Single-molecule reaction-diffusion

Abstract

We propose capturing reaction-diffusion on a molecule-by-molecule basis from the fastest acquirable timescale, namely individual photon arrivals. We illustrate our method on the intrinsically disordered human protein linker histone H1.0 and its chaperone prothymosin $\alpha$, as these diffuse through an illuminated confocal spot and interact, forming larger ternary complexes on millisecond timescales. Most importantly, single-molecule reaction-diffusion (smRD) reveals single-molecule properties without the requirement to trap or otherwise confine molecules to surfaces. We achieve smRD within a Bayesian paradigm and term our method Bayes-smRD. Bayes-smRD is further free of the average, bulk results inherent to analyzing long photon arrival traces by fluorescence correlation spectroscopy. In learning from mere thousands of photon arrivals, continuous spatial positions, and discrete conformational and photophysical state changes, Bayes-smRD estimates kinetic parameters on a molecule-by-molecule basis with two to three orders of magnitude less data than tools such as fluorescence correlation spectroscopy, thereby also dramatically reducing sample photodamage.

Significance

This research presents the first method capable of concurrently analyzing a single molecule’s reaction kinetics and diffusion dynamics based solely on single-photon arrival data. Utilizing single-molecule, single-photon confocal Förster resonance energy transfer data, we trace the continuous spatial trajectory and discrete state trajectories—encompassing conformational and photophysical changes—of individual molecules. This approach allows us to observe the reaction-diffusion dynamics of single molecules in real time without immobilizing them on surfaces, trapping them, or averaging signals across populations, thereby preserving single-molecule resolution.

Introduction

Individual biochemical reactions are the basis of information communication and signaling in cells (1,2,3,4,5). As such, developing methods to track and study individual biochemical reactions is the key to unraveling the mechanisms of life on a molecule-by-molecule basis. To capture individual molecular interactions at the fastest possible acquirable timescale, one may analyze photon arrivals derived from single molecules as these traverse a confocal spot (6,7).

Typical methods of analysis of confocal data, such as fluorescence correlation spectroscopy (FCS), indeed analyze photon arrival data (8,9) and conclude events at or below microsecond timescales (10,11,12). However, to derive diffusion coefficients (13,14) as well as other dynamical quantities (13,14,15), correlative methods rely on averaging over multiple (often thousands and more) single-molecule traversals through the confocal spot (termed bursts; see Fig. 1) (8,9,10,11,12,13,14,15). In doing so, correlative methods provide bulk, averaged properties. Yet, information at the single-molecule level is encoded in individual bursts. This includes information on the heterogeneity of pairwise interactions of interest here, for example, interactions of pairs of intrinsically disordered proteins (IDPs) (16).

Figure 1.

An illustration of a molecule labeled with a FRET pair undergoing state transitions while traversing a confocal spot. (a) The molecule’s 3D spatial trajectory. Darker colors indicate the molecule is further away from the spot center. (b) The molecule’s state trajectory. State 1 (2) corresponds to the low (high) FRET efficiency state. State 1 also has a slower diffusion coefficient. (c) The molecule’s probability of being excited by a laser pulse versus time. This trajectory encodes information on the excitation rate and spatial distance with respect to the confocal spot’s center. (d) A photon detection trace coming from these trajectories. A region with dense photon detections is often referred to as a burst.

To resolve either reaction kinetics or diffusion dynamics at the single-molecule level from individual photon arrivals, attempts have been made toward obtaining reaction rates with Förster resonance energy transfer (FRET) pairs (17) as molecules diffuse through confocal volumes. However, these methods cannot simultaneously resolve diffusion dynamics or track a molecule’s trajectory. These methods include works from Gopich and Szabo (18,19,20) and ${\mathrm{H}}^2$MM (7). Other tools exist for tracking molecules through confocal spots in the absence of reaction kinetics, exemplified by our lab’s previous work (21,22,23) and other confocal-based methods (24,25). For example, in this recent publication (26), diffusion dynamics are extracted but with no reaction (species are noninterconverting). Similarly, numerous methods exist for extracting FRET state transitions, albeit for complexes tied to surfaces (27,28,29,30,31). Yet tying substrates to surfaces can lead to changes in conformation and reactivity (32,33), and this motivates our ambition to investigate single-molecule reaction-diffusion (smRD) for freely diffusing substrates.

In other words, we seek a method capable of extracting both reaction rates and diffusion dynamics at the single-molecule level from the fastest acquirable timescale, namely photon arrivals. As such, we propose smRD dynamics using the following information: 1) photon arrival times (i.e., excited-state lifetimes) for pulses resulting in photon arrivals, 2) whether pulses yield detectable photons (i.e., which pulses are “empty”), and 3) donor or acceptor channels from which each photon is detected.

To avoid biases in our estimates for smRD parameters, we must also incorporate factors including background photon shot noise, fluorophore photophysical transitions (such as photoblinking and nonradiative decays), the shape of instrumental response functions (IRFs), direct acceptor excitation probability, and the bleed through between donor and acceptor detection channels.

Given that we aim to capture both reaction kinetics and diffusion dynamics from single bursts, accounting for the factors highlighted above, our smRD framework, Bayes-smRD, must efficiently extract information from every photon. To achieve this goal, we operate within the Bayesian paradigm (21,22,23,34,35,36,37), from which we build a hierarchical mathematical model that establishes probabilistic connections between the reaction-diffusion dynamics and experimental observations. Our framework is highly efficient, as all temporal correlations in the data are leveraged within this mathematical model without data preprocessing.

Concretely, in the Bayesian paradigm, our objective is to calculate complete probability distributions for the trajectories of the molecule states (including conformational, photophysical, and spatial) as well as their coinciding dynamical parameters (transition kinetics and diffusion coefficients) self-consistently and simultaneously. This study considers discrete conformational and photophysical states directly influencing FRET efficiency. We also consider continuous spatial states unrelated to FRET efficiency. For clarity, we use the term “state” to refer specifically to the joint conformational and photophysical state. In contrast, we use “spatial position” or “spatial trajectories” instead of speaking of a continuous spatial state.

When analyzing single-photon arrivals, an important concern is the possibility that the duration of a traversal, a “burst,” determined by the confocal volume size and diffusion coefficient, might be shorter than the state lifetimes. In such cases, the analyzed burst may not capture any transitions. To address this issue, our framework also allows for analyzing multiple bursts with independent state and spatial trajectories while assuming they share the same reaction-diffusion parameter values. Despite considering multiple bursts, the amount of data analyzed is still two to three orders of magnitude less than the data typically analyzed in (38,39). This is a critical advantage of the method we propose in avoiding photodamage, with the potential to probe reactions of photosensitive biomolecules (40,41).

In the subsequent sections, we demonstrate Bayes-smRD on both synthetic and experimental data involving the necessarily heterogeneous pairwise interaction of intrinsically disordered human proteins prothymosin $\alpha$ (ProT$\alpha$) to linker histone H1.0 (H1) at a single-molecule level.

Materials and methods

Mathematical formulation

We begin by considering equally spaced pulses at times $t_{1:K}=\left(t_1,t_2,\dots ,t_K\right)$ with an interpulse interval of $T$ and total pulse number $K$. At time point $t_k$, we have three observations: 1) whether there is a photon detection; 2) the photon arrival time after the pulsed laser ${\delta }_k$, termed the microtime; and 3) the detector in which the photon is detected $d_k$ (correlated to the color of the photon). In case pulse $k$ does not produce photon detection, both ${\delta }_k$ and $d_k$ take the value of the void $(\varnothing )$. These three types of observations can be combined into two arrays: ${\delta }_{1:K}=\left({\delta }_1,{\delta }_2,\dots ,{\delta }_K\right)$ and $d_{1:K}=\left(d_1,d_2,\dots ,d_K\right)$.

Now, both ${\delta }_{1:K}$ and $d_{1:K}$ are used as the input data to construct a posterior over the quantities we care about, namely $\theta =\left({\mathrm{Q}}_c,D_{1:M},c_{1:K},x_{1:K}\right)$. Here, ${\mathrm{Q}}_c$ is the state transition rate matrix of size $M\times M$, where $M$ is the number of states, $D_{1:M}$ contains all the diffusion coefficients of each state, and $c_{1:K}$ and ${\mathit{x}}_{1:K}$ are the state and spatial trajectories, respectively.

We assume that states and spatial positions remain relatively constant during an interpulse interval of approximately $50$ $\text{ns}$. To validate our assumption, we perform two quick back-of-envelope calculations. After transitioning into a state with a lifetime of $5$ $\mu \mathrm{s}$, the probability of the system leaving this state within $50$ $\text{ns}$ is less than 1%. Furthermore, if a particle in this state has a diffusion coefficient of $1000$ $\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$, there is a 95% chance that the particle’s displacement will be less than $0.02$ $\mu \mathrm{m}$, which is more than 10 times smaller than the size of a confocal spot.

Forward model

To simulate and later infer $\mathit{\theta }$, we must first prescribe the system’s evolution in terms of its state and spatial trajectories.

$$c_k|c_{k-1},\text{Π}\sim {\text{categorical}}_{1:M}\left({\text{π}}_{c_{k-1}}\right)$$ (1)

gives the state trajectory. Here, ${\pi }_{c_{k-1}}$ is the row $c_{k-1}$ of the transition probability matrix $\text{Π}$, and $\text{Π}$ is obtained from $\mathrm{\Pi }=\exp \left(T{Q}_c\right)$. Next, the evolution of the spatial trajectory is given by

$${\mathit{x}}_k|{\mathit{x}}_{k-1},D_{c_{k-1}}\sim \text{normal}\left({\mathit{x}}_{k-1},2D_{c_{k-1}}T\right)$$ , (2)

which follows from the solution of the diffusion equation with open boundaries.

With the state and spatial trajectories, we can briefly describe how they give rise to observations ${\delta }_k$ and $d_k$. As mentioned earlier, $d_k$ has three possible outcomes: no photon detected ($d_k=\varnothing$), a photon detected in donor channels $\left(d_k=D\right)$, or a photon detected in acceptor channels $\left(d_k=A\right)$, so $d_k$ is sampled from

$$d_k|{\mathit{x}}_k,f_D,f_A,{\lambda }_E,{\lambda }_F^{c_k}\sim {\text{categorical}}_{\varnothing ,D,A}\left(P_{d_k=\varnothing },P_{d_k=D},P_{d_k=A}\right)\text{.}$$ (3)

Here, $P_{d_k=\varnothing },P_{d_k=D}$, and $P_{d_k=A}$ are probabilities of each possible outcome whose complete formulas can be found in Eqs. S44 and S46, and $f_D$ and $f_A$ are two binary variables marking the existence of the donor and acceptor, respectively. These probabilities are calculated based on the state $c_k$, the illumination at position ${\mathit{x}}_k$, a Bernoulli random variable that denotes whether an active donor/acceptor fluorophore is present or not $f_{D/A}$, the effective excitation rate ${\lambda }_E$, and the FRET rates of each state ${\lambda }_F^{1:M}=\left({\lambda }_F^1,{\lambda }_F^2,\dots ,{\lambda }_F^M\right)$. Here, we define the effective excitation rate as the number of molecule-emitted photon detections per unit of time. Also, in our model, the actual FRET efficiency of the state $k$ is defined as ${\lambda }_F^k/\left({\lambda }_F^k+{\lambda }_R^D\right)$, where ${\lambda }_R^D$ is the predefined donor emission rate.

In addition, calculating these probabilities also involves the consideration of spectral bleed through, uniform background photons, and the direct excitation of acceptor dyes (see supporting material section C for details). For bleed through, we predefine probabilities connecting photon emission channels to detection channels. Uniform background photons are treated as independent photon sources with fixed emission rates. Furthermore, the direct excitation of acceptor dyes is dealt with by introducing a multiplicative factor that multiplies the donor excitation rate. All predefined parameters can be obtained based on the wavelengths of photon emissions and optical filters used in actual experiments.

Next, we write down a likelihood for the microtime observation ${\delta }_k$. When no photon is detected $\left(d_k=\varnothing \right)$, ${\delta }_k$ is identically void $(\varnothing )$ with probability 1. Otherwise, ${\delta }_k$ is sampled from

$${\delta }_k|d_k,c_k,{\mathit{x}}_k,f_D,f_A,{\lambda }_E,{\lambda }_F^{1:M}\sim \mathbb{P}\left(d_k,c_k,{\mathit{x}}_k,f_D,f_A,{\lambda }_E,{\lambda }_F^{1:M}\right)\text{.}$$ (4)

The full expression for this distribution can be found in Eqs. S21–S23. Eq. 4 is derived mainly from the convolutions of exponential distributions and the IRF. The exponential distributions reflect waiting times in the excited states of both the donor and acceptor, so they depend on $c_k$, $f_D$, $f_A$, ${\lambda }_E$, and ${\lambda }_F^{1:M}$, as well as the precalibrated donor and acceptor emission rates ${\lambda }_R^D$ and ${\lambda }_R^A$. Moreover, the IRF, whose mathematical form is precalibrated from experiments, is characterized by two fixed parameters: offset ${\tau }_{\delta }$ and width ${\sigma }_{\delta }$.

Inverse model

From the previous section, we see that to infer $\mathit{\theta }$, we must also learn the latent variables that coincide with the state trajectory $c_{1:K}$ and the spatial trajectory ${\mathit{x}}_{1:K}$. As such, the complete set of quantities to be learned can now be expanded to $\mathrm{\Theta }=\left(c_{1:K},x_{1:K},f_D,f_A,\mathrm{\Pi },D_{1:M},{\lambda }_E,{\lambda }_F^{1:M}\right)$. Given all the equations above and the expanded likelihoods accounting for bleed through, uniform background photons, acceptor direct excitation, and IRFs (see Eqs. S47–S49), we can now, alongside priors, construct a full posterior probability distribution over $\mathit{\theta }$.

To sample this high-dimensional posterior, $P\left(\mathrm{\Theta }|d_{1:K},{\delta }_{1:K}\right)$, we opt for specialized Markov chain Monte Carlo schemes that we design herein. The sampling of this posterior is achieved by invoking a global Gibbs sampling scheme (42). Within this Gibbs sampler, for $f_D$, $f_A$, each row of $\text{Π}$, and each element of $D_{1:M}$, we select the priors listed in supporting material section C6. Briefly, for computational convenience (detailed in the supporting material), directly sampling $f_D$ and $f_A$ requires Bernoulli priors, each row of $\text{Π}$ requires a Dirichlet prior, and each element of $D_{1:M}$ requires an inverse gamma prior. The corresponding conditional posteriors can be found in Eqs. S52, S53, S116, and S125. As for $c_{1:K}$, we place a categorical distribution as a prior on $c_1$ and then apply the forward-filtering backward-sampling algorithm (43) to sample all the states recursively, as shown in Eq. S111.

Other quantities, ${\lambda }_F^{1:M}$, ${\mathit{x}}_{1:K}$, and ${\lambda }_E$, cannot be sampled directly. As a result, we use the Metropolis-Hastings algorithm (44,45). We use gamma distributions as proposal and prior probability distributions to sample ${\lambda }_F^{1:M}$. As ${\mathit{x}}_{1:K}$ and ${\lambda }_E$ are slow to converge, on account of the fact ${\mathit{x}}_{1:K}$ contains many intercorrelated continuous variables and inferring ${\lambda }_E$ heavily depends on ${\mathit{x}}_{1:K}$, we therefore opt for Hamiltonian Monte Carlo steps (46) within our broader Gibbs sampling scheme to help speed up the overall convergence. Detailed sampling schemes for ${\lambda }_F^{1:M}$, ${\mathit{x}}_{1:K}$, and ${\lambda }_E$ are covered in supporting material sections D3, D7, and S8.

Burst selection

Burst selection is achieved by first binning the single-photon arrival data and setting a threshold on the number of photons per bin. We locate bursts’ start and end times for bins above the threshold. Within each burst, we calculate the FRET efficiencies and stoichiometries with the corrections often regarding background photon emission rate, detection efficiencies, and bleed-through ratios provided by our experimental collaborators. Corrections are applied for the experimental data sets analyzed in this paper using the route correction matrix (47,48). FRET efficiencies and stoichiometries are checked to rule out any burst containing uneven numbers of donor and acceptor fluorophores.

Results

Within the Bayesian paradigm, we compute full joint probability distributions (termed posteriors) over a molecule’s state trajectories, spatial trajectories, state transition rates, diffusion coefficients, and other quantities of interest (including FRET efficiencies and excitation rate). The breadth of these posterior probability distributions, especially critical in single-molecule settings, reflects uncertainty propagated from the finiteness of the data available (as molecules come in and out of the confocal volume) but also other experimental parameters (background noise, breadth of IRF, and detector bleed through). This paper’s posterior distributions, shown as histograms, are normalized as probability densities (with unit area). In addition to distributions, specifically for state and spatial trajectories, we provide the maximum a posteriori trajectories as point estimates.

In this work, we make the assumption (though not required) of a three-dimensional Gaussian confocal volume (49,50). As discussed in the materials and methods section, the framework can accommodate any precalibrated (known) confocal volume shape. The confocal volume’s spatial symmetries result in multiple spatial trajectories sharing the same probability density, leading to degeneracy in a molecule’s absolute spatial positions. Although this degeneracy does not pose a problem during the inference step, it introduces ambiguity in visualizing results. Therefore, we can only represent the molecule’s “excitation probability trajectory” instead of its absolute spatial trajectory when visualizing results. The excitation probability trajectory, defined in Eq. S5, combines statistically equivalent spatial trajectories and provides a combined measure of the molecule’s excitation rate and its distance from the confocal spot center (see Fig. 1 c).

Along these same lines, we report each state’s escape rates (inverse lifetime) in what follows, reflecting how fast the system leaves its current state as an alternative to reporting concentration-dependent association rates otherwise less meaningful at the single-molecule level.

In the following two subsections, we apply Bayes-smRD to experimental data collected on the interaction of IDP fragments ProT$\alpha$ and H1. Then, we validate our framework’s robustness using synthetic data where ground truth is accessible for comparison.

Experimental data and analysis description

In the experimental analysis, we monitor the interactions of human proteins ProT$\alpha$ (net charge −44) and linker histone H1 (net charge +53). As both are highly and oppositely charged, ProT$\alpha$ and H1 appear in bound states while retaining disordered features (51,52,53,54), as illustrated in Fig. 2. Experimental data from single-molecule, circular dichroism, and NMR spectroscopy, as well as molecular simulations (51,52,53), support the hypothesis of nonspecific ProT$\alpha$ and H1 binding with expected diffusion-limited and concentration-dependent associations and rapid concentration-independent off-rates, despite their high and opposite charges (51,52,53). It has been hypothesized that this (dis)association may help retain regulatory cell network responsiveness (52).

Figure 2.

Some kinetic schemes involving ProT$\alpha$ and H1 molecules (P and H in this figure, respectively) shown with snapshots from coarse-grained molecular dynamics simulations (51,52). A freely diffusing ProT$\alpha$ and an H1 may form a ProT$\alpha$-H1 (PH) dimer, compactifying their conformation. This dimer may putatively continue binding other ProT$\alpha \mathrm{s}$and H1s if available, forming larger ternary complexes including ProT${\alpha }_2$-H1 and ProT$\alpha$-H1₂ (PPH and PHH in this figure, respectively). Possible further reactions, e.g., the formation of tetramers, are excluded herein. At each step, this figure illustrates only one of many candidate structures for this highly dynamic and intrinsically disordered system.

A fluorescent FRET pair (Alexa Fluor 488 as donor and 594 as acceptor) is attached to ProT$\alpha$ at positions 56 and 110, respectively. A free doubly labeled ProT$\alpha$, with two dyes typically far apart, is expected to coincide with a low FRET efficiency state, whereas bound states of the IDP pair bring the two fluorophores on ProT$\alpha$ closer together, yielding higher FRET efficiencies (51,52,53).

The concentration of labeled ProT$\alpha$ in the data sets analyzed is either $50$ or $75$ $\text{pM}$. Given the confocal volume’s dimensions (see Table S2), the expected number of labeled ProT$\alpha$ within the volume is less than 0.05% at any instant. As such, a good approximation is to assume that either zero labeled ProT$\alpha$ or, at most, one is present within the illuminated region. As a result, we anticipate data with bursts of photon arrivals that indicate the presence of one ProT$\alpha$ diffusing within the volume.

From (51,52), we know that some complexes, e.g., the ProT$\alpha$-H1 dimer, have lifetimes longer than the average duration of a burst (about $5$ $\text{ms}$). Therefore, for all concentrations discussed in this paper, as mentioned in the introduction, we analyze several bursts together, assuming that they have independent state and spatial trajectories but otherwise share the same reaction-diffusion dynamics (e.g., same diffusion coefficients for each conformational state even if each state is not visited in each trajectory). Specifically, from a roughly 8-min-long time trace, we select 12 short time intervals at different times, half containing bursts (the burst group) and the other half containing burst-free regions (the burst-free group). Each group has its reaction-diffusion dynamics. See burst selection for more details on how we select bursts.

Direct head-to-head comparison between inferences drawn from Bayes-smRD and those from conventional methods is nontrivial, as no method other than Bayes-smRD infers reaction-diffusion dynamics, state trajectories, and spatial trajectories simultaneously from the same data set, namely single-photon arrivals. When we compare the results we infer using Bayes-smRD to other methods, we must compare our results to values drawn from various experimental setups. For example, although we extract diffusion coefficients and reaction rates from photon arrivals, we have to compare the results we obtain for diffusion coefficients from those reported by NMR or those we calculate ourselves by FCS, as discussed in greater depth later. As for reaction rates to which we compare, they are those otherwise inferred from single-photon arrivals.

$75$$\text{pM}$ labeled ProT$\alpha$ only

The first data set analyzed contains only $75$ $\text{pM}$ doubly labeled ProT$\alpha$. The burst group contains six bursts that are $34$ $\text{ms}$ long, amounting to 5517 photon detections. From this data set, we expect to encounter only one conformational state (and thus one FRET state), though blinking of both dyes (55,56) may introduce apparent FRET transitions. For this reason, we run a two-state model on this data set. Suppose our model considers more states than are present in the data. In that case, we find (shown in Fig. S5) that the additional states have, as expected, much higher uncertainty (broader posterior probability distributions) associated with state trajectories, escape rates, diffusion coefficients, and FRET efficiencies than those of the real states. Moreover, these additional states are often hardly visited. For instance, in Fig. S5, the additional state (state 3) is visited for less than 2% of the time.

The results of our analysis are shown in Fig. 3. In particular, we show the state trajectory of one burst in Fig. 3 a and the corresponding excitation probability trajectory in Fig. 3 b from the burst shown in Fig. 3 c. We also obtain both states’ escape rates (equivalent to transition rates in a two-state system), diffusion coefficients, and FRET efficiencies in Fig. 3, d–f, respectively, jointly from all six bursts analyzed, for the reason explained in the introduction. With high certainty, Fig. 3 a shows that the system visits both states, suggesting one more state than expected. Fig. 3 f provides more evidence for the existence of this extra state, as it shows two very sharp and well-separated FRET efficiencies.

Figure 3.

Results from the experimental data set with $75$$\text{pM}$ labeled ProT$\alpha$ only. (a) A purple line highlights the learned (maximum a posteriori [MAP]) state trajectory. In the main text, we explain why state 1 is attributed to acceptor photophysics (and, as such, is expected to have a diffusion coefficient similar to state 2 within error). In contrast, state 2 is a freely diffusing labeled ProT$\alpha$. The color map indicates confidence in the learned trajectory in terms of probability at each pulse. Darker colors indicate higher confidence. (b) The corresponding excitation probability trajectory offers a joint measure of the spatial trajectory and the excitation rate. Here, the excitation probability is defined as the “probability of getting some photon detection from a single pulse through donor excitation.” (c) The corresponding experimental data trace containing 458 detected photons within nearly $3$$\text{ms}$. Each green or red bar represents photon detection in the donor or acceptor channel. The height of each bar represents its arrival time. (d–f) The escape rates, diffusion coefficients, and FRET efficiencies learned from six bursts (only one shown in a–c). All gray histograms are from state 1, whereas all blue histograms are from state 2. The reference diffusion coefficient (denoted by a vertical line in e) derived from FCS (see main text) is equal to $28$$\mu m^2{\mathrm{s}}^{-1}$. The reference FRET efficiency (52) is 0.36 in (f).

One state here should coincide with the freely diffusing doubly labeled ProT$\alpha$ molecule. We attribute state 2 (blue in Fig. 3, d–f) to this freely diffusing ProT$\alpha$ because its FRET efficiency, $0.317$–$0.359$ as the 95% credible interval (CI), is very close to that of a free ProT$\alpha$ reported in (52). On the other hand, state 1 (gray in Fig. 3, d–f) has zero FRET efficiency with very low uncertainty (95% CI is $0.000$–$0.029$). This low FRET efficiency state may putatively be attributed to 1) a conformation yielding a large distance between the donor and the acceptor, 2) free donor fluorophores in solution, and 3) the acceptor’s dark state (donor dye’s photophysics typically decreases donor channel photon counting rate and yields higher FRET efficiencies).

Here, option 1 is unlikely as it is not supported by any known simulation or experiment (51). We also have reason to believe that option 2, free dye, is not very likely as the 95% CI for the diffusion coefficient of state 1 is $(12.6$–$47.4)$ $\mu m^2{\mathrm{s}}^{-1}$, agreeing with state 2’s diffusion coefficient 95% CI $(22.4$–$42.0)$ $\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$. Thus, we conclude that state 1 likely originates from the acceptor’s dark state (option 3).

We first look into existing literature to compare Bayes-smRD’s diffusion estimate to a reference value. In (51), the diffusion coefficient for free ProT$\alpha$ is reported to be $55\pm 1$ $\mu m^2{\mathrm{s}}^{-1}$. This value was determined through pulsed-field gradient NMR at $283$ $\mathrm{K}$, utilizing $100$ $\mu \mathrm{M}$ ProT$\alpha$. The experimental conditions in this study differ from the data set under analysis ($75$ $\text{pM}$ ProT$\alpha$ at room temperature, single-photon confocal FRET microscopy (52)).

For this reason, we conducted FCS on the same data set from which a small portion was analyzed using Bayes-smRD. Besides FCS being a conventional method for estimating diffusion coefficients using single-photon arrival data (13,15,21,39,51,57,58,59,60,61), FCS also offers a more direct comparison, given that both FCS and Bayes-smRD necessitate the same PSF size calibration in estimating diffusion coefficients. As posited earlier, a simple, single-state FCS fit is warranted in this data set as all states share the same conformation. Consequently, we conducted FCS under the same confocal volume calibration invoked for Bayes-smRD (see Table S2) using the identical data set, albeit with the complete time trace, capturing over $3\times {10}^7$ photons (approximately $5\times {10}^3$ times more than the quantity analyzed by Bayes-smRD). Fig. S6 shows our FCS curves and associated fits. The best least-square fit diffusion coefficient by FCS is $28$ $\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$, falling within Bayes-smRD’s CIs. A possible explanation for the discrepancy between the NMR results and those of Bayes-smRD (consistent with those of FCS) lies in the fact that, as depicted in Fig. S6, the $y$-intercepts of the FCS fits are higher than those observed in the actual FCS autocorrelation curves. This suggests that the PSF calibration employed in FCS, as well as Bayes-smRD, may slightly underestimate the PSF’s true volume (13), assuming the concentration of ProT$\alpha$ at $75$ $\text{pM}$.

Furthermore, in Fig. S5, where we run Bayes-smRD on the same data set but with a three-state model, state 2’s diffusion coefficient CI is $(21.4$–$28.8)$ $\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$. These agreements suggest that our method learns diffusion coefficients consistently, given accurate and confocal volume calibration.

Having interpreted the states and diffusion dynamics in our analysis, we now turn to the transition rates between them. As shown in Fig. 3 d, the transition rate from state 2 to state 1 (i.e., the rate at which an acceptor transitions to a dark state) has 95% CI $(0.12$–$0.64)$ ${\mathrm{s}}^{-1}$. At the same time, that of the reverse process is $(0.67$–$4.02)$ ${\mathrm{s}}^{-1}$ based on all six bursts analyzed.

Here, we list the estimates from the same six bursts using a three-state model (Fig. S5) to further support the reaction-diffusion dynamics estimates reported. States 1 and 2 in Fig. S5 maintain their interpretations, whereas state 3 is intended to be redundant (and therefore has no meaningful interpretation) for the reason explained at the beginning of this section. The diffusion coefficient CIs of states 1 and 2 are $(14.5$–$39.5)$ and $(21.4$–$28.8)$ $\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$, respectively. As for the transition rates from state 1 to state 2 and from state 2 to state 1 in Fig. S5, we have $(0.09$–$0.77)$ and $(0.16$–$4.07)$ ${\text{ms}}^{-1}$, respectively. Therefore, the two- and three-state models yield consistent reaction-diffusion dynamics estimates.

$75$$\text{pM}$ labeled ProT$\alpha$ + $10$$\text{nM}$ H1: ProT$\alpha$-H1 dimer should dominate

The second data set we analyzed contains $75$ $\text{pM}$ labeled ProT$\alpha$ and $10$ $\text{nM}$ H1. As H1’s concentration is much higher than that of labeled ProT$\alpha$, as illustrated in Fig. 2, we expect to start seeing another state corresponding to the ProT$\alpha$-H1 complex with a higher FRET efficiency than the free ProT$\alpha$ state (51,52,53), as ProT$\alpha$ compactifies upon binding. At least four states are anticipated in this data set: 1) free ProT$\alpha$ with a bright acceptor, 2) free ProT$\alpha$ with a dark acceptor, 3) ProT$\alpha$-H1 with a bright acceptor, and 4) ProT$\alpha$-H1 with a dark acceptor. Other possibilities, such as a burst created by free dyes, can be excluded during burst selection (see burst selection).

However, as states 2 and 4 have the same FRET efficiency (zero) and similar diffusion coefficients (51) (the difference is less than the breadth of the posteriors in Fig. 3 e), they are challenging to distinguish from the observables provided. Therefore, we run a three-state model instead of treating them as separate states.

As before, we selected a group of six bursts (5930 photons in total) for analysis. Our results are shown in Fig. 4 following the same layout as the earlier Fig. 3. Immediately, from Fig. 3 f, we detect a high FRET efficiency state (state 3, green) with $0.618$–$0.701$, which (when comparing to the results of Fig. 3) we ascribe as originating from the ProT$\alpha$-H1 dimer (making note that the FRET efficiency estimates depend on the precalibrated background photon rates; see Table S2 for the value used in our analyses). Moreover, Fig. 3 a shows that the system spends the most time in state 3, consistent with (52). As for state 3’s diffusion coefficient, Fig. 4 e gives a 95% CI of $(32.0$–$41.9)$ $\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$.

Figure 4.

Results from the experimental data set with $75$$\text{pM}$ labeled ProT$\alpha$ and $10$$\text{nM}$ H1. The layout is the same as in Fig. 3. We discuss in the main body why state 3 is most likely attributed to the ProT$\alpha$-H1 complex. (a) A purple line highlights the MAP state trajectory. The color map indicates confidence in the learned trajectory in terms of probability at each pulse. Darker colors indicate higher confidence. (b) The corresponding excitation probability trajectory offers a joint measure of the spatial trajectory and the excitation rate. (c) The corresponding experimental data trace of nearly 7 ms. Each green or red bar represents photon detection in the donor or acceptor channel. The height of each bar represents its arrival time. (d–f) The escape rates, diffusion coefficients, and FRET efficiencies learned from six bursts (only one shown in (a–c)). All gray histograms are from state 1, all blue histograms are from state 2, and all green histograms are from state 3. The reference FRAET efficiency (52) is 0.55.

Although, again, no literature has specifically measured the diffusion coefficient of the ProT$\alpha$-H1 complex under identical conditions for direct comparison, (51) reports a diffusion coefficient of $47\pm 3$ $\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$ for a mixture containing $100$ $\mu \mathrm{M}$ ProT$\alpha$ and $100$ $\mu \mathrm{M}$ H1 (saturating concentration). This measurement used pulsed-field gradient NMR at $283$ $\mathrm{K}$. Similar to our rationale from the previous data set, we attribute the observed difference in diffusion coefficient estimates to PSF calibration, aside from the variations in experimental conditions. Lastly, as this data set now involves interconversions between an unknown number of complex species, naive comparison to FCS with a one-state fit similar to Fig. 3 e is no longer feasible.

We also note that Bayes-smRD’s estimate of state 2’s diffusion coefficient from the posterior density in Fig. 4 e appears to exhibit an almost bimodal pattern. This observation parallels what is seen in Fig. 3 e. Nevertheless, we argue that this apparent bimodality is simply a byproduct of an imperfectly characterized posterior due to the paucity of data inherent to analyzing small collections of bursts, as further detailed in the subsequent section on the analysis of synthetic data.

State 1 (gray), on the other hand, has a FRET efficiency range of $0.001$–$0.094$, which is very similar to the state interpreted as acceptor photophysics in Fig. 3. To confirm this, we look at the transition rates between state 1 and state 3. From state 3 to state 1, the transition rate has 95% CI $(0.003$–$0.38)$ ${\text{ms}}^{-1}$, whereas the reverse transition has $(0.09$–$5.96)$ ${\text{ms}}^{-1}$. Both are similar to the photophysical rate ranges reported in Fig. 3, and subtle discrepancies may originate from variations in the acceptors’ photophysics as they are brought closer to the donor dyes in the ProT$\alpha$-H1 complex.

On the flip side, interpreting state 2 is trickier, as Fig. 4, a and d, indicates that the system hardly spends any time in state 2 (average lifetime is about $0.01$ $\text{ms}$). However, state 2’s parameter posterior probability distributions (Fig. 4, d–f) despite being broad, do not closely follow the shape of the corresponding priors (for this case, see Fig. S5, state 3), meaning the analyzed bursts probably carry enough data supporting state 2’s existence, but its reaction-diffusion dynamics as well as FRET efficiency can hardly be pinpointed due to its short lifetime (see Fig. S4 for how short lifetimes affect our analysis). We also speculate that state 2 may originate from the photoblinking of acceptor fluorophores, as fluorophores may exhibit photophysics dependent upon variations in ProT$\alpha$ and H1 concentrations. Another speculation is that state 2 corresponds to a free ProT$\alpha$ molecule, and its short lifetime is caused by the abundance of H1 in the surrounding environment. Though this speculation is less likely, as state 2’s escape rate (Fig. 4 d, blue) is much higher than the association rates (with $10$ $\text{nM}$ H1) reported in (52), we mention it nonetheless due to the high uncertainty in state 2’s escape rate.

$50$$\text{pM}$ labeled ProT$\alpha$ + $20$$\mu \mathrm{M}$ H1 + $84$$\mu \mathrm{M}$ unlabeled ProT$\alpha$: ProT${\alpha }_2$-H1 trimer should be present

To further demonstrate the profound data efficiency of Bayes-smRD, we now move to the third date set in which unlabeled ProT$\alpha$ is added ($50$ $\text{pM}$ labeled ProT$\alpha$ $20$ $\mu \mathrm{M}$ H1 and $84$ $\mu \mathrm{M}$ unlabeled ProT$\alpha$). Under this set of concentrations, we expect both ProT$\alpha$-H1 and ProT${\alpha }_2$-H1 to be present (52).

Following our argument in the previous subsection, we still run a three-state model, expecting to see three states with different FRET efficiencies. We subsequently interpret these states based on their reaction-diffusion dynamics. Immediately, from Fig. 5, a and f, we can see that, as expected, Bayes-smRD indeed recovers three distinct states whose posteriors over FRET efficiency differ substantially from the prior and captures transitions among them. Moreover, Fig. 5 d shows that the average lifetimes of the zero FRET (state 1, gray) and the low FRET state (state 2, blue) are both around $0.2$ $\text{ms}$. This indicates that binning-based FRET analysis tools must have a bin size no greater than $0.2$ $\text{ms}$ to capture these transitions. However, in the burst shown in Fig. 5 c, there are 760 photon detections in total; hence, setting $0.2$ $\text{ms}$ means each bin contains less than 40 photons. Such a low number of detections will result in a very low signal/noise ratio, thereby greatly undermining the effectiveness of binning-based tools (31) and highlighting the importance of the direct photon-by-photon analysis used herein.

Figure 5.

Results from the experimental data set with $50$$\text{pM}$ labeled ProT$\alpha$$20$$\mu \mathrm{M}$ H1 and $84$$\mu \mathrm{M}$ unlabeled ProT$\alpha$. The layout is the same as Fig. 3. (a) A purple line highlights the MAP state trajectory. The color map indicates confidence in the learned trajectory in terms of probability at each pulse. Darker colors indicate higher confidence. (b) The corresponding excitation probability trajectory offers a joint measure of the spatial trajectory and the excitation rate. (c) The corresponding experimental data trace of nearly 4 ms. Each green or red bar represents photon detection in the donor or acceptor channel. The height of each bar represents its arrival time. (d-f) The escape rates, diffusion coefficients, and FRET efficiencies learned from six bursts (only one shown in (a-c)). All gray histograms are from state 1, all blue histograms are from state 2, and all green histograms are from state 3. The reference value from (52) in (d) is $1.9$${\text{ms}}^{-1}$. This value does not offer a direct comparison as photophysical states are not considered in this literature.

In previous subsections, we have presented how the zero-FRET state (Figs. 4 and S5, state 1) and the low-FRET state (Fig. S5, state 2) are interpreted as acceptors’ dark state and free ProT$\alpha$, respectively. The same arguments still hold for states 1 and 2 in Fig. 5. Though state 2’s diffusion coefficient and FRET efficiency in this data set differ from the estimates presented in Figs. 3 and S5, we attribute this discrepancy to the finiteness of data, as only about 7% of the 4548 (about 320) photons collected in all six bursts originate from state 2. One may argue against this statement by saying the finiteness of data should result in higher uncertainties represented by wider posteriors, unlike what is shown in Fig. 5 e. However, a posterior probability distribution’s breadth reflects uncertainty when sufficient samples are present. In the next section, we will show that posteriors over diffusion coefficients typically require over 1000 photons for the 95% CIs to always cover the ground truths, which is three times more than the currently available number of photons from state 2. Therefore, the number of photons and posterior breadths are necessary for evaluating our estimates’ confidence.

On the other hand, it is still worth discussing what contributes to state 3 in Fig. 5. As mentioned earlier, under this set of concentrations ($50$ $\text{pM}$ labeled ProT$\alpha$ $20$ $\mu \mathrm{M}$ H1 and $84$ $\mu \mathrm{M}$ unlabeled ProT$\alpha$), according to (52), we anticipate complexes including not only ProT$\alpha$-H1 but also larger ternary complexes such as ProT${\alpha }_2$-H1 or ProT$\alpha$-H1₂ to form (see Fig. 2).

To proceed, we first calculate state 3’s escape rate from Fig. 5 d, whose 95% CI is $(0.31$–$1.07)$ ${\text{ms}}^{-1}$, and the mean value is $0.63$ ${\text{ms}}^{-1}$. These numbers are smaller than the off rate of ProT${\alpha }_2$-H1 reported in (52), which is $1.9$ ${\text{ms}}^{-1}$. However, we argue this is not a discrepancy. The key point here is that ProT$\alpha$-H1 and ProT${\alpha }_2$-H1 do not show a noticeable difference in their FRET efficiencies (52); hence, they can hardly be separated during burst selection. Therefore, part of the six bursts analyzed may contain ProT$\alpha$-H1, whereas others contain ProT${\alpha }_2$-H1. Consequently, state 3 in Fig. 5 can represent a mixture of these two complexes. Since ProT$\alpha$-H1 has a much lower off rate, $1.7\times {10}^{-3}$ ${\text{ms}}^{-1}$ (52), it is reasonable that the overall escape rate of state 3 is lower than $1.9$ ${\text{ms}}^{-1}$. This explanation highlights the importance of having a tool capable of inferring reaction-diffusion dynamics from single bursts.

Synthetic data

To validate Bayes-smRD, we generate synthetic data by computer simulation (62,63,64,65,66), representing the Brownian motion of a single molecule in a confocal volume while undergoing transitions. These simulations incorporate real-life complications, including background noise, IRF, direct excitation of the acceptor fluorophores, excitation probabilities dependent on the light intensity at the fluorophore’s instantaneous spatial location, and detector bleed through. These parameters mimic those in real experiments (51,52) and are tabulated in Table S2.

As illustrated in Fig. 6, Bayes-smRD can simultaneously capture a single molecule’s state and spatial trajectories in good agreement with ground truth. Fig. 6, d and e, shows Bayes-smRD’s performance in learning reaction rates (escape rates) as well as diffusion coefficients. All ground-truth values fall within the 95% CIs and usually lie close to the posterior probability distribution maxima. Moreover, Fig. 6 shows that Bayes-smRD avoids binning artifacts introduced in FRET analysis methods, thereby reducing temporal resolution as transitions occurring on timescales shorter than the bin size are averaged out.

Figure 6.

Results from synthetic data, along with the corresponding ground truths, priors, and 95% credible intervals (CIs). Our simulation assumes a two-state model. The layout is the same as in Fig. 3. (a) A purple line highlights the MAP state trajectory. The color map indicates confidence in the learned trajectory in terms of probability at each pulse. Darker colors indicate higher confidence. (b) The corresponding excitation probability trajectory offers a joint measure of the spatial trajectory and the excitation rate. (c) The corresponding experimental data trace of about 3 ms. Each green or red bar represents photon detection in the donor or acceptor channel. The height of each bar represents its arrival time. (d–f) The escape rates, diffusion coefficients, and FRET efficiencies learned from six bursts (only one shown in (a–c)). All blue histograms are from state 1 and all green histograms are from state 2. Ground-truth values are $1.95$${\text{ms}}^{-1}$ for state 1 and $2.92$${\text{ms}}^{-1}$ for state 2 in (d), $60$$\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$ for state 1 and $30$$\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$ for state 2 in (e), and 0.33 for state 1 and 0.67 for state 2 in (f).

Earlier, we highlighted in Figs. 3 e and 4 e that the posterior probability densities of the diffusion coefficient appear bimodal. This bimodal trend is again evident in Fig. 6 e for state 2. Given that Fig. 6 is generated through simulation without any inherent bimodality in the diffusion coefficient, we assert that this apparent bimodal behavior arises in situations where relatively high uncertainty is associated with the diffusion coefficient due to the finiteness of data.

To further demonstrate the robustness and validity of Bayes-smRD, we assess our method by systematically varying key parameters one at a time using the baseline parameter values listed in the caption of Fig. 6. We observe how these parameter variations influence the results of Bayes-smRD. The initial parameter we vary is the number of bursts included in the analysis, and the corresponding results are depicted in Fig. S3, a–c.

These figures illustrate that analyzing two bursts together yields 95% CIs covering half of the quantities’ actual values under a realistic excitation rate. Conversely, including all five bursts is necessary to ensure that the 95% CIs all cover their respective ground-truth values. Given that these bursts share the same set of reaction-diffusion dynamics, exhibit similar lengths, and each contains approximately 10 state transitions, Bayes-smRD requires a minimum of around 20 transitions, with a preference for 50 for all ground-truth values to lie within the respective 95% CI. Consequently, if a known factor slows down the reaction kinetics of a system, the length of the time trace analyzed should be extended by the same factor. This extension can be achieved either through the analysis of longer bursts or by incorporating more independent bursts. This result underscores the necessity of jointly analyzing six bursts for experimental data analysis.

Although Bayes-smRD targets systems with both diffusion dynamics and reaction kinetics, we also ran Bayes-smRD on two-state data sets without diffusion dynamics or reaction kinetics to demonstrate its applicability. In Fig. S1, where diffusion dynamics is absent, Bayes-smRD captures the ground-truth reaction kinetics, $1.95$ and $2.92$ ${\text{ms}}^{-1}$, by producing the corresponding 95% CIs as $1.35$–$3.52$ and $2.25$–$5.47$ ${\text{ms}}^{-1}$, respectively. Bayes-smRD also reports the diffusion coefficient 95% CI for both states as $3.41–5.47$ and $3.53–5.16$ $\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$, respectively. Similarly, for the data set without reaction kinetics in Fig. S2, the escape rates’ 95% CIs estimated by Bayes-smRD are $0.00$–$0.39$ and $0.26–1.76$ ${\text{ms}}^{-1}$, respectively. As for the diffusion dynamics, our method reports $25.9–33.5$ and $35.0–48.5$ $\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$, whereas the ground truths are $30$ and $40$ $\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$, respectively.

The next robustness test involves multiplying the entire transition rate matrix used in Fig. 6 by a multiplicative factor while keeping all other parameters fixed. The results from this test are depicted in Fig. S3, d–f. These figures show that Bayes-smRD’s results deviate from the ground-truth values when transitions occur roughly 10 times over a millisecond. Given the parameters listed in the caption of Fig. 6, 10 transitions in one millisecond means that roughly 10 photons are detected between two transitions. In other words, Bayes-smRD remains valid for transition timescales 10 times slower than the average interphoton time.

Similarly, we test Bayes-smRD’s robustness by increasing a state’s lifetime while keeping the other state’s lifetime constant. This former test simulates systems where states have substantially different lifetimes. Its results, shown in Fig. S4, a–c, are consistent with our conclusion so far that Bayes-smRD requires at least 10 photon detections within a state’s lifetime to resolve the system’s reaction-diffusion dynamics accurately.

In the last robustness test, we conduct analyses on progressively longer burst segments centered around the time frame when the molecule is closest to the center of the confocal volume. This test aims to illustrate which burst parts should be included in the analysis. As depicted in Fig. S4, d–f, we obtained results similar to those shown in Fig. S3, a–c, where the number of bursts included varies. This outcome is anticipated because altering the length of each burst segment and adjusting the number of bursts impact the number of photons and transitions similarly. In Fig. S4, d–f, we can also see that increasing the burst length will only add computational burden if sufficient transitions are included. Therefore, for computational convenience, but not necessity, a burst can exclude regions where the excitation probability is less than 0.5%.

Discussion

The current focus on unraveling smRD dynamics at the highest temporal resolution requires analyzing sequential photon arrivals. Single-photon analysis, already common in FRET (7,18,19,20), is key toward resolving rapid processes including protein folding (6) and interactions of transcription factors with nucleotides critical to cellular regulation (67). On the other hand, a single-molecule data analysis method capable of learning reaction-diffusion dynamics can be coupled with deep-learning-based tools predicting biochemical reactions (68) to provide training and verification data. Although many single-molecule processes can be probed by FRET-labeled molecules tagged on surfaces (28,29,52), in addition to force spectroscopy (69), or both combined (70,71), these setups also raise questions pertaining to whether the presence of molecules near surfaces affects their reactivity (33).

However, diffusion-limited single-molecule reactions, with both molecular actors freely diffusing, introduce unique complexities that have limited our ability to resolve smRD models from photon arrivals. For instance, to interpret smRD data, severe approximations must be invoked, including assuming that the illumination is roughly uniform over the diffusion volume (7,18), thereby eliminating our ability to infer spatial trajectories containing information on state switching. Alternatively, tens of millions of photons must be used in the analysis, limiting our ability to resolve events at the single-molecule level (6,52). At its core, smRD presents the unique mathematical challenge that a molecule diffuses simultaneously in continuous space across an inhomogeneously illuminated volume while evolving in discrete state space.

By addressing this mathematical challenge, Bayes-smRD achieves nearly three orders of magnitude higher data efficiency by constructing a comprehensive spatiotemporal framework that captures the intrinsic correlations between all physical quantities. This approach ensures that every detected photon meaningfully contributes to inference, maximizing the extraction of relevant information. In addition, Bayes-smRD imposes priors on only a minimal set of variables, including a molecule’s initial state, spatial position, transition rates, and FRET efficiency for each state (see supporting material section D1). Moreover, these priors are chosen primarily for algebraic convenience, where applicable, and are designed to be flat and unbiased. As a result, the robust data-informed likelihoods in Bayes-smRD enable the simultaneous determination of both diffusive and kinetic parameters, even in cases where data are sparse or otherwise challenging to analyze.

As we work within the Bayesian paradigm, Bayes-smRD propagates all sources of error, including spectral bleed through and background photons, into estimates of each state’s reaction rates, diffusion coefficients, and FRET efficiencies. Inevitably, Bayes-smRD has a higher computational cost than traditional correlative methods such as FCS (72,73). Bayes-smRD’s computational time scales roughly linearly with the number of pulses considered. However, the number of states, i.e., the dimensionality of the transition rate matrix, does not appreciably contribute to Bayes-smRD’s computational cost. Nonetheless, resolving reaction-diffusion dynamics for systems with more states requires more photon detections from each state. Therefore, Bayes-smRD requires more bursts and is more computationally expensive for such systems.

To accurately resolve a system’s reaction kinetics, Bayes-smRD requires tens of transitions within each burst and at least 10 photons between consecutive transitions. For diffusion dynamics, approximately 1000 photons per state are necessary. Given these requirements, analyzing data sets like those presented in the manuscript typically takes 2–3 days on a scientific desktop computer.

Despite this computational cost, no other known methods extract from single-molecule single-photon FRET what Bayes-smRD can, namely simultaneous inference of reaction-diffusion dynamics, state trajectories, and spatial trajectories. Moreover, numerous optimizations can be applied to enhance the efficiency of our existing code. For example, Bayes-smRD could significantly gain from parallelism, which has not yet been fully exploited. Put differently, since independent bursts exhibit independent molecular trajectories in terms of both state and spatial aspects, these trajectories could, in principle, be concurrently inferred. Subsequently, a central processor can compile statistics from these trajectories and deduce the shared reaction-diffusion dynamics. Through this approach, the computational cost of Bayes-smRD can be substantially decreased, primarily because trajectory inference tends to be the most resource-intensive component to infer. This reduction is approximately proportional to the number of processors used.

Another limitation of Bayes-smRD is its sensitivity with respect to PSF size shared by FCS as well as all other analysis methods dependent on particles of interest diffusing across an inhomogeneously illuminated volume. This is because the PSF essentially provides a spatial reference frame. In Fig. S7, we specifically demonstrate FCS’s sensitivity by showing that the diffusion coefficient estimated can vary by a factor of two upon a 30% change in the PSF size. Therefore, we encourage users to use an experimentally calibrated PSF rather than simple theoretical models (74,75) if absolute diffusion coefficients are of interest.

Beyond providing a smRD picture, Bayes-smRD can be further generalized to include any shape of the confocal volume by inserting its mathematical form into Eqs. S26 and S27. Furthermore, the method can also treat any pulse shape or sequence by appropriate modification of Eqs. S44–S46.

Despite advancements brought by Bayes-smRD, several challenges remain (beyond reactions occurring on experimentally inaccessible timescales): 1) finite data, 2) multiple labeled particles present in the observation volume, and 3) an unknown number of system states.

Challenge 1 may arise for different reasons, including, but not limited to, fast diffusion dynamics regarding the size of observation volumes and rapid photobleaching. This paper analyzes several bursts together, assuming all selected bursts share the same reaction-diffusion dynamics to ease the issue. An alternative would be to rely on increasingly photostable labels (76) and detectors with shorter dead times to help maximize the information in each burst. However, theoretical tools can be leveraged to partially beat detector dead times or even integration times in the case of binned photon analysis (77,78). Another possible workaround is to apply or develop more advanced burst selection tools (79) capable of bursts with higher chances of containing a sufficient number of photons and state transitions.

On the other hand, challenges 2 and 3 can potentially be resolved theoretically. In the Bayesian paradigm, rather than estimating the number of molecules or states using methods such as Bayesian information criteria (72), these numbers can both be treated as parameters to learn. This approach requires extending Bayes-smRD to the Bayesian nonparametric paradigm, placing priors on an infinite candidate number of states and molecules, i.e., thereby requiring a doubly nonparametric framework. Despite technical challenges, the ability to theoretically recruit multiple states and molecules within a computational framework, as warranted by the data, would provide a powerful generalization of Bayes-smRD. Here, it is important to emphasize that, as discussed earlier in this section, reliable inference depends on detecting a sufficient number of photons from each state. As the number of states increases, meeting this requirement within a single traversal time becomes increasingly challenging. Therefore, we currently recommend limiting the analysis to no more than three states to ensure accurate inference.

Although Bayes-smRD has been applied here to protein binding interactions by monitoring labels from just one binding partner, it raises broader questions. In particular, it places within conceptual reach that dynamics may be extracted from multiple molecular actors, all simultaneously operating within a small diffraction-limited region, such as a gene locus under active transcription (80), from photon arrivals.

Data and code availability

Synthetic data sets are generated with the code provided in the previous section; experimental data sets come from (52).

Our code is available at https://github.com/LabPresse/Bayes-smRD.

Acknowledgments

The authors thank Benjamin Schuler, Daniel Nettels, and Andrea Sottini for their interesting discussions and for the data they provided. We also thank Ioannis Sgouralis, Mohamadreza Fazel, Ayush Saurabh, Shep Bryan, Camille Moyer, Matthew Safar, and Max Schweiger for helpful discussions. S.P. acknowledges support from the NIH (grant nos. R01GM134426, R01GM130745, and R35GM148237).

Author contributions

S.P. conceived and supervised the project, provided all computational resources, and revised the manuscript. S.J. and Z.K. provided codes from their previous works, co-supervised the project, and revised the manuscript. L.W.Q.X. wrote the code, performed data analyses, and wrote the manuscript.

Declaration of interests

The authors declare no competing interests.

Editor: Florence Tama.