Stochastic approach to the molecular counting problem in superresolution microscopy

Abstract

Superresolution imaging methods--now widely used to characterize biological structures below the diffraction limit--are poised to reveal in quantitative detail the stoichiometry of protein complexes in living cells. In practice, the photophysical properties of the fluorophores used as tags in superresolution methods have posed a severe theoretical challenge toward achieving this goal. Here we develop a stochastic approach to enumerate fluorophores in a diffraction-limited area measured by superresolution microscopy. The method is a generalization of aggregated Markov methods developed in the ion channel literature for studying gating dynamics. We show that the method accurately and precisely enumerates fluorophores in simulated data while simultaneously determining the kinetic rates that govern the stochastic photophysics of the fluorophores to improve the prediction's accuracy. This stochastic method overcomes several critical limitations of temporal thresholding methods.

Keywords: counting problem; fluorescence; protein complexes; single molecule; superresolution.

Fig. 1.

Kinetic model for PA-FP blinking. This kinetic model has four states inactive (I), active (A), dark (D), and photobleached (B). The only fluorescent state is A. We name the transitions between states this way: activation $\left(I\to A\right)$, blinking $\left(A\to D\right)$, recovery $\left(D\to A\right)$, and photobleaching $\left(A\to B\right)$.

Fig. 2.

Histogram of bootstrapping results from simulated data set 1. Histogram of bootstrapping results from simulated data set 1 with 200 traces (200 bootstrap iterations). The theoretically expected results are shown in the dotted line.

Fig. 3.

Histogram of bootstrapping results from simulated data set 2. Histogram of bootstrapping results from simulated data set 2 with 200 traces (200 bootstrap iterations). An overall bias toward slow $k_d$ is observed in the $k_d$ distribution. We will discuss how to correct for missed transitions, which gives rise to this bias, later. The theoretically expected results are shown in the dotted line.

Fig. 4.

Histogram of bootstrapping results from simulated data set 2 with 5-ms time bins. We followed the same procedure as in Fig. 3 except we now used 5-ms—as opposed to 50-ms—time bins.

Fig. 5.

Histogram of N obtained by assuming rates. We demonstrate that we can accurately determine N if the rates are known. We use parameters from set 1 from Table 1 and use the simulated data to determine N (and not the rates). (Left) Shown with 50-ms time resolution (and therefore underestimated N). (Right) Improvement afforded by smaller time resolution (5 ms).

Fig. 6.

Histogram of bootstrapping results from in vitro data. Histogram of bootstrapping results from in vitro data with 1,000 time traces (300 bootstrap iterations). The dashed lines show the parameter values from Lee et al. (40).

Fig. 7.

Increasing the dead time, $t_d$, helps approximately correct for missed transitions. We combine 10 $N=1$ in vitro traces and use the combined traces to determine rates and N (whose theoretical value should be $N=10$). We generated 1,000 of such traces (with 200 bootstrap iterations). The distribution over N is shown for increasing values of $t_d$. Dendra2 has a dwell time in the bright state, which is on average approximately as long as the time resolution of experiment (50 ms). Thus, when $t_d$ is zero, approximately half the transitions to the bright state are missed. As $t_d$ increases, our estimate for N improves. Fig. 4 shows that PA-FPs with kinetics slower than the integration time should improve the estimate for N.

Fig. 8.

Using intensity measurements can improve the estimate for N. (A) When $N=5$ we can observe, in principle, a fluorescence intensity ranging from one to five bright (if all PA-FPs are on simultaneously). Here we show how many states coincide with the different fluorescence intensities up to three simultaneously bright. (B) We show that we can take advantage of the intensity measurements to compute a correctly centered distribution for N (using 200 synthetic traces with 200 bootstrap iterations).

Fig. 9.

Intensity measurements become particularly useful when estimating rates and N for larger N. Here we used simulated set 1 with $N=10$ (200 synthetic traces with 200 bootstrap iterations). If the data are instead focused on determining only N, rather than N and the rates, the distribution over N becomes dramatically sharper even for larger N (Fig. 5).

Fig. 10.

Likelihood function. An idealized time trace. Each dwell is color-coded with its corresponding term in the likelihood function.