Accuracy of distance distributions and dynamics from single-molecule FRET

Abstract

Single-molecule spectroscopy combined with Förster resonance energy transfer is widely used to quantify distance dynamics and distributions in biomolecules. Most commonly, measurements are interpreted using simple analytical relations between experimental observables and the underlying distance distributions. However, these relations make simplifying assumptions, such as a separation of timescales between interdye distance dynamics, fluorescence lifetimes, and dye reorientation, the validity of which is notoriously difficult to assess from experimental data alone. Here, we use experimentally validated long-timescale, all-atom explicit-solvent molecular dynamics simulations of a disordered peptide with explicit fluorophores for testing these assumptions, in particular the separation of the relevant timescales and the description of chain dynamics in terms of diffusion in a potential of mean force. Our results allow us to quantitatively assess the resulting errors; they indicate that, even outside the simple limiting regimes, the errors from common approximations in data analysis are generally smaller than the systematic uncertainty limiting the accuracy of Förster resonance energy transfer efficiencies. We also illustrate how the direct comparison between measured and simulated experimental data can be employed to optimize force field parameters and develop increasingly realistic simulation models.

Significance

For many biomolecular systems, it is essential to obtain information about their dynamics and internal distance distributions to understand their conformational properties and functional roles. Important examples include disordered proteins or single-stranded nucleic acids, many of which do not assume a well-defined folded structure. A method that has turned out to be particularly useful for studying such conformationally heterogeneous systems is single-molecule Förster resonance energy transfer (FRET) between two fluorophores attached to the molecule of interest. However, inferring the distribution of distances and distance dynamics between the dyes usually requires simplifying assumptions in the analysis, such as a separation of timescales between the different physical processes affecting the transfer process. Here, we use atomistic molecular dynamics simulations that allow us to dissect these contributions and quantify the implications for the accuracy of the quantitative interpretation of FRET experiments.

Introduction

The combination of single-molecule fluorescence spectroscopy with Förster resonance energy transfer (FRET) has become widely used for measuring distances, distance distributions, and distance dynamics in biomolecular systems (1,2), and a broad range of timescales from nanoseconds to days has been probed (3,4). Owing to this versatility in accessible timescales, and by avoiding ensemble averaging, single-molecule FRET is particularly powerful for investigating conformationally heterogeneous systems that can be challenging to resolve with other methods. However, quantitative applications of Förster’s theory (5) usually invoke several approximations. One is the rapid orientational averaging of the fluorophores during the excited-state lifetime of the donor, required for using the average value of the orientational factor κ² of 2/3 (6,7,8,9). Another commonly employed assumption is that the interdye distance dynamics are slow relative to the excited-state lifetime of the donor dye (10,11). In many systems, however, the separation between the timescales of dye rotation, excited-state lifetime, and distance dynamics is not large, making it difficult to assess the validity of the approximations.

An interesting case in point is the dynamics of unfolded and disordered proteins, which are essential for understanding protein folding and the function of disordered proteins. Important aspects in protein folding are, e.g., the elementary process of searching for the native state as a diffusive process on a free-energy surface (12,13,14), the preexponential factor or “speed limit” of protein folding (15), and the effects of internal friction (16,17,18) in the polypeptide chain. Proteins that remain disordered in their biological context also pose many interesting questions. Some of these intrinsically disordered proteins (IDPs) fold upon binding a partner (19), others remain disordered in the bound state (20). The dynamics of IDPs have a direct impact on the range and speed of their intermolecular interactions (21,22), and on the mechanisms and stoichiometry of binding (20). Single-molecule FRET experiments have provided important insights into these questions (23,24), e.g., by measuring the distances between specific sites within a protein and their fluctuations (3). The experimental observables in these studies of IDPs are primarily the transfer efficiency and the variance of the transfer efficiency distribution. From these, we can infer the distribution of distances that a molecule samples, reflecting its conformational landscape. Additionally, fluorescence intensity correlations are crucial for assessing the dynamics of these distance fluctuations, providing information on the timescales of chain reconfiguration (25).

However, interpreting experimental signals requires appropriate models. Significant advances in understanding the underlying molecular processes have come from theory and simulations (26,27,28,29,30,31). For unfolded and disordered proteins, concepts from polymer theory have been particularly useful in conceptualizing their conformational distributions and chain dynamics (25). An aspect that is often underappreciated is that data interpretation with these models typically relies on simplifying approximations, especially with respect to the separation of timescales mentioned above. On the other hand, the increasing timescales accessible in molecular simulations, owing to increased computational power (32) and the use of multiscale modeling (33), along with the development of force fields suitable for simulating unfolded and disordered proteins (29,34) now provide a detailed description of the dynamics, without requiring such approximations. The growing overlap between the timescales accessible in experiments and simulations allows us to use molecular simulations directly to interpret experimental data. In the context of FRET, including an explicit representation of fluorophores in the simulations is particularly useful: it provides a realistic representation of the structural properties of the molecular system; it allows the simulation of photophysical processes such as energy transfer with minimal assumptions, enabling direct comparisons with experimental observables; and it helps identify dye-related effects. In turn, experimental data provide stringent benchmarks for the optimization of molecular models and force fields (35,36).

Here, we take advantage of this synergy and employ microsecond-long, experimentally validated, all-atom simulations of a disordered peptide with an explicit representation of donor and acceptor dyes to obtain a detailed picture of combined dye and chain dynamics and to assess in quantitative detail the validity of common approximations in the analysis of single-molecule FRET measurements. Specifically, we use a disordered peptide whose dynamics are so rapid that the timescales discussed above are only moderately separated, which provides a challenging yet realistic test case and allows us to examine the resulting influence on the experimental observations and the interpretation based on simple models.

Relevant timescales in single-molecule FRET of protein dynamics

In FRET (5,6), nonradiative energy transfer occurs between a fluorescent donor (D) and an acceptor (A), usually with different emission wavelengths (Fig. 1 a). Upon selective excitation of the donor, energy is either emitted as a photon by the donor, dissipated in a nonradiative decay, or transferred to the acceptor, which can then emit a photon. In the simplest approximation, the transfer rate, $k_{\mathrm{T}}\left(r\right)$, depends on the inverse sixth power of the distance, r, between donor and acceptor. (Note that we use the point dipole approximation inherent in Förster’s theory (5) and ignore higher-order moments, which can change the distance dependence at very short separations between the dyes (37). However, the contribution of quenching via processes such as photoinduced electron transfer is expected to dominate over such effects at short distances (4).) Consequently, by quantifying the emission from both donor and acceptor, most commonly expressed in terms of the transfer efficiency, we can infer detailed information about distances within the molecule (Fig. 1 f) (38). The quantitative interpretation of the experimentally observed transfer efficiency can be complex, because it is affected by several processes that occur on different but potentially overlapping timescales (Fig. 1): the first important timescale is given by the photophysics of the FRET process. Of particular importance is the excited-state lifetime of the donor, which is well approximated by ${\tau }_{\text{DA}}=1/\left(k_{\mathrm{T}}\left(r\right)+k_{\mathrm{D}}\right)$ for a single fixed distance r between D and A (Fig. 1 c). $k_{\mathrm{D}}=1/{\tau }_{\mathrm{D}}$ is the intrinsic fluorescence decay rate of the donor when attached to the protein but in the absence of the acceptor; ${\tau }_{\mathrm{D}}$ is the corresponding fluorescence lifetime, which is in the range of a few nanoseconds for commonly used fluorophores and can be measured with high precision using time-correlated single-photon counting (39).

Figure 1.

Processes and timescales relevant for FRET of a disordered protein. (a) Representative snapshots of the disordered AGQ peptide from MD simulations with explicit donor and acceptor dyes, Alexa Fluors 488 (D, green) and 594 (A, red) with dipole moments (μ_D, μ_A, black arrows), indicated with various interdye distances r and orientational factors κ², which result in different transfer rate coefficients, k_T(r,κ²). After donor excitation (blue arrow), the donor can either emit a photon (green arrow) or transfer its excitation energy to the acceptor, which can then emit a photon (red arrow). (b) Short segment of a trajectory of the orientational factor κ² from an all-atom MD simulation, reflecting rapid rotational motion of the relative dye orientations. (c) Jablonski diagram of the FRET process. S_iS_j denotes the combined state in which D and A are in the singlet states S_i and S_j, respectively, including the double-excited state S₁S₁ and its nonradiative depopulation by singlet-singlet annihilation (SSA) (74) with rate coefficient k_SSA. (d) Short segment of an interdye distance trajectory from an all-atom MD simulation. (e) Time-resolved fluorescence anisotropy decays of donor (green) and acceptor (red) from MD simulations with fits (dashed lines) (see materials and methods). (f) Distance distribution, P(r), of the AGQ peptide from MD simulations with urea (blue), with the root mean-squared end-to-end distance (r_rms, blue solid line) and the distance dependence of the energy transfer efficiency ($\epsilon \left(r\right)$, solid black line), with Förster radius R₀ (black dotted line). (g) Normalized dye-to-dye distance correlation function, G_r(τ), calculated from the distance trajectory of MD simulations in urea. The commonly assumed separation of timescales and the equation used for relating the average transfer efficiency to $\epsilon \left(r\right)$ and P(r) in the isotropic limit are shown in the center.

${\tau }_{\text{DA}}$ is a key timescale in FRET, relative to which we need to assess whether processes such as dye rotation or chain relaxation appear static or dynamic. The second important timescale is that of rotational diffusion of the fluorophores, which influences FRET because $k_{\mathrm{T}}$ depends not only on the distance between donor and acceptor but also on the relative orientation of their transition dipole moments, quantified in terms of the orientational factor κ² (Fig. 1 b, see materials and methods) (6). $k_{\mathrm{T}}$ is affected both by the distribution of κ² and the timescale on which κ² fluctuates relative to τ_DA. In the simple case, in which the dyes rapidly sample an isotropic orientational distribution, indicated by rapid fluorescence anisotropy decays of donor and acceptor (Fig. 1 e), we can use the approximation ⟨κ²⟩ ≈ 2/3, and the dyes' rotational dynamics is well approximated by a single rotational correlation time, τ_rot. In practice, τ_rot is often in the range of hundreds of picoseconds if there are no pronounced fluorophore-protein interactions (2,39,40). Finally, in the context of unfolded proteins and IDPs, the dynamics of the polypeptide chain yield the third important timescale. We quantify these dynamics in terms of the chain reconfiguration time, τ_r, defined as the distance decorrelation time between pairs of residues in the chain (Fig. 1 g), which can be obtained experimentally from the fluctuations in donor and acceptor emission with nanosecond fluorescence correlation spectroscopy (nsFCS) (25,41,42,43).

The focus of our interest here is to assess how the separation of these three timescales—the rotational decorrelation time of the dyes, τ_rot, the donor fluorescence lifetime, τ_DA, and the chain reconfiguration time, τ_r—influences experimental observables. Specifically, we aim to elucidate the impact of these timescales on the interpretation of intramolecular distance distributions and chain dynamics. Based on the relative magnitudes of τ_rot, τ_DA, and τ_r, three physically plausible limits of different averaging regimes, for which analytic expressions are available, are commonly considered for relating the average transfer efficiency to the distance distribution between the dyes (10,44).

• 1.If τ_rot << τ_DA and τ_DA << τ_r, i.e., the dye orientations average out during the excited-state lifetime of the donor, and the distance dynamics are much slower than the fluorescence lifetime of the donor (“isotropic limit” (11)):

$${⟨\epsilon ⟩}_{\text{iso}}={\langle \frac{k_T\left(r\right)}{k_D+k_T\left(r\right)}\rangle }_t={⟨\epsilon \left(r\right)⟩}_r=\underset{0}{\overset{\infty }{\int }}\epsilon \left(r\right)P\left(r\right)dr,\text{with}\epsilon \left(r\right)=\frac{1}{1+{\left(\frac{r}{R_0}\right)}^6}\text{,}$$ (1)

where $\epsilon \left(r\right)$ is the transfer efficiency, P(r) is the normalized interdye distance probability density function, ⟨·⟩_r indicates averaging over distance, and ⟨·⟩_t averaging over a time much longer than τ_r, respectively. This is the approximation most commonly used.

• 2.If τ_rot << τ_DA and τ_r << τ_DA (“dynamic limit”):

$${⟨\epsilon ⟩}_{\text{dyn}}=\frac{{⟨k_T⟩}_r}{k_D+{⟨k_T⟩}_r}=\frac{\underset{0}{\overset{\infty }{\int }}{\left(R_0/r\right)}^{6}P\left(r\right)dr}{1+\underset{0}{\overset{\infty }{\int }}{\left(R_0/r\right)}^{6}P\left(r\right)dr}\text{,}$$ (2)

where ${⟨k_T⟩}_r$ is the transfer rate averaged over P(r). In cases 1 and 2, an average value of 2/3 is assumed for κ².

• 3.If τ_DA << τ_rot and τ_DA << τ_r, the distributions of both r and κ² need to be taken into account explicitly (“static limit”):

$${⟨\epsilon ⟩}_{\text{stat}}=\underset{0}{\overset{4}{\int }}\underset{0}{\overset{\infty }{\int }}\epsilon \left(r,{\kappa }^2\right)P\left(r\right)P\left({\kappa }^2\right)drd{\kappa }^2$$ (3)

$$\mathrm{with}P\left({\kappa }^2\right)=\{\begin{array}{cc}\frac{1}{2\sqrt{3{\kappa }^2}}\mathit{ln}\left(2+\sqrt{3}\right)& 0<{\kappa }^2\le 1\\ \frac{1}{2\sqrt{3{\kappa }^2}}\mathit{ln}\left(\frac{2+\sqrt{3}}{\sqrt{{\kappa }^2}+\sqrt{{\kappa }^2-1}}\right)& 1\le {\kappa }^2\le 4\end{array}\text{and}\epsilon \left(r,{\kappa }^2\right)=\frac{1}{1+{\left(\frac{r}{R_0\left({\kappa }^2\right)}\right)}^6}$$

However, the validity of these approximations is often difficult to assess. For sufficiently large unfolded proteins, τ_r is typically more than an order of magnitude greater than τ_DA (25), but the separation of timescales between τ_DA and τ_rot is often less clear, which calls into question the use of the simple relations above.

In sufficiently detailed molecular simulations, the processes that determine τ_rot, τ_DA, and τ_r can be modeled explicitly, and the average transfer efficiency can be obtained without assuming any of the limiting cases above (Eqs. 1, 2, and 3). For the corresponding general case of fluctuating $k_T\left(t\right)$, Gopich and Szabo (45) defined the average energy transfer ${⟨\epsilon ⟩}_{\ast }$ as the probability that the excited-state of the donor, $D^{\ast }$, decays by transferring its energy to the acceptor:

$${⟨\epsilon ⟩}_{\ast }=1-k_D{\int }_0^{\infty }{p}_{D^{\ast }}\left(t\right)dt\text{,}$$ (4)

where $p_{D^{\ast }}\left(t\right)$ is the decay of the excited state given by the path average:

$$p_{D^{\ast }}\left(t\right)={\langle \exp \left(-k_{D}t-{\int }_0^t{k}_T\left(t^{\text{'}}\right)d{t}^{\text{'}}\right)\rangle }_{r,{\kappa }^2}\text{.}$$ (5)

${⟨\epsilon ⟩}_{\ast }$ can be calculated for simulated data as described in materials and methods and compared with the simple limiting regimes (Eqs. 1, 2, and 3). Note that, in our notation, we distinguish between $\epsilon$, the instantaneous transfer efficiency of the molecular system, and the ratiometric transfer efficiency obtained from a FRET measurement, $E=N_A/\left(N_D+N_A\right)$, where $N_D$ and $N_A$ are the number of donor and acceptor photons observed in a time bin or a fluorescence burst. It has been shown that $⟨E⟩={⟨\epsilon ⟩}_{\ast }$ (45). Apart from the above considerations, several other processes may have to be taken into account for interpreting single-molecule FRET results quantitatively. Two of them are photophysics and photochemistry, such as triplet blinking and radical formation, which affect many fluorophores and are detectable as relaxation components in fluorescence correlation functions in the range of a few microseconds and above (46). Here, we focus on chain dynamics in the submicrosecond range and will thus ignore those contributions for simplicity, but they can be addressed experimentally and included in the model (47,48,49,50). Another potentially interfering process is the quenching of fluorophores by aromatic residues or dye-dye stacking (46,49,50,51), which can occur on similar timescales as chain dynamics. Such contributions can also be quantified experimentally and included quantitatively in the model if necessary (49,50). There are two more timescales that can be important to consider. One is the average interphoton time, which is usually >1 μs. All processes we consider here are much faster, so we neglect its influence (44). Finally, in experiments on freely diffusing molecules, the average diffusion time through the confocal observation volume may need to be taken into account. However, this timescale is typically in the range of 0.1–1 ms, so its influence on the rapid chain dynamics we are interested in here is also negligible. We will thus ignore processes whose timescales are much longer than those of chain dynamics. Aspects such as the detection characteristics of the instrument or the time binning used in data analysis can also influence experimental observables, but they predominantly affect the signal-to-noise ratio of the measurements. In this work, we focus on fundamental quantities at the molecular level that are largely independent of the measurement process.

Materials and methods

Molecular dynamics simulations

All-atom explicit-solvent molecular dynamics (MD) simulations were performed essentially as described previously (52). In short, the initial all-atom configuration of the AGQ peptide with explicit Alexa 488 and Alexa 594 dyes was generated using CHARMM (53). The structure was placed in a 7.5-nm truncated octahedral box, and the energy of the system was minimized with the steepest-descent algorithm. For simulations in denaturant, 1450 urea molecules were inserted into the simulation box (urea concentration: ∼7.4 M) and the system was energy minimized. Subsequently, the simulation box was filled with TIP4P/2005s water (54). Twenty-eight sodium and 23 chloride ions were added to the simulation box (ionic strength: ∼128 mM) to match the ionic strength of the buffer used in the experiment and to keep the system electrostatically neutral. Subsequently, the system was again energy minimized and equilibrated in a 10-ps simulation run with the Berendsen barostat (55) (τ = 5 ps), followed by a 100-ns run with the Parrinello-Rahman barostat (56) (τ = 5 ps), resulting in box sizes of ∼7.0 and ∼7.25 nm for the systems with and without urea, respectively. The final structures from these equilibrations were used as starting structures for the first set of 15 independent production runs (52), performed using GROMACS (57) 2020.3, the Amber99SBws (54) force field, and the TIP4P/2005s water model (54), with protein-dye interaction parameters (58) as well as protein-urea, urea-urea, and water-urea parameters (59) as reported previously (52), except as described below.

Following the previous optimization of protein-water interaction strengths against experimental FRET data, we extended our simulations to investigate the effect of varying dye-water interaction strengths. Specifically, we performed two more sets of simulations in which the Lennard-Jones ϵ parameter for the interactions between all dye atoms and the water oxygen was scaled relative to the values determined by the standard Lorentz-Berthelot combination rules, as implemented in the Amber99SBws force field and the TIP4P/2005s water model. In our previous work, and as the default for Amber99SBws and TIP4P/2005s, the Lennard-Jones ϵ parameter for all dye atoms was scaled by a factor of 1.10 when interacting with the water oxygen (52). Here, we conducted two additional sets of simulations, where the Lennard-Jones ϵ parameter for all dye atoms was increased by factors of 1.15 and 1.20 when interacting with water oxygen. From the first of the 15 runs with the default dye-water parameters, we wrote out structural snapshots every 50 ns, which were then used as starting points for an additional 15 independent runs for each of the new dye-water scaling factors (1.15 and 1.20). The temperature was kept constant at 295.15 K in all simulations using a stochastic velocity rescaling thermostat (60) (τ = 1 ps), and the pressure was kept at 1 bar with the Parrinello-Rahman barostat (56). Long-range electrostatic interactions were modeled using the particle-mesh Ewald method (61). Dispersion interactions and short-range repulsion were described by a Lennard-Jones potential with a cutoff at 1 nm. H-bond lengths were constrained using the LINCS algorithm (62). The length of the 15 production simulations for each of the 3 sets of dye-water scaling factors was approximately 1.1 μs, resulting in a total simulation time of around 16.5 μs, with a timestep of 2 fs. The first 20 ns of each simulation were excluded from the analysis to eliminate initial structure bias. System coordinates were saved every 5 ps.

Simulations without urea were performed almost identically to the simulations with urea, but in a slightly smaller simulation box (7.0 nm truncated octahedron) containing 25 sodium and 20 chloride ions (ionic strength: ∼128 mM). For the default dye-water interactions, 17 independent production simulations were conducted, with a total length of 16.14 μs. For the dye-water scaling factors of 1.15 and 1.20, we performed 15 independent simulations, each approximately 1.1 μs in length, resulting in a total simulation time of around 16.5 μs. The first 20 ns of each simulation were excluded from the analysis. The simulations are available from Zenodo: https://zenodo.org/records/15223221.

The simulations were previously validated by direct comparison with experimentally determined transfer efficiencies and variances of the transfer efficiency distribution from lifetime information; nsFCS measurements probing chain dynamics; and time-resolved fluorescence anisotropy measurements probing the rotational motion of the fluorophores (52). Here, we further refined the force field parameters for the dye-water interactions by comparison with time-resolved fluorescence anisotropy measurements as well as nsFCS amplitudes and changes in quantum yields probing the influence of dye-dye quenching (see main text and materials and methods below for details). We note that the anisotropy decays in the simulations with urea are roughly a factor of 1.5 slower than the corresponding experimental ones, indicating a slightly increased viscosity.

Brownian dynamics simulations of diffusive dye rotation

We used Brownian dynamics simulations of a random walk on a sphere (63) to model the rotational motion of the transition dipole moments of the fluorophores independent of distance dynamics (Fig. 2). We initiated trajectories from initial spherical coordinates with polar and azimuthal angles (${\theta }_i,{\varphi }_i$). The rotational diffusion coefficient, $D_{rot}$, and the time step, $\Delta t$, determine the standard deviation of the rotational displacement,

$$\sigma =\sqrt{4D_{rot}\Delta t}\text{.}$$ (6)

Each step involves generating a random rotational displacement, using a normal distribution with mean zero and standard deviation σ for the displacement Δr on the surface of the unit sphere, and a uniform distribution in the interval [–π,π] for the direction $\alpha$ of the displacement. The iterative updates are given by the trigonometric relations for spherical triangles:

$$\begin{gathered} {\theta }_n={\cos }^{-1}\left(\cos \left({\theta }_{n-1}\right)\cos \left(\Delta r\right)+\sin \left({\theta }_{n-1}\right)\sin \left(\Delta r\right)\cos \left(\alpha \right)\right), \\ {\varphi }_n={\varphi }_{n-1}-{\sin }^{-1}\left(\frac{\sin \left(\alpha \right)\sin \left(\Delta r\right)}{\sqrt{1-{\cos }^2\left({\theta }_n\right)}}\right) \end{gathered}$$ (7)

The updated spherical coordinates, ${\theta }_n$ and ${\varphi }_n$, were converted to Cartesian coordinates to obtain a unit vector

$$\hat{\mathbf{u}}=\left(\cos \left({\varphi }_n\right)\sin \left({\theta }_n\right),\sin \left({\varphi }_n\right)\sin \left({\theta }_n\right),\cos \left({\theta }_n\right)\right)\text{.}$$ (8)

Figure 2.

Influence of the distribution and dynamics of dye orientations on the average FRET efficiency. (a) Snapshot of a Brownian dynamics simulation modeling rotational diffusion of a donor (μ_D, green arrow) and an acceptor transition dipole (μ_A, red arrow) separated by a fixed distance r = R₀, the Förster radius for κ² = 2/3. The black to gray lines illustrate examples of trajectories of the tips of the $\overrightarrow{{\mu }_D}$ and $\overrightarrow{{\mu }_A}$ vectors on a sphere over the course of the donor lifetime, τ_D, for the case of τ_rot/τ_D = 0.1 (see Video S1). (b) Distribution of κ² from the MD simulations of the AGQ peptide in urea for all simulation frames (gray line) and only emissive frames (blue line), compared with the theoretical distribution for fully isotropic orientations of the transition dipoles (6) (black line), and (c) corresponding time-resolved anisotropy decays of Alexa 488 (green) and 594 (red). (d) Dependence of the average transfer efficiency ${⟨\epsilon ⟩}_{\ast }$ (Eq. 4) on τ_rot/τ_D for fixed r = R₀. The dynamic and static limits (Eqs. 2 and 3) and corresponding transfer efficiencies are indicated as horizontal dashed lines. The lighter blue squares indicate ${⟨\epsilon ⟩}_{\ast }$ and τ_rot/τ_D obtained using the dye orientation dynamics from the MD simulations but assuming constant r = R₀. The gray shaded area indicates the range 0.50 ± 0.04, corresponding to the estimated systematic experimental uncertainty in single-molecule FRET (92,93). Error bars show the standard deviations based on 15 independent simulation trajectories. (e) Donor (green) and acceptor (red) steady-state anisotropy as a function of τ_rot/τ_D for the system shown in (a). The gray-shaded area corresponds to steady-state anisotropies that yield ${⟨\epsilon ⟩}_{\ast }$ values in the range 0.50 ± 0.04, as indicated in (d). The colored squares indicate the results using the dye dynamics from the MD simulations analogous to (d). Solid lines in (d) and (e) are empirical interpolation functions linking the limiting regimes. (f) Ratio of τ_rot/τ_D required for ${⟨\epsilon ⟩}_{\ast }$ to remain below a deviation ${\Delta ⟨\epsilon ⟩}_{\ast }$ of ±0.04 from the transfer efficiency obtained in the limit τ_rot << τ_DA (blue shaded area) as a function of r/R₀ (blue line from simulations as shown in (d) at different values of r/R₀), reflecting that short interdye distances require lower τ_rot for accurately quantifying the distance between the fluorophores with the approximation τ_rot << τ_DA. (g) Corresponding steady-state anisotropy for which $\left|{\Delta ⟨\epsilon ⟩}_{\ast }\right|$ < 0.04 (blue shaded area) as a function of r/R₀ (blue line from simulations as shown in (e) at different values of r/R₀).

The decay of the fluorescence anisotropy, ρ, with rotational correlation time, ${\tau }_{rot}=1/6D_{\mathrm{rot}}\text{,}$ in the simulated system was evaluated from the decay of the correlation function (64):

$$\rho \left(t\right)={\rho }_0⟨P_2\left(\hat{\mathbf{u}}\left(t\right)·\hat{\mathbf{u}}\left(0\right)\right)⟩\text{,}$$ (9)

where $P_2\left(x\right)=\left(3x^2-1\right)/2$ is the second-order Legendre polynomial. ${\rho }_0$ corresponds to the fundamental anisotropy of 2/5 if the excitation and emission dipole moments are colinear. Here, we use for ${\rho }_0$ a value of 0.38, the measured limiting anisotropy of Alexa 488 and 594 (65). To mimic a FRET dye pair with dipole orientations (${\hat{\mathbf{u}}}_i,{\hat{\mathbf{u}}}_j$), we performed two independent dye rotational diffusion simulations; the rate of rotational diffusion was varied systematically by changing $D_{rot}$. To relate the simulated rotational dynamics to FRET efficiency, we calculated the orientation factor κ² as (6)

$${\kappa }^2\left({\hat{\mathbf{u}}}_i,{\hat{\mathbf{u}}}_j,\hat{\mathbf{r}}\right)={\hat{\mathbf{u}}}_i·{\hat{\mathbf{u}}}_j-3\left({\hat{\mathbf{u}}}_i·\hat{\mathbf{r}}\right)\left({\hat{\mathbf{u}}}_j·\hat{\mathbf{r}}\right)\text{,}$$ (10)

where $\hat{\mathbf{r}}$ denotes the normalized separation vector between the dyes, which we treat as point dipoles. These κ² values were used for photon simulations of the system (see calculating transfer efficiencies from simulations).

Rouse chain simulations

We use a Rouse chain (66) as a simple model of a polymer to investigate the effect of chain dynamics, characterized either by the Rouse time, ${\tau }_R$, or by the end-to-end distance decorrelation time, ${\tau }_r$, on transfer efficiency. The Rouse model represents the polymer as a series of $N$ segments, where each segment has a root mean square end-to-end distance of $b$, such that $r_{rms}=\sqrt{N{b}^2}$ is the root mean-square end-to-end distance of the polymer. Stretching a segment results in a harmonic force of entropic origin with spring constant $k_0=3k_{B}T/b^2$. We simulate the time evolution of a Rouse chain by using

$$\mathbf{A}\left(t+\Delta t\right)=\mathbf{A}\left(t\right)+\frac{D\Delta t}{k_{B}T}\mathbf{K}\mathbf{A}\left(t\right)+\mathbf{X}\left(t\right)$$ (11)

where $\mathbf{A}\left(t\right)$ is a matrix containing as rows the transposed position vectors of the $N+1$ segment ends at time $t,\Delta t$ is the time step, and $D$ is the diffusion coefficient of a segment. $\mathbf{K}$ is an $\left(N+1\right)\times \left(N+1\right)$ matrix with matrix elements ${\mathbf{K}}_{i,i+1}={\mathbf{K}}_{i+1,i}=k_0$, all other off-diagonal elements are equal to zero, and the diagonal elements are chosen so that the elements of each column of $\mathbf{K}$ sum to zero. $\mathbf{X}\left(t\right)$ is a $\left(N+1\right)\times 3$ matrix whose elements are randomly drawn in each iteration from a normal distribution with zero mean and standard deviation $\sigma =\sqrt{2D\Delta t}$. The simulation proceeds by iteratively updating the positions of all beads according to the above equation, over a specified number of frames (see below), thereby generating a trajectory that reflects the polymer chain’s temporal evolution. For each frame of the simulation, we calculated the end-to-end vector, $\mathit{r}\left(t\right)$, from the difference between the first and last position vectors in $\mathbf{A}\left(t\right)$, and the end-to-end distance, r, is obtained from the Euclidean norm (magnitude) of r. Combining the simulation of the Rouse chain with simulations of dye rotational diffusion allowed us to vary ${\tau }_R$ and ${\tau }_{rot}$ independently and systematically investigate the effects of chain dynamics and dye mobility on FRET efficiency. To ensure that both the fastest and slowest processes are resolved accurately, the time step was set to be two orders of magnitude smaller than the shortest relevant timescale (${\tau }_R$, ${\tau }_{rot}$, or ${\tau }_D$), and the number of frames was chosen so that the total length of the simulation exceeded the longest relevant timescale by two orders of magnitude to achieve sufficient sampling.

Parametrization of the Rouse chain based on all-atom MD simulations

The Rouse model was parametrized based on the all-atom MD simulations of the AGQ peptide in presence of 7.4 M urea. From the simulation data, we can calculate the mean-squared end-to-end distance, $⟨{\mathit{r}}^2⟩=⟨\mathit{r}·\mathit{r}⟩$, and the average decay time of the autocorrelation function of the end-to-end distance vector defined by

$$\overline{\tau }={\int }_0^{\infty }\frac{⟨\mathit{r}\left(0\right)·\mathit{r}\left(t\right)⟩}{⟨{\mathit{r}}^2⟩}dt$$ (12)

According to the Rouse model, the following relations hold for these quantities: $⟨{\mathit{r}}^2⟩=N{b}^2$ and $\overline{\tau }=\frac{{\pi }^2}{12}{\tau }_R$, where the Rouse time, ${\tau }_R$, is given by

$${\tau }_R=\frac{N^2{b}^2}{3{\pi }^{2}D}\text{.}$$ (13)

For the simulations, we need to choose the number of segments. Based on $b=⟨{\mathit{r}}^2⟩/L_c$ and $N=L_c^2/⟨{\mathit{r}}^2⟩$, with $⟨{\mathit{r}}^2⟩$ from the MD simulations and the contour length of the chain, L_c, we chose $N=9$. Given the values $⟨{\mathit{r}}^2⟩=15{\text{nm}}^2$ and $\overline{\tau }=30\text{ns}$ from the MD simulations, we obtain the remaining Rouse chain model parameters as $b=1.3\text{nm}$, ${\tau }_R=37\text{ns}$, and $D=0.13{\text{nm}}^2/\text{ns}$. This parametrization enables a systematic translation of the all-atom MD simulation results into the simplified parameters of the Rouse model, and thus the computationally efficient exploration of the effect of chain dynamics on energy transfer. We are not aware of an analytical relation between the end-to-end vector correlation time, ${\tau }_R$, and the scalar end-to-end distance correlation time (or chain reconfiguration time), ${\tau }_r$, but simulations with different chain lengths and diffusion coefficients show that ${\tau }_r$/ ${\tau }_R\approx 0.5$ (67).

Calculating transfer efficiencies from simulations

Transfer efficiencies were calculated from the MD simulations, Rouse chain simulations, and dye rotational diffusion simulations using the corresponding ε(r), and the photophysical parameters measured previously (52). In a first step, the survival probability of the donor excited state with a fluctuating transfer rate, averaged over all possible time origins, t₀, along the simulation trajectory was calculated according to (45,58)

$$p_{D^{\ast }}\left(t\right)={\langle \exp \left(-{\int }_{t_0}^{t_0+t}\left(k_D+k_T\left(t^{\prime }\right)\right)d{t}^{\prime }\right)\rangle }_{t_0}\text{,}$$ (14)

where

$$k_T\left(t\right)=k_D\frac{{\kappa }^2\left(t\right)}{2/3}{\left(\frac{R_0}{r\left(t\right)}\right)}^6\text{.}$$ (15)

κ² denotes the orientational factor for an instantaneous relative orientation of donor and acceptor dye, r is the interdye distance, and R₀ the Förster radius for κ² = 2/3. The time-dependent values of κ² and r were obtained from the MD trajectories, Rouse simulation trajectories, and dye rotational diffusion simulations using snapshots spaced by 5 ps. The corresponding fluorescence lifetime of the donor in the presence of the acceptor, ${\tau }_{DA}$, was obtained using

$${\tau }_{DA}={\int }_0^{\infty }{p}_{D^{\ast }}\left(t\right)dt\text{.}$$ (16)

In the all-atom MD simulations, the dyes were treated as nonemissive when the distance between any two atoms of donor and acceptor was less than 0.4 nm, corresponding to van der Waals contact (68,69). The average FRET efficiency, ${⟨\epsilon ⟩}_{\ast }$, was calculated as (45,58)

$${⟨\epsilon ⟩}_{\ast }=1-k_D{\int }_0^{t_{\mathit{max}}}{p}_{D^{\ast }}\left(t\right)dt\text{,}$$ (17)

where $t_{\mathit{max}}$ = 20 ${\tau }_D$, a time after excitation by which the fluorescence intensity has essentially decayed to zero (58).

Determining the variance of the transfer efficiency, ${\sigma }_{\epsilon }^2$, from simulations

Gopich, Szabo, and Chung derived the following relations for the donor lifetime in the presence of the acceptor, ${\tau }_{\text{DA}}$, and the acceptor lifetime after donor excitation, ${\tau }_{\text{AD}}$ (Fig. 3, b–e and f) (45,70):

$$\begin{gathered} \frac{{\tau }_{\text{DA}}}{{\tau }_{\mathrm{D}}}=1-⟨\epsilon ⟩+\frac{{\sigma }_{\epsilon }^2}{1-⟨\epsilon ⟩}\text{and} \\ \frac{{\tau }_{\text{AD}}-{\tau }_{\mathrm{A}}}{{\tau }_{\mathrm{D}}}=1-⟨\epsilon ⟩-\frac{{\sigma }_{\epsilon }^2}{⟨\epsilon ⟩}\text{.} \end{gathered}$$ (18)

These relations are applicable for the isotropic limit (τ_rot << τ_DA and τ_DA << τ_r) and the static limit (τ_DA << τ_rot and τ_DA << τ_r). In the first case, the variance is ${\sigma }_{\epsilon }^2=\underset{0}{\overset{\infty }{\int }}{\left(\epsilon \left(r\right)-⟨\epsilon ⟩\right)}^{2}P\left(r\right)dr$, and in the second case, ${\sigma }_{\epsilon }^2={\int }_0^4{\int }_0^{\infty }\left(\epsilon \left(r,{\kappa }^2\right)-{⟨\epsilon ⟩}_{r,{\kappa }^2}\right)P\left(r\right)P\left({\kappa }^2\right)drd{\kappa }^2$. The relations between lifetimes and transfer efficiency are shown in Fig. 3. The dynamic FRET lines (71) for the Rouse chain were approximated by varying the root mean-square distance of a corresponding Gaussian chain. For the SAW $\nu$ model (72), the dynamic FRET lines were obtained by varying the scaling exponent, $\nu$. In the case of the MD simulations, we have only one P(r), but we can obtain the curves by varying R₀. The static FRET line with ${\sigma }_{\epsilon }^2=0$, resulting in ${\tau }_{\mathrm{DA}}/{\tau }_{\mathrm{D}}=\left({\tau }_{\mathrm{AD}}-{\tau }_{\mathrm{A}}\right)/{\tau }_{\mathrm{D}}=1-⟨\epsilon ⟩$, applies for fixed interdye distances combined with fast dye rotation (${\kappa }^2=2/3$). We are using Eq. 18 also in cases where the dynamics (${\tau }_r$ and/or ${\tau }_{\text{rot}})$ are faster than the exited state lifetime of the donor; in this case, we obtain an apparent variance, ${\sigma }_{\epsilon ,\text{app}}^2$.

Figure 3.

Influence of distance distributions and chain dynamics on observables. (a) Dye-to-dye distance distributions, P(r), from MD simulations of the donor/acceptor-labeled AGQ peptide in urea (light blue and gray). The gray peak originates from nonemissive configurations, where donor and acceptor are in van der Waals contact, and the light blue histogram shows emissive conformations that exclude dye-dye contacts. The resulting distance distribution is compared with the P(r) from SAWν (72) (purple) and Rouse chain simulations (dark blue) parameterized so that they yield the same average transfer efficiency as the MD simulations. The dashed vertical lines indicate the R₀ values that were chosen for each distribution so that ${⟨\epsilon ⟩}_{\text{iso}}$ = 0.50, and the solid vertical lines indicate root mean-squared distances, r_rms. The inset shows the corresponding transfer efficiency distributions assuming κ² = 2/3. (b) The black line shows the dependence of the relative lifetimes versus transfer efficiency (see materials and methods) for fluorophores at a fixed distance (static line). The colored lines represent the dynamic FRET lines for the SAWν distribution (purple), the AGQ peptide MD simulations in urea (light blue), and a Gaussian chain (corresponding to the Rouse model; dark blue) assuming τ_rot << τ_D << τ_r; upper lines, donor lifetimes; lower lines, acceptor lifetimes (see materials and methods). Data points show relative donor (green circle) and acceptor lifetimes (red circle) from the MD simulations and using the photophysical parameters observed experimentally for simulating photon emission (52) (see materials and methods). Error bars show the standard deviations based on 15 independent simulation trajectories. The gray shaded area (here and in other panels) indicates the transfer efficiency range 0.50 ± 0.04, corresponding to the estimated systematic experimental uncertainty in single-molecule FRET (92, 93). (c) Comparison of amplitude-normalized correlations of the scalar dye-to-dye distance from MD (light blue) and Rouse chain simulations (dark blue, see Video S2) with resulting reconfiguration times, τ_r. (d) Dependence of ${⟨\epsilon ⟩}_{\ast }$ on τ_r/τ_D upon variation of τ_r for a Rouse chain with κ² = 2/3 (blue symbols) and for a constant ratio of τ_r/τ_rot = 23, corresponding to the value observed in the MD simulations in urea, to illustrate the approach to the static limit for both τ_r and τ_rot (gray symbols, Eq. 3). R₀ was chosen so that E = 0.50 for τ_rot << τ_D << τ_r. The dynamic and static limits (Eqs. 2 and 3) are indicated as horizontal dashed lines. Solid lines are empirical interpolation functions between the limiting regimes. The inset shows the range of τ_r/τ_D required for ${⟨\epsilon ⟩}_{\ast }$ to remain below a deviation ${\Delta ⟨\epsilon ⟩}_{\ast }$ of ±0.04 from the transfer efficiency of 0.50 obtained assuming τ_rot << τ_DA << τ_r (blue shaded area) as a function of r_rms/R₀ (blue line from Rouse chain simulations as in main panel at different values of r_rms/R₀). The results from the MD simulations are shown as light blue symbols (circles: κ² = 2/3; squares: dye orientational dynamics from MD). Error bars show the standard deviations based on 15 independent simulation trajectories. (e) Analogous to (b), with dynamic FRET lines for a Rouse (or Gaussian) chain (dark blue) for τ_rot << τ_D << τ_r and for a constant value of τ_r/τ_rot = 23 (gray line). Data points show the values from the Rouse chain simulations (same simulations and color code as in (d); upper values [green boundary]: donor lifetimes; lower values [red boundary]: acceptor lifetimes). (f) Apparent variance of the transfer efficiency distribution, ${\sigma }_{\epsilon ,\text{app}}^2$, obtained from the analysis in (e) as a function of τ_r/τ_D for a Rouse chain (same simulations as in (d)). Solid lines are empirical interpolation functions between the limiting regimes. Error bars for the Brownian diffusion simulations represent the standard deviations from three independent simulation runs. The inset shows the range of τ_r/τ_D required for ${\sigma }_{\epsilon ,\text{app}}^2$ to deviate less than 20% from ${\sigma }_{\epsilon }^2$ as a function of r_rms/R₀ (blue line from Rouse chain simulations as in (d) at different values of r_rms/R₀).

Simulating photon emission and obtaining nsFCS from all-atom simulations

Photon emission was simulated based on the MD trajectories following the approach previously described (50,73,74). Briefly, a rate matrix K describes the transitions between the four states of the underlying photophysical model of the FRET-labeled system: donor and acceptor in the ground state (DA), donor excited/acceptor in the ground state (D^∗A), donor in the ground state/acceptor excited (DA^∗), and both donor and acceptor excited (D^∗A^∗) (Fig. 4). For low excitation rates, the population of D^∗A^∗ is negligible (74), but in view of the high excitation rates in zero-mode waveguides (ZMWs) (52,75,76), we explicitly accounted for D^∗A^∗, which can be populated via DA^∗ → D^∗A^∗ and D^∗A → D^∗A^∗ with the rate coefficients $k_{exD}$ and $k_{exA}$. D^∗A^∗ is depopulated by singlet-singlet annihilation, where the energy is transferred from the excited donor to the excited acceptor with a rate coefficient $k_{SSA}\left(r,{\kappa }^2\right)$, which is similar in magnitude to that of the regular FRET process (74). The evolution of the populations of these four states (DA, D^∗A, DA^∗, D^∗A^∗) with time, p(t), is given by the rate equation dp/dt = Kp(t), with the rate matrix (74)

$$\mathbf{K}=\left(\begin{array}{cccc}-k_{exD}-k_{exA}& k_D& k_A& 0\\ k_{exD}& -k_D-k_{exA}-k_T\left(r,{\kappa }^2\right)& 0& k_A\\ k_{exA}& k_T\left(r\right)& -k_A-k_{exD}& k_D+k_T\left(r,{\kappa }^2\right)\omega \\ 0& k_{exA}& k_{exD}& -k_D-k_A-k_T\left(r,{\kappa }^2\right)\omega \end{array}\right)\text{,}$$ (19)

where $\omega$ = ${k_{SSA}\left(r,{\kappa }^2\right)/k}_T\left(r,{\kappa }^2\right)$ = 0.95; k_exD = 0.02/ns · $\xi$ (with $\xi =3.7$ in ZMWs (52)), and k_exA = d k_exD are the excitation rates (41); k_D = 1/ $⟨{\tau }_D⟩$ and k_A = $1/⟨{\tau }_A⟩$ are the decay rates of the excited states for donor and acceptor, respectively. The acceptor direct excitation coefficient, d, and the mean fluorescence lifetimes of the donor, $⟨{\tau }_D⟩$, and the acceptor, $⟨{\tau }_A⟩$, are experimentally determined values (52), and k_T(r, κ²) = $k_{\mathrm{D}}{\kappa }^2/\left(2/3\right){\left(R_0/r\right)}^6$, with κ² and r obtained from the MD simulations with a time resolution of 5 ps. With the detection matrices ${\mathbf{V}}_{\mathrm{D}}$ and ${\mathbf{V}}_{\mathrm{A}}$, which contain the rate constants for the monitored transitions,

$${\mathbf{V}}_{\mathrm{D}}=\left(\begin{array}{cccc}0& k_D& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& k_D\\ 0& 0& 0& 0\end{array}\right),{\mathbf{V}}_{\mathrm{A}}=\left(\begin{array}{cccc}0& 0& k_A& 0\\ 0& 0& 0& k_A\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)\text{,}$$ (20)

the fluorescence intensity correlation between the detection channels i and j (i,j = A,D) of a single molecule with photon emission rates described by the rate matrix K can be computed as (41,74)

$$g_{ij}\left(\tau \right)=\frac{1^{\mathrm{T}}{\mathbf{V}}_j{e}^{\mathbf{K}\tau }{\mathbf{V}}_i{\mathbf{p}}_{\text{ss}}}{\left(1^{\mathrm{T}}{\mathbf{V}}_i{\mathbf{p}}_{\text{ss}}\right)\left(1^{\mathrm{T}}{\mathbf{V}}_j{\mathbf{p}}_{\text{ss}}\right)}\text{.}$$ (21)

$e^{\mathbf{K}\tau }$ is the matrix exponential of $\mathbf{K}\tau$, ${\mathbf{1}}^{\mathrm{T}}$ = (1,1,1,1) denotes the transposed vector of ones, and p_ss is the normalized steady-state solution of dp/dt = K p(t) (74). The dyes were treated as nonemissive when the distance between any two atoms of donor and acceptor was less than 0.4 nm, corresponding to van der Waals contact (68,69). Fluorescence intensity correlations from simulations with and without urea were computed from the individual simulation runs; the resulting individual correlation curves were weighted according to the trajectory length, averaged, normalized at $\tau$ = 50 ns, and fitted in the same way as the experimental nsFCS curves (52) (see FCS). For nsFCS simulations of direct acceptor excitation, k_exD was set to 0 and k_exA to 0.02/ns.

Figure 4.

Simulating a FRET experiment and its analysis. (a) Concatenated trajectories of dye-to-dye distance, r (blue), and κ² (light blue) from 16 MD simulation runs (∼1 μs each, indicated by vertical lines) of the AGQ peptide in 7.4 M urea with corresponding r and κ² distributions (right-hand side). (b) Jablonski diagram of the FRET process. S_iS_j denotes the combined state in which D and A are in the singlet states S_i and S_j, respectively, including the double-excited state S₁S₁ and its nonradiative depopulation by singlet-singlet annihilation (SSA) (74) with rate coefficient k_SSA. The evolution of the populations of the four states (DA, D^∗A, DA^∗, D^∗A^∗) with time, p(t), is given by the rate equation dp/dt = K(t) p(t), with the time-dependent rate matrix K(t) and a Förster radius of R₀ = 5.4 nm. (c) Zoom-in of the distance, r (blue) and κ² (light blue) trajectories (a) with emitted donor (green) and acceptor photons (red) based on the photophysical model (b), with representative MD snapshots illustrating rapid distance fluctuation and dye reorientation. (d) Transfer efficiency histogram obtained from photon emission simulations based on (a) and (b). (e) Distance distributions (see Fig. 3a) from which the potentials of mean force are calculated (f). (f) Potentials of mean force (PMFs) derived from the MD simulations (blue) compared with the PMF inferred from the SAWν distribution (purple) with effective end-to-end diffusion coefficients used for 1D Brownian dynamics simulations. (g) Donor (green) and acceptor (red) fluorescence autocorrelations and donor-acceptor cross correlations (blue) from photon simulations based on all-atom MD trajectories and experimentally observed photophysical parameters (measured in ZMWs (52)) with global fits (black solid lines, see materials and methods) and resulting intensity relaxation time, τ_c. (h) Comparison of end-to-end distance autocorrelation functions, G_r(τ), calculated directly from the MD simulations (blue) or from 1D diffusion in the PMFs derived from the MD simulations (black) and SAWν (purple).

Single-molecule fluorescence spectroscopy

Single-molecule fluorescence experiments were performed on a four-channel MicroTime 200 confocal instrument (PicoQuant, Berlin, Germany) equipped with an Olympus UplanApo 60×/1.20 W objective. Alexa 488 was excited with a diode laser (LDH-D-C-485, PicoQuant) at an average power of 100 μW (measured at the back aperture of the objective). The laser was operated in continuous-wave mode for nsFCS experiments and in pulsed mode with interleaved acceptor excitation for fluorescence lifetime measurements (77). The wavelength range used for acceptor excitation was selected with two band pass filters (z582/15 and z580/23, Chroma, Bellows Falls, VT) from the emission of a supercontinuum laser (EXW-12 SuperK Extreme, NKT Photonics, Birkerød, Denmark) driven at 20 MHz (45 μW average laser power after the band pass filters), which also triggered interleaved pulses from the 488-nm diode laser. Sample fluorescence was collected by the microscope objective, separated from scattered light with a triple band pass filter (r405/488/594, Chroma) and focused on a 100-μm pinhole. After the pinhole, fluorescence emission was separated into two channels, either with a polarizing beam splitter for fluorescence lifetime measurements, or with a 50/50 beam splitter for nsFCS measurements to avoid the effects of detector deadtimes and afterpulsing on the correlation functions (41). Finally, the fluorescence photons were separated by wavelength into four channels by dichroic mirrors (585DCXR, Chroma), additionally filtered by bandpass filters (ET525/50M and HQ650/100, Chroma), and focused onto one of four single-photon avalanche detectors (SPCM-AQRH-14-TR, Excelitas, Waltham, MA). The arrival time of every detected photon was recorded with a HydraHarp 400 counting module (PicoQuant). The disordered AGQ peptide G(AGQ)₆AGC was purchased from GL Biochem (Shanghai, China), purified, and labeled with Alexa Fluors 488 and 594 as previously described (47).

Fluorescence correlation spectroscopy

Fluorescence correlation spectroscopy (FCS) measurements were performed on freely diffusing Alexa 488/595-labeled AGQ peptides in two different solution conditions. One set of measurements was performed in 50 mM sodium phosphate buffer (pH 7.4) in the presence of 7.4 M urea, and a second set in 50 mM sodium phosphate buffer (pH 7.4) in the absence of urea (referred to as "with urea" and "without urea," respectively, in the text). Additionally, we included 140 mM β-mercaptoethanol as a photoprotectant (78) and 0.01% Tween 20 (Pierce/ThermoFisher, Switzerland) to minimize surface adhesion (79). The sample chamber used for ZMW measurements was a perfusion chamber with a volume of 40 μL (CoverWell, Grace Bio-Labs, Bend, OR). To prevent evaporation, sample chambers were sealed with adhesive tape (Scotch). The concentration of labeled AGQ peptide was 310 nM for measurements in ZMWs (52).

The correlation between two time-dependent intensity signals $I_i\left(t\right)$ and $I_j\left(t\right)$ measured on two detectors $i$ and $j$ is defined as

$$G_{ij}\left(\tau \right)=\frac{⟨I_i\left(t\right)I_j\left(t+\tau \right)⟩}{⟨I_i\left(t\right)⟩⟨I_j\left(t\right)⟩}-1\text{,}$$ (22)

where the pointed brackets indicate time averaging. In our experiments, we use two acceptor channels and two donor channels, resulting in the autocorrelations $G_{AA}\left(\tau \right)$ and $G_{DD}\left(\tau \right)$, and cross correlations $G_{AD}\left(\tau \right)$ and $G_{DA}\left(\tau \right)$. By correlating detector pairs, and not the signal from a detector with itself, contributions to the correlations from dead times and afterpulsing of the detectors are eliminated (41,80). The full FCS curves upon acceptor direct excitation of the labeled AGQ peptide with and without urea with logarithmically spaced lag times ranging from nanoseconds to seconds, normalized in the limit of $\left|\tau \right|\ll {\tau }_D^{ij}$, are shown in Fig. 5 e. The FCS curve obtained in the absence of urea was fitted with (47,81)

$$G_{ij}\left(\tau \right)=a_{ij}\frac{\left(1-c_{ab}^{ij}{e}^{-\left|\tau \right|/{\tau }_{ab}^{ij}}\right)\left(1+c_{cd}^{ij}{e}^{-\left|\tau \right|/{\tau }_{cd}}\right)\left(1+c_Q^{ij}{e}^{-\left|\tau \right|/{\tau }_Q}\right)\left(1+c_T^{ij}{e}^{-\left|\tau \right|/{\tau }_T^{ij}}\right)}{\left(1+\frac{\left|\tau \right|}{{\tau }_D}\right){\left(1+\frac{\left|\tau \right|}{{s^2\tau }_D}\right)}^{1/2}}\text{.}$$ (23)

Figure 5.

Optimizing simulation parameters. (a and b) Comparison of time-resolved anisotropy decays from experiment (acceptor: red; donor: green) and simulations with different scaling factors of the dye-water interaction strength (1.10: pink; 1.15: blue; 1.20: orange; see materials and methods) without urea. (c) Comparison of nsFCS curves upon acceptor direct excitation from experiment and simulations without urea, with corresponding timescale for dye-dye quenching, τ_Q, and fraction of dark states indicated (r ≤ 0.4 nm in simulations, see materials and methods). The simulations using a scaling factor of 1.10 (^∗) have a selection bias owing to lack of convergence, see main text. (d) Ensemble brightness of double-labeled AGQ peptide upon acceptor excitation at 585 ± 7 nm measured via ensemble fluorescence emission spectra with and without urea, indicating fluorescence quenching in the absence of urea. (e) Normalized acceptor autocorrelation functions upon direct acceptor excitation from picoseconds to seconds of the labeled AGQ peptide measured with (orange) and without urea (blue). The fit (see materials and methods) of the correlation curve obtained without urea is shown as a black solid line; the black dashed line shows the same fit without the quenching component, which agrees well with the amplitude observed with urea, indicating the absence of quenching in the presence of urea.

The four terms in the numerator with amplitudes c_ab, c_cd, c_Q, c_T, and timescales ${\tau }_{ab},{\tau }_{cd},{\tau }_Q,{\tau }_T$ describe photon antibunching, chain dynamics, quenching, and triplet blinking, respectively. ${\tau }_D$ is the translational diffusion time of the labeled molecules through the confocal volume; a point spread function of three-dimensional (3D) Gaussian shape was assumed, with a ratio of axial over lateral radii of s = ω_z/ω_xy ($s$ = 5.3) (52). τ_cd, τ_Q, and τ_D were shared parameters for all G_ij. The fit to the data without urea is shown in Fig. 5 e. Note that in nsFCS with direct acceptor excitation, we do not observe chain dynamics, accordingly c_ab was set to 0. Additionally, to illustrate the absence of quenching, c_Q was set to 0 (dashed black line in Fig. 5 e).

To study the fast dynamics in more detail, donor and acceptor fluorescence autocorrelation and crosscorrelation curves were computed and analyzed over a linear range of lag times from $\tau =-50$ to +50 ns (17,82). For this analysis, we used only photons of bursts with E in the range of ±0.2 of the mean transfer efficiency of the population of molecules double labeled with active donor and acceptor fluorophores, which reduces the contribution of donor-only and acceptor-only signal to the correlation. To aid their direct comparison, correlation curves were normalized to unity at $\tau =$ 50 ns. After normalization and in the limit of $\left|\tau \right|\ll {\tau }_T^{ij}$ and $\left|\tau \right|\ll {\tau }_D^{}$, Eq. 23 reduces to

$$\begin{gathered} G_{ij}\left(\tau \right)=b_{ij}\left(1-c_{ab}^{ij}{e}^{-\frac{\left|\tau \right|}{{\tau }_{ab}^{ij}}}\right) \\ \left(1+c_{cd}^{ij}{e}^{-\frac{\left|\tau \right|}{{\tau }_{cd}}}\right)\left(1+c_Q^{ij}{e}^{-\frac{\left|\tau \right|}{{\tau }_Q}}\right)\text{,} \end{gathered}$$ (24)

where $b_{ij}$ is a normalization constant. In the presence of urea, the correlation curves showed no signs of quenching (Fig. 5), so the third term on the right-hand side of Eq. 24 could be set to unity. We note that nsFCS measurements performed with and without ZMWs yield consistent results in terms of amplitudes and timescales (52).

Fluorescence lifetime measurements

To determine the relevant timescales for fluorescence lifetime analysis, we performed polarization-resolved ensemble lifetime measurements of double-labeled AGQ peptide with a custom-built fluorescence lifetime spectrometer (47), which allowed us to determine the fluorescence anisotropy decays of Alexa 488 and 594 conjugated to the AGQ peptide. The measurements were performed with and without urea with sample concentrations of 200 nM (Fig. 5). Alexa 488 was excited by a picosecond diode laser (LDH DC 485) at 488 nm with a pulse repetition rate of 40 MHz using a donor-only-labeled sample. Alexa 594 was excited by a supercontinuum light source (SC450-4, Fianium, Southampton, UK), with the wavelength selected using a z582/15 band-pass filter and a pulse frequency of 40 MHz in a donor/acceptor-labeled sample. The emitted donor fluorescence was filtered with an ET 525/50 filter (Chroma Technology), and the acceptor fluorescence with an HQ 650/100 filter (Chroma Technology). The emitted photons were detected with a microchannel plate photomultiplier tube (R3809U-50; Hamamatsu City, Japan) and the arrival times recorded with a PicoHarp 300 photon-counting module (PicoQuant). Intensity decays, $I_{VH}\left(t\right)$ and $I_{VV}\left(t\right)$, with horizontal and vertical polarizer orientations for fluorescence detection, respectively, were measured with vertically polarized excitation. The anisotropy decay, ρ(t), was calculated from $I_{VV}\left(t\right)$ and $I_{VH}\left(t\right)\text{:}$

$$\rho \left(t\right)=\frac{I_{VV}\left(t\right)-{GI}_{VH}\left(t\right)}{I_{VV}\left(t\right)+2G{I}_{VH}\left(t\right)},$$ (25)

where G accounts for the different detection efficiencies of vertically and horizontally polarized light and was obtained for the donor and acceptor intensities from the ratio of the vertical and horizontal emission upon horizontally polarized excitation, $G=I_{HV}/I_{HH}$. Average anisotropy decay times, ${\tau }_{rot}$, were obtained from ${\int }_0^{\infty }\rho \left(t\right)/{\rho }_{0}dt$ both for experiments and MD simulations, with ρ₀ = 0.38 (65).

Fluorescence brightness measurements

The measurements of emission spectra of the Alexa 488/594-labeled AGQ peptide in the presence and absence of urea were performed on a Fluorolog-3 (HORIBA) at an excitation wavelength of 585 nm, an excitation slit width of 3.5 nm, an emission slit width of 2 nm and integration time of 0.5 s. A range of 600–750 nm was detected for the fluorescence emission of the acceptor dye. The measurements were performed at 22°C using a quartz cuvette, with identical sample concentrations of 300 nM in the presence and absence of urea. The emission spectra were corrected for their different extinction coefficients at 585 nm due to a spectral shift under the two measurement conditions, obtained from amplitude-normalized excitation spectra. The change in fluorescence brightness was obtained by integrating the corrected emission spectra from the measurement with and without urea from 600 to 750 nm.

Results

MD simulations

To investigate the complex interrelation between timescales in single-molecule FRET experiments, we capitalize on simulations that allow us to assess the dynamics of the relevant processes. Since we require absolute timescales, we employ all-atom simulations with explicit solvent and with an explicit representation of the fluorophores. Such simulations allow us to quantify both the dynamics of the polypeptide chain and the fluorophores. We had previously performed multiple-microsecond simulations of the disordered peptide G(AGQ)₆AGC (47,59,83,84) (referred to as “AGQ peptide” here) in the absence and presence of 7.4 M urea (52), with an explicit representation of C-terminal Alexa Fluor 488 as the donor dye and N-terminal Alexa Fluor 594 as the acceptor dye, with the Amber99SBws (54) force field and the TIP4P/2005s water model (54), a combination that has been successful in describing IDPs (67,85,86,87,88,89,90,91), with optimized protein-dye (58) as well as protein-urea, urea-urea, and water-urea interaction parameters (59). The simulations of this AGQ peptide were in good agreement with the experimental results in terms of chain dimensions and dynamics (52), validating the simulation force field used. As detailed in force field optimization based on experimental data, we used here a slightly modified set of force field parameters for the fluorophores that provide even better agreement with experimental fluorescence anisotropy and quenching results. To ensure extensive sampling of all configurations, we simulated a total of 16 μs, three orders of magnitude longer than the chain reconfiguration time. From these simulations, we can thus calculate for every simulation frame the instantaneous value of κ² and the interdye distance, r, and from these two values the corresponding instantaneous Förster transfer rate, k_T, the key quantity for relating the simulations to the FRET measurements. For the direct comparison with experimental data, we complement the MD simulations with Monte Carlo simulations of photon emission based on the Förster radius and the known rate coefficients of the FRET process (Fig. 1 c; see materials and methods). Altogether, we can thus treat the simulations as a detailed model of a FRET experiment and explore the effects of the different timescales and their separation on experimental observables. Moreover, we can analyze simulated experimental data in the same way as we would analyze actual measured data and compare the inferred results regarding distance distributions and chain dynamics with the ground truth of the simulations. We complement the atomistic simulations with simple Brownian dynamics and Rouse chain simulations, which allow us to systematically vary the rotational motion of the dyes and the chain dynamics, respectively.

Influence of the distribution and dynamics of dye orientations

The distribution and dynamics of the dye orientations play a crucial role in interpreting the transfer efficiency observed in FRET experiments (6,7,8). The common assumption of κ² = 2/3 relies on the premise that all dye orientations are sampled isotropically and rapidly, with a rotational correlation time, τ_rot, much shorter than the donor excited-state lifetime. The timescale of dye reorientation is quantified in time-resolved fluorescence anisotropy measurements, where a subnanosecond decay to zero anisotropy is expected for rapid and isotropic dye motion (39) (Fig. 2 c). For unfolded and disordered proteins, this behavior is often observed to good approximation (25). More difficult to assess experimentally is the distribution of relative dye orientations, quantified by the distribution of κ², but we can inspect it based on the experimentally validated all-atom simulations. Fig. 2 b illustrates that the distribution of κ² from the MD simulations closely aligns with the function expected for isotropic dye orientations, with a mean value of 0.64. The slight deviation from the ideal value of 2/3 results in an error of less than 2% in R₀.

Video S1. Brownian dynamics simulation of rotational diffusion of a donor (green arrow) and an acceptor transition dipole (red arrow) separated by a fixed distance corresponding to the Förster radius

The black to gray lines illustrate trajectories of the tips of the vectors on a sphere over the course of the donor lifetime (see Fig. 2a).

To systematically analyze the effect of the separation of timescales between τ_DA and τ_rot on the average transfer efficiency, ${⟨\epsilon ⟩}_{\ast }$, we employ Brownian dynamics simulations to model diffusive dye rotation with different effective rotational diffusion coefficients, D_rot (see materials and methods). We first isolate the impact of τ_rot on ${⟨\epsilon ⟩}_{\ast }$ for a fixed interdye distance (Fig. 2); we choose r = R₀, the Förster radius for κ² = 2/3, the limiting case of rapid isotropic dye reorientation. Fig. 2 d shows the dependence of ${⟨\epsilon ⟩}_{\ast }$ on the ratio τ_rot/τ_D. (Note that we plot the results in terms of τ_rot/τ_D rather than τ_rot/τ_DA since τ_DA itself depends on κ², distance, and dynamics, and is thus not a convenient independent variable.) As expected, for τ_rot/τ_D << 1, the dynamic limit of ${⟨\epsilon ⟩}_{\text{dyn}}=1/2$ is approached, and for τ_rot/τ_D >> 1, the static limit, with ${⟨\epsilon ⟩}_{\text{stat}}\approx 0.38$. A reduction in ${⟨\epsilon ⟩}_{\ast }$ relative to the dynamic limit by 0.04, roughly the estimated systematic uncertainty limiting the accuracy of smFRET experiments (92,93), is reached at a ratio of τ_rot/τ_D ≈ 1, corresponding to a steady-state anisotropy of ∼0.14 (Fig. 2 e; note that the statistical uncertainty of single-molecule FRET measurements is much lower than the systematic uncertainty, in favorable cases <0.01 (94)). Even for anisotropies in this range, which are usually considered acceptable to justify the assumption of κ² ≈ 2/3, a significant effect on the observed transfer efficiency is obtained. However, if we accept realistic bounds on the accuracy that can be achieved in FRET-based distance measurements, this result illustrates that even a relatively small separation of timescales between τ_rot and τ_D yields results close to the commonly assumed dynamic limit, τ_rot << τ_D. Fig. 1 f and g illustrate how the corresponding requirements on τ_rot/τ_D and anisotropy depend on the interdye distance.

To assess the effect of the separation of timescales between τ_rot and τ_D on ${⟨\epsilon ⟩}_{\ast }$ for more realistic fluorophore motion, we again calculated ${⟨\epsilon ⟩}_{\ast }$ for a single fixed distance with r = R₀, but now using the orientational dynamics of the dyes from the all-atom MD simulations (but ignoring the distance fluctuations between donor and acceptor). We obtained τ_rot/τ_D of 0.16 and 0.45 based on the simulations in the absence and presence of urea, respectively (Fig. 2, d and e), corresponding to steady-state donor anisotropies of 0.08 and 0.12. As a result, the distance inferred from ${⟨\epsilon ⟩}_{\ast }$ assuming τ_rot/τ_D << 1 would be overestimated by 1 and 3%, respectively. Obviously, the impact of this effect on the observed transfer efficiency and the inferred distance depends on the interdye distance. Fig. 2 f and g illustrate the results for a range of distances above and below R₀.

Distance distributions and chain dynamics

An important aspect for the analysis of disordered and unfolded proteins is the presence of broad intramolecular distance distributions, P(r) (Fig. 3 a). The end-to-end distance distribution of the AGQ peptide from the MD simulations is shown in Fig. 3 a. The component of the distribution where the dyes are within van der Waals contact is indicated in gray and is treated as nonemissive in the fluorescence analysis because of contact quenching (46,49,50,51,52,68) (see materials and methods). An important experimental test for the presence of a distance distribution in a sample is the combined analysis of transfer efficiencies obtained from the average numbers of donor and acceptor photons on the one hand and of the corresponding fluorescence lifetimes on the other (Fig. 3 b). For a distance distribution that is sampled rapidly compared with the burst time but slowly compared with τ_DA, the populations corresponding to the double-labeled molecules do not lie on the diagonal line expected for static distances (the static FRET line), but above (for donor emission) (95) or below (for acceptor emission) (70), with a displacement from the static line that is proportional to the variance of the transfer efficiency distribution, ${\sigma }_{\epsilon }^2$ (45). It is worth emphasizing that here we do not refer to the variance of the transfer efficiency histogram (Fig. 4 d), which is dominated by shot noise (96) for most IDPs, since their dynamics are faster than the typical fluorescence burst durations or binning times in the millisecond range (25); rather, we refer to the variance of the underlying transfer efficiency distribution sampled by the labeled molecule, P(ε) (Fig. 3 a). If τ_rot << τ_DA and τ_DA << τ_r, ${\sigma }_{\epsilon }^2$ can be directly related to the distance distribution between the dyes, P(r) (45,70). Fig. 3 b shows the lifetime-versus-transfer efficiency analysis for simulated photon emission data based on the MD simulations of the AGQ peptide. The deviations from the diagonal static line reflect its broad interdye distance distribution (Fig. 3 a). Note that, for this analysis, we rescaled R₀ such that ${⟨\epsilon ⟩}_{\text{iso}}$ = 0.50 to facilitate direct comparison with Fig. 2.

In the absence of reliable simulation data, a common procedure for the analysis of measured single-molecule FRET data from unfolded or disordered proteins is the use of analytical polymer models with a single free parameter to infer P(r) from ${⟨\epsilon ⟩}_{iso}$ by inverting Eq. 1 (36,97). A semiempirical distribution that has been shown to be quite reliable for unfolded and disordered proteins is the self-avoiding walk with adjustable scaling exponent, SAWν (72), but several other distributions have been used (25,26). Fig. 3 a shows the distance distribution inferred from ${⟨\epsilon ⟩}_{\ast }$ of the MD simulations based on the SAWν model in comparison with the true distance distribution. The inferred value of the root mean-squared dye-to-dye distance, r_rms, is within 3% of the true value, and the shape of the SAWν distribution is reasonably close to the true distribution. Fig. 3 b shows the dynamic FRET lines (45,70,95) calculated based on P(r) from the MD simulations and SAWν assuming τ_rot << τ_D << τ_r, illustrating that they provide similar estimates of ${\sigma }_{\epsilon }^2$, albeit at somewhat higher donor lifetimes and lower acceptor lifetimes for SAWν than expected for the P(r) from the MD simulations.

To investigate the effect of τ_r on ${⟨\epsilon ⟩}_{\ast }$ and the apparent variance, ${\sigma }_{\epsilon ,\text{app}}^2$, more systematically, we employ simulations of a Rouse chain, which allow us to vary τ_r over a wide range (see materials and methods). The agreement between the end-to-end distance autocorrelations from the MD and Rouse simulations illustrates the similarity of the overall chain relaxation dynamics (Fig. 3 c). To single out the contribution of τ_r on ${⟨\epsilon ⟩}_{\ast }$, we set κ² to 2/3 for each simulation frame, corresponding to rapidly sampled isotropic dye orientations (τ_rot << τ_DA), and varied the ratio τ_r/τ_D by changing the friction coefficient in the Rouse chain simulations. Fig. 3 d shows the expected transition from ${⟨\epsilon ⟩}_{\text{iso}}=0.50$ for a chain that is static relative to the donor lifetime (Eq. 1) to transfer efficiencies approaching ${⟨\epsilon ⟩}_{\text{dyn}}=1$ for a chain with extremely fast distance dynamics (Eq. 2); in this extreme, P(r) is sampled so rapidly that the transfer rate is dominated by the shortest distances (98). If τ_r is at least ∼6 times greater than τ_D, the deviation from ${⟨\epsilon ⟩}_{\text{iso}}=0.50$ is <0.04, indicating that even a moderate separation of timescales between τ_r and τ_D suffices for employing Eq. 1 in the analysis of distance distributions with reasonable accuracy. For more rapid chain dynamics, substantial averaging of the transfer rate occurs during τ_D, and the effect of distance fluctuations needs to be taken into account explicitly (45,98).

The Rouse chain simulations further allow us to illustrate the change in the apparent variance of the transfer efficiency distribution, ${\sigma }_{\epsilon ,\text{app}}^2$, as τ_r decreases relative to τ_D (Fig. 3, e and f). We find a notable decrease in ${\sigma }_{\epsilon ,\text{app}}^2$ by 16% compared with the limit assuming τ_rot << τ_D << τ_r already for an order-of-magnitude separation of timescales, τ_r/τ_D = 10. To quantify the effect of τ_r on the observed ${⟨\epsilon ⟩}_{\ast }$ and ${\sigma }_{\epsilon ,\text{app}}^2$ for the MD simulations of the AGQ peptide, we again set κ² to 2/3 for each simulation frame, corresponding to rapidly sampled isotropic dye orientations (τ_rot << τ_D). With a realistic value of τ_D = 4 ns, τ_r exceeds τ_D by only a factor of ∼4 for the simulations in urea, yielding ${⟨\epsilon ⟩}_{\ast }$ = 0.53 instead of the limiting value of 0.50 (Fig. 3 d). However, this increase in ${⟨\epsilon ⟩}_{\ast }$ caused by rapid chain dynamics happens to be counteracted by the decrease in ${⟨\epsilon ⟩}_{\ast }$ owing to the orientational dynamics of the dyes (Fig. 2 d). Indeed, if we include the more realistic orientational dye dynamics from the MD simulations, we obtain ${⟨\epsilon ⟩}_{\ast }$ = 0.50 (Fig. 3 d). To further dissect the combined influence of chain dynamics and dye rotation on ${⟨\epsilon ⟩}_{\ast }$ and ${\sigma }_{\epsilon ,\text{app}}^2$, we use Rouse chain simulations together with Brownian diffusion simulations of the dye rotation, with τ_r/τ_rot set to the value of 23 observed in the AGQ peptide MD simulations with urea, as an example for a relatively low value of this ratio, and vary τ_r and τ_rot in parallel. As expected, we approach the fully static limit (Eq. 3) for large values of τ_r/τ_D, where τ_rot >> τ_D and τ_r >> τ_D (Fig. 3 d). With slower reorientation of the dyes, less of the κ² distribution is sampled while the donor is in the excited state, resulting in an increase in ${\sigma }_{\epsilon ,\text{app}}^2$, and we thus also approach the variance expected for the fully static limit (Eq. 3) for large values of τ_r/τ_D (Fig. 3 f).

Overall, our results demonstrate the importance of considering dye orientation dynamics and chain dynamics in the quantitative analysis of FRET experiments of unfolded and disordered proteins. However, results that are accurate to within an error of ∼0.04 in transfer efficiency can be obtained for realistic values of chain reconfiguration times and rotational correlation times of the fluorophores even for the short peptide with its rapid dynamics that we simulated with all-atom MD simulations. For larger IDPs with typical chain reconfiguration times of ∼100 ns (25), the separation of timescales between τ_r and τ_D is substantially greater, and the applicability of Eq. 1 more obviously justified.

Simulating a FRET experiment and its quantitative analysis

A complete assessment of the assumptions justifying the use of Eqs. 1, 2, and 3 is difficult or impossible in experiments, but the availability of experimentally validated all-atom simulations with explicit fluorophores enables us to generate photon emission data that realistically mimic a single-molecule FRET measurement (Fig. 4). These simulated photon emission data can then be analyzed using the same methods applied to actual experimental data to infer distance distributions and chain dynamics. This, in turn, provides a powerful means of comparing the results from standard data analysis with the known ground truth from the MD simulations, offering insights into the accuracy of the assumptions commonly made during the interpretation of FRET data. For this direct comparison, we can complement the MD simulations with Monte Carlo simulations of photon emission based on the experimentally quantified photophysical rate constants of the fluorophores (Fig. 4b, see materials and methods). Consequently, we can simulate experimental observables such as fluorescence lifetimes, time-resolved fluorescence anisotropies, transfer efficiencies, and fluorescence correlation functions (Fig. 4). This approach provides detailed photon statistics and avoids the simplifications related to the relative timescales of photophysics, dye rotation, and chain dynamics required for the limiting cases of Eqs. 1, 2, and 3. The time resolution of the MD simulations, with a step size of 2 fs and a saving frequency of (5 ps)⁻¹, allows us to cover the entire range of the photophysical and dynamic processes we need to consider. Similar approaches have been used previously (99,100), but here we chose a molecular system whose dynamics are sufficiently fast to enable extensive sampling in the MD simulations, so that converged intramolecular distance distributions are obtained, and no slower processes are relevant that would have to be modeled or approximated beyond the simulations.

A strength of single-molecule FRET experiments is the ability to characterize the chain dynamics in terms of distance relaxation times, or reconfiguration times, τ_r, between any two residues in the chain, and thus the contribution of different relaxation modes expected from polymer dynamics (17,25,66,101,102). The key experimental observable in single-molecule FRET experiments on flexible biopolymers is the fluorescence intensity decorrelation time from nsFCS (41,42) (Fig. 4 g). Although several molecular processes can contribute to fluorescence correlation functions in the submicrosecond regime, including quenching (103), triplet state dynamics (46), and rotational motion (104,105,106), distance fluctuations can be identified unequivocally by the anticorrelated signal in the donor-acceptor cross correlation function with a relaxation time identical to that of the donor-donor and acceptor-acceptor autocorrelation functions (82). The intensity relaxation times observed in nsFCS experiments, τ_c, are close to the reconfiguration time of the chain, τ_r, defined as the relaxation time of interdye distance autocorrelation (82). But to relate the two quantities accurately, we need to take into account the distance dependence of the transfer rate and the shape of the distribution within which the distance fluctuations occur. A commonly used approach (25) is to represent the chain dynamics in terms of 1D diffusion in the corresponding potential of mean force (PMF) (107) and identify the effective interdye diffusion coefficient or relaxation time that matches the experimentally observed value of τ_c (Fig. 4 f) (41,82). The AGQ peptide simulations provide an opportunity to assess this approximation, which does not take into account aspects such as the multimodal relaxation dynamics of polymers (17,108).

To assess the accuracy of the 1D diffusion model, we use the photon simulations based on the MD trajectories, which in this context represent the ground truth, to imitate the experimental procedure commonly used. First, we simulate photon emission based on the MD trajectories (Fig. 4, a–c) and calculate mean transfer efficiencies, $⟨E⟩$, and nsFCS curves. In the absence of reliable simulation data, the common procedure is to infer P(r) from the measured $⟨E⟩$ based on analytical polymer models; here we use the SAWν model (72), but other polymer models yield similar results (25). The fluorescence relaxation times, τ_c, from nsFCS, combined with the distance distribution, and the known distance dependence of the FRET efficiency (Fig. 1 f) fully define the dynamics of the chain in the framework of diffusion in a PMF (82) in terms of the chain reconfiguration time, τ_r (Fig. 5 c), or by the effective end-to-end diffusion coefficient (see materials and methods). We then used the resulting effective end-to-end diffusion coefficient as input for a 1D Brownian dynamics simulation in the PMF corresponding to the inferred P(r) (Fig. 5 d) to obtain end-to-end distance trajectories. This procedure enables a direct comparison of the interdye distance correlation function calculated from the MD simulations with the correlation function obtained from 1D diffusion (Fig. 5 h). Although subtle differences between the two correlation functions are visible, the overall agreement is remarkable. The resulting inferred values of τ_r deviate by 7% from the true reconfiguration time calculated directly from the MD simulations, which is less than the typical experimental uncertainty in τ_r. The good overall agreement indicates that the simple 1D diffusion approach offers a reasonable approximation for quantifying chain dynamics in disordered and unfolded proteins.

For the analysis described above, we used Eq. 1 to infer P(r), which is based on the assumption that τ_rot << τ_D << τ_r. However, although the separation of timescales between τ_rot ≈ 0.4 ns, τ_D ≈ 4 ns, and τ_r ≈ 15 ns for the simulations of the AGQ peptide is not large, quantifying chain dynamics in terms of 1D diffusion in the PMF is remarkably robust: variations in $⟨E⟩$ by ±0.04 lead to changes in the inferred P(r), but the deviations from the true value of τ_r remain below 12% (see materials and methods). A contribution that is more difficult to evaluate without detailed molecular simulations is the effect of the fluorophores on chain dynamics. For large polypeptide chains, the size of the dyes is small compared with the protein, but especially for shorter chains like the AGQ peptide their contribution is not negligible. MD simulations with and without fluorophores indicate that τ_r for the unlabeled peptide is ∼30% lower than for the labeled peptide (see materials and methods) (52), illustrating how optimized simulations provide a detailed and reliable interpretation of experimental results (35,109,110). However, not only does the interpretation of experimental results benefit from MD simulations; the development and refinement of MD force fields in turn benefits from the benchmarking with experimental data. The MD simulations described and analyzed above were performed with improved parameters for the dye-water interactions compared with previous work (52). In the following section, we describe the corresponding optimization process.

Force field optimization based on experimental data

In our previous study (52), we used the Amber 99SBws force field and the TIP4P/2005s water model, which had been optimized for simulations of IDPs by tuning the strength of the protein-water interactions by means of an increase in the Lennard-Jones ϵ parameter for the interaction strength between protein atoms and the oxygen atom of water (54). The short AGQ peptide, with its reconfiguration times between 5 and 10 ns, provides an excellent system for converging all-atom explicit-solvent simulations with dyes and using the direct comparison with experiment to further refine the simulation parameters. At 7.4 M urea concentration, when the AGQ peptide predominantly adopts expanded conformations, we found remarkable agreement between experiments and simulations (52). However, upon inspection of simulations without urea, persistent dye-dye and dye-protein interactions were detected, suggesting that the dye-dye and/or dye-protein interactions may be slightly too strong. To enable as rigorous a comparison with experiment as possible, we calculated four different observables from the simulations performed both with and without urea and compared them with the experimental values: 1) mean transfer efficiency, 2) nsFCS upon direct excitation of the acceptor, 3) time-resolved anisotropy decays, and 4) percentage of dark states where the fluorophores are in van der Waals contact (see materials and methods) (Fig. 5). Since the force field had previously been optimized by tuning the protein-water interactions (54,58), we aimed to decrease attractive dye-dye and dye-protein interactions by slightly increasing the strength of the dye-water interactions. We thus multiplied the Lennard-Jones ϵ between the oxygen atom of water and each atom of the dyes, as determined by the force field using the Lorentz-Berthelot combination rules, by a scaling factor of 1.10, a value previously optimized for protein-water interactions (54), and subsequently tested the effect of increasing this factor. We found that a slight increase from 1.10 to 1.15 improves the agreement between simulated and experimental anisotropy decays (Fig. 5, a and b), an important observable influenced by protein-dye interactions. This observation is also consistent with an earlier study with this force field that found that a factor of 1.15 improved the agreement of anisotropy decays with experiment in certain cases (58).

Another useful comparison is the effect of dye-dye quenching, assessed experimentally by nsFCS measurements with direct acceptor excitation (25), and included in simulations by assuming donor and acceptor to be nonemissive when in van der Waals contact (52) (Fig. 5 c). Since both fluorophores are dark while in contact, dye-dye quenching is not expected to influence the observed transfer efficiency, but the fluorescence intensity fluctuations on the timescale of ∼100 ns typical of the lifetime of dye stacking are detected in FCS measurements (111). It is important to note that, in the case of the simulations with a dye-water scaling of 1.10, 53% of the simulation runs ended in long-lived dark states, which do not contribute to the nsFCS curve, thus the timescale for quenching, τ_Q, obtained from the simulations represents a lower bound, and is therefore difficult to compare quantitatively with experiment. However, with scaling factors above 1.15, the amplitude of the simulated acceptor emission autocorrelation function becomes unrealistically low (Fig. 5 c). Moreover, a comparison of fluorescence emission spectra measured in the presence and absence of urea suggests that 34% of the molecules are quenched without urea owing to the short chain length and the correspondingly short distance between donor and acceptor (Fig. 5 d), similar to the fraction of nonemissive configurations in the simulations for a dye-water scaling of 1.15 and lower values. Correspondingly, the acceptor autocorrelation functions indicate the presence of a quenching component in the absence of urea and no detectable quenching in the presence of urea (Fig. 5 e). The mean transfer efficiencies and reconfiguration times from the simulations remained invariant within the statistical uncertainty of the simulations for scaling factors between 1.10 and 1.20. For the case of simulations with water-dye scaling factors of 1.10–1.20 in presence of urea, none of the observables changed significantly. We note, however, that even though the timescale for dye-dye quenching improved with a scaling factor of 1.15 compared with experiment, converging the long timescales of dye-dye interaction kinetics with simulations remains a challenge.

Discussion

The goal of this work was to test common approximations in the analysis of single-molecule FRET experiments, especially for disordered biopolymers, such as unfolded or disordered proteins as well as single-stranded nucleic acids. The validity of many of these approximations is difficult to assess based on simple analytical models, and our intention was thus to include as many of the relevant aspects of the underlying processes as possible in simulations. These include: 1) the detailed MD simulations of the biopolymer in all-atom, explicit solvent simulations with a state-of-the-art force field suitable for unfolded and disordered proteins; 2) the details of the structure and interactions of the FRET dyes with an explicit atomistic representation fully accounting for excluded volume effects, other attractive and repulsive interactions with the protein as well as with the other fluorophore, and for the influence of the dyes on the chain dynamics; 3) a comprehensive description of the photophysics of FRET for simulating photon emission, including not only the dependence on the interdye distance but also the relative dye orientation, as well as additional effects such as singlet-singlet annihilation and dye-dye quenching by photoinduced electron transfer, which leads to nonemissive states when the dyes are in close proximity or form stacked complexes.

Most importantly, this approach allows us to assess the role of all relevant timescales that affect the observed photon emission and other derived observables, especially the absolute and relative magnitudes of the fluorescence lifetimes, the rotational correlation times of the fluorophores, and the chain dynamics. The most common analysis procedures make simple assumptions regarding these timescales, such as very rapid dye reorientation relative to the fluorescence lifetime of the donor, or chain reconfiguration times that are much slower than the donor fluorescence lifetime. Although some of these approximations are plausible in many applications, they are unlikely to be fully met, and frequently we are in a regime where their validity is highly uncertain. For instance, the timescale of rotational decorrelation of the dyes is often close to the fluorescence lifetime. Moreover, the analysis of FRET experiments on disordered polymeric systems commonly involves the use of simple polymer models for describing intramolecular distance distributions. Here, we tested such assumptions based on extensive all-atom simulations validated by experimental fluorescence data, including intensity and lifetime-based FRET, time-resolved anisotropy, nsFCS, and dye-quenching measurements.

The specific system we investigate is particularly challenging for single-molecule FRET experiments: a short 22-residue peptide with correspondingly high transfer efficiency and chain reconfiguration times of ∼15 ns or even below, depending on solution conditions. The relevant timescales are thus not highly separated, the applicability of simple polymer models is far from obvious, and the size of the fluorophores relative to the peptide is large. Nevertheless, we find that the common approximations and analysis procedures work surprisingly well. The distance distributions inferred based on simple polymer models can provide a good approximation even in this case, and fluorescence lifetime measurements are a valuable approach for assessing whether the width of the inferred distribution is realistic (Fig. 3). One of the most frequently criticized and highly debated aspects of FRET is the validity of the popular approximation ⟨κ²⟩ ≈ 2/3 (6). Our analysis indicates that, even in a case where τ_rot and τ_DA are of similar magnitude, the effect of incomplete orientational sampling during the fluorescence lifetime yields transfer efficiency values based on ⟨κ²⟩ ≈ 2/3 that deviate from the true value by less than 0.04, comparable with the systematic experimental uncertainty in single-molecule FRET (92,93). Especially with the typical anisotropies observed in experiments on disordered proteins, ⟨κ²⟩ ≈ 2/3 is thus often likely to be a reasonable approximation. We emphasize, however, that measuring anisotropy for each sample is essential for assessing dye mobility and, especially in folded proteins, multiple components of the observed anisotropy decays can complicate analysis and interpretation (65,112). Finally, we find that the simplified analysis of chain dynamics in terms of diffusion in a PMF provides timescales within ∼10–30% of the true value even for such a short peptide, although the donor fluorescence lifetime is only a factor of 2–3 shorter than the chain reconfiguration time. For longer chains, where the separation of timescales between the two processes is much greater (25), the agreement is expected to be even better. In cases where fluorescence lifetimes and distance dynamics are less well separated, such as for the diffusive motion of the dyes within their accessible volumes (94,113,114,115), it can be important to take the influence of distance fluctuations into account explicitly (94,98,113).

Although the simple types of analysis that are commonly employed in the absence of detailed simulations often work remarkably well, molecular simulations provide a wealth of additional information. For instance, they allow us to assess the detailed origins of internal friction (89,101), the role of solvent dynamics (18,116), the influence of the fluorophores on the dynamics of the system (52), and many other aspects (35). Molecular simulations are also particularly important for systems where approximations with simple polymer models are likely to fail, such as sequences with pronounced residual structure or highly polarized charge distributions (117). In such cases, simulations validated by direct comparison with experimental data are the method of choice for a detailed mechanistic interpretation.

In conclusion, our integrated approach, which combines single-molecule FRET spectroscopy and all-atom MD simulations with simulations of photon emission, offers a comprehensive perspective on the interplay between the relevant timescales affecting the quantitative interpretation of single-molecule FRET data. We find that many of the simplifying assumptions commonly employed in the analysis of single-molecule FRET experiments are reasonable approximations even for systems with very rapid chain dynamics if we accept realistic bounds on the accuracy that can be achieved in FRET-based distance measurements. Simple simulation models, such as the Rouse model or diffusion in a potential of mean force, are useful tools for interpreting chain dynamics from single-molecule FRET and nsFCS data and for quantifying the quality of specific approximations. Directly comparing observables from experiments and simulations is a valuable approach for optimizing simulation models, including the force fields of MD simulations. Our results illustrate the growing synergy between experiment and simulation for achieving an increasingly quantitative understanding of molecular mechanisms.

Acknowledgments

We thank Dmitrii Makarov, Eitan Lerner, and Jan-Philipp Günther for discussions and valuable comments on the manuscript. This work was supported by the Swiss National Science Foundation (200021_219629, 310030_197776, to B.S.), the Intramural Research Program of the National Institute of Diabetes and Digestive and Kidney Diseases at the National Institutes of Health (to R.B.B.), and the Forschungskredit of the University of Zurich (to M.T.I.). We utilized the computational resources of Piz Daint and Eiger at the CSCS Swiss National Supercomputing Center.

Author contributions

All authors designed the research. M.N. and M.T.I. performed the research. All authors contributed analytic tools. M.N., D.N., and M.T.I. analyzed the data. R.B.B. and B.S. supervised the research. M.N., D.N., and B.S. wrote the manuscript with the help from all authors.

Declaration of interests

The authors declare no competing interests.

Editor: Elizabeth Rhoades.