Modeling Non-additive Effects in Neighboring Chemically Identical Fluorophores

Abstract

Quantitative fluorescence analysis is often used to derive chemical properties, including stoichiometries, of biomolecular complexes. One fundamental underlying assumption in the analysis of fluorescence data—whether it be the determination of protein complex stoichiometry by super-resolution, or step-counting by photobleaching, or the determination of RNA counts in diffraction-limited spots in RNA fluorescence in situ hybridization (RNA-FISH) experiments—is that fluorophores behave identically and do not interact. However, recent experiments on fluorophore-labeled DNA origami structures such as fluorocubes have shed light on the nature of the interactions between identical fluorophores as these are brought closer together, thereby raising questions on the validity of the modeling assumption that fluorophores do not interact. Here, we analyze photon arrival data under pulsed illumination from fluorocubes where distances between dyes range from 2 to 10 nm. We discuss the implications of non-additivity of brightness on quantitative fluorescence analysis.

Graphical Abstract

INTRODUCTION

Fluorescent labels have been critical in allowing us to discriminate between homogeneous background and biomolecules of interest.¹ For instance, they have been useful in determining the locations of molecules within neighboring regions by exploiting the nonlinear response (fluorophore activation) of fluorophores to incoming light.^2,3 They have also allowed us to determine the stoichiometry of protein complexes based on a spot’s emission intensity.^4–7 In particular, we note that photobleaching event counting experiments, where fluorophores are stochastically deactivated, lead to step-like patterns in brightness traces.⁸

Quantitative fluorescence analysis experiments have been used extensively throughout biophysics to quantify the stoichiometry of a number of complexes involved, for example, in bacterial flagellar switch,⁹ eukaryotic flagella,¹⁰ point centromere,^11,12 mammalian neurotransmitter receptors,¹³ human calcium channels,¹⁴ transmembrane α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptor-regulatory proteins,¹⁵ T4 bacteriophage helicase loader protein,¹⁶ bacterial oxidative phosphorylation complexes,¹⁷ microRNAs in processing bodies,^18,19 RNAs in a bacteriophage DNA-packaging motor,²⁰ and other membrane proteins and protein complexes.^21,22

However, a quantitative analysis of the stoichiometry of a protein complex^7,23 or the enumeration of the number of fluorophores within a diffraction-limited spot,⁵ or other chemical properties of a system tagged with identical fluorophore labels²⁴ unavoidably requires simplifying assumptions. One critical assumption pervading such analyses is that chemically identical fluorophores do not interact and therefore are photo-emissively identical as well. Here, we reassess this fundamental assumption on the basis of data derived from modern techniques such as PALM,^2,25,26 STORM,²⁷ photobleaching event counting,⁸ RNA-FISH,²⁸ and others.^29,30

To carefully isolate and analyze the non-additive effects of fluorophore interactions, we must account for a number of factors including noise generated by acquisition devices (cameras or single photon detectors),^31,32 the quantized nature of photon emission (shot noise), and background. Therefore, an accurate assessment of non-additive effects as well as a quantitative treatment of fluorescence data requires a hierarchical mathematical treatment of the stochastic effects arising from the contributions mentioned above.

Previous studies have quantified some of these contributions, for example, instrument response function (IRF),^33–35 shot noise,^5,36 and noise from background fluorescence.^7,37 However, incorporating the effects of unintended interactions among identical fluorophores and biomolecules remains an open challenge. As a result, a forward model is required in order to develop an inverse strategy to infer protein complex stoichiometry or even the enumeration of multiply labeled RNA, especially when these RNA are in closely spaced physical regions, from RNA-FISH experiments³⁸ is yet to be proposed. In particular, the primary goal of this paper is not to elucidate the quantum mechanical origin of interactions among fluorophores but to devise a strategy to analyze imaging data.

Recent studies on fluorescently labeled DNA origami structures^39–42 have shown to what degree some identical fluorophores interact when separated by distances ranging from 1 to 10 nm, leading to unique antibunching characteristics,³⁹ improved photostability,⁴⁰ and increased photoblinking.⁴⁰ The statistical signatures of these phenomena have very recently been proposed as a tool to determine molecular distances below 10 nm.⁴² Furthermore, such experiments provide building blocks that may help in constructing models of interacting fluorophores. While antibunching experiments analyze inter-photon arrival times, in this paper, we specifically investigate the effects of interactions on excited state lifetimes and excitation probabilities.

In particular, we study fluorocubes,⁴⁰ which are constructed using four 16 base-pair (bp) long double-stranded DNA (dsDNA), as a highly photostable option for collecting long trajectories of labeled biomolecules. The four DNA helices are assembled together in such a way that the fluorophores attached to the ends of the helices form the corners of a cuboid (see Figure 1) separated by 2–6 nm. One of the corner sites is usually kept reserved for a tag that binds to proteins at specific locations, and the last remaining one is left unlabeled.

Figure 1.

Fluorocube. In a fluorocube, six fluorescent dyes (red) are attached to the ends of the four 16 bp DNA helices. One position is usually reserved for a functional tag (blue) to be linked to a molecule of interest⁴⁰ and the last remaining corner is left unlabeled.

The photophysical properties of these fluorocubes were studied and contrasted with cubes labeled with only one dye (single-dye cubes) and with six dyes that are separated by larger distances of 6–10 nm (large cubes) as well.⁴⁰ It was found that these properties significantly vary among fluorocubes depending on the species of the dyes used. Overall, most fluorocube configurations blinked less and exhibited reduced photobleaching rates compared to their single-dye counterparts. Even the DNA scaffolding itself resulted in improved photostability as single-dye cubes emitted many fold more photons in many cases with longer lifetimes by comparison to a single dsDNA helix coupled to one dye.⁴⁰ Concretely, increasing the dye separations in such cubes from ~2 to 6 nm to ~6 to 10 nm showed that photobleaching time decreases with size, indicating that dye–dye interactions play an important role in the fluorocube photophysics.

Here, we make an attempt at building a model of fluorophore interactions in order to derive quantities such as transition rates between states and excitation probabilities from single photon arrivals. These quantities are then estimated for the lowest energy states of fluorocubes from experimental data. We also consider cubical structures of different sizes (single-dye cubes and large cubes) to demonstrate the changes in photophysics resulting from changes in interactions between dyes. We finally illustrate the consequences of these interactions by analyzing synthetically generated photobleaching event counting traces.

METHODS AND MATERIALS

State Space.

We first develop a quantum mechanical model to describe and label all possible nondegenerate states of a fluorocube. In an effort to simplify our modeling, we make the following physical assumptions based on typical time scales and data collection procedure (see the Experimental Methods section):

1. fluorophores are well approximated by a two state system with a ground and an excited state,

2. the excitation pulse is infinitesimally narrow (~12 ps wide) compared to other time scales of the problem,

3. the duration between pulses is long enough (~100 ns) for the fluorocubes to return to the ground state before the next pulse (<4 ns average lifetime),

4. photoblinking effects, induced by visiting rarely occupied states such as triplet states, are ignored for simplicity as we are primarily interested in interactions among fluorophores in their ground and first excited states,⁴³ and

5. distances between the dyes of a fluorocube do not vary during the experiments, and the upper and lower faces of a fluorocube as seen in Figure 1 are squares with equal separation between adjacent dyes. This may not always be true as the distances between the dyes are affected by the surrounding chemical environment⁴⁰ and cannot be accurately determined dynamically with currently available techniques.

The quantum mechanical nature of interactions between fluorophores is well known in the case of Förster resonance energy transfer (FRET).⁴⁴ For this reason, we use a similar approach to develop a model for fluorocubes and postulate the following Hamiltonian

$$H(t)=\sum _{i=1}^6{H}_i+H_{int}+\delta H(t)$$ (1)

where H_i is the Hamiltonian for individual dyes, H_int is the Hamiltonian for interactions among dyes, and ΔH(t) is the time-dependent perturbation Hamiltonian that models excitation with light. This Hamiltonian is symmetric under a transformation swapping the dye labels, (F₁, F₂, F₃) → (F₆, F₅, F₄), as seen in Figure 1. More intuitively, different combinations of dyes have different pair-wise separations, which implies many different possibilities for the interaction energy. The label swapping symmetry, therefore, determines which combinations of dyes have similar/dissimilar interaction energies and can be used to group all the energy eigenstates of the fluorocube into a collection of degenerate/nondegenerate energy eigenstates. This approach also avoids any need to identify all specific interactions entering the full Hamiltonian for a complex multimolecular system.

In total, we have 2⁶ = 64 energy eigenstates for a collection of six two-state systems. For a large cube with distances between dyes varying from 6 to 10 nm, H_int ≈ 0. Therefore, in this case alone, all fluorophores act effectively and independently leading to maximum allowable degeneracy among the system’s energy eigenstates.

The 64 states of such a large cube can be grouped according to the number of dyes excited at any instant: 6 degenerate states for one and five-dye excitations each, 15 degenerate states for two and four-dye excitations each, 20 degenerate states for three-dye excitations, and one state for ground and six-dye excitations each.

Interactions among fluorophores, however, reduce this degeneracy. Such splitting among the degenerate states of the system is akin to level splitting in the Zeeman effect⁴⁵ and is fully dictated by the symmetries of the interaction Hamiltonian. Here, we focus on the geometrical symmetries of a fluorocube where interactions are expected to be significant and do not consider fine-splitting induced by the asymmetries of the individual molecules in the assembly.

From the geometrical symmetries of a fluorocube (see Figure 1), it can be seen that the degenerate states will split into multiple levels; 6 degenerate one-dye excitation states split into 3 levels, 15 degenerate two-dye excitation states split into 9 levels, 20 degenerate three-dye excitation states split into 10 levels, 15 degenerate four-dye excitation states split into 9 levels, and 6 degenerate five-dye excitation states split into 3 levels.

We can mathematically represent these 35 nondegenerate excitations as

$$\sigma =\left\{\begin{array}{l}{|1\rangle }^{(1)},{|1\rangle }^{(2)},{|1\rangle }^{(3)},\hfill \\ {|2\rangle }^{(1)},{|2\rangle }^{(2)},{|2\rangle }^{(3)},\dots ,{|2\rangle }^{(9)}\hfill \\ {|3\rangle }^{(1)},{|3\rangle }^{(2)},{|3\rangle }^{(3)},\dots ,{|3\rangle }^{(10)},\hfill \\ {|4\rangle }^{(1)},{|4\rangle }^{(2)},{|4\rangle }^{(3)},\dots ,{|4\rangle }^{(9)}\hfill \\ {|5\rangle }^{(1)},{|5\rangle }^{(2)},{|5\rangle }^{(3)}\hfill \\ {|4\rangle }^{(1)}\hfill \end{array}\right\}$$ (2)

where the labels inside the bracket notation indicates the number of dyes excited by the pulses and the superscripts only run over the nondegenerate states. This labeling simplification can be done because degenerate states cannot be distinguished from each other on account of similar lifetimes. We keep the ground state 0 separate from this set for convenience as it does not require a stochastic treatment, as discussed later.

In an experiment, a sample containing a collection of fluorescent probes of the same type is illuminated by a total of N laser pulses. We label the interpulse windows between excitation pulse times t_{n − 1} and t_{n − 1} + Δ with n, where Δ is the window size. Since we assume that fluorocubes are always in the ground state at the beginning of these interpulse windows, the photon arrival times can be considered independent and identically distributed (iid). We represent these measurements with μ_n.

Now that we have defined our state space and measurements, we postulate a forward model to generate these measurements and then design an inverse strategy to learn the parameters of interest. Similar mathematical formulations for fluorescence data have been recently developed and applied extensively for the modeling and analysis of experimental findings.^46–52

Forward Model.

A detected photon may have originated from some background source (dark current, scattering, etc.) or from a fluorescent probe in the sample. Therefore, in addition to the quantum states for the fluorescent probes, we introduce a background state b that allows for a simple mathematical treatment of the background photon counts. The photons from such a state are predominantly uniformly distributed over the interpulse window with a probability of detection π_b. This uniform assumption is validated from experimental data under the assumption that most photons whose arrivals exceed 40 ns are mostly dark counts generated by the detector; see photon count histograms (in the results section) with almost equal photon counts per bin beyond 40 ns.

If the fluorescent probes get excited, the system’s states before photons are emitted are represented by s_n, which take values from the set of states in eq 2. Given the infinitesimal width of the excitation laser, the probabilities of exciting the fluorocubes to states σ can be collected into a constant probability vector π_σ. These probabilities are labeled π_i (where i = 1, …,35 in order of increasing excitation level) corresponding to the excited states in eq 2. Additionally, the probability of staying in the ground state is labeled as π₀, and since it can be computed directly from the number of interpulse windows without photon detections, it is not treated as a random variable.

We emphasize here that, since our single photon data is collected in a raster-scanned manner, the probability of no excitation π₀ depends on the location of the beam over the sample and is not entirely an inherent property of the fluorocube. However, since the excitation beam spends approximately the same amount of time over regions of equal size, the ratios π_i/π_j of excitation probabilities remain unaffected.

Now, following excitation, the system’s state is randomly chosen from a Categorical distribution (a generalization of the Bernoulli distribution for many options)

$$s_n\mid {\pi }_{\sigma },{\pi }_b~{\text{Categorical}}_{\sigma ,b}\left({\pi }_{\sigma },{\pi }_b\right).$$ (3)

On the other hand, interpulse windows can either be empty with no photons or have photon arrival times that are either exponentially or uniformly distributed (for background) continuous random variables

$${\mu }_{\text{n}}|{\text{s}}_{\text{n}},{\lambda }_{\mathit{\sigma }}~\{\begin{array}{ll}\varphi ,\hfill & s_n=0\hfill \\ \text{Exponential}\left({\lambda }_{s_n}\right),\hfill & s_n\ne 0,s_n\ne b\hfill \\ \text{Uniform}\left(0,\Delta \right),\hfill & s_n=b,\hfill \end{array}$$ (4)

where ${\lambda }_{s_n}$ is the emission rate for the corresponding excited state s_n. All emission rates corresponding to the 35 nondegenerate excited states of eq 2 are labeled λ_i for the states in eq 2 and collected in the set λ_σ.

The forward model described above can be visualized using the graphical model shown in Figure 2.⁵³ Based on this formulation, the likelihood function for the arrival time of a photon is given by a mixture of probability distributions

$$\begin{gathered} p\left({\mu }_n\mid {\lambda }_{\sigma },{\pi }_0,{\pi }_{\sigma },{\pi }_b\right)={\pi }_0{\delta }_{\varphi }\left({\mu }_n\right) \\ +{\pi }_1\text{Exponential}\left({\mu }_n;{\lambda }_1\right) \\ +\dots \\ +{\pi }_{35}\text{Exponential}\left({\mu }_n;{\lambda }_{35}\right) \\ +{\pi }_b\text{Uniform}\left({\mu }_n;0,\Delta \right) \end{gathered}$$ (5)

and for a set of N pulses, the likelihood becomes a product of many such mixtures

$$p\left({\mu }_{1:N}\mid {\lambda }_{\mathit{\sigma }},{\pi }_0,{\pi }_{\mathit{\sigma }},{\pi }_b\right)=\prod _{n=1}^{N}p\left({\mu }_n\mid {\lambda }_{\mathit{\sigma }},{\pi }_0,{\pi }_{\mathit{\sigma }},{\pi }_b\right)$$ (6)

Figure 2.

Graphical model. A graphical model depicting the random variables and parameters involved in the generation of photon arrival data for fluorocubes. Circles represent the random variables involved in the inverse model. Circles shaded in blue correspond to the parameters of interest (rates and probabilities), and the one shaded in gray corresponds to the measurements, while the unshaded circles represent the hidden/latent variables of the model. The arrows represent conditional dependence among variables. The two plates enclosing the excitation rates/probabilities and hidden variables/measurements indicate that these variables are repeated over indices i and n, respectively.

Inverse Model.

To learn these rates and excitation probabilities from the photon arrival data, we use Bayesian inference where a parameter probability distribution based on preliminary beliefs (called prior) is updated through a likelihood function as more experimental data is provided. More formally, using Bayes’ theorem, the updated probability distribution over the transition rates and excitation probabilities^53–55 (termed the posterior) is given as

$$p\left({\lambda }_{\sigma },{\pi }_{\sigma },{\pi }_b\mid {\mu }_{1:N},{\pi }_0\right)\propto p\left({\mu }_{1:N}\mid {\lambda }_{\sigma },{\pi }_0,{\pi }_{\sigma },{\pi }_b\right)\times p\left({\lambda }_{\sigma }\right)p\left({\pi }_{\sigma },{\pi }_b\right)$$ (7)

Those distributions multiplying the likelihood, p(λ_σ) and p(π_σ, π_b), are priors and are typically selected on the basis of computational convenience and in whose domains the associated parameters exist.⁵⁴ In many situations, we may simply choose a wide prior to represent our lack of knowledge regarding the values that a parameter may take. Additionally, when large amounts of data are incorporated, the likelihood provides the majority of the contribution to the posterior, diminishing the influence of the prior (as desired).

We use the following priors for the transition rates and excitation probabilities

$${\lambda }_i~\text{Gamma}\left(A,{\lambda }_{ref}/A\right)\text{ and }$$ (8)

$${\pi }_{\sigma },{\pi }_{\mathit{b}}~\text{Dirichlet}{}_{\sigma ,\mathit{b}}\left({\alpha }_{\sigma ,\mathit{b}}\right)$$ (9)

where we choose A = 1 and λ_ref = 1 ns⁻¹ and set all elements of the vector α_{σ, b} to be 1.

To sample transition rates and excitation probabilities from the posterior distribution above, we use the Gibbs algorithm,⁵³ which involves sampling individual parameters from their respective conditional posteriors

$${\lambda }_i~p\left({\lambda }_i\mid {\lambda }_{\sigma }\backslash {\lambda }_i,{\pi }_0,{\pi }_{\sigma },{\pi }_b,{\mu }_{1:N}\right)\text{ and }$$ (10)

$${\pi }_{\sigma },{\pi }_b~p\left({\pi }_{\sigma },{\pi }_b\mid {\lambda }_{\sigma },{\pi }_0,{\mu }_{1:N}\right),$$ (11)

where the parameters following a backslash “\″ are excluded from the preceding set. Sampling parameters individually, in many situations, allow the conditional posteriors to assume analytical forms amenable to highly efficient direct sampling, where a probability distribution is first integrated and then inverted to finally generate samples using a random number generator.

In its present form, however, the Gibbs sampling scheme above still requires brute-force Metropolis–Hastings algorithm for each of the three steps as none of the associated conditional distributions have analytical forms allowing for direct sampling. The computational cost of brute-force sampling helps motivate the (more complex) sampling scheme now presented below.

To generate conditional posteriors in a Gibbs sampling scheme from which we can sample directly, we now exploit the conjugacy of Exponential-Gamma and Multinomial-Dirichlet likelihood-prior pairs. To exploit these properties, we must demarginalize the distribution in eq 7 over the system’s states s_n as follows

$$p\left({\lambda }_{\sigma },{\pi }_{\sigma },{\pi }_b\mid {\mu }_{1:N},{\pi }_0\right)\propto \left(\sum _{s_{1:N}}p\left({\mu }_{1:N},s_{1:N}\mid {\lambda }_{\sigma },{\pi }_0,{\pi }_{\sigma },{\pi }_b\right)\right)\times p\left({\lambda }_{\sigma }\right)p\left({\pi }_{\sigma },{\pi }_b\right)$$ (12)

This de-marginalization creates a new set of variables, s_{1 : N}, which we must now also sample by supplementing the Gibbs algorithm presented from eqs 10 and 11.

On account of these conjugacy conditions, the updated Gibbs sampling algorithm is now as follows

$${\text{s}}_{\text{n}}~\text{p}\left({\text{s}}_{\text{n}}\mid {\lambda }_{\sigma },{\pi }_0,{\pi }_{\sigma },{\pi }_{\text{b}},{\text{s}}_{1:\text{N}}{\text{s}}_{\text{n}},{\mu }_{1:\text{N}}\right)\propto {\text{Categorical}}_{\sigma ,b}\left(\text{Exponential}\left({\mu }_n\mid {\lambda }_1\right){\pi }_1,\dots ,\text{Exponential}\left({\mu }_n\mid {\lambda }_{35}\right){\pi }_{35},{\pi }_b/\Delta \right)$$ (13)

$${\pi }_{\sigma },{\pi }_b~p\left({\pi }_{\sigma }\mid {\lambda }_{\sigma },{\pi }_0,s_{1:N},{\mu }_{1:N}\right)\propto \text{Dirichle}{t}_{\mathit{\sigma },b}\left({\alpha }_{\mathit{\sigma },b}+{\eta }_{\sigma ,b}\right),\text{ and}$$ (14)

$${\lambda }_i~p\left({\lambda }_i|{\lambda }_{\sigma }\backslash {\lambda }_i,{\pi }_0,{\pi }_{\sigma },{\pi }_b,s_{1:N},{\mu }_{1:N}\right)\propto \text{Gamma}\left(A+\sum _{n=1}^N{\Delta }_i\left(s_n\right),{\lambda }_{ref}/A+\sum _{n=1}^N{\mu }_n^{{\Delta }_i\left(s_n\right)}\right),$$ (15)

where η_{σ, b} is a vector containing the number of pulses exciting a fluorocube to a given state in σ and to the background state b, and Δ_i(s_n) is 1 whenever s_n = i and otherwise 0, respectively.

Experimental Methods.

We acquired data to be analyzed with our methods using an SP8 point-scanning system (Leica) with a Picoquant TCSPC module and excitation delivered by a WLL at a 10 MHz repetition rate and maximum available power of 3.3 μW. The emission spectrum was split onto two SMD HyD detectors: 581–603 and 608–687 nm for 561 nm excitation or 646–670 and 675–752 nm for 640 nm excitation. We used an HC PL APO CS2 100×/1.4NA objective (lateral PSF width of approximately 120 nm), a pinhole size of 1 AU, 9.5 nm pixels, and a dwell time of 97.66 microseconds. Data was extracted from .ptu files using a parser written in python.

We assembled the large cubes⁵⁶ and the fluorocubes as described by Niekamp et al.⁴⁰ Briefly, for each six-dye fluorocube and single-dye cube we used four 32 bp long oligonucleotide strands, each modified either with dyes or biotin (sequences shown in Table S1). We mixed each of the four oligos (for the fluorocubes) or the 28 oligos (for the large cubes, sequences shown in Table S1) to a final concentration of 10 μM in folding buffer (5 mM Tris pH 8.5, 1 mM EDTA and 40 mM MgCl2). We then annealed the oligos by denaturation at 85 °C for 5 min, followed by cooling from 80 to 65 °C with a decrease of 1 °C per 5 min, followed by further cooling from 65 to 25 °C with a decrease of 1 °C per 20 min. Afterward, the samples were held at 4 °C. We then analyzed the folding products by 3.0% agarose gel electrophoresis in TBE (45 mM Tris-borate and 1 mM EDTA) with 12 mM MgCl₂ at 70 V for 2.5 h on ice. We finally purified the samples by extraction and centrifugation in Freeze’N Squeeze columns (Bio-Rad Sciences, 732–6165).

We prepared flow cells as described by Niekamp et al.⁴⁰ First, we used a laser cutter to cut custom three-cell flow chambers out of double-sided adhesive sheets (Soles2dance, 9474–08x12 – 3M 9474LE 300LSE). We cleaned 170 μm-thick coverslips (Zeiss, 474,030–9000–000) in a 5% v/v solution of Hellmanex III (Sigma, Z805939–1EA) at 50 °C overnight and washed extensively with Milli-Q water afterward. Then, we used three-cell flow chambers together with glass slides (Thermo Fisher Scientific, 12–550–123) and coverslips to assemble the flow cells.

The preparation method for flow cells is the same for the fluorocubes and the large cubes.⁴⁰ Briefly, we first added 10 μL of 5 mg/mL Biotin-BSA (Thermo Scientific, 29130) in BRB80 (80 mM Pipes (pH 6.8), 1 mM MgCl₂, 1 mM EGTA) to the flow cell and incubated for 2 min. We then added an additional 10 μL of 5 mg/mL Biotin-BSA in BRB80 and incubated for another 2 min. Afterward, we the flow cell was washed with 20 μL of fluorocube buffer (20 mM Tris pH 8.0, 1 mM EDTA, 20 mM Mg-Ac and 50 mM NaCl) with 2 mg/mL of β-casein (Sigma, C6905) and 0.4 mg/mL κ-casein (Sigma, C0406). Next, we added 10 μl of 0.5 mg/mL Streptavidin (Vector Laboratories, SA-5000) in PBS (pH 7.4), incubated for 2 min and then washed with 20 μL of fluorocube buffer with 2 mg/mL β-casein and 0.4 mg/mL κ-casein. Afterward, we added either fluorocubes or large cubes in fluorocube buffer with 2 mg/mL β-casein and 0.4 mg/mL κ-casein and incubated for 5 min. Next, the flow cell was washed with 30 μL of fluorocube buffer with 2 mg/mL β-casein and 0.4 mg/mL κ-casein. Finally, we added the protocatechuic acid (PCA)/protocatechuate-3,4-dioxygenase (PCD)/Trolox oxygen scavenging system (3, 4) in fluorocube buffer with 2 mg/mL β-casein, and 0.4 mg/mL κ-casein to the flow cell. For the PCA/PCD/Trolox oxygen scavenging system we used 2.5 mM of PCA (Sigma, 37580) at pH 9.0, 5 U of PCD (Oriental Yeast Company Americas Inc., 46852004) and 1 mM Trolox (Sigma, 238813) at pH 9.5.

RESULTS AND DISCUSSION

Analysis of ATTO-647N Fluorocubes.

We now use the formulation described above to learn how photophysical properties of fluorocubes and large cubes may change as the distances between dyes are varied. The photon arrival data is acquired as a collection of these cubical fluorescent probes is illuminated by a pulsating laser around every 100 ns. The experiment is repeated many times under different intensities of the laser (see the Experimental Methods section). Moreover, in order to be able to compare the probabilities π_{0 : 35} directly for different cases, we choose to analyze equally sized regions containing the same number of well-separated fluorescent probes (i.e., three) as seen in Figure 3a.

Figure 3.

Experimental data analysis. In row (a), we have three 256 × 256 pixels raster-scanned images of samples containing three cubes labeled with ATTO 647N dyes, each. The images on the left and the right are for single-dye cubes and fluorocubes illuminated with 80% maximum laser power, respectively. However, the center image for large cubes was produced using only 30% laser power. The laser power is varied in order to obtain approximately equal photon numbers per probe per experimental run. Histograms (log-scale) for photon arrival times (microtimes) recorded from these images are shown in row (b). In row (c), each panel shows a bivariate posterior generated from samples over all the emission rates λ_i (log scale) and corresponding rescaled probabilities π_i/(1 − π₀) (as in eq 5) collected together. The left panel shows the distribution for single-dye cubes. The distribution is concentrated about λ_i ≈ 0.25 ns⁻¹ or a lifetime of around 1/λ_i = 4.0 ns. π_b ≈ 0.10 and π₀ ≈ 0.49 in this case. The plot for large cubes in the center panel shows no change in lifetime, suggesting insignificant interactions. π_b ≈ 0.08 and π₀ ≈ 0.37 for these cubes. In the rightmost panel for fluorocubes, a strong peak for a significantly shorter lifetime of around 1.5 ns and a small secondary peak at 3.7 ns appear. The background probability π_b here is 0.12, slightly higher than that of non-interacting cubes, and π₀ ≈ 0.59. We smooth out the bivariate plots for illustrative purposes only using the kernel density estimation (KDE) tool available in Julia.

In Figure 3b,c, we show microtime histograms and three bivariate plots of the learned distributions for transition rates (lifetimes) and excitation probabilities for three different types of cubes labeled with ATTO-647N dyes. Since the rates λ_i and probabilities π_i for the excited states in the likelihood (eq 5) have label switching symmetry, that is, any exchange of labels λ_i ↔λ_j in combination with π_i ↔ π_j leads to the same likelihood, individual labels do not represent specific states. Therefore, only distinctly different numerical values for rates and excitation probabilities can identify separation among states. This allows us to conveniently combine the samples generated for all the rates and excitation probabilities together to plot the bivariate posteriors in Figure 3c. We also note that these results closely match with the ones generated using brute-force Metropolis–Hastings algorithm albeit at greater computational cost (see Section S1).

In the plots for single-dye cubes and large cubes—illuminated with 80% and 30% of the maximum laser power, respectively—we expect the interactions to not affect the photophysics significantly and that is what we observed (since large cubes are not affected by interactions, they are expected to emit far more photons per probe than fluorocubes and are therefore illuminated with less power to have the same number of photon emissions per probe). As shown in the left and center panels of Figure 3c, the most frequently sampled rates are of around 0.25 ns⁻¹, which are equivalent to lifetimes of around 4.0 ns, close to the value specified by the manufacturer (≈3.5 ns)⁵⁷ and therefore used as an independent measurement here to compare parameters when interactions are present. This provides one form of validation for the formalism we are using here. Additionally, the probability of excitation is much higher for the large cubes compared to the single-dye cubes, which is expected since a large cube has 5 additional fluorophores that can be independently excited at the same time.

The most frequently sampled rate in the case of fluorocubes illuminated with 80% laser power (rightmost panel in Figure 3c), where interactions are expected, is around 0.65 ns⁻¹ or an equivalent lifetime of 1.54 ns. This clearly indicates that photophysical properties of a collection of fluorophores change significantly when they are in close proximity. Even though we do not see very sharp splitting here, the presence of a small secondary peak at around λ_i = 0.27 ns⁻¹ indicates the possibility of exciting multiple dyes simultaneously or even that the single dye excitations are nondegenerate. To make sure that this is not an artifact of the computational algorithm, we demonstrate the algorithm’s robustness via distinctly visible splitting for synthetically generated data for six states with precisely known lifetimes and probabilities of excitations (see Section S2).

We also note that excitation probabilities (normalized for laser power and the number of fluorophores per probe) for the smaller ATTO647N fluorocubes are significantly lower compared to its single-dye and the larger six-dye counterparts. This change in excitation probability as distance between fluorophores changes plays a crucial role in determining the brightness (π_i/Δ) of the fluorescence signal. This is also confirmed in the experiments by Niekamp et al.⁴⁰ (see Figures 3 and 8 of their supplementary document), where they observe that single-dye ATTO647N cubes are typically as bright as the six-dye ATTO647N fluorocubes or brighter if the illumination is strong (~1.7 times brighter at exposure values of 2.0 μJ/μm²). This suggests that interactions significantly dampen the brightness per fluorophore when a collection of fluorophores comes closer together. This is possibly the result of new escape mechanisms from excited states that become available due to interactions, and affect quantum yield (quenching) and fluorescence lifetime.

This dampening effect (per fluorophore) is observed to not depend on the species of dye used to label the cubes.⁴⁰ To demonstrate this, we repeated our experiments with large cubes and fluorocubes labeled with Cy3 dyes. We again observe (see Figure S3) significantly smaller excitation probability per fluorophore for Cy3 fluorocubes compared to Cy3 large cubes.

Effects of Non-additivity.

Photobleaching occurs when fluorophores are chemically deactivated upon repeated excitations. This phenomenon finds applications in event counting experiments where the brightness decreases in a step-like pattern⁸ as fluorophores successively photobleach.

In typical experiments, the accurate determination of the number and size of the steps is affected⁵⁸ by instrumental noise, high numbers of fluorophores, overlapping photobleaching events, spatially varying intensity profile, and interactions between fluorophores and the surrounding molecules in the sample.

In the absence of interactions, the brightness should decrease by approximately the same amount every time a fluorophore photobleaches. However, interactions can affect many aspects of physics dictating photobleaching event counting, including brightness of fluorophores,^40,41 the stability of the chemical bonds against repeated excitations,^40,41 and lifetimes of the excited states (as demonstrated earlier). For instance, it has been observed that, in the case of ATTO647N fluorocubes, the brightness is significantly reduced by interactions.⁴⁰ On the other hand, photostability of these fluorocubes improves dramatically and many fold more photons are emitted⁴⁰ resulting in delayed photobleaching, one contributing factor for which may simply be the highly dampened emission rate (brightness) per fluorophore, however, in some cases ~43 fold more photons are emitted⁴⁰ compared to single dyes, which cannot be explained without a full quantum mechanical treatment.

The effects of the interactions noted above may manifest themselves in the form of irregular steps sizes (non-additive brightness) in photobleaching traces, which is commonly observed when dealing with binned data on account of shot noise, background, and detector electronics.⁵ This raises doubt on the assumption currently underlying all step counting methods that rely on equal average step sizes.⁸

To illustrate non-additivity effects in step counting, we generate synthetic photobleaching traces using the recipe in ref 5. We generate photobleaching traces for a collection of interacting and non-interacting fluorophores. The region of interest (ROI) under study contains one interacting cluster of six fluorophores (a fluorocube) and another cluster of two sparsely distributed non-interacting fluorophores.

The algorithm used here for generating photobleaching traces requires us to specify brightness values for the fluorescent probes (isolated dyes and fluorocubes), which can be computed from the excitation probabilities as shown earlier. As the brightness values recorded by a camera are typically noisy, we model such measurements made by an EMCCD camera with a Gamma distribution^5,32

$${\mu }_d~\text{Gamma}\left({\mu }_b/2+{\mu }_e,2G\right)$$ (16)

where μ_e is the number of photons emitted in a second by the emitters, μ_b is the number of background photons emitted in a second, μ_d is the number of ADUs recorded by the detector in the same time interval, and G is the camera gain. In the equation above, excess noise added by the factors of 2 inside the Gamma distribution results from the statistical properties of gain produced by the electron multiplying register.^32,59

Now, from the probability of excitation, we can compute the mean μ_e used in eq 16 for both independent fluorophores and fluorocubes as π_i/Δ. We recover that fluorocubes and independent fluorophores have roughly the same brightness (consistent with Figures S3, S5, and S8 of Niekamp et al.⁴⁰).

More concretely, to compute the brightness in our experiments with 80% maximum laser power and three probes of the same type in the field of view, both single-dye cubes and fluorocubes have similar effective excitation probabilities of around 0.14 and 0.10 per pulse per probe (see Figure 3), respectively. With an interpulse time, Δ, of around 100 ns, this corresponds to an emission rate, μ_e, of around 1.4 × 10⁶ and 1.0 × 10⁶ photons per second per probe. Additionally, the probabilities for background counts, π_b, in both cases are similar and equal to approximately 0.10 and 0.12, which are equivalent to background count rates, μ_b, of 1.0 × 10⁶ and 1.2 × 10⁶ photons per second for the whole sample consisting of all the fluorescent probes. To generate the traces using the model from eq 16, we need to specify a camera gain and set it to 10 (a value, for instance, similar to that used by ref 5).

Figure 4 illustrates what can be expected to happen when a single fluorophore (not interacting with any other) photobleaches (panel a) as compared to when an entire fluorocube photobleaches (panel b).

Figure 4.

Photobleaching traces. Here, we show two sections of a synthetically generated photobleaching counting trace assuming an EMCCD camera model for a collection consisting of one ATTO647N fluorocube (with six tightly packed interacting dyes each) and two well-separated ATTO647N (non-interacting) dyes. In (a), we see that brightness is reduced when one of the non-interacting dyes photobleaches by around 14 × 10⁶ ADU per second. We note in (b) that the brightness again reduces by a similar amount even though six fluorophores in the fluorocube photobleach at the same time. This dampened reduction in brightness may lead to significant undercounting of subunits of a biomolecular complex under investigation.

As is apparent from Figure 4, the dampened brightness of the fluorocube has immediate implications: the step size cannot be considered an independent and identically distributed random variable for each fluorophore and may strongly depend on its local environment.

CONCLUSIONS

Interactions among fluorophores and the surrounding chemical environment play a consequential role in the quantitative analysis of fluorescence imaging data. From our preliminary analysis of the photon arrival data for DNA origami structures labeled with ATTO647N dyes, we observe significant changes in the excitation probabilities and emission rates (lifetimes) of the excited states when interactions are involved. Interactions are also observed to significantly dampen the brightness of an interacting cluster of identical fluorophores. The DNA scaffolding itself contributes to modification of the dye photophysics.⁴⁰ These changes play an important role in the analysis of photobleaching event counting methods,⁸ where brightness traces for samples with large number of densely packed fluorophores cannot be modeled accurately within the existing non-fluorophore–interaction paradigm.

What is more, this finding also suggests that determination of a protein complex’s stoichiometry using PALM^2,25,26 or photobleaching event counting⁸ or related methods³⁸ may be impacted by the local distance between complex subunits.

So far, we have demonstrated that distances between fluorophores have an effect on the heights and lengths of steps in a photobleaching trace. However, in principle, assuming that we know the number of molecules, we can begin quantifying intermolecular distances from accurate lifetime measurements in a manner similar to FRET⁴⁴ as well as from changes in emission frequencies.

More precisely, we may design a means by which to precisely learn the interaction Hamiltonian, H_int, in eq 1 (and parameters such as emission rates, excitation probabilities, and photobleaching times) as a function of fluorophore separation, orientation, and dipolar coupling. This calls for further experiments into precisely characterizing the relationship between photophysics and spatial distributions of identical fluorophores.