Conformational dynamics of biomolecules are of fundamental importance for their function. Single-molecule studies of Förster Resonance Energy Transfer (smFRET) between a tethered donor and acceptor dye pair are a powerful tool to investigate the structure and dynamics of labeled molecules. However, capturing and quantifying conformational dynamics in intensity-based smFRET experiments remains challenging when the dynamics occur on the sub-millisecond timescale. The method of multiparameter fluorescence detection addresses this challenge by simultaneously registering fluorescence intensities and lifetimes of the donor and acceptor. Together, two FRET observables, the donor fluorescence lifetime τ_{D} and the intensity-based FRET efficiency *E*, inform on the width of the FRET efficiency distribution as a characteristic fingerprint for conformational dynamics. We present a general framework for analyzing dynamics that relates average fluorescence lifetimes and intensities in two-dimensional burst frequency histograms. We present parametric relations of these observables for interpreting the location of FRET populations in *E*–τ_{D} diagrams, called FRET-lines. To facilitate the analysis of complex exchange equilibria, FRET-lines serve as reference curves for a graphical interpretation of experimental data to (i) identify conformational states, (ii) resolve their dynamic connectivity, (iii) compare different kinetic models, and (iv) infer polymer properties of unfolded or intrinsically disordered proteins. For a simplified graphical analysis of complex kinetic networks, we derive a moment-based representation of the experimental data that decouples the motion of the fluorescence labels from the conformational dynamics of the biomolecule. Importantly, FRET-lines facilitate exploring complex dynamic models via easily computed experimental observables. We provide extensive computational tools to facilitate applying FRET-lines.

## I. INTRODUCTION

Many experimental techniques provide information on biomolecular structural heterogeneity and can be utilized to resolve ensembles of structures through integrative modeling.^{1} However, few techniques simultaneously inform on the structure and dynamics from picoseconds to seconds and offer the option for live-cell and *in vivo* measurements. Current advanced fluorescence spectroscopy has a broad dynamic range and can inform on local motions (femtosecond to nanosecond timescales), chain dynamics in disordered systems (nanoseconds to microseconds), and large-scale conformational changes (milliseconds to seconds),^{2–4} and it can be applied to a variety of *in vitro*, in live cells,^{5–8} and *in vivo* samples.^{9} Thus, there is considerable interest to exploit fluorescence spectroscopic information for integrative modeling of biological processes.^{3,10}

A typical fluorescence spectroscopic modality is single-molecule Förster resonance energy transfer (smFRET). smFRET studies opened the possibility to interrogate the structure and conformational dynamics of individual fluorescently labeled biomolecules directly by the distance-dependent dipolar coupling of fluorophores,^{11–15} provided mechanistic insights in diverse areas of biological research, and could pave the way toward dynamic structural biology.^{2} Examples of biomolecular processes studied by smFRET are folding and unfolding transitions,^{16–19} dynamics of intrinsically disordered proteins,^{20–24} conformational dynamics of nucleic acids^{25–28} and proteins,^{29–32} multidomain structural rearrangements,^{33,34} and membrane receptors.^{35,36} The need for accurate and precise distance information for integrative modeling motivated a previous inter-laboratory benchmark study^{37} and the current effort of the smFRET community to establish standards for accurate processing of smFRET data.^{38} However, as will be exemplified in this manuscript, the complex conformational dynamics of multi-state systems with fast exchange kinetics can be overlooked. To this end, we generalize our previous approach that jointly interprets different spectroscopic observables to detect conformational dynamics^{39} to a general framework to highlight conformational dynamics and facilitate the interpretation of smFRET data of dynamic multi-state systems.

In smFRET experiments, a broad range of fluorescence spectroscopic observables, such as absorption and emission spectra,^{40} brightness and quantum yields,^{41–43} fluorescence lifetimes,^{44,45} and anisotropies,^{12,46} can be registered. However, the most used quantifier for FRET is the FRET efficiency, *E*, which is usually estimated by average fluorescence intensities. The FRET efficiency is the yield of the FRET process, i.e., the fraction of excited donor molecules that transfer energy to an acceptor molecule due to dipolar coupling. Besides intensities, fluorescence spectroscopy offers the anisotropy and the time-evolution information as quantifiers for FRET.^{11,47–49} Here, we provide a simple framework that combines information from fluorescence intensities and time-resolved observables. While we focus on revealing and interpreting conformational dynamics in smFRET experiments, our framework can be, in principle, applied to all FRET experiments where intensity and time-resolved information are registered simultaneously, such as fluorescence lifetime imaging (FLIM).

SmFRET experiments are performed on either freely diffusing molecules or molecules tethered to surfaces. In experiments on freely diffusing molecules, the molecules are excited and detected by confocal optics with point detectors.^{50} In experiments on surface-immobilized molecules, the molecules are typically excited by total internal reflection fluorescence (TIRF) and detected by cameras.^{51} The readout time limits the time resolution in camera-based detection to ∼1–10 ms.^{52} When paired with time-correlated single-photon counting (TCSPC), point detectors offer a significantly higher time-resolution with picosecond timing precision that enables accurate measurements of the fluorescence lifetimes. Fluorescence spectroscopy provides multidimensional observables that can be registered in parallel. A parallel spectral-, polarized-, and time-resolved registration of photons is called MFD (multiparameter fluorescence detection). Simultaneous registration of multiple fluorescence parameters by MFD has been widely applied to study the conformational dynamics of biomolecules in our and other groups.^{50,53–59}

Due to its time resolution, confocal detection is particularly well-suited to study fast biomolecular dynamics. Various approaches have been developed to reveal and quantify dynamics by analyzing fluorescence intensities in confocal smFRET experiments. Different maximum likelihood approaches take advantage of the color information and the arrival time of single photons to determine kinetic rates from the unprocessed photon streams.^{60,61} An analysis of FRET efficiency histograms (FEHs) of single molecules reveals and informs on single-molecule kinetics. By variation of the integration time, dynamics are identified by changes of the FEH shapes.^{39,62,63} FEHs can be described by a combination of Gaussian distributions to reveals kinetic rate constants.^{64} For more accurate analysis, the shot noise in FEHs is explicitly accounted for in (dynamic) photon distribution analysis (PDA).^{39,65} Alternatively, variance analysis of the FRET efficiencies of single molecules reveals heterogeneities, e.g., by comparing the average photon arrival times in the donor and FRET channels.^{56,66} In burst variance analysis (BVA), the variance of the FRET efficiency is estimated, and dynamics are detected if the variance exceeds the shot noise limit.^{67} The two-channel kernel density estimator (FRET-2CDE filter) method applies a similar approach to detect anticorrelated fluctuations of the donor and acceptor signal.^{68} Finally, very fast conformational dynamics on the sub-millisecond timescale can be determined by fluorescence correlation spectroscopy,^{15,69} where the donor and FRET-sensitized acceptor fluctuations in the signal result in a characteristic anti-correlation in the cross-correlation function.^{15,32} For a robust estimation of the timescales of exchange, the contrast in fluorescence correlation spectroscopy (FCS) can be amplified in filtered-FCS by statistical filters that use spectral, lifetime, and anisotropy information registered in MFD experiments.^{70,71}

Even though various analysis methods have been developed for intensity-based FRET experiments, interpreting the data of systems with fast kinetics remains challenging. Two factors contribute to the problem. First, many analysis approaches require the kinetic model to be defined *a priori*. Second, most analysis methods are applied to reduced, one-dimensional representations of the experimental data, such as the FEH, the fluorescence decay, or the correlation function, which alone often do not provide sufficient information to distinguish between competing models. Hence, the model selection often remains ambiguous on the level of the individual data representations while also being the deciding factor for the correct interpretation and quantification of the observed dynamics. A solution to this problem is to exploit the multidimensionality of the smFRET data in MFD experiments, where it has early been recognized that conformational dynamics could be detected in two-dimensional burst frequency histograms of the FRET efficiency, *E*, and donor fluorescence lifetime, τ_{D}.^{72} The different averaging behavior of the two observables produces distinct dynamic fingerprints in the two-dimensional plots. To describe these patterns, parametric relationships have been introduced to describe static molecules,^{72} those undergoing dynamic exchange between two^{39} or three^{73} states, folding–unfolding transitions,^{74,75} and disordered systems described by idealized polymer models.^{20} Here, we call the guidelines defined by these parametric relationships “FRET-lines.” Despite their wide application by expert users, there is a lack of a comprehensive overview and description of a general formalism to compute FRET-lines, especially if experimental complications, such as the dynamics of the flexibly coupled fluorophores, have to be considered. We present a detailed discussion of the different theoretical and practical aspects of FRET-lines. To interpret two-dimensional burst frequency histograms computed using average intensities and lifetimes, we first introduce the average observables and relate them to conformational heterogeneities (Fig. 1, Concepts). Using a simple two-state system, we describe how model parameters can be recovered from the FRET-lines. Next, we present the definition of the FRET-lines and provide a rigorous framework for their calculation. We present transformations that can be applied to experimental data that directly visualize conformational heterogeneity and can be used to resolve the species population of exchanging states and generalize the concepts presented for two-state systems to multi-state systems (Fig. 1, Concepts). The second part of this manuscript assembles the most relevant equations needed to interpret data of static and dynamic systems for dyes that are fixed stiffly to the molecule of interest (Fig. 1, Fixed dyes), and presents conformational heterogeneity caused by flexibly coupling dyes is accounted for (Fig. 1, Flexible dyes). Finally, we present FRET-lines that can inform on an order-disorder transition (Fig. 1, Disordered systems).

## II. THEORY

### A. Förster resonance energy transfer

*D*) to an acceptor (

*A*) fluorophore by dipolar coupling that depends strongly on the interdye distance

*R*

_{DA}. The rate constant of the energy transfer from

*D*to

*A*,

*k*

_{RET}, depends on the distance between the donor and the acceptor transition dipole moments,

^{11}

*k*

_{F,D}is the radiative rate constant of the donor, Φ

_{F,D}is the fluorescence quantum yield of the donor,

*R*

_{0}is the dye-pair specific Förster radius, and

*R*

_{DA}is the

*DA*-distance. The Förster radius,

*R*

_{0}, depends on the mutual orientation of the fluorophore dipoles, captured by the orientation factor

*κ*

^{2}. Moreover,

*R*

_{0}depends on the spectral overlap integral $J\lambda $, the refractive index of the medium,

*n*, and Φ

_{F,D}, the quantum yield of the donor fluorophore,

*N*

_{A}is Avogadro’s constant. The spectral overlap integral is defined by $J\lambda =\u222bfD\lambda \epsilon A\lambda \lambda 4d\lambda $, where $fD\lambda $ is the normalized emission spectrum of the donor and $\epsilon A\lambda $ is the extinction of the acceptor at wavelength

*λ*. For simplicity, we focus on dyes that reorient fast compared to the FRET rate constant. For such a case,

*κ*

^{2}can be approximated by the isotropic average, $\kappa 2iso=2/3$. This approximation applies to freely rotating dyes that are flexibly coupled to biomolecules via long linkers.

^{33,37,76,77}

*E*, by

*τ*

_{D(0)}and

*τ*

_{D(A)}are the donor fluorescence lifetimes in the absence and presence of the acceptor. The FRET efficiency is related to the interdye distance,

*R*

_{DA}, by

^{11}

### B. Time-resolved fluorescence

*t*of photons emitted by a donor with a fluorescence lifetime

*τ*

_{D(0)}that is quenched by FRET with a rate constant,

*k*

_{FRET}, follows an exponential decay,

*τ*

_{D(A)}is the fluorescence lifetime of the donor in the presence of FRET. Here, the superscript of the time-resolved fluorescence signal $fD|DDA$ denotes that the sample is a FRET sample, and the subscript indicates that the fluorescence is measured in the donor channel after donor excitation. If the radiative rate constant,

*k*

_{F,D}, is independent of the FRET rate, the fluorescence decay of a mixture of species with different fluorescence lifetimes with the lifetime distribution

*p*(

*τ*

_{D(A)}) is given by

*R*

_{DA}, directly as

*p*(

*R*

_{DA}) from the FRET-sensitized donor fluorescence decay. The interpretation hereby depends on the choice of the model function for

*p*(

*R*

_{DA}); thus, it is important to consider the broadening introduced by the flexible linkers, which will be discussed in detail in Sec. IV B.

In this work, it is assumed that the properties of the fluorophores do not vary for different conformational states of the molecule (homogeneous approximation). In practice, this assumption does often not hold when the environment of the fluorophores changes, leading to local quenching by aromatic residues, spectral shifts, or sticking interactions with the biomolecular surface. For details on how to account for a correlation between the photophysical and conformational states, the reader is referred to Ref. 48.

### C. Intensity-based observable: FRET efficiency

*F*that is fully corrected for the quantum yields and detection efficiencies, the FRET efficiency is given by

*DA*is a FRET sample), and the subscripts refer to the excitation, (…|

*X*), and detection, (

*X*|…), channels.

*D*and

*A*refer to the donor and acceptor fluorophore, respectively. For instance, $FA|D(DA)$ is the fluorescence intensity of the acceptor ($A|\u2026$ of a FRET molecule (

*DA*), given that the donor was excited (…|

*D*). In practice, the detected raw signals in the donor, (

*I*

_{D|D}), and acceptor, (

*I*

_{A|D}), channels need to be corrected (for details, see Ref. 37) to yield fluorescence intensities,

*F*.

*F*is determined by integrating the fluorescence intensity decay $ft$. For the distribution of fluorescence lifetimes,

*p*(

*τ*

_{D(A)}), the integrated donor fluorescence intensity is given by

*t*is equivalent to the fluorescence lifetime, $\u222be\u2212t/\tau DAdt=\tau DA$, and the fluorescence intensity is, hence, proportional to the species-averaged fluorescence lifetime, $\tau DAx$,

*E*to the time-resolved fluorescence decays of the donor in the presence and absence of FRET, $fD|D(DA)t$ and $fD|D(D0)t$,

### D. Lifetime-based observable: Average delay time

In smFRET experiments with pulsed excitation, the detected photons are characterized by their delay time with respect to the excitation pulse. Due to the limited number of photons available in a single-molecule experiment, it is impossible to recover the distribution of fluorescence lifetimes *p*(*τ*_{D(A)}). However, an average delay time, $t$, can be determined reliably.

*p*(

*τ*

_{D(A)}), the average $t$ is then

*F*(

*τ*

_{D(A)}), of a species with a lifetime

*τ*

_{D(A)}is proportional to its fluorescence lifetime,

So far, we have assumed that the fluorescence is excited by an ideal *δ*-pulse. Experimentally, the analysis is complicated due to the finite width of the laser excitation pulse and characteristics of the detection electronics, defining the instrument response function (IRF). In the analysis, the IRF is accounted for by convolution with the ideal decay model. For low photon numbers, accurate lifetimes are best extracted using a maximum likelihood estimator (MLE) that correctly accounts for the noise characteristics of the photon detection, anisotropy effects, and the presence of the background signal.^{78} The fluorescence lifetime obtained by maximizing the likelihood function is equivalent to the intensity-averaged fluorescence lifetime, i.e., $\tau MLE=\tau DAF$ (see the supplementary material, Note 1).

## III. CONCEPTS

### A. Detecting dynamics using the fluorescence lifetime information

The detection and analysis of fast conformational dynamics in smFRET experiments remains challenging due to the limited signal collected for each single molecule event [Fig. 2(a)]. Here, kinetics is considered fast if the associated exchange of states happens on a timescale comparable to or faster than the observation time. In confocal experiments, the upper limit of the observation time is set by the diffusion time of a molecule in the confocal volume. The photon detection rate determines the lower limit of the observation time. In a typical confocal smFRET experiment, usually less than 500 photons are detected per single molecule in an observation time of a few milliseconds. For each molecule, the FRET efficiency, *E*, is calculated from the integrated fluorescence intensities. As most a few hundred photons are registered, only an average FRET efficiency, *E*, can be estimated reliably for each molecule, and the kinetic information is partially lost [Fig. 2(a)].

*τ*

_{D(A)}. Importantly,

*t*corresponds to the intensity weighted average fluorescence lifetime $\tau DAF$. The fluorescence lifetime of the donor in the presence of FRET fluctuates with the FRET efficiency [Fig. 2(b)]. For a donor dye with a mono-exponential fluorescence decay and a fixed distance between the dyes, the quantities

*E*,

*τ*

_{D(A)}and $\tau DAF$ are related as follows:

*E*and $\tau DAF$ follow the linear dependence: $\tau DAF=\tau D0(1\u2212E)$. We call the reference line described by this relation the ideal static FRET-line, as it is valid for single molecules and ensembles with a single FRET efficiency.

^{78}In this case, the FRET efficiency relates to the species average of lifetimes $\tau DAx$. On the other hand, the intensity-weighted average fluorescence lifetime $\tau DAF$ is determined by the donor intensity, and species with a smaller FRET efficiency contribute more photons to the donor fluorescence decay. Therefore, the estimated average lifetime, $t=\tau DAF$, is biased toward longer fluorescence lifetimes compared to the species average $\tau DAx$ [Fig. 1(b)],

*E*and $\tau DAF$ correspond to different averages, the pair of experimental observables $E,\tau DAF$ reveals sample dynamics and heterogeneities through a deviation from the ideal behavior. In single-molecule counting histograms of $E,\tau DAF$, heterogeneities are identified by a shift of populations from the reference static FRET-line [Fig. 2(c)].

There are many ways to compute other reference lines that relate a FRET efficiency, *E*, to an average fluorescence weighted lifetime, $\tau DAF$. We call any parametric relation between the FRET observables a “FRET-line.” FRET-lines can serve as valuable guides to interpret experimental distributions because they relate model parameters to experimental observables, identify dynamic populations, and allow to understand the dynamic exchange in complex kinetic networks encoded as an experimental fluorescence fingerprint.

### B. Detecting dynamics by intensity-based approaches

In purely intensity-based approaches, the average inter-photon time limits the ability to detect conformational dynamics. We demonstrate this limitation by simulations of smFRET experiments of molecules that undergo conformational dynamics between distinct states at increasing interconversion rates. We simulate typical smFRET experiments with a count rate per molecule of 100 kHz. The simulated smFRET data were processed using the popular burst variance analysis (BVA)^{67} procedure.

*E*

_{i}is the FRET efficiency of a sample,

*M*is the total number of samples, and

*E*is the average FRET efficiency of the single-molecule event obtained by Eq. (9). The standard deviation

*σ*

_{E}is then plotted against the average FRET efficiency

*E*[Fig. 3(b)]. The lower boundary for the standard deviation of the FRET efficiency is given by the theoretical shot noise limit, determined by the number of photons per sample

*N*,

^{79}

The simulated data were processed by BVA with a photon window of *N* = 5 [Figs. 3(b)–3(f)]. A standard deviation *σ*_{E} observed in BVA that exceeds the shot noise limit decreases as the dynamics become faster [Figs. 3(b)–3(f)]. In the simulations, the average inter-photon time was 10 *µ*s, and the time resolution is further reduced due to the need to average over a given photon number (typically, *N* = 5).^{67} All faster processes than this limit will be averaged over the sampling time and, thus, not detected as dynamic [Figs. 3(e) and 3(f)]. The dependency on the timescale of dynamics makes it difficult to predict the exact shape of the observed distributions, which requires taking into account the experimental photon count distribution.^{67} Hence, they have mainly been used as qualitative indicators of conformational dynamics. It should be noted that dynamics on timescales faster than the inter-photon time can still be detected by fluorescence correlation spectroscopy (FCS), wherein the effective time resolution is determined mainly by the signal-to-noise ratio. However, in contrast to the single-molecule analysis, it is challenging to directly identify states or their connectivity from the FCS curves.

On the other hand, using the relation between the FRET efficiency *E* and the intensity-averaged donor fluorescence lifetime $\tau DAF$, conformational dynamics are identified even if they are fast [Figs. 3(g)–3(k)]. This lifetime-based indicator is independent of the detection count rate because it relies only on the deviation of the fluorescence decay from the ideal single-exponential behavior. Hence, all dynamic processes that are slower than the fluorescence lifetime (>ns) are detected, and no decrease of the dynamic shift is observed at increasing timescales of the dynamics. Combining fluorescence lifetimes and intensities is, thus, superior in detecting and visualizing fast conformational dynamics than approaches that rely on intensities alone.

In practice, one must consider potential artifacts that lead to a false-positive detection of dynamics. Examples include dark states of the acceptor (e.g., due triplet states). Acceptor dark states always affect intensity-based indicators of dynamics as they result in fluctuations of the apparent FRET efficiency. The effect of dark acceptor states on the donor fluorescence lifetime depends on the nature of the photophysical change. Triplet states often still act as FRET acceptors with a similar Förster radius as the single state; a similar situation is found for the *cis–trans* isomerization of cyanine dyes, such as Cy5.^{80–82} Radical or ionic dark states, on the other hand, often are not viable FRET acceptors. In this case, the donor lifetime will fluctuate as a function of the photophysical state of the acceptor.^{83}

### C. FRET-lines of static and dynamic molecules

*E*and $\tau DAF$, the static FRET-line is defined as

*E*–$\tau DAF$ histogram.

*k*

_{12}and

*k*

_{21}are the microscopic interconversion rates between the two states that define the probability that a molecule spends a fraction of time

*x*

^{(i)}in state

*i*during the observation time. The fractions

*x*

^{(i)}are stochastic quantities and change from one observation to another. For now, we are not interested in the exact distribution of the state occupancy

*x*

^{(1)}and treat it as the independent parameter of the model. This is equivalent to the assumption of a uniform distribution for

*x*

^{(1)}. The effect of the distribution of state occupancies is discussed in detail in Paper II.

*x*

^{(1)}and

*x*

^{(2)}= 1 −

*x*

^{(1)},

*E*and $\tau DAF$, we find the line that describes all values of

*x*

^{(1)}by combining Eqs. (24) and (25), relating the species-weighted average lifetime to the intensity-weighted average lifetime,

*x*

^{(1)}being in the interval [0, 1]. Because Eq. (27) describes the FRET-line for a binary system in dynamic exchange, we call it the

*dynamic FRET-line*. Dynamic FRET-lines connect two static states. They were first introduced by Kalinin

*et al.*,

^{39}and Gopich and Szabo

^{73}later described analogous relations.

Figure 4 illustrates the concept of static and dynamic FRET-lines. Static FRET-lines describe pure states, which are described by sharp distributions (*δ*-functions) in terms of the lifetime distribution $p(\tau DA)$ [Fig. 4(a)]. In contrast, dynamic FRET-lines describe the mixing of two pure states as a function of the state occupancy *x*^{(1)}. The corresponding donor fluorescence decays are single-exponential for pure states and bi-exponential in the case of mixing between pure states [Fig. 4(b)]. In the $E\u2212\tau DAF$ plot, the dynamic FRET-line connects the two points of the contributing pure states on the static FRET-line by a curved line [Fig. 4(d)].

*dynamic shift*, ds, orthogonal to the static FRET-line [Fig. 4(d)]. Like the dynamic FRET-line, the value of the dynamic shift depends only on the FRET efficiencies of the limiting states

*E*

_{1}and

*E*

_{2}, and is given by (see the supplementary material, Note 2)

*E*against the

*normalized*intensity-averaged donor fluorescence lifetime, $\tau DAF/\tau D0$. Exemplary dynamic FRET-lines for different FRET efficiencies

*E*

_{1}and

*E*

_{2}are shown in Fig. 4(e) with their corresponding dynamic shifts. By visualizing the dynamic shift as a function of the FRET efficiencies of the limiting states [Fig. 4(f)], one can define sensitive and insensitive regions depending on a given detectability threshold for the dynamic shift. This threshold depends on how well the experimental setup is calibrated, the accuracy of the fluorescence lifetime estimation, and the measurement statistics that typical threshold values for the detectability of dynamic shifts are on the order of 0.05 or less, potentially reaching a sensitivity of 0.01 for well-calibrated setups and carefully performed experiments. This places the purple dynamic FRET-line shown in Fig. 4(e) on the border of the insensitive region, while the other two examples with a dynamic shift above 0.1 are clearly in the sensitive region.

### D. General definition of FRET-lines

FRET-lines are idealized relations between the FRET-related experimental observables *E* and $\tau DAF$ for different physical models of the system. Before considering more specific scenarios, such as the effect of the flexible dye linkers or disordered systems, we first present a general definition of FRET-lines.

*p*(Λ),

*E*and $\tau DAF$ as a function of the model parameters Λ. If we choose a fixed value for all model parameters, we obtain a single point on the $(E,\tau DAF)$ plane. If, instead, we vary a single parameter, a defined curve—the FRET-line—is obtained. Let the variable parameter be

*λ*and the fixed values for the remaining model parameters be Λ

_{f}. Then, the parametric relation between

*E*and $\tau DAF$ for a given model is obtained from the moments of the lifetime distribution by the following equations:

*x*

^{(1)}as the free parameters ($\lambda =x(1)$) and kept the FRET efficiencies constant ($\Lambda f={E1,E(2)}$).

*E*and $\tau DAF$ (or any related observable) as a function of the variable parameter

*λ*. Finally, to obtain the explicit form of the FRET-line, the free parameter

*λ*can be eliminated by substitution, and the resulting expression defines a direct relation between the observables

*E*and $\tau DAF$. A detailed description of this general formalism is given in the supplementary material, Note 3.

### E. Experimental observables and moments of the lifetime distribution

*p*(

*τ*

_{D(A)}) is equal to the expected value of the fluorescence lifetime. The second moment is given by the expected value of the square of the fluorescence lifetime. The variance Var(

*τ*

_{D(A)}) is the second

*central*moment, defined as the average squared deviation from the mean, which is related to the first and second moments by

*variance*of the lifetime distribution. Using the relations between the experimental observables

*E*and $\tau DAF$ and the moments of the lifetime distribution, we obtain

*E*

_{τ}defined as

*E*and $\tau DAF$, denoted as $\tau DA\nu x$, to make a clear distinction between the expected theoretical quantities and the experimental estimates.

*E*and $\tau DAF$. The result is identical to the expression obtained in Ref. 73, relating $\tau DAF$ to the variance of the FRET efficiency distribution,

*δ*,

*v*th moment is then given by $\tau DA\nu $, and the variance of the distribution, as given by Eq. (33), is zero, defining the equivalent static FRET-line. Thus, the static FRET-line [Eq. (22)] corresponds to the particular case of lifetime distributions with vanishing variance. For two-component lifetime distributions, the distribution of lifetimes is given by the weighted sum of two

*δ*-functions, leading to the following expression for the moments of the lifetime distribution:

*x*

^{(1)}. For the mixing between two states [Eq. (39)], the variance is then given by

*x*

^{(1)}to obtain the relation between $Var\tau DA$ and $\tau DAx$,

To illustrate that, we can indeed estimate the variance of the FRET efficiency distribution from the two experimental observables *E* and $\tau DAF$, we compare the variance estimate with that obtained from burst variance analysis (BVA) for a simulated dataset (Fig. 5).

BVA correctly identifies the presence of conformational dynamics between the two states at FRET efficiencies of 0.25 and 0.8 [Fig. 5(a)]. The variance estimate obtained from BVA, however, includes the contribution of photon shot noise Eq. (21) [black line in Fig. 5(a)], and the dynamics is shown as excess variance beyond the shot noise limit.

To obtain the contribution to the variance due to conformational dynamics ($Varc(E)$), we subtract the shot noise contribution given by $\sigma SN2=E(1\u2212E)/N$, where *N* = 5 is the photon window used for the analysis [Fig. 5(b)]. Compared to the expected variance given by Eq. (41) (pink line), BVA underestimates the variance of the FRET efficiency caused by the averaging over the photon window used in the calculation. It must also be considered that BVA measures the combined variance of the FRET efficiency caused by the contributions of shot noise and dynamics. However, these contributions are not strictly additive. The simple subtraction of the shot noise contribution performed here is, thus, only approximative. In the (*E*, $\tau DAF$) representation, the same dataset shows a dynamic shift from the diagonal line that is described by the dynamic FRET-line [Fig. 5(c)]. From the experimental observables, we calculate the variance of the FRET efficiency distribution. Unlike the variance obtained by BVA, this variance estimate represents the pure contribution of the conformational dynamics and follows the expected dynamic FRET-line. Note, however, that the molecule-wise distribution of the variance estimated from the observables *E* and $\tau DAF$ generally shows a broader distribution compared to BVA. Conceptual static and two-state dynamic FRET-lines for the mean-variance representation of the data are shown in Fig. 6(b).

### F. Alternative representation of dynamic lines

*δ*-functions, the moments of the lifetime distribution are simply given by the linear combination of the moments of the pure components [compare Eq. (39)],

*moment difference*Γ is related to the experimental observables

*E*and $\tau DAF$ by

*E*, directly connecting the two points belonging to the pure states [Fig. 6(c)].

In the difference between the first and second normalized moments Γ, we found a parameter that linearizes the dynamic mixing while retaining a simple relation for the static FRET-line. The linearization of dynamics in this *moment representation* dramatically simplifies the graphical analysis of kinetic networks by providing a direct visualization of the kinetic connectivity. To highlight its usefulness, we show the moment representation together with the histogram of the experimental observables, *E* and $\tau DAF$, in the following discussions of more complex scenarios. The moment representation resembles the analysis of fluorescence lifetimes in the phasor approach to fluorescence lifetime imaging (Phasor-FLIM).^{84} In both approaches, single-exponential fluorescence decays are found on a curve, a parabola in the moment representation, and a circle in Phasor-FLIM. Moreover, bi-exponential decays are shifted inward from the curve and lie on the line connecting the coordinates of the pure components. The phasor calculation only requires fluorescence decays and, thus, is also applicable to study quenching without FRET. In principle, the moment representation could thus be combined with the phasor information to add another dimension to the analysis. The different transformations of the observables *E* and $\tau DAF$ and their theoretical equivalents are summarized in Table I.

Model . | . | Experiment . |
---|---|---|

Probability distribution | ↔ | Random realization |

Expected value | ↔ | Experimental observable |

Probability density function | ↔ | Histogram |

FRET-lines | ↔ | Distribution of FRET efficiency, fluorescence lifetime, or related quantities |

Expectation value of FRET efficiency | ↔ | Species-averaged FRET efficiency |

$E=1\u2212\tau DA\u0304\tau D0$ | $=\u2227$ | $E=FA|DFA|D+FD|D=1\u2212\tau DAx\tau D0$ |

First moment of the lifetime distribution | ↔ | Species-averaged lifetime |

$\tau D(A)\u0304=\u222b0\u221e\tau D(A)p\tau D(A)d\tau D(A)$ | $=\u2227$ | $\tau DAx=1\u2212E\tau D0$ |

Second moment of the lifetime distribution | ↔ | Species-averaged squared lifetime |

$\tau D(A)2\u0304=\u222b0\u221e\tau D(A)2p\tau D(A)d\tau D(A)$ | $=\u2227$ | $\tau DA2x=\tau DAF1\u2212E\tau D(0)$ |

Ratio of the second and first moments of the | ↔ | Intensity-weighted average fluorescence lifetime, average delay time |

lifetime distribution | ||

$\tau D(A)2\u0304\tau D(A)\u0304$ | $=\u2227$ | $\tau DAFort$ |

No equivalent | ↔ | Intensity-weighted average FRET efficiency |

$1\u22121\tau D0\tau D(A)2\u0304\tau D(A)\u0304$ | $=\u2227$ | $E\tau =1\u2212\tau DAF\tau D(0)$ |

Variance of the lifetime distribution | ↔ | No equivalent |

$Var\tau DA=\tau D(A)2\u0304\u2212\tau DA\u03042=VarE\tau D(0)2$ | $=\u2227$ | $1\u2212EE\u2212E\tau \tau D02$ |

Difference between the normalized first and second | ↔ | No equivalent |

moments of the lifetime distribution | ||

$\Gamma =\tau D(A)\u0304\tau D0\u2212\tau D(A)2\u0304\tau D02$ | $=\u2227$ | (1 − E)E_{τ} |

Model . | . | Experiment . |
---|---|---|

Probability distribution | ↔ | Random realization |

Expected value | ↔ | Experimental observable |

Probability density function | ↔ | Histogram |

FRET-lines | ↔ | Distribution of FRET efficiency, fluorescence lifetime, or related quantities |

Expectation value of FRET efficiency | ↔ | Species-averaged FRET efficiency |

$E=1\u2212\tau DA\u0304\tau D0$ | $=\u2227$ | $E=FA|DFA|D+FD|D=1\u2212\tau DAx\tau D0$ |

First moment of the lifetime distribution | ↔ | Species-averaged lifetime |

$\tau D(A)\u0304=\u222b0\u221e\tau D(A)p\tau D(A)d\tau D(A)$ | $=\u2227$ | $\tau DAx=1\u2212E\tau D0$ |

Second moment of the lifetime distribution | ↔ | Species-averaged squared lifetime |

$\tau D(A)2\u0304=\u222b0\u221e\tau D(A)2p\tau D(A)d\tau D(A)$ | $=\u2227$ | $\tau DA2x=\tau DAF1\u2212E\tau D(0)$ |

Ratio of the second and first moments of the | ↔ | Intensity-weighted average fluorescence lifetime, average delay time |

lifetime distribution | ||

$\tau D(A)2\u0304\tau D(A)\u0304$ | $=\u2227$ | $\tau DAFort$ |

No equivalent | ↔ | Intensity-weighted average FRET efficiency |

$1\u22121\tau D0\tau D(A)2\u0304\tau D(A)\u0304$ | $=\u2227$ | $E\tau =1\u2212\tau DAF\tau D(0)$ |

Variance of the lifetime distribution | ↔ | No equivalent |

$Var\tau DA=\tau D(A)2\u0304\u2212\tau DA\u03042=VarE\tau D(0)2$ | $=\u2227$ | $1\u2212EE\u2212E\tau \tau D02$ |

Difference between the normalized first and second | ↔ | No equivalent |

moments of the lifetime distribution | ||

$\Gamma =\tau D(A)\u0304\tau D0\u2212\tau D(A)2\u0304\tau D02$ | $=\u2227$ | (1 − E)E_{τ} |

### G. Multi-state systems

*x*

^{(i)}of the measured molecule, which in the limit of fast dynamics (or long observation time) tend to the equilibrium fractions. Using the representation of the moment difference [Fig. 7(c)], it is possible to determine the state occupancies of the different states graphically from the two-dimensional plot [Fig. 7(d)]. As an example, we consider the high-FRET state 1 [red circle in Fig. 7(d)]. The red line connecting state 1 to the mixed population (orange) intersects the binary exchange line between states 2 (green) and 3 (blue) at a given point (turquoise). Then, the state occupancy

*x*

^{(1)}is obtained from the length of the segments of the red line,

*a*

^{(1)}and

*b*

^{(1)}, defined by the position of the mixed population along the line, by

*x*

^{(2)}and

*x*

^{(3)}are obtained analogously as described for

*x*

^{(1)}above, as indicated by the dashed lines in Fig. 7(c). A detailed derivation of this expression is given in the supplementary material, Note 4.

To draw such multi-state FRET-lines, it is important that the parameters of the limiting states are known. How easily this information is gathered depends on the system at hand. For slow dynamics, residual populations of pseudo-static molecules, i.e., molecules that by chance did not convert during the observation time, clearly indicate the location of the limiting states in the two-dimensional plots. On the other hand, in the case of fast dynamics, only a single population might be observed, which deviates from the static FRET-line. In this case, the fluorescence lifetimes of the limiting states might be identified from a sub-ensemble TCSPC analysis. However, distinguishing between different numbers of states in such an analysis is challenging if a low number of photons is detected, and it is recommended that a network of FRET pairs is globally analyzed.^{85} Often, the conformational equilibrium can also be modulated, allowing the populations to be shifted toward or locked into a certain conformation. Finally, prior information on the structure of stable conformational states, e.g., obtained from x-ray crystallography or nuclear magnetic resonance (NMR) spectroscopy, can be used to predict the parameters of the limiting states.

In multi-state systems, FRET-lines are especially helpful in identifying the minimal set of states and their kinetic connectivity. This information can reduce the complexity of the kinetic model by eliminating exchange pathways, providing crucial information for further quantitative analysis of the dynamic network by dynamic photon distribution analysis or fluorescence correlation spectroscopy. This aspect of FRET-lines is illustrated in detail in Paper II.

## IV. PRACTICAL ASPECTS AND APPLICATION

### A. Multi-exponential donor decays

Up until now, we have assumed that the fluorescence decay of the donor dye in the absence of the acceptor is single exponential. Experimentally, however, this condition is often violated due to the effect of the local environment on the tethered dyes. The most common mechanisms that affect the quantum yield of tethered dyes are the quenching of rhodamine or xanthene based dyes by electron-rich amino acids, such as tryptophane, through photo-induced electron transfer (PET)^{86–88} and the enhancement of the fluorescence of cyanine-based dyes due to steric restriction and dye–surface interactions that modulate the cis–trans isomerization.^{89–91} In addition, the used organic dyes may consist of a mixture of isomers with distinct fluorescence properties. The effect of multi-exponential fluorescence decays of the donor fluorophore on the static and dynamic FRET-lines depends on the timescale of the dynamic exchange between the different donor states. This exchange may be fast (e.g., in the case of dynamic quenching by PET), on a similar timescale as the observation time of a few milliseconds (e.g., for sticking of the fluorophore to the biomolecular surface), or non-existent (e.g., in the case of an isomer mixture).

*k*

_{RET}does not depend on the donor-only lifetime

*τ*

_{D(0)}. In this case, the donor fluorescence decay in the absence of FRET is described by a distribution of fluorescence lifetimes

*p*(

*τ*

_{D(0)}),

*p*(

*τ*

_{D(0)}) and $pkRET$ correspond to the donor-only lifetimes and FRET rate distributions, respectively. Note that due to the homogeneous approximation, we have factored the joint distribution of donor and FRET states, that is, $p\tau D0,kRET=p(\tau D(0))pkRET$.

*j*) and FRET (

*i*) states, respectively, and $\tau DAi,j=\tau D0(j)\u22121+kRET(i)\u22121$. From the moments, the observable $\tau DAF$ is then readily calculated according to Eq. (17).

*E*because the fluorescence intensities obtained for the different donor states are weighted by their respective quantum yields. Consequently, it becomes impossible to define a single distance-related FRET efficiency. Instead, we define the proximity ratio

*E*

_{PR}in analogy to Eq. (9) based on the average fluorescence intensities detected in the donor and acceptor channels $FD|DDA\u0304$ and $FA|DDA\u0304$ by

*γ*′ is given by the ratio of the quantum yields of the acceptor and donor fluorophores, $\gamma \u2032=\Phi F,A\Phi F,D$. See the supplementary material, Note 5, for a derivation of Eq. (55). For the moment representation, the moment difference Γ in the case of a mixture of donor states is then defined as

The effect of a mixture of two distinct photophysical states of the donor is illustrated in Fig. 8 for the (*E*-$\tau DAF$) parameter space [(a)–(c)] and in the moment representation [(d)–(f)]. We consider two different donor lifetimes of $\tau D01$ = 4 ns and $\tau D02$ = 1 ns that correspond to distinct donor quantum yields of $\Phi F,D(1)$ = 0.8 and $\Phi F,D(2)$ = 0.2. When separate measurements are performed [Figs. 8(a) and 8(d)], accurate FRET efficiencies *E* can be calculated for each measurement, and the ideal static and dynamic FRET-lines are obtained. For the dynamic exchange, we assume equilibrium fractions of $xD0(1)$ = 0.25 and $xD0(2)$ = 0.75 for the two donor states. In the case of exchange on a timescale much slower than the observation time [Figs. 8(b) and 8(e)], an individual correction of the different populations is not possible. For the proximity ratio *E*_{PR}, curved static FRET-lines are obtained for the two species as an effect of the averaging in Eq. (54). In the moment representation, this effect shows as an increased (for the species with $\tau D02$ = 4 ns) or decreased (for the species with $\tau D02$ = 1 ns) curvature of the static FRET-lines, while the linearity of the dynamic FRET-lines is retained. The effect of fast exchange between the different donor states (i.e., complete averaging during the observation time) is illustrated in Figs. 8(c) and 8(f). For the (*E*-$\tau DAF$) parameter space, a single convex static FRET-line is obtained. This line falls between the curved static FRET-lines obtained for the slow exchange and intersects the $\tau DAF$ axis at the intensity-weighted average donor fluorescence lifetime $\tau D0F$ = 2.71 ns. In the moment representation [Fig. 8(f)], the static FRET-line shows a higher curvature than the ideal static FRET-line (dashed gray line). Notably, even in the case of fast exchange between different donor states, the dynamic FRET-line in the moment representation remains linear (see the supplementary material, Note 5). Here, we have not considered the calculation of accurate FRET efficiencies for distributions of donor and acceptor states and instead introduced the proximity ratio. Using the general formalism introduced here, however, reference static and dynamic FRET-lines can still be defined even for uncorrected data if the corrections are instead accounted for in the FRET-lines.

Experimentally, the presence of multiple states of the donor fluorophores can be detected by a careful analysis of the donor-only population or, preferentially, a separate donor-only sample. For slow dynamics, the histogram of the molecule-wise donor fluorescence lifetimes will reveal clearly separated peaks. The FRET sample can then be analyzed in two ways. If it is possible to separate the two donor states, distinct values for the correction factor *γ* can be applied to the two populations based on the different donor quantum yields [Figs. 8(a) and 8(d)]. If such a separation is not possible, the proximity ratio, *E*_{PR}, should be used instead of the FRET efficiency, *E*, to calculate the FRET-lines corresponding to the different photophysical states of the donor fluorophore [Figs. 8(b) and 8(e)]. For fast exchange, the presence of multiple donor states is detected from the number of lifetime components in a TCSPC analysis. In this case, a common FRET-line should be computed in the *E*_{PR}-$\tau DAF$ parameter space as described above, since the donor states will be averaged for all FRET states [Figs. 8(c) and 8(f)].

### B. Dye-linker dynamics

So far, we have assumed that a conformational state of the molecule is described by a single donor fluorescence lifetime and will be represented by a point lying on the ideal diagonal static FRET-line. A heterogeneous mixture of molecules with different FRET efficiencies, i.e., different donor–acceptor distances, would then follow this static FRET-line. This line does, however, not describe experimental data accurately. It is consistently observed that the population mean of static molecules deviates from the ideal static-FRET-line, exhibiting a bias toward longer fluorescence-weighted donor lifetimes, $\tau DAF$. The deviation from the ideal static FRET-line is caused by the use of long flexible linkers of 10–20 Å length that tether the fluorophore to the biomolecules.^{39,76,92} Fast variations of the donor–acceptor distance *R*_{DA} during the observation time result in a distribution of donor lifetimes $p(\tau DA)$ that are sampled in each single-molecule event [Figs. 9(a) and 9(b)]. Due to the finite width of the distribution, the population is, thus, shifted toward longer donor fluorescence lifetimes, whereby the deviation from the ideal static FRET-line increases with the increasing linker length and, thus, distribution width *σ*_{DA} [Fig. 9(c)]. Recently, we estimated that the translational diffusion coefficient of dyes tethered to proteins is on the order of 5–10 Å^{2}/ns.^{48} Assuming free three-dimensional diffusion, this estimate of 10 Å^{2}/ns would translate to an expected root-mean-square displacement $x=6Dt$ of ∼10 Å per 2 ns, resulting in significant changes of the inter-dye distance during the excited state lifetime.^{93,94} However, it is to be expected that the effective displacement is reduced due to the restriction of the dye’s movement by the linker. Under the assumption that the diffusion of the fluorophore is slow compared to the fluorescence lifetime, the fluorescence decays may be approximated by a static distribution of distances.^{48,76,95} The observation time for every single molecule on the order of milliseconds is long compared to the diffusional motion of the dyes, resulting in complete averaging of the spatial distribution of the dyes around their attachment during the observation time. Under these assumptions, we can calculate the averaged quantities and moments of the lifetime distribution based on the equilibrium distance distribution.

Different approaches for modeling the spatial distribution of tethered fluorophores have been developed.^{76,96–98} In the accessible volume (AV) approach,^{92} the possible positions of the fluorophore in the three-dimensional space are identified through a geometric search algorithm. By considering all possible combinations of donor–acceptor distances, the inter-dye distance distribution can be obtained from the accessible volumes of the donor and acceptor dyes. Extensions of the AV approach have incorporated surface trapping of fluorophores^{3,48} or accounted for the energetic contributions of linker conformation.^{8,99} More accurate models of the spatial distribution of tethered dyes are obtained by coarse-grained^{48,98,100} or all-atom^{101,102} molecular dynamics simulations, from which explicit inter-dye distance distributions may be obtained. However, as will be discussed below, the contribution of the linker flexibility is mainly defined by the width of the inter-dye distance distribution, *σ*_{DA}, and shows only a weak dependence on the explicit shape of the distribution. Experimentally, the width of the linker distribution may be obtained from the fluorescence decay of the donor by modeling the fluorescence decays with a model function that includes a distribution of distances.^{48,99} Alternatively, by using the two-dimensional histogram of $\tau DAF$ vs *E*, one can vary the width parameter of the static FRET-line such that it intersects with the population of static molecules. Typically, we consider a fixed standard deviation of $\sigma DA\u223c6A\u030a$ that satisfies benchmarking experiments on rigid DNA molecules.^{39,76}

#### 1. General description in the presence of distance fluctuations

*T*

_{obs}(limited by the diffusion time

*t*

_{diff}), allowing us to treat the distance distribution as stationary,

*R*

_{DA}, by

*p*(

*R*

_{DA}), are obtained from Eq. (32) by a change of variables $\tau DA\u2192\tau DARDA$ given by Eq. (58),

*R*

_{mp}(here defined as the distance between the mean positions) and a set of parameters Λ that describe the shape of the distribution (e.g., its width). To construct the static FRET-line for a given distance distribution model

*p*(

*R*

_{DA}|

*R*

_{mp}, Λ), we vary the mean donor–acceptor distance,

*R*

_{mp}, and compute the fluorescence averaged lifetime and the FRET efficiency from the moments of the lifetime distribution given by Eq. (59). The integrals in Eq. (59) are difficult to solve analytically, even for the simple case of a normal distribution of distances, due to the sixth-power dependence between $\tau DA$ and

*R*

_{D(A)}. However, they can be calculated numerically for arbitrary models of the distribution. The shape of the distribution may potentially also depend on the conformation of the biomolecule and, thus, the donor–acceptor distance

*R*

_{mp}, in which case the shape parameters would depend on the conformation, Λ → Λ(

*R*

_{mp}). From the experimental observables

*E*and $\tau DAF$, we can only access the first and second moments of the lifetime distribution. Consequently, it is not possible to address the shape of the lifetime distribution $p(\tau DA)$ explicitly. The same dynamic shift can thus be observed for different distributions, as long as their mean and variance (or equivalently, their first and second moments) are identical.

#### 2. FRET-lines of flexibly linked dyes

*R*

_{mp}and show no interaction with the biomolecule. In this case, the inter-dye distance vector,

*R*_{DA}, is also normally distributed with width

*σ*

_{DA}= $\sigma D2+\sigma A2$, where

*σ*

_{D}and

*σ*

_{A}are the width of the spatial distributions of the donor and acceptor dyes, respectively. The distribution of inter-dye distances,

*R*

_{DA}, is then given by the non-central χ-distribution with the distance between the mean positions of the dyes,

*R*

_{mp}, as the non-centrality parameter and

*σ*

_{DA}as the width parameter,

*R*

_{μ}and a width

*σ*

_{DA}taken at non-negative values

*R*

_{DA}≥ 0 (positive truncation) to avoid non-sensical distance values below zero. At small variance-to-mean ratios (i.e., at large distances), the

*χ*-distribution tends to the normal distribution. Therefore, the distribution

*p*(

*R*

_{DA}) may be approximated by a normal distribution with mean inter-dye distance

*R*

_{mp},

*R*

_{DA}≥ 0 results in a significant deviation from the

*χ*-distribution at small inter-dye distances [see Fig. 10(b)]. Compared to the

*χ*-distribution, the truncated Gaussian distance model overestimates the contribution of small distances (corresponding to high FRET efficiencies). Overall, this results only in minor deviations of the generated static FRET-lines compared to the

*χ*-distribution, which are most pronounced at large distribution widths and high FRET efficiencies [Fig. 10(c)]. However, the two models show significant deviations in terms of the average FRET efficiency at identical center distances

*R*

_{mp}. To illustrate this effect, we plot the change of the average FRET efficiency at constant

*R*

_{mp}and increasing

*σ*

_{DA}for the Gaussian and

*χ*distance distributions in Fig. 10(c) (see solid blue and red lines, respectively). The deviation of the average FRET efficiencies between the two models increases with increasing width

*σ*

_{DA}. Notably, the interpretation of average FRET efficiencies in terms of the distance between the mean positions of the dyes

*R*

_{mp}is, thus, biased by choice of the model function for the linker distribution.

In summary, the choice of the distance distribution model function has only a minor effect on the shape of the static FRET-lines, which is mainly determined by the width parameter. However, we propose that the *χ*-distribution should be preferred for the interpretation of linker-averaged FRET efficiencies in terms of physical distances when broad linker distributions are expected.

### C. Conformational dynamics in the presence of linker broadening

^{(dyn)}is the set of parameters describing the conformational dynamics, i.e., the transition rate matrix. The linker distributions in the different conformational states are given by $p(RDA|Rmpc,\Lambda lc)$, whereby the parameters of the linker distance distribution, $\Lambda l(c)$, may potentially be different for the conformational states. The combined distance distribution $pRDA$ then takes the following general form:

*χ*-distribution characterized by the mean inter-dye distance $Rmp(c)$ and its corresponding width, $\sigma DA(c)$. For the case of two conformational states, the combined distribution of inter-dye distances integral in Eq. (62) then simplifies to the discrete sum

*x*

^{(1)}and numerically calculating the moments, as described in Eq. (59).

#### 1. Separating the contributions of linkers and conformational dynamics

*c*, $\tau DAl(c)$ and $\tau DA2l(c)$, defined as

*E*and $\tau DAF$ of the pure states. The linker-averaged moments then replace the corresponding powers of the pure state lifetimes in the calculation of dynamic FRET-lines [Eq. (39)]. Thus, the moments of the lifetime distribution for two-state dynamic exchange, i.e.,

*c*∈ {1, 2}, are given by

*x*

^{(1)}as described before. Therefore, the linearity of the dynamic mixing of the moments for conformational dynamics is still valid in the presence of linker fluctuations. Dynamic FRET-lines, thus, stay linear in the moment representation.

Dynamic FRET-lines in the presence of flexible linkers are illustrated in Fig. 11. In the $(E,\tau DAF)$ representation, it is not possible to perform a simple graphical construction of the dynamic FRET-line for flexible linkers. In the moment representation, however, the dynamic FRET-line for flexible linkers is simply obtained by connecting the linker-averaged coordinates of the two states. This simplification has important implications for the accurate description of dynamic FRET-lines in complex experimental systems, where no model for the linker distribution is available. Consider, for example, the case that the width of the linker distribution depends on the inter-dye distance in an unknown manner. In this case, it is not possible to obtain a general static FRET-line. However, with the presented formalism, we only require the knowledge of the positions of the limiting static states, which are sufficient to fully describe the corresponding dynamic FRET-line. For the $(E,\tau DAF)$ representation, the linker-averaged first and second moments of the limiting states, $\tau DAli$ and $\tau DA2li$, can be determined from the averaged FRET observables *E* and $\tau DAF$ of the static populations, from which the dynamic FRET-line is obtained by a linear combination of the moments [Eq. (68)] and conversion back into the $(E,\tau DAF)$ parameter space. In the moment representation, the dynamic FRET-line is simply obtained graphically by connecting the conformational states with a straight line. Thus, for the construction of the dynamic FRET-line, it is generally not required to know the linker distance distribution in analytical form. If structural information is available, the linker distribution may also be obtained from the accessible volumes of the dyes in distinct conformations. In a three-state system, the dynamic FRET-lines in the presence of flexible linkers are shifted toward the center of the area enclosed by the limiting lines (Fig. 12).

### D. FRET-lines of flexible polymers

In Secs. IV B and IV C, we have described the contributions of the flexible linkers to the static and dynamic FRET-lines. Through the stationary distance distribution, the effects of the fast linker dynamics could be accounted for. In principle, the linkers are equivalent to short flexible polymers, which may be treated analogously to the procedure described above when a model for the equilibrium distance distribution is available. In the following, we present FRET-lines for different polymer models in the context of the potential application to the study of flexible biological polymers, such as unfolded or intrinsically disordered proteins.

#### 1. Disordered states

Single-molecule FRET measurements are particularly suited to characterize biomolecules with partial or lack of stable tertiary structures, such as unfolded proteins, intrinsically disordered proteins (IDPs), and proteins with intrinsically disordered regions (IDRs).^{103–108} In the one-dimensional analysis of FRET efficiency histograms, the information about the fast dynamics of these systems is hidden, and complementary methods, such as small-angle x-ray scattering (SAXS), have to be employed to assert the presence of disorder.^{104–106,109} In contrast, the knowledge of the fluorescence weighted average lifetime $\tau DAF$ in addition to the FRET efficiency, *E* allows dynamics to be identified directly from the single-molecule FRET experiment. As described above, these quantities allow one to address the mean and variance of the distribution of fluorescence lifetimes and, thus, contain information about the mean and variance of the distribution of inter-dye distances [Figs. 13(a)–13(c)]. Here, we outline how to exploit this information to characterize IDPs and proteins with IDRs by means of FRET-lines of polymer models.

*μ*s.

^{110–112}In the measurement, a single population is then observed at a position that corresponds to the average over the continuous distribution of conformations. We first consider that the disordered system is described by a Gaussian chain (GC) model. This model approximates the conformational space by a quasi-continuum of states and has previously been applied to the description of experimental single-molecule FRET histograms of

*E*and $\tau DAF$ of IDPs.

^{20}The distribution of interdye distances is given by the central

*χ*-distribution and depends only on the variance of the interdye distance, $\sigma DA2$,

*σ*

_{DA}), only a single Gaussian chain FRET-line may be constructed. This FRET-line describes all polymers that behave like an ideal Gaussian chain [red line in Figs. 13(d) and 13(e)]. It can be thought of as a reference line for polymers that describes how ideal the studied system behaves, analogous to the static FRET-line for structured systems. More realistically, a disordered peptide chain may be described by the worm-like chain (WLC) model (see the supplementary material, Note 6).

^{113,114}The parameters defining the inter-dye distance distribution of the WLC model are the total chain length

*L*and the persistence length

*l*

_{p}that define the stiffness of the chain by $\kappa =lpL$. In principle, the total length of the chain is known

*a priori*from the protein sequence. From the experimentally observed position of the population in the two-dimensional histogram, the stiffness of the chain can then be estimated. FRET-lines for the WLC model are shown for different combinations of the parameters

*κ*and

*L*in Figs. 13(c) and 13(d). Notice that different combinations of

*κ*and

*L*can result in identical FRET efficiencies, as indicated by the horizontal line in the plot. To determine both parameters, in addition, $\tau DAF$ needs to be known.

#### 2. Order–disorder transitions

*x*

^{(f)}is the species fraction of the molecules in the folded state, $pfRDA|Rmp(f)$ describes the linker distribution in the folded state around the average distance $Rmp(f)$ [Fig. 14(b)], and $p(u)RDA|\Lambda (u)$ describes the distance distribution in the unfolded state, dependent on the parameters of the polymer model, Λ

^{(u)}[Figs. 14(a)–14(c)]. By varying

*x*

^{(f)}while keeping the parameters of the distance distributions [$Rmp(f)$ and Λ

^{(u)}] constant, the dynamic FRET-line is obtained. These FRET-lines are conceptually identical to dynamic FRET-lines describing the exchange between two folded states under the assumption that the sampling of the distance distribution in the unfolded state is fast compared to the transition rate to the folded state (see Sec. IV C 1). The broad distance distribution of the unfolded state shifts the endpoint of the resulting folding FRET-line far from the static FRET-line [Figs. 14(d) and 14(e)]. Dynamic transitions between a single folded state, characterized by $Rmp(f)$, and different unfolded states, each described by the WLC model with varying stiffness at constant length ($\Lambda (u)={\kappa ,L}$), are illustrated in Figs. 14(c) and 14(d) in the $(E,\tau DAF)$ and moment representations. Notice how all unfolded states are described by a single curve defined by the total chain length. Even though both folded and unfolded states are described by a distribution of distances, the folding FRET-line in the moment representation remains linear [Fig. 14(d)]. Dynamic unfolding FRET-lines describe folding/unfolding transitions of proteins similar to binary dynamic FRET-lines.

^{75,115}The position of the population on the folding/unfolding FRET-lines informs on kinetic rate constants of the folding/unfolding events (see Paper II). For fast-folding/unfolding transitions on the microsecond timescale, the position of the population along the folding/unfolding FRET-line may thus be used to determine the equilibrium constant of the folding process.

#### 3. Application of FRET-lines to experimental data

In this section, we review the application of FRET-lines and the moment representation to experimental data by revisiting published data on three different proteins as prototypic examples for static multi-state dynamic and disordered systems.

As a first example, we consider the protein Syntaxin-1, a member of the SNARE (soluble N-ethylmaleimide-sensitive factor attachment protein receptors) family of proteins that play a central role in membrane fusion.^{116} We have previously shown that Syntaxin-1 fluctuates between a closed and open conformation with a detached SNARE motif on the sub-millisecond timescale, while the Habc domain with a three helix bundle remains stable.^{32} Placing the donor and acceptor fluorophores at different positions on this stable Habc domain, a single FRET population is observed that falls onto the linker corrected static FRET-line in both the *E*–$\tau DAF$ and moment representations [Fig. 15(a), magenta line]. Note that dye-linker correction is needed to describe the FRET population as it deviates from the ideal static FRET line [black lines in Fig. 15(a)].