Estimation of the number of single-photon emitters for multiple fluorophores with the same spectral signature

Abstract

Fluorescence microscopy is of vital importance for understanding biological function. However most fluorescence experiments are only qualitative inasmuch as the absolute number of fluorescent particles can often not be determined. Additionally, conventional approaches to measuring fluorescence intensity cannot distinguish between two or more fluorophores that are excited and emit in the same spectral window, as only the total intensity in a spectral window can be obtained. Here we show that, by using photon number resolving experiments, we are able to determine the number of emitters and their probability of emission for a number of different species, all with the same measured spectral signature. We illustrate our ideas by showing the determination of the number of emitters per species and the probability of photon collection from that species, for one, two, and three otherwise unresolvable fluorophores. The convolution Binomial model is presented to model the counted photons emitted by multiple species. And then the Expectation-Maximization (EM) algorithm is used to match the measured photon counts to the expected convolution Binomial distribution function. In applying the EM algorithm, to leverage the problem of being trapped in a sub-optimal solution, the moment method is introduced in finding the initial guess of the EM algorithm. Additionally, the associated Cramér-Rao lower bound is derived and compared with the simulation results.

Figures (8)

• Figure 1: (a) Schematic of using PNRD to discriminate three species of emitters. The excitation laser (blue beam) passes through the dichroic mirror and focuses on three species $(M_1,p_1)$, $(M_2,p_2)$, $(M_3,p_3)$ under the diffraction limit locating within one Gaussian focal spot in (c). The emitting light (red beam) is detected by PNRD and the synthetic photon number resolving signal is shown in (b) as bar charts in three colors. With the measurement time increases the probability of detecting the photon numbers would follow Poisson distribution, shown as the black smooth curves. The area underneath each poisson curve or bar chart is always one, indicating that in a practical measurement the photon counts or the intensity from each species is the same, therefore they cannot be identified by conventional intensity-only based microscopy. However they generate distinguishing PNRD signals, building on which we introduce the MLE approach to tell them apart.
• Figure 2: The comparison between the relative occurrence obtained from synthetic measurement from photons with different $\nu$, (a) $\nu = 100$ and (b) $\nu = 1\text{e}7$, where $\bm{\theta}=[8,0.1,10,0.2,12,0.3]$. (c) shows the Binomial PMF for each species. As an additional comparison, the Normal approximation for the expected PMF is given. It can be seen that the histogram obtained from the synthetic data converges to the expected distribution of photons with the increasing of the $\nu$.
• Figure 3: Given different number of experiments $\nu$, the comparison between the $\sqrt{\text{CRLB}}$ and sampled standard deviation when the number of species is $m=1$.
• Figure 4: The required number of experiments, $\nu_{exp}$, to attain $\text{CRLB}(M_1)/M_1=1\%$ while $M_1$ varies from $1$ to $20$ and $p_1$ from $0.05$ to $0.95$. Please note that $\nu$ is plotted in $\log_{10}$ scale.
• Figure 5: Given different value of $\nu$, the comparison between the $\sqrt{\text{CRLB}}$ and sampled standard deviation when the number of species is $j=2$.

Theorems & Definitions (1)

• Remark 1