Pitching Single-Focus Confocal Data Analysis One Photon at a Time with Bayesian Nonparametrics

Fluorescence time traces are used to report on dynamical properties of molecules. The basic unit of information in these traces is the arrival time of individual photons, which carry instantaneous information from the molecule, from which they are emitted, to the detector on timescales as fast as microseconds. Thus, it is theoretically possible to monitor molecular dynamics at such timescales from traces containing only a sufficient number of photon arrivals. In practice, however, traces are stochastic and in order to deduce dynamical information through traditional means—such as fluorescence correlation spectroscopy (FCS) and related techniques—they are collected and temporally autocorrelated over several minutes. So far, it has been impossible to analyze dynamical properties of molecules on timescales approaching data acquisition without collecting long traces under the strong assumption of stationarity of the process under observation or assumptions required for the analytic derivation of a correlation function. To avoid these assumptions, we would otherwise need to estimate the instantaneous number of molecules emitting photons and their positions within the confocal volume. As the number of molecules in a typical experiment is unknown, this problem demands that we abandon the conventional analysis paradigm. Here, we exploit Bayesian nonparametrics that allow us to obtain, in a principled fashion, estimates of the same quantities as FCS but from the direct analysis of traces of photon arrivals that are significantly smaller in size, or total duration, than those required by FCS.

Popular Summary

The basis of all spectroscopy is the detection of photons. Photon arrivals encode complex information on dynamical processes, and methods that analyze individual photon arrivals can, in principle, reveal information on such processes at the fastest available timescale. Here, we turn our attention to methods that illuminate one bright spot (confocal methods) and use individual photon arrivals (obtained from fluorescent molecules traversing this spot) to reveal dynamical models for single molecules.

While photon detections may reveal dynamics at millisecond timescales or faster, data drawn from confocal methods are typically collected over minutes to obtain sufficient data for traditional analysis methods from which dynamical models are deduced. Here, instead, we reduce the required amount of data by several orders of magnitude by exploiting “Bayesian nonparametrics,” a set of mathematical statistical tools largely unknown in the physical sciences, to learn models directly from single-photon arrivals.

Confocal methods have been the workhorse of biophysics for half a century. Our work broadens the applicability of confocal methods by learning models from minimal amounts of data, which is relevant to probing rare or nonequilibrium processes. Furthermore, our work has deep implications in reducing phototoxic damage, inherent in illuminating samples, as our novel mathematical tools are highly efficient at extracting data and therefore require exposing samples to fewer photons.

Figure 1

Photon arrival times can characterize dynamical properties of molecules on fast, photon-detection, timescales. (a) Schematic of an illuminated confocal volume (blue) with fluorescent molecules emitting photons based on their location within that volume. (b) Synthetic trace containing $\approx 1500\mathrm{photon}$ arrivals produced by four molecules diffusing at $1\mu {\mathrm{m}}^2/\mathrm{s}$ for a total time of 30 ms under background and molecule photon emission rates of ${10}^3\mathrm{photons}/\mathrm{s}$ and $4\times {10}^4\mathrm{photons}/\mathrm{s}$, respectively. (c) Autocorrelation curve $G(\tau )$ of the trace in (b), binned at $100\mu \mathrm{s}$. On account of the limited data available in the trace, any reasonable fit is impossible. Normally, in FCS analysis, much longer traces are used to generate smoother $G(\tau )$ that are fitted to determine a diffusion coefficient. In Fig. 14 of the Appendix, we show that the quality of the fit does not improve considerably by fitting to a semilogarithmic curve. (d) Comparison between diffusion-coefficient estimates using our proposed method (detailed later) and FCS as a function of the number of photon arrivals in the analyzed trace. Since by $1.5\times {10}^4\mathrm{photon}$ arrivals our method has converged, we avoid analyzing larger traces.

Figure 2

Estimates of diffusion coefficients from photon arrival traces strongly depend on the number of molecules assumed to be contributing to the trace. The trace analyzed contained $\approx 1800\mathrm{photon}$ arrivals produced by four molecules diffusing at $1\mu {\mathrm{m}}^2/\mathrm{s}$ for a total time of 30 ms under background and molecule photon emission rates of ${10}^3\mathrm{photons}/\mathrm{s}$ and $4\times {10}^4\mathrm{photons}/\mathrm{s}$, respectively. To estimate $D$ parametrically, we assume a fixed number of molecules, $N=1$ (a); $N=2$ (b); $N=3$ (c); $N=4$ (d); and $N=5$ (e). The correct estimate in (d)—and the mismatch in all others—underscores why it is critical to estimate the number of molecules contributing to the trace to deduce quantities such as diffusion coefficients from single-photon arrivals.

Figure 3

BNP formulation used for the analysis of photon arrival traces. Molecules, indexed $n=1,2,\dots$, evolve over the experimental time course which is indexed by $k=1,2,\dots ,K$. Here, $R_k^n=(x_k^n,y_k^n,z_k^n)$ indicates the location of molecule $n$ at time $t_k$. During the experiment, only a single observation (interarrival time) $\mathrm{\Delta }{t}_k$ is recorded, thereby combining photon emissions from every molecule and the background. The diffusion coefficient $D$ determines the evolution of the molecular positions which influence the photon emission rates and eventually the recorded $\mathrm{\Delta }{t}_k$. The indicator variables $b^n$ are introduced to infer the unknown molecule population size. In the graphical model, the measured data are highlighted by gray shaded circles and the model variables, which require priors, are designated by blue circles.

Figure 4

A higher number of total photon arrivals provide more photons per unit time and sharper diffusion-coefficient estimates. (a1) Instantaneous molecule photon emission rates ${\mu }_k^n$, normalized by ${\mu }_{\mathrm{mol}}$. (a2) Photon arrival trace resulting from combining photon emissions from every molecule and the background. This synthetic trace contains $\approx 2000$ photon arrivals produced by four molecules diffusing at $1\mu {\mathrm{m}}^2/\mathrm{s}$ for a total time of 30 ms under background and molecule photon emission rates of ${10}^3\mathrm{photons}/\mathrm{s}$ and $4\times {10}^4\mathrm{photons}/\mathrm{s}$, respectively. The dashed lines show the initial 30%, 50%, 80%, and 100% portions of the original trace containing $\approx 600$, $\approx 1000$, $\approx 1600$, $\approx 2000\mathrm{photon}$ arrivals, respectively. (b1)–(b4) Posterior probability distributions drawn from traces with differing length [shown in (a2)]. As expected, for the longer traces, the peak of the posterior matches with the exact value of $D$ (dashed line). Gradually, as we decrease the total number of photon arrivals analyzed, the estimation becomes less reliable.

Figure 5

A higher molecular concentration provides more photons per unit time and sharper diffusion-coefficient estimates. (a1),(b1),(c1) Instantaneous molecule photon emission rates ${\mu }_k^n$, normalized by ${\mu }_{\mathrm{mol}}$. (a2),(b2),(c2) Photon arrival traces resulting from combining photon emissions from every molecule and the background. These are produced by ten molecules containing $\approx 3000\mathrm{photon}$ arrivals (a2), four molecules containing $\approx 2000\mathrm{photon}$ arrivals (b2), and one molecule containing $\approx 1000\mathrm{photon}$ arrivals (c2), diffusing at $1\mu {\mathrm{m}}^2/\mathrm{s}$ for a total time of 30 ms under background and molecule photon emission rates of ${10}^3\mathrm{photons}/\mathrm{s}$ and $4\times {10}^4\mathrm{photons}/\mathrm{s}$, respectively. (a3),(b3),(c3) Posterior probability distributions drawn from traces with differing number of molecules [shown in (a2),(b2),(c2)]. As expected, for the traces with higher number of molecules, the peak of the posterior matches with the exact value of $D$ (dashed line). Gradually, as we decrease the total number of molecules the estimation becomes less reliable.

Figure 6

A lower diffusion coefficient provides more photons per unit time and sharper diffusion-coefficient estimates. Posterior probability distributions drawn from traces containing $\approx 2000\mathrm{photon}$ arrivals produced by four molecules diffusing at $D=0.01,0.1,1,10\mu {\mathrm{m}}^2/\mathrm{s}$ for a total time of 30 ms under background and molecule photon emission rates of ${10}^3\mathrm{photons}/\mathrm{s}$ and $4\times {10}^4\mathrm{photons}/\mathrm{s}$, respectively. For molecules diffusing at $D=100\mu {\mathrm{m}}^2/\mathrm{s}$, under similar conditions, we use a trace containing $\approx 3000\mathrm{photons}$ for a total time of 50 ms, since we needed a longer trace to gather sufficient information for drawing a posterior.

Figure 7

A higher molecule photon emission rate provides more photons per unit time and sharper diffusion-coefficient estimates. (a)–(d) Posterior probability distributions drawn from traces produced by four molecules diffusing at $1\mu {\mathrm{m}}^2/\mathrm{s}$ for a total time of 30 ms under a background photon emission rate of ${10}^3\mathrm{photons}/\mathrm{s}$ and molecule photon emission rates $4\times {10}^5$, $4\times {10}^4$, $4\times {10}^3$, ${10}^3\mathrm{photons}/\mathrm{s}$, respectively. As expected, under higher molecule photon emission rates, the peak of the posterior matches sharply with the exact value of $D$ (dashed line). Gradually, as we decrease the molecule photon emission rate, the estimation becomes less reliable.

Figure 8

Higher molecular concentrations in experimental traces provide more photons per unit time resulting in sharper diffusion-coefficient estimates. Estimates shown are drawn from experimental traces with a low (100 pM) (a) and high (1 nM) (b) concentration of Cy3 dye molecules and 75% glycerol at a fixed laser power of $100\mu \mathrm{W}$. Similarly to Fig. 1, we compare our method’s diffusion-coefficient estimate (circle green dots) to FCS (blue asterisk) as a function of the number of photons used in the analysis. Since by $1.5\times {10}^4\mathrm{photon}$ arrivals our method has converged, we avoid analyzing larger traces. The red dashed line is the FCS estimate produced from the entire 5-min trace containing $\approx 3\times {10}^6\mathrm{photon}$ arrivals.

Figure 9

Lower diffusion coefficients in experimental traces provide more photons per unit time and sharper diffusion-coefficient estimates. Estimates shown are drawn from experimental traces with 99% glycerol (a), 94% glycerol (b), 75% glycerol (c), 67% glycerol (d), 50% glycerol (e), and 0% glycerol (f) with fixed concentration 1 nM of Cy3 dye molecules and laser power of $100\mu \mathrm{W}$. Similarly to Fig. 1, we compare our method’s diffusion-coefficient estimate (circle green dots) to FCS (blue asterisk) as a function of the number of photons used in the analysis. Since by $1.5\times {10}^4\mathrm{photon}$ arrivals our method has converged, we avoid analyzing larger traces. The red dashed line is the FCS estimate produced from the entire 5-min trace containing $\approx 3\times {10}^6\mathrm{photon}$ arrivals.

Figure 10

Higher laser powers in experimental traces provide more photons per unit time and sharper diffusion-coefficient estimates. Estimates shown are drawn from experimental traces with high ($100\mu \mathrm{W}$) (a) and low ($25\mu \mathrm{W}$) (b) laser power with fixed concentration 1 nM of Cy3 dye molecules and 75% glycerol. Similarly to Fig. 1, we compare our method’s diffusion-coefficient estimate (circle green dots) to FCS (blue asterisk) as a function of the number of photons used in the analysis. Since by $1.5\times {10}^4\mathrm{photon}$ arrivals our method has converged, we avoid analyzing larger traces. The red dashed line is the FCS estimate produced from the entire 5-min trace containing $\approx 3\times {10}^6\mathrm{photon}$ arrivals.

Figure 11

Background photon emission rates are artificially added to experimental traces yielding challenging imaging conditions and broader diffusion-coefficient estimates. Experimental traces with fixed concentration 1 nM of Cy3 dye molecules and 67% glycerol and fixed laser power $100\mu \mathrm{W}$. The same total number of photons analyzed under differing (artificially increased) background photon emission rates [0 (a1), 500 (b1), 1000 (c1) photons/s] (a2),(b2),(c2) Similarly to Fig. 1, we compare our method’s diffusion-coefficient estimate (green dots) to FCS (blue asterisk) as a function of the number of photons used in the analysis. Since by $1.5\times {10}^4\mathrm{photon}$ arrivals our method has converged, we avoid analyzing larger traces. The red dashed line is the FCS estimate obtained from the entire 5-min trace containing $\approx 3\times {10}^6\mathrm{photon}$ arrivals.

Figure 12

Diffusion-coefficient estimates of labeled protein. Estimates shown are drawn from experimental traces with fixed concentration 1 nM of Cy3-labeled streptavidin molecules and laser power $100\mu \mathrm{W}$. Similarly to Fig. 1, we compare our method’s diffusion-coefficient estimate (green dots) to FCS (blue asterisk) as a function of the number of photons used in the analysis. Since by $1.5\times {10}^4\mathrm{photon}$ arrivals our method has converged, we avoid analyzing larger traces. The red dashed line is the FCS estimate obtained from the entire 5-min trace containing $\approx 3\times {10}^6\mathrm{photon}$ arrivals.

Figure 13

Diffusion-coefficient estimates of 5-TAMRA dye. Estimates shown are drawn from experimental traces with fixed concentration 20 nM of 5-TAMRA dye molecules. Similarly to Fig. 1, we compare our method’s diffusion-coefficient estimate (green dots) to FCS (blue asterisk) as a function of the number of photons used in the analysis. Since by $1.5\times {10}^4\mathrm{photon}$ arrivals our method has converged, we avoid analyzing larger traces. The red dashed line is the FCS estimate obtained from the entire, 10-min, trace containing $\approx 6\times {10}^6\mathrm{photon}$ arrivals.

Figure 14

FCS curves resulting from exceedingly short traces (same synthetic data as Fig. 1) with linear (a) and semilogarithmic (b) binning. Because of the limited data, the quality of the fitted autocorrelation curve $G(\tau )$ does not improve considerably for (b) as compared with (a).

Figure 15

FCS curves resulting from exceedingly short traces. Shown are autocorrelation curves $G(\tau )$ of 5-TAMRA experimental traces, binned at $10\mu \mathrm{s}$, for 100 ms and $\approx 500\mathrm{photon}$ arrivals (a); 200 ms and $\approx 1000\mathrm{photon}$ arrivals (b); 300 ms and $\approx 3000\mathrm{photon}$ arrivals (c); 2 s and $\approx 15000\mathrm{photon}$ arrivals (d); 30 s and $\approx 15\times {10}^5\mathrm{photon}$ arrivals (e); 100 s and $\approx 15\times {10}^6\mathrm{photon}$ arrivals (f). Even a visual inspection illustrates how poorly FCS applies on traces as sort as those analyzed by our BNP method.

Figure 16

A larger molecule photon emission rate provides more photons per unit time and sharper diffusion-coefficient estimates. (a1),(b1) Instantaneous molecule photon emission rates ${\mu }_k^n$, normalized by ${\mu }_{\mathrm{mol}}$. (a2),(b2) Photon arrival trace resulting from combining photon emissions from every molecule and the background. These traces are produced by 10 molecules diffusing at $10\mu {\mathrm{m}}^2/\mathrm{s}$ for a total time of 50 ms under background photon emission rate of ${10}^3\mathrm{photons}/\mathrm{s}$ and molecule photon emission rate $4\times {10}^5\mathrm{photons}/\mathrm{s}$ containing $\approx 3000\mathrm{photon}$ arrivals (a2), and molecule photon emission rate $4\times {10}^4\mathrm{photons}/\mathrm{s}$ containing $\approx 2000\mathrm{photon}$ arrivals (b2). (a3),(b3) Posterior probability distributions drawn from traces with differing molecule photon emission rates [shown in (a2),(b2)]. As expected, for the traces with higher molecule photon emission rate, the peak of the posterior sharply matches with the exact value of $D$ (dashed line). Gradually, as we decrease the molecule photon emission rate, the estimation becomes less reliable.

Figure 17

A higher molecule photon emission rate provides more photons per unit time and sharper emission rate estimates. (a1),(b1),(c1) Instantaneous molecule photon emission rates ${\mu }_k^n$, normalized by ${\mu }_{\mathrm{mol}}$. (a2),(b2),(c2) Photon arrival traces resulting from combining photon emissions from every molecule and the background. These traces produced by 10 molecules diffusing at $10\mu {\mathrm{m}}^2/\mathrm{s}$ for a total time of 50 ms under a background photon emission rate of ${10}^3\mathrm{photons}/\mathrm{s}$ and molecule photon emission rate $4\times {10}^5\mathrm{photons}/\mathrm{s}$ containing $\approx 3000\mathrm{photon}$ arrivals (a2), molecule photon emission rate $4\times {10}^4\mathrm{photons}/\mathrm{s}$ containing $\approx 2000\mathrm{photon}$ arrivals (b2), and molecule photon emission rate $4\times {10}^3\mathrm{photons}/\mathrm{s}$ containing $\approx 1000\mathrm{photon}$ arrivals (c2). (a3),(b3),(c3) Posterior probability distributions drawn from traces with differing molecule photon emission rates [shown in (a2),(b2),(c2)]. As expected, for the traces with a higher molecule photon emission rate, the peak of the posterior sharply matches with the exact value of ${\mu }_{\mathrm{mol}}$ (dashed line). Gradually, as we decrease the molecule photon emission rate, the estimation becomes less reliable.

Figure 18

Estimation of the diffusion coefficient and molecule photon emission rate for Cy3 dyes. (a) Experimental intensity trace (binned at $100\mu \mathrm{s}$) with concentration 1 nM of Cy3 dye molecules and 61% glycerol. A background photon emission rate of $600\mathrm{photons}/\mathrm{s}$ is known from calibration. (b) Analyzed portion of the trace containing $\approx 3000\mathrm{photon}$ arrivals. (c) Posterior probability distributions and the value (red dashed line) of molecule photon emission rate determined by the photon counting histogram (PCH) method on the entire trace [59]. (d) Similarly to Fig. 1, we compare our method’s diffusion-coefficient estimate (green dots) to FCS (blue asterisk) as a function of the number of photons used in the analysis. Since by $1.5\times {10}^4\mathrm{photon}$ arrivals our method has converged, we avoid analyzing larger traces. The red dashed line is the FCS estimate obtained from the entire 5-min trace containing $\approx 3\times {10}^6\mathrm{photon}$ arrivals.

Figure 19

Estimation of the diffusion coefficient and molecule photon emission rate for 5-TAMRA dyes. (a) Experimental intensity trace (binned at $10\mu \mathrm{s}$) with concentration 20 nM of 5-TAMRA dye molecules. A background photon emission rate of $300\mathrm{photons}/\mathrm{s}$ is known from calibration. (b) Analyzed portion of the trace containing $\approx 8000\mathrm{photon}$ arrivals. (c) Posterior probability distributions and the value (red dashed line) of molecule photon emission rate determined by the photon counting histogram (PCH) method on the entire trace [59]. (d) Similarly to Fig. 1, we compare our method’s diffusion-coefficient estimate (green dots) to FCS (blue asterisk) as a function of the number of photons used in the analysis. Since by $1.5\times {10}^4\mathrm{photon}$ arrivals our method has converged, we avoid analyzing larger traces. The red dashed line is the FCS estimate obtained from the entire, 10-min, trace containing $\approx 6\times {10}^6\mathrm{photon}$ arrivals.