assumed to be a 3D Gaussian31 (please note that this is not required to obtain eqs 1-2, or the general theory of PDA). The brightness of the species (Qi) was taken comparable to experimental values measured with our setup (ca. 100 kHz). Moreover, Q was set to be equal for all states, except in paragraph 4.5 where the brightness effects are investigated. The simulated fluorescence was collected by “green” and “red” detection channels, which covered the fluorescence peaks of the donor and the acceptor, respectively. The mean number of molecules in the “focus” (defined as in FCS, i.e., via the correlation amplitude, G(tc ) 0) ) 1/NFCS32) was NFCS ) 0.002 unless stated otherwise. The diffusion times (td) of all species was 3 ms with the exception of the series where the effect of td was investigated (section 4.4). Poisson-distributed background signal consisting of detector dark counts and scatter contribution was added. The following common parameters were used for all simulations unless stated otherwise: time step, 0.005 ms; green background BG ) 2 kHz; red background BR ) 1.2 kHz; crosstalk from donor signal into the “red” detection channel, R ) 0.01. Simulations of conformational dynamics were performed via modeling s1 f s2 and s2 f s1 transitions. Initial fractions of molecules in states 1 and 2 were determined by the corresponding rate constants according to k2/(k1 + k2) for s1 and k1/(k1 + k2) for s2. The times the molecules remained in s1 and s2 states (t1 and t2, respectively) were exponentially distributed with density functions p1(t1) ) k1-1 exp(-k1t1) and p2(t2) ) k2-1 exp(-k2t2). To resemble as much as possible a real measurement, a third state representing molecules labeled only with the donor dye was added. Simulated data were saved as a stream of detected fluorescence photons, in SPC-132 data format (Becker & Hickel GmbH, Berlin, Germany) for standard

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analyses using published procedures to calculate all fluorescence parameters.33 3.2. Data Fitting by PDA. The recorded sequence of photon events was used to compute the intensity trace of equal noninterleaving TWs (or time bins) of the length ∆t. Typically, the whole photon trace was analyzed because this allows for rigorous correction for brightness variations and multiple molecule events.14 In some cases, a burst search algorithm34 was employed, and TWs only within selected bursts were considered. The data are typically presented as one-dimensional (1D) histograms of green to red signal intensity ratio (SG/SR). The model function used to fit PDA histograms of simulated data included two interconverting states (s1 and s2; eqs 1-2) and a donor-only (D-only) species as often encountered in real experiments. Hence, five fitting parameters were needed: the individual FRET efficiencies E1 and E2, the corresponding offrates k1 and k2, and the fraction of the D-only species. Convolution of the model distribution (eqs 1-2) with shot noise and background produced 2D distributions P(SG, SR), from which SG/SR histograms (i.e., the model functions) were generated and fitted to the simulated data.7,26 The quality of fits was judged by reduced χ2 values (χr2) as well as by examining weighted residuals plots. 4. Results and Discussion The choice to test the applicability and the performance of dynamic PDA using simulated data is dictated primarily by the need to have standards with exactly known rate constants against which we can benchmark our method, and secondarily to give us the possibility to study the parameter dependence of the fit outcomes. Reanalysis of simulated data allows us to characterize both random (section 4.3) and systematic (sections 4.4-4.6) errors inherent to dynamic PDA. Systematic deviations are expected because the proposed theoretical model (eqs 1-2) is based on several approximations, in particular, on a critical assumption of constant brightness during the selected TW. The validity of these approximations is discussed in sections 4.4-4.6. To test the usefulness of our new tools, we apply them to simulated and experimental data. 4.1. Qualitative Description of the Method by Using 1D and 2D Fluorescence Parameter Histograms. Effect of TW Length. At first we want to introduce the general concepts and give a purely qualitative and visual description of the method that will be used throughout the paper. Let us consider the dynamic system described in section 3.1, where a molecule, depending on its diffusion coefficient, spends a certain amount of time in the laser focus. During this period, it will switch a certain number of times between the two states. Depending on the number of the switching occurrences, the fluorescence bursts of the interconverting molecules will show a different degree of mixing of the two states, s1 and s2 (Figure 1). A burstwise (BW) analysis of sm events for three distinct samples consisting of D-only and interconverting FRET molecules is presented in Figure 1, to demonstrate the influence of the dynamics on the fluorescence parameters for these three cases (panels A-C). The 2D frequency histograms of the green to red signal intensity ratio, SG/SR, against donor lifetime in the presence of acceptor (τD(A)) are shown in the left panels of Figure 1 in the order of increasing interconversion rates. The number of molecules (fluorescence bursts) in each bin is gray scale colored from white (lowest) to black (highest). The corresponding 1D (τD(A)) histograms of the three cases are overlaid as projection in the top diagram, while the SG/SR histograms are

Figure 1. Influence of the dynamics on the fluorescence parameter histograms for the BW analysis of sm events considering three different dynamic regimes (A-C). The characteristic diffusion time of all species is 1 ms. The data are analyzed in 2D frequency histograms of SG/SR vs τD(A), whereby the number of molecules (fluorescence bursts) in each bin is gray scale colored from white (lowest) to black (highest). The simulated data represent a mixture of three species: D-only, FRET state s1, and FRET state s2, with total number of molecules in the focus of 0.001, 0.0005, and 0.0005, respectively. The simulated lifetimes are 4, 2, and 0.8 ns for D-only, FRET s1 (E1 ) 0.5), and FRET state s2 (E2 ) 0.8), respectively. A green and red background of 4.2 kHz and 1.2 kHz, respectively, is also added. (D) Effect of the application of TWs of different lengths on the SG/SR histogram for case B. The black solid line represents the average SG/SR that can be devised from BW analysis, while the black dashed lines indicate the SG/SR value of states s1 and s2.

presented individually on the right. Considering a mean FCS diffusion time of 1 ms, three cases are presented for the switching constants ki: quasi-static case (i.e., very slow averaging) with k1 ) k2 ) 0.1 ms-1 (Figure 1A), slow averaging with k1 ) k2 ) 1 ms-1 so that the dynamics can be resolved in real time (Figure 1B), and very fast averaging with k1 ) k2 ) 10 ms-1 (Figure 1C). In Figure 1A, the three populations are clearly visible as a result of the slow quasi static dynamics. With increasing switching occurrences, the FRET populations shift: in Figure 1B there is a broad diagonal peak visible, while in Figure 1C the FRET peaks have merged to one round peak with a mean SG/SR ratio. Let us focus our attention on the middle panel of Figure 1. If we take the photon trace within a burst and we divide it in TWs of equal length (Figure 1D), it becomes obvious that separation of the peaks increases with decreasing TW length (i.e., increasing time resolution). By comparing the different TWs in Figure 1D, it is even possible to obtain a rough estimate of the rate constants (e.g., ref 35). We want to point out that in dynamic



PDA, the ratio between rate constants and the TWs is the most important factor. Of course the TWs have their limits, they cannot be indefinitely small because we will not have enough photons to have a good statistics, nor can they be indefinitely long because, when the TW approaches or exceeds the diffusion time, we will have problems in estimating the rate constants (see section 4.4). Different AWeraging of Fluorescence Lifetime and Intensity. The second dimension of fluorescence histograms in Figure 1, the apparent donor lifetime, could be also exploited. Two cases will be considered: (I) the static case, where the heterogeneous ensemble does not change within the observation time, and (II) the dynamic case, where the individual populations interconvert, with relaxation times being smaller than the characteristic diffusion time and larger than the fluorescence lifetime. To study the correlation between intensities and lifetimes, the fluorescence counts F of the signal must be calculated from the total signal counts S, taking distinct detection efficiencies g, mean background signal 〈B〉 and spectral crosstalk R between D and A into account (eqs 7a and 7b). For the static case, one expects a correlation between the donor lifetime in presence of FRET (τD(A)) and quantities derived from fluorescence intensities FD and FA of the donor and the acceptor such as FD/FA (eq 7c) and E (eq 7d).36

FD )

FA )

SG - 〈BG〉 FG ) gG gG

(7a)

(7b)

( ) Estatic )

(7c)

FA /ΦFA τD(A) )1FD /ΦFD(0) + FA /ΦFA τD(0)

(7d)

In eqs 7c and 7d, τD(0) is the lifetime of the donor in the absence of FRET. Figure 2 demonstrates that mixing of populations results in a deviation from the expected correlation (blue solid line) of the static case, which is caused by the different averaging of intensities and related parameters (E, FD/FA, or SG/SR) and the apparent fluorescence lifetime. In the dynamic averaging, which results in a multiexponential fluorescence decay of the donor the FRET efficiency is related to the speciesaverage lifetime 〈τ〉x23 (eq 8a). Clearly for a fluorescence burst having at most few 100 photons, it is impossible to resolve multiple donor lifetimes and determine 〈τ〉x. Thus, the maximum likelihood estimator (MLE)37-39 that is commonly used to compute a single lifetime for sm fluorescence bursts provides approximately the fluorescence-weighted average lifetime, 〈τ〉f (eq 8b):36

〈τ〉x ) x1τ1 + (1 - x1)τ2

In eq 8, τ1 and τ2 denote the lifetimes of the donor in the presence of FRET (with τ1 < τ2) for the states s1 and s2, respectively. x1 is the species fraction, which is equivalent to the time spent in state s1 (x1 ) T1/∆t), and the FRET efficiencies of the individual states are calculated by E1 ) 1 - τ1/τ0 and E2 ) 1 - τ2/τ0. To take dynamic averaging into account, we express the species-average lifetime 〈τ〉x as a function of the calculated fluorescence-weighted average lifetime, 〈τ〉f (eq 8c).

〈τ〉x )

τ1τ2 τ1 + τ2 - 〈τ〉f

(8c)

The mean FD/FA ratio for the dynamic case, (FD/FA)dyn, is

( ) FD FA

)

dyn

ΦFD(0) 〈τx〉 · ΦFA τD(0) - 〈τx〉

(9a)

Using the observable 〈τ〉f, one finally obtains eq 9b:

( ) FD FA

) dyn

ΦFD(0) τ1τ2 ΦFA τD(0)[τ1 + τ2 - 〈τ〉f] - τ1τ2

Edyn ) 1 - 〈τ〉x /τD(0) Edyn ) 1 -

ΦFD(0) τD(A) ) · ΦFA τD(0) - τD(A)

static

(8b)

(9b)

or, alternatively, the corresponding FRET efficiencies given by eqs 9c and 9d

(SR - 〈BR〉) - R(SG - 〈BG〉) FR - RFG ) gR gR

FD FA

x1τ12 + (1 - x1)τ22 〈τ〉f ) x1τ1 + (1 - x1)τ2

(8a)

τ1τ2 τD(0)[τ1 + τ2 - 〈τ〉f]

(9c)

(9d)

Equations 9b and 9d are defined for any 〈τ〉f ∈ [τ1, τ2]. Considering a low-FRET state in slow exchange with a highFRET state, it is visible in Figure 2A that the fluorescence lifetimes determined by MLE do not follow eqs 7 (static case, solid line) but rather follow the “alternative” dashed line for the dynamic case (eq 9b) Thus, fluorescence lifetimes that are longer than the values expected for the static FRET case (eqs 7c and 7d) can be considered as another qualitative indication of the presence of FRET dynamics. Special Case: Linker Dynamics. An important special case is fast dynamics of dye linkers, which leads to a distribution of FRET efficiencies and donor lifetimes40 that can be directly measured by ensemble TCSPC. This process is usually too fast to be quantitatively characterized by dynamic PDA, because the fluorescence intensities are averaged in time bins that are usually larger than 100 µs. Nevertheless, the theory for the dynamic FRET (eqs 9) holds, and a quasi-static shift of FRET populations toward longer lifetimes is expected. In other words, if linker dynamics is slower than nanoseconds, it results in a nonexponential donor fluorescence decay with 〈τ〉f > 〈τ〉x (eqs 8). The width of donor-acceptor distance (RDA) distribution due to dye linker dynamics can be estimated in an ensemble lifetime experiment40 on a rigid molecule. This width may partly reflect the reorientational dynamics of the dye (κ2-effects), but for the following discussion, the exact reason for the observed distribution of FRET rates on the nanosecond time scale is unimportant.

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Figure 2. Effects of dynamic exchange on the correlation between the donor and acceptor fluorescence intensity ratio, FD/FA, and the apparent donor lifetime, τD(A). (A) Interconversion of states. The blue solid line represents the correlation expected for a static distribution of two states (eq 7c), while the orange-black dashed line shows the correlation for two interconverting states (eq 9b). Simulation parameters: E1 ) 0.25 (τ1 ) 3 ns), E2 ) 0.75 (τ2 ) 1 ns), k1 ) k2 ) 0.05 ms-1; ΦFD(0)/ΦFA ) 1; NFCS ) 0.02. The diffusion time is td ) 3 ms. (B-C) Correction function for linker dynamics. Linker dynamics was simulated by first calculating the donor-acceptor distance of each state in Figure 2A for R0 of 50 Å (RDA1 ) 60.05 Å and RDA2 ) 41.63 Å) and then simulating a fast interconversion (equal rate constants of 10 ms-1) between RDA + σDA and RDA - σDA, with σDA ) 6 Å, as described in the text and the SI, section 2.1. (B) 〈τ〉x is plotted as a function of 〈τ〉f (black squares) and fitted by a third order polynomial (red line, 〈τ〉x ) -0.0462 + 0.463〈τ〉f + 0.295 〈τ〉2f - 0.0395〈τ〉3f) (eq 10). The black dashed line, 〈τ〉x ) 〈τ〉f, is plotted to illustrate the region were the two averaged lifetimes differ the most. (C) The blue solid line represents the static FRET line expected in the absence of state mixing (eq 7c). The blue dashed line shows the dependence given by the dynamic FRET line (eq 9a), using the 〈τ〉x (〈τ〉f) correction function (eq 10). (D) Simultaneous species interconversion and linker dynamics (combination of cases A and B). The orange-black dashed line shows the FD/FA(〈τ〉x) dependence expected for this case obtained as described in the section entitled “Simultaneous Linker Dynamics and State Interconversion” (eq 9b). The 〈τ〉x (〈τ〉f) conversion function is approximated with eq 10 (〈τ〉x ) 4.14 - 7.11〈τ〉f + 5.79〈τ〉2f - 2.0〈τ〉3f + 0.273〈τ〉4f). The solid blue (static FRET, eq 7c) and the dashed blue (FRET with linker dynamics, eq 9a) lines have the same meaning as in panel C. (E) Interconversion between the open and closed conformations of the protein syntaxin-1 (reanalyzed data from ref 35). The blue line represents the static FRET line (eq 7c with the parameters τD(0) ) 4.0 ns, ΦFD(0) ) 0.80, and ΦFA ) 0.65). The orange-black dashed line represents the dynamic FRET line (eq 9b with the parameters τD(0) ) 4.0 ns, ΦFD(0) ) 0.80, ΦFA ) 0.65, τ1 ) 3.7 ns (open state), and τ2 ) 0.8 ns (closed state)). (F) Mixture of double-labeled double-stranded DNA (dsDNA) molecules (for sequences, see section 1 of the SI). The FRET and the D-only populations follow the line that takes into account linker motions (blue dashed line with the same 〈τ〉x (〈τ〉f) correction function used in panel B) rather than the static FRET line (eq 7d, solid blue line).

Experimental studies on the various effects, such as dye quenching, sticking, intercalation, and cis-trans isomerization, can be found in refs 13 and 41-46. It is notable that MFD is especially useful to detect and characterize the individual subpopulations arising from these effects. In this work we use Cy5 as the acceptor. We note that, in the literature, there are great discrepancies on the fluorescence properties of Cy5, because two quite chemically different compounds are sold under the same name: (I) the phosphoamidite derivative of Cy5, which is frequently used for end-labeling DNA, contains no sulfonic acid groups so that the dye tends to stick to DNA,43,47 and (II) the NHS-ester of Cy5, which is used to internally label DNA, has two sulfonic acid groups so that sticking to DNA is minimized.46,48 In this work we use the nonsticking NHS-ester of Cy5. In the case of linker dynamics, a conversion of fluorescenceaveraged to species-averaged lifetimes can be performed as follows. In our experience, the use of C6-linkers leads to an apparent distance distribution width of σDA ) 5-6 Å for dyes that do not stick to DNA (measured using dsDNA postlabeled with the NHS-esters of Alexa488 and Cy5).49 We illustrate in

a step-by-step procedure in section 2.1 of the SI and in Figure S1, how the lifetime distribution for any mean donor-acceptor distance 〈RDA〉 can be generated in two steps. In step 1, we assume a Gaussian distance distribution of distances with a half width of σDA. In practice, this distribution is approximated by a large number of discrete distances Ri (e.g., 100 values in the range of 〈RDA〉 - 4σDA to 〈RDA〉 + 4σDA) with corresponding probabilities p(Ri) ∝ exp[-(Ri - 〈RDA〉)2/2σDA2]. For each Ri, the donor lifetime τi is calculated (τi ) τD(0)[1 + (R0/Ri)6]-1), and thus the fluorescence and species averaged lifetimes are calculated for each 〈RDA〉 (cf. eqs 8a and 8b). In step 2, we use the distinct values of the species and fluorescence weighted averaged lifetimes, which were computed in step 1 for a range of approximately 100 distinct values for 〈RDA〉, to compute a 〈τ〉x - 〈τ〉f dependence, which can be further used to correct for systematic deviation of 〈τ〉f due to linker dynamics. Because it is impossible to describe this case analytically, the relation between the different averaged lifetimes, to obtain the 〈τ〉x that has to be substituted in eq 9a or 9c, we plot it as a function of 〈τ〉f and fit the dependence with an empirical polynomial with the coefficients ci:



n

〈τ〉x )

∑ ci(〈τ〉f)i

(10)

i)0

This procedure is depicted in Figure 2B. The deviation between the static FRET line (solid line, eq 7c) and the dynamic FRET line (dashed line, eq 9b) for simulated data in Figure 2C demonstrates that the correction for linker dynamics is indeed necessary. This approach has been also successfully applied in a recent study of nucleosome disassembly.15 Simultaneous Linker Dynamics and State InterconWersion. The theory presented above can be easily extended to this case (Figure 2D). The procedure is illustrated in Figure S1 (SI). We consider two Gaussian distance distributions p1(Ri) and p2(Ri) to account for linker dynamics in each interconverting state. For these distributions, we assume the same σDA estimated as discussed in the previous section, distinct 〈RDA〉(1) and 〈RDA〉(2), which correspond to E1 and E2, respectively, and discrete Ri varying from 〈RDA〉(1) - 4σDA to 〈RDA〉(2) + 4σDA. A mixed state with species fractions x1 and 1 - x1, respectively, is then described by a distribution p(Ri) ) x1p1(Ri) + (1 - x1)p2(Ri). Varying x1 in ∼100 steps between 0 and 1, a set of 〈τ〉x and 〈τ〉f is calculated for the distributions of x1p1(Ri) + (1 - x1)p2(Ri) (Figure S1), to obtain a 〈τ〉x - 〈τ〉f conversion function similar to that described above (see SI section 2 for details). Clearly, in the presence of linker dynamics, mixed populations are expected to shift further toward longer lifetimes (Figure 2D). Experimental Data Supports Dynamic Lifetime Shifts. To demonstrate applications of the presented theory to experimental data, we consider two examples. First we reanalyzed data obtained for syntaxin.35 Syntaxin-1, a member of the SNARE superfamily, must adopt various conformations (open50 and closed;51 gray lines in Figure 2E) during its functional cycle, which depend on its interaction partners. Considering smFRET data of the uncomplexed free Syntaxin-1, the 2D FD/FA versus τD(A) plot (Figure 2E) clearly shows that both the open and closed conformation already coexist in solution. It is also apparent that an additional elongated island situated between the two states is also present, which follows the dynamic FRET line (eq 9b, orange dashed line in Figure 2E) rather than the static dependence given by eq 7c (blue line). This fact clearly indicates interconversion between the open and closed conformations on the submillisecond time scale, in agreement with previously published results.35 Second, we studied a mixture of two dsDNA molecules and D-only molecules (sequences are given in the SI) labeled with Alexa488 (donor) and Cy5 (acceptor). A 2D histogram of E versus τD(A) obtained for this dsDNA mix is presented in Figure 2F. One can notice that the simple static equation (eq 7d; straight solid line in Figure 2F) is not fulfilled. Taking into account linker dynamics (dashed curve in Figure 2F) is necessary and sufficient to explain the observed shifts toward longer experimental fluorescence lifetimes. 4.2. Distinguishing between a Static and a Dynamic System. Unfortunately, a possibility to fit the dynamic model (eqs 1-2) to the data does not prove the presence of dynamics. In other words, for any fixed TW, it is possible to find also a static distribution P(E) (or an equivalent distribution of distances) that would adequately describe the data, for example, by using model-free deconvolution.14 In particular, in the case of fast interconversion rates, the distribution given by eq 2c can be very well approximated by a Gaussian distribution. This fact is not surprising. Since eq 2c is a probability function describing a statistical sampling process (in this case, the number

of flips between states 1 and 2 is being indirectly sampled), it is not surprising that, in agreement with the central limit theorem, the distribution approaches a Gaussian as the number of samples (flips) increases, which in our case corresponds to fast interconversion rates. For the same reason it would not be surprising if many different signal broadening mechanisms approached Gaussian distributions in some limit, and care must be taken to provide evidence for the dynamical behavior of the system before fitting the model (eqs 1-2) to the data. Detection of Dynamics by Comparing TWs. To overcome the above-mentioned problem, we will make use of the obvious fact that, for a static system, the P(E) distribution is independent of the TW length. On the other hand, fluctuations on the time scale of selected TWs (typically milliseconds) produce such dependence (see eqs 1-2 and Figure 1). Thus, to distinguish between a static and a dynamic system, one should try to fit the experimental data using the same static model distribution P(E) for a set of TWs with different lengths. As shown below, it is impossible to fit a dynamic data set using the same static P(E) for all ∆t values (and vice versa), which allows one to judge whether any processes leading to FRET fluctuations on the millisecond time scale take place. Analogous to Figure 1, a quantitative strategy to distinguish static and dynamic systems is illustrated in Figure 3 for a single data set displayed in TWs of different lengths. Using a TW of ∆t ) 1 ms (Figure 3A) and fitting with the dynamic model (eqs 1-2), one obtains the two FRET states with E1 ) 0.252, E2 ) 0.747, the rate constants k1 ) 0.87 ms-1, k2 ) 0.89 ms-1, and a fraction of 40.8% for the D-only species. The fit is good, as indicated by the equally distributed weighted residuals and the small reduced χ2, χr2 ) 1.19. However, the same histogram can be equally well fitted (χr2 ) 1.20) with various complex static models (for example, four static states; see Figure 3 legend). Thus, dynamic PDA using a single TW does not allow us to judge whether our system undergoes conformational dynamics or whether a static model is more relevant (Figure 3A). To distinguish between these cases, we keep the recovered model parameters constant and try to approximate the same data but using ∆t ) 0.3 ms. As shown in Figure 3B, this results in a dramatic difference in the fit quality of the dynamic and the static models. Misfit of the static model yielding large deviations in the weighted residuals and a large χr2 ) 58 clearly indicates the presence of FRET fluctuations on the time scale of milliseconds and allows one to rule out the static distance distribution model. On the basis of these observations, we recommend performing a global analysis of several TWs to unequivocally detect dynamics by PDA. Moreover, global analysis can reveal the existence of a superposition of static and dynamic components. 4.3. Quantitative Dynamic PDA. Having verified that the system of interest exhibits conformational dynamics and the minimal two-state model (eqs 1-2) is applicable, the interconversion rates can be extracted from the data by fitting the corresponding PDA histograms. In this section, we investigate mainly the stability and suitability of dynamic PDA as a quantitative method as well as the influence of the TW length on its accuracy. For this purpose, we simulated various data sets and tested whether the fits allow us to recover the parameters used for the simulation. The results of this analysis are summarized in Figure 4 (see also Table S1, SI) by plotting the recovered versus simulated rate constants. The broad gray area represents the region of calculated rate constants being within (10% of the simulated values. The mean burst duration is characterized by

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Figure 3. Global fit of SG/SR histograms obtained for a simulated dynamic system (gray area), using two-state dynamic (solid black lines) and static (dashed red lines) models. TW duration ∆t is (A) 1 ms and (B) 0.3 ms. Reduced χ2 values are 1.19 (∆t ) 1 ms, dynamic model; k1 ) 0.89 ms-1; k2 ) 0.87 ms-1), 1.20 (∆t ) 1 ms, static model), 1.43 (∆t ) 0.3 ms, dynamic), and 58 (∆t ) 0.3 ms, static). Weighted residuals plots are displayed above main graphs. Simulation parameters are E1 ) 0.25, E2 ) 0.75, k1 ) k2 ) 1 ms-1; 40% of D-only species; NFCS ) 0.002. The static model used to fit the data needs four states: D-only (40.2%), two Gaussian distributions of distances with 〈R1〉 ) 1.18R0 (〈E1〉 ) 0.27), σ1 ) 0.03R0 (18.3%) 〈R2〉 ) 0.98R0 (〈E2〉 ) 0.53), σ2 ) 0.09R0 (27.6%), and one state with fixed R3 ) 0.84R0 (〈E3〉 ) 0.74; 13.9%) (R0 stands for the Fo¨rster radius). These parameters are given only to demonstrate the possibility to fit a static model to dynamic data and otherwise have no meaning.

Figure 4. Interconversion rates k1 and k2 recovered by dynamic PDA, plotted against their simulated values. TW lengths are 0.3 ms (circles), 1 ms (squares), and 3 ms (diamonds). Filled and open symbols represent k1 and k2 values, respectively. Inset: the values of E1 and E2 have been fixed in the analysis. The gray area indicates the region of calculated rate constants being within (10% of the simulated values. Error bars represent expected confidence intervals calculated from k1 vs k2 χ2 surfaces, which does not necessarily reflect cross-correlation between all model parameters. Other simulation parameters: E1 ) 0.25, E2 ) 0.75, 40% of D-only species; total NFCS ) 0.002. The diffusion time is td ) 3 ms. The total number of photons is 3 × 107, which corresponds to ca. 3 × 104 useful 1 ms TWs. Data for this figure are presented as a table in the SI (Table S1). A series of simulations with k1 * k2 revealed no effects specific for this case (data not shown).

an FCS translational diffusion time of 3 ms (exchange rate 0.33 ms-1). As expected, the maximum accuracy is achieved when the TW duration ∆t is comparable with the relaxation time (k1 + k2)-1, which essentially means that the molecules undergo a few transitions during the dwell time in the observation volume (central part of the diagram). Otherwise the histogram peaks are either completely separated (quasi-static border case) or completely merged (very fast dynamics), yielding a weighted average of E1 and E2. For both cases, the shape of SG/SR histograms is thereby practically independent of the rates (Figure 1B), making it impossible to determine their exact values. Thus, the range of measurable k1 and k2 values depends on the TW length ∆t, which is in turn limited by the diffusion time and low photon numbers for short TWs. In general, by choosing optimal ∆t, relaxation times on a time scale of 0.1∆t to 10∆t can be recovered with a reasonable accuracy, as shown in Figure 4. Qualitatively, dynamic behavior can be detected outside this region. Considering the data shown in Figure 4, meaningful qualitative results could be expected in the range of 0.03∆t to

30∆t, especially if a global analysis using multiple ∆t values is performed (section 4.2) However, in quantitative analysis, a global approach is not expected to improve the accuracy of results because of systematic errors correlated with ∆t (mainly due to diffusion), as discussed below in section 4.4. Another expected result is that random errors (shown as error bars in Figure 4) become smaller for longer TWs, which contain large number of photons. This observation is fully consistent with our previous results on detecting heterogeneities by PDA,26 clearly showing the advantages of using long TWs for resolving multiple states. The same holds for the qualitative analysis: for example, for k1 ) k2 ) 0.03 ms-1 and ∆t ) 0.3 ms zero rates are within statistical uncertainty (Table S1), whereas using ∆t ) 3 ms allows one at least to detect the presence of FRET fluctuations. Instabilities in the range of high interconversion rates (cf. Figure 4) appear to be mainly due to a cross-correlation between the fitted rates and efficiencies E1 and E2. Fixing the values of E1 and E2 significantly improves the precision of dynamic PDA in determining the rates, especially in the range of fast dynamics (see Figure 4 inset). An independent estimation of the individual FRET efficiencies is experimentally possible if the equilibrium can be shifted to one side, i.e., the system can be “locked” in one state (e.g., by adding a ligand, see ref 36). In addition, systematic deviations of the recovered rates from the simulated values are observed in Figure 4. The most obvious effect is a considerable systematic underestimation of the rates when using long TWs (diamonds). We believe that this effect is mainly due to shortening of the effective observation time due to diffusion on the ∆t time scale, which will be discussed in the following section. 4.4. Fast Diffusion Effects. The theoretical model presented in the Theory section requires that the fluorescence intensity does not change during a time interval ∆t. In practice, this is not always the case, for two reasons. First, if Q1 is not equal to Q2, the brightness would obviously change upon an s1 T s2 transition. What is more important, the TW lengths (∆t) used in sm analysis are often comparable with the diffusion time (td). As a result, the observation time, or the “effective” TW length becomes shorter than the specified one. This effect may result in a significant underestimation of the interconversion rates if our theory is applied without modifications. The problem is especially pronounced when long TWs are used to study slow dynamics and/or to take advantage of good photon statistics (cf. the ∆t ) 3 ms series shown in Figure 4). To illustrate the influence of diffusion on dynamic PDA, we have simulated and reanalyzed a series of data sets using various



Figure 5. Underestimation of rate constants by dynamic PDA as a consequence of diffusion on the ∆t time scale (here ∆t ) 1 ms), which results in shortening of the observation time. Two series are simulated with k1 ) k2 ) 3 ms-1 (A) and k1 ) k2 ) 0.3 ms-1 (B), with the diffusion time td increasing from 0.5 to 16 ms. For both time regimes, the results of TW analysis within selected bursts and of the whole photon trace are displayed. The gray area indicates the range of rate constants, which differ from their simulated values by no more than 10%. Solid curves show the empirical correction of eq 11 derived in the Appendix. Simulation parameters: NFCS ) 0.002, ca. 30 000 useful 1 ms TWs for PDA analysis.

diffusion times, whereas all other parameters have been kept constant. The results for the whole photon trace of a long measurement (squares in Figure 5) clearly support the explanation presented above: dynamic PDA underestimates the rates unless ∆t , td. Strictly speaking, diffusion also leads to a distribution of observation times, which is inconsistent with eqs 1-2 and thereby results in generally higher χr2 values for long TW series. It is clear that the discussed effects must be taken into account if possible.18 Unfortunately, rigorous treatment of diffusion effects for an arbitrary ∆t/td ratio is far from simple.18 Moreover, it implies that exact knowledge of the spatial intensity distribution profile would be needed, which would require the assumption of an appropriate model and an experimental calibration of descriptive parameters. Selection of Bright Bursts. To overcome this problem and to suppress diffusion effects, Gopich and Szabo proposed that one could “impose a sufficiently high threshold to eliminate events where the molecule transiently leaves and immediately re-enters the laser spot”.18 Please note that the use of an empirical burst search algorithm or a strict burst selection algorithm together with a high photon threshold may change the distribution of observation times in a completely unpredictable way, so that no analytical description is available. Figure 5 confirms the predicted behavior, that the restriction of the analysis to TWs within intensive bursts reduces the diffusionrelated systematic deviations significantly (triangles in Figure 5). However, random errors (shown as error bars in Figure 5) increase, mainly because too many photons are disregarded during burst selection. Moreover, setting a high photon threshold has a least three partly counteracting effects: (I) the signal of dimmer molecules is suppressed, (II) at the same time we underestimate the fraction of molecules in a dim state by the high threshold, and (III) as we prefer brighter events, the apparent contribution of multimolecular events is increased (see section 4.6 and ref 52). Corrections: Empirical or by Simulations. The main idea of the empirical correction is to find a shorter “effective” TW ∆t′, representing the mean observation time, for which the fluorescence intensity is constant and eqs 1-3 are exact. By assuming that the change of the fluorescence intensity is solely due to diffusion, this effect is experimentally accessible by measuring the diffusion term Gd(tc) of the normalized fluorescence autocorrelation function (values between 0 and 1) with the correlation time tc. The correction is simply achieved by

multiplying the values of interconversion rates kfit by ∆t/∆t′ (eq 11; for details, see the Appendix).

kcorrected ≈ kfit × ∆t/∆t′ ) kfit ×

4 - Gd(tc ) ∆t) 3

(11) with

( )(

tc Gd(tc) ) 1 + td

-1

ω02 tc 1+ 2 z td 0

)

-1/2

The diffusion term describes the observation volume by a three-dimensional Gauss function with radial and axial 1/e2 radii, ω0 and z0, respectively. The characteristic diffusion time td for the fluorescent molecules is related to the translational diffusion coefficient D by D ) ω02/4td. Equation 11 is a crude but convenient approximation and is useful for the range ∆t/td < 0.5 (see solid curve in Figure 5). The obtained result can be considered as empirical and used when simulations are impractical. In a general case, or if a better accuracy is required, corrections can be obtained by simulations, as follows. The rates recovered from experimental data by using eqs 1-2 are taken as the first approximation, and td can be measured by means of FCS. These parameters are used to perform a simulation, which is then reanalyzed in order to estimate to what extent dynamic PDA is biased by diffusion, at given conditions. Calculated rates are expected to deviate from their simulated values. As discussed before in this section, relative errors of the rate constants show only a weak dependence on the absolute values of k1 and k2 (cf. Figure 5A,B, open and filled squares). In other words, the errors are determined by the ∆t/td ratio rather than by interconversion rates. Thus, the ratio between simulated and recovered rates can be used as an alternative correction factor. 4.5. Brightness Effects. So far we have assumed that the total brightness (i.e., the registered donor and acceptor fluorescence) of states s1 and s2 is the same. In practice this would rarely be the case because of the usually different quantum yields of the donor and the acceptor dyes and wavelength-dependent detection efficiency. This problem manifests itself in two related ways. At first, in this case the average FRET efficiency is not simply a time-weighted average (eq 1), but rather is shifted

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Figure 6. Brightness effects in dynamic PDA. The brightness of the acceptor is taken to be 0.375 of that of the donor. Simulation parameters: E1 ) 0.25, E2 ) 0.75; the other parameters are the same as for Figure 3. The gray area indicates the range of rate constants that differ from their simulated values by no more than 10%.

toward that of the brighter species. Thus, the weights of E1 and E2 become dependent on the corresponding brightness values Q1 and Q2 in eq 1. Second, one cannot use the same (overall) fluorescence intensity distribution P(F) for all species, as discussed in details in ref 14. Brightness variations can significantly bias dynamic PDA if not taken into account, as illustrated in Figure 6. Unlike other artifacts, which lead to under- or overestimation of both apparent rates by roughly the same factor (see Figure 5), brightness variations also affect the k1/k2 ratio, i.e., the equilibrium constant (Figure 6). For a realistic difference in fluorescence quantum yields, including saturation effects (e.g., 0.80 and 0.32, as for the widely used FRET pair Alexa 488-Cy5) and equal detection efficiency errors in the range of 30-50% are expected. Fortunately, a combination of eq 1 and brightness correction methods (sections 2.3-2.5 in ref 14) enables unbiased estimation of interconversion rates irrespective of brightness effects. Figure 6 clearly shows that corrected k1 and k2 values (shown as circles) can be calculated with the same precision as in the absence of brightness variations (Figure 4). 4.6. Multiple Molecule Events. It has been recently shown that the shape of FRET histograms in sm measurements can be affected by multimolecule events even at relatively low concentrations, which may lead to severe misinterpretations.14,52 In the case of dynamic PDA, multimolecular events can result in bursts with mixed species, which generate histogram points in between the major peaks. Thus, this effect leads to higher fitted rate constants, which makes it important to correct for these events. Clearly, multiple molecule events are expected to have the most pronounced effect on systems showing slow interconversion rates and distant peaks. To study the influence of multimolecule events on dynamic PDA, the behavior of such a system has been simulated with E1 ) 0.1, E2 ) 0.9, k1 ) k2 ) 0.1 ms-1, and varying total concentration. Surprisingly, multimolecule events might have a considerable (∼20%) effect on the recovered rates already at a “typical sm” concentration of NFCS ) 0.01 (Figure 7). At higher concentrations, an overestimation of the rate constants by more than a factor of 2 can be expected (Figure 7, squares). To correct for multimolecular events, we applied the procedure described in ref 14 (eqs 19-20). Briefly, the correction involves calculation of P′(F) and P′(FG, FR) corresponding to an sm concentration. Convolution of P′(FG, FR) with itself and further convolution with the background provides the desired model function14 (cf. eq 6). As shown in Figure 6, this correction yields excellent results also in the case of dynamic states with recovered parameters,

Kalinin et al.

Figure 7. Multiple molecule events correction, the recovered rate constants are plotted as a function of NFCS. Simulation parameters: E1 ) 0.1; E2 ) 0.9; k1 ) k2 ) 0.1 ms-1; no D-only species; the other parameters are the same as for Figure 3. The gray area indicates the range of rate constants that differ from their simulated values by no more than 10%. The upper axis shows the total concentration of FRET species in a 3 fL detection volume.

in good agreement with the simulated ones. “Corrected” rates show no significant dependence on the concentration of the species at least up to NFCS ) 0.1 (see Figure 7, circles), extending the usable concentration range by an order of magnitude. This possibility to perform measurements at slightly elevated concentrations could help to reduce unwanted contributions of impurities and hardware instabilities. Simultaneous correction for multiple molecule events and brightness variations is also possible (section 2.8 in ref 14). 4.7. Possible Applications to the Case of More than Two Interconverting FRET States. For the case of more than two (n) interconverting states, no analytical solution similar to eqs 2 exists.18 However, a simple and relatively fast computational procedure can be employed to address this problem. Essentially, a state trajectory is simulated assuming exponentially distributed lifetimes (level durations) of the involved states and then used to calculate the distributions p(Ti) (see section 2.2. in the SI for details). An example of an E-histogram calculated for a three-state system is given in the SI (Figure S2). One should also notice a quickly growing number of variables (n(n - 1) rates and n FRET efficiencies), that is, nine fit parameters for a three-state model. In a general case (no a priori knowledge), it might be very difficult or impossible to unambiguously determine all nine model parameters. Moreover, it is very likely that at least some of n(n - 1) rates would be too fast or to slow to be resolved by PDA. For instance, the data simulated assuming three-state kinetics can be very well fitted using the two-state model (eqs 2) with only four variables (see SI Figure S2C) for a single TW. However, if several times are fitted globally (see section 4.2), it becomes clear that a dynamic two-state model is insufficient to describe all histograms (see SI Figure S2D-F). If there is no previous knowledge for a mechanism, it might be impossible to find a unique solution, because there are several options for more complex models besides the three-state model (e.g., a combination of two-state dynamics with a single static state). Thus a rational model and some external constraints (e.g., known FRET efficiencies; some processes are known to be quasi-static on the millisecond time scale) must be applied to extract valuable information by PDA. Because of a very large number of possibilities for such constraints, detailed discussion of multiplestate models is beyond the scope of this paper. At this point, we notice that PDA is capable of detecting more complex exchange kinetics, and a prediction of a FRET


Figure 8. (A) 2D histogram of FD/FA vs τD(A) generated from simulated data of a mixture of three states: slowly interconverting high-FRET (E1 ) 0.75) and low-FRET (E2 ) 0.25) and static mid-FRET (E3 ) 0.5; 23%). The simulated interconversion rates are k12 ) k21 ) 0.05 ms-1, NFCS ) 0.013. 1D histograms are shown in projections. The dashed orange-black line represents the dynamic FRET line given by eq 9b, and the solid line shows the static FD/FA (τD(A)) dependence (eq 7c). (B) Normalized GG (solid black line) and GR (dashed orange line) correlation curves obtained for the same simulated parameters but with NFCS ) 1.3. (C) PDA of the data shown in panel A. Extracted parameters: E1 ) 0.749; E2 ) 0.251; E3 ) 0.5 (22%); k12 ) 0.066 ms-1; k21 ) 0.065 ms-1; χr2 ) 0.86. The blue squares show the model function, and the vertical red line highlights the static state (E3 ) 0.5).

histogram for known parameters is feasible. It is also worth mentioning that all qualitative approaches proposed here to identify dynamic states (visual analysis of 2D plots (section 4.1) and by using a TW series, section 4.2) can be directly applied to more complex cases. 4.8. Dynamic PDA and Visual Analysis of 2D Plots Complement FCS. To illustrate the benefits of using both PDA and FCS, we consider the following example. Imagine that, in addition to the dynamic exchange between two states (highFRET with E1 ) 0.25 and low-FRET with E2 ) 0.75), a static state is present (mid-FRET with E3 ) 0.5; Figure 8A), which is not involved in the dynamic equilibrium. A simple visual inspection of 2D plots allows one to immediately identify interconverting states (high-FRET and low-FRET species are connected with a dynamic FRET line, orange dashed curve)

J. Phys. Chem. B, Vol. 114, No. 23, 2010 7993 and design a kinetic model, which can be further used in both PDA and FCS analysis. However, if we use FCS alone (Figure 8B), two major problems arise: (i) complexity of the kinetic model, (ii) overlap of diffusion and dynamic time scales. At first we consider the choice of appropriate kinetic models. In FCS, it is possible to compute three distinct correlation curves for the two spectral channels (green (G) and red (R): GG, GR, and RR). Figure 8B shows the normalized GG(tc) and GR(tc) correlation curves obtained for the case of Figure 8A. By global analysis of the three correlation curves, we obtain a global kinetic relaxation time and three bunching amplitudes (we assume that the number of molecules NFCS is fitted globally; see ref 35). This fact limits the number of variables in a FCS fit to four, irrespective of the data quality. Even if the kinetic scheme is more complex, FCS alone can at best detect an additional complexity of the case shown in Figure 8A, where six parameters are needed to describe the system (three efficiencies, two rates, and the fraction of the static state). In this case, FCS would most likely extract meaningless FRET efficiencies if a two-state model22 is applied. In principle, PDA alone can handle this case because the extension of the twostate exchange model (blue squares) to include a static state (vertical red line) is obviously trivial (Figure 8C). For FCS, such a modification is more complicated53 and generally requires preknowledge of the brightness of the states, which has to be determined by 2D FIDA.54 Moreover, PDA is generally useful in the case of slow dynamics where the dynamic term overlaps with the diffusion time td (Figure 8A). Figure 8B shows that FRET dynamics on this time scale results only in a small shift between the GG and GR FCS curves with no extra correlation or anticorrelation term. Besides FRET dynamics, such shifts (Figure 8B) can be due to slightly different “green” and “red” detection volumes or different diffusion coefficients of “green” and “red” species, whereas the amplitudes are affected by the background55,56 and impurities. On the other hand, for dynamic PDA, this time scale is nearly optimal (section 4.3), and the interconversion rates are readily extracted (Figure 8C). To summarize, we believe that FCS and dynamic PDA perfectly complement each other. FCS is a powerful technique with a huge dynamic range, but it cannot be applied blindly (cf. Figure 8A). PDA, FIDA, and visual analysis of 2D plots are most useful for getting an overview of the system and characterizing each state in detail. In particular, PDA is especially suited to additionally detect extra broadening of each state.7 5. Conclusions With the present work we showed how PDA theory can be extended to dynamic cases. With the help of simulations, we were able to identify the range of applicability for this method. Dynamic PDA works well if the dynamic processes have relaxation times that are 0.1-10 times the selected TW ∆t. It is important to note, that the TW’s lower limit is defined by the photon statistic and the upper limit by the diffusion time. In combination with 2D-fluorescence parameter histograms dynamic PDA has the advantage of an intuitive graphical visualization, which is important for the detection and analysis of more complex kinetic scenarios. Using these 2D-parameter histograms for high-resolution FRET studies, one can also exploit the different averaging regimes of fluorescence lifetime and intensity to detect dynamics. Furthermore, we demonstrated the usefulness of correction methods to account for different brightness of the states and the occurrence of multimolecule

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events. Finally, the results reported in this paper show how dynamic PDA complements FCS when the dynamic term is superimposed to the diffusion time in the correlation. Acknowledgment. S.K. is grateful to the Alexander von Humboldt Foundation for an AvH-fellowship for starting the project. C.A.M.S and S.K. thank the German Science foundation (DFG) in the priority program SPP 1258 “Sensory and regulatory RNAs in prokaryotes” for funding this work. Derivation of the Empirical Correction for Fast Diffusion (Eq 11) We propose a simple approximate correction procedure, which can, to some extent, reduce systematic errors due to the discussed effect of fast diffusion, but still extracts all necessary information from measurable parameters. The main idea of this correction is to find a shorter “effective” TW ∆t′, representing the mean observation time, for which the fluorescence intensity is constant and eqs 1-3 are exact. The derivation given below involves clear oversimplifications and assumptions that are certainly violated in reality. The obtained result can thus be considered as empirical and used when simulations are impractical. We consider a step-like fluorescence intensity with the number of photons F′(t), which has the same mean and variance as the normalized experimental fluorescence intensity with the number of photons F(t). If the same number of photons are detected in a smaller TW ∆t′, F′(t) can be expressed as

F′(t) ) ∆t/∆t′, F′(t) ) 0,

0 e t e ∆t′

∆t′ < t e ∆t

(A7a) (A7b)

It is clear that 〈F′(t)〉 ) 1, and the variance of F′(t) is

〈F′2(t)〉 - 1 ) ∆t/∆t′ - 1 ) 〈F2(t)〉 - 1

(A8)

Next, we will try to find an approximate expression for 〈F2(t)〉. Let us consider a linear change of the fluorescence intensity with time, for which

F(t) ) 1 - f + 2ft/∆t

(0 e t e ∆t)

(A9)

where f is a constant, and 〈F(t)〉 ) 1. The corresponding variance of the fluorescence intensity is a function of f, given by

〈F2(t)〉 - 〈F(t)〉2 ) 〈F2(t)〉 - 1 ) f 2 /3

(A10)

Now we have to estimate the value of f from measurable parameters. The mean change of fluorescence intensity during the TW can be accessed by FCS. Formal substitution of eq A9 into the definition of correlation function for t ) ∆t yields

G(∆t) ≡

〈F(t)F(t + ∆t)〉 ) 1 - f2 2 〈F(t)〉

(A11)

Assuming that the change of the fluorescence intensity in eq A9 is solely due to diffusion, only the diffusion term Gd(∆t) of the correlation function is relevant. Assuming a 3D Gaussian detection volume, it is given by31,32,57

(

∆t G(∆t) ) Gd(∆t) ) 1 + td

)

-1

(

ω02 ∆t 1+ 2 z td 0

)

-1/2

(A12)

where the 1/e2 radii of the laser focus in the xy and z directions are referred to as ω0 and z0, respectively, and td is the diffusion time. By combining eqs A8, A10, and A11, we finally obtain

∆t′/∆t ) 〈F2(t)〉-1 ) (1 + f 2 /3)-1 )

3 4 - Gd(∆t) (A13)

Gd(∆t) is measurable in a FCS experiment. Correction is achieved simply by multiplying the values of interconversion rates k by ∆t/∆t′, which is equivalent to using the value of ∆t′ (instead of ∆t) in eqs 1-3. As shown in section 4.4 this simple correction works satisfactory at least for ∆t/td < 0.5, provided that Nmin is sufficiently high. For a more general case (or if a better accuracy is required) corrections can be obtained by simulations. Supporting Information Available: Details of the materials, computational procedures, and supporting results. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Fo¨rster, T. Ann. Phys. 1948, 2, 55. (2) Ha, T.; Enderle, T.; Ogletree, D. F.; Chemla, D. S.; Selvin, P. R.; Weiss, S. Proc. Natl. Acad. Sci. U.S.A. 1996, 93, 6264. (3) Weiss, S. Science 1999, 283, 1676. (4) Roy, R.; Hohng, S.; Ha, T. Nat. Methods 2008, 5, 507. (5) Nettels, D.; Gopich, I. V.; Hoffmann, A.; Schuler, B. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 2655. (6) Chung, H. S.; Louis, J. M.; Eaton, W. A. Proc. Natl. Acad. Sci. U.S.A. 2009, 106, 11837. (7) Antonik, M.; Felekyan, S.; Gaiduk, A.; Seidel, C. A. M. J. Phys. Chem. B 2006, 110, 6970. (8) Nir, E.; Michalet, X.; Hamadani, K. M.; Laurence, T. A.; Neuhauser, D.; Kovchegov, Y.; Weiss, S. J. Phys. Chem. B 2006, 110, 22103. (9) Haas, E. ChemPhysChem 2005, 6, 858. (10) Kask, P.; Palo, K.; Ullmann, D.; Gall, K. Proc. Natl. Acad. Sci. U.S.A. 1999, 96, 13756. (11) Chen, Y.; Mu¨ller, J. D.; So, P. T. C.; Gratton, E. Biophys. J. 1999, 77, 553. (12) Felekyan, S.; Ku¨hnemuth, R.; Kudryavtsev, V.; Sandhagen, C.; Becker, W.; Seidel, C. A. M. ReV. Sci. Instrum. 2005, 76, 083104. (13) Neubauer, H.; Gaiko, N.; Berger, S.; Schaffer, J.; Eggeling, C.; Tuma, J.; Verdier, L.; Seidel, C. A. M.; Griesinger, C.; Volkmer, A. J. Am. Chem. Soc. 2007, 129, 12746. (14) Kalinin, S.; Felekyan, S.; Valeri, A.; Seidel, C. A. M. J. Phys. Chem. B 2008, 112, 8361. (15) Gansen, A.; Valeri, A.; Hauger, F.; Felekyan, S.; Kalinin, S.; To´th, K.; Langowski, J.; Seidel, C. A. M. Proc. Natl. Acad. Sci. U.S.A. 2009, 106, 15308. (16) Palo, K.; Mets, U.; Loorits, V.; Kask, P. Biophys. J. 2006, 90, 2179. (17) Gopich, I. V.; Szabo, A. J. Phys. Chem. B 2003, 107, 5058. (18) Gopich, I. V.; Szabo, A. J. Phys. Chem. B 2007, 111, 12925. (19) Elson, E. L.; Magde, D. Biopolymers 1974, 13, 1. (20) Palmer, A. G., III; Thompson, N. L. Biophys. J. 1987, 51, 339. (21) Nettels, D.; Hoffmann, A.; Schuler, B. J. Phys. Chem. B 2008, 112, 6137. (22) Torres, T.; Levitus, M. J. Phys. Chem. B 2007, 111, 7392. (23) Lakowicz, J. R. Principles of Fluorescence Spectroscopy, 3rd ed.; Springer: New York, 2006. (24) Berezhkovskii, A. M.; Szabo, A.; Weiss, G. H. J. Chem. Phys. 1999, 110, 9145. (25) Gopich, I.; Szabo, A. J. Chem. Phys. 2005; 014707. (26) Kalinin, S.; Felekyan, S.; Antonik, M.; Seidel, C. A. M. J. Phys. Chem. B 2007, 111, 10253. (27) Enderlein, J.; Robbins, D. L.; Ambrose, W. P.; Goodwin, P. M.; Keller, R. A. Bioimaging 1997, 5, 88. (28) Laurence, T. A.; Kapanidis, A. N.; Kong, X. X.; Chemla, D. S.; Weiss, S. J. Phys. Chem. B 2004, 108, 3051.

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