Analysis of photon trajectories from diffusing single molecules

Abstract

In single-molecule free diffusion experiments, molecules spend most of the time outside a laser spot and generate bursts of photons when they diffuse through the focal spot. Only these bursts contain meaningful information and, therefore, are selected using physically reasonable criteria. The analysis of the bursts must take into account the precise way they were chosen. We present new methods that allow one to accurately determine the brightness and diffusivity of individual molecule species from the photon arrival times of selected bursts. We derive analytical expressions for the distribution of inter-photon times (with and without burst selection), the distribution of the number of photons in a burst, and the distribution of photons in a burst with recorded arrival times. The theory accurately treats the bias introduced due to the burst selection criteria. We use a Maximum Likelihood (ML) method to find the molecule’s photon count rate and diffusion coefficient from three kinds of data, i.e., the bursts of photons with recorded arrival times (burstML), inter-photon times in bursts (iptML), and the numbers of photon counts in a burst (pcML). The performance of these new methods is tested on simulated photon trajectories and on an experimental system, the fluorophore Atto 488.

Unraveling Burst Selection Bias in Single-Molecule FRET of Species with Unequal Brightness and Diffusivity

“…This number decreases when selecting bursts above the threshold N th . Approximating the distribution of photon counts in a burst as exponential, the probability of surpassing threshold can be estimated as exp(−α i N th ), where α i –1 is the mean number of photons in bursts (of any size) of species i . These arguments and the inverse relationship between burst duration and diffusion coefficient lead to the following fraction of bursts originating from species 1 (detailed derivation is given in the Supporting Information): p 1 b = p 1 D 1 e − ξ 1 p 1 D 1 e − ξ 1 + p 2 D 2 e − ξ 2 where ξ i = α i N th is the threshold parameter that determines the threshold effect.…”

“…The normalization factor Z in eq is obtained by integrating eq with respect to all possible interphoton times τ i and summing over colors c i = D, A and the total numbers of photons N : , Z = p f T M N th − 1 false( bold-italicI − bold-scriptM ) − 1 bold-scriptN p in where I is the identity matrix. The time integral matrix bold-scriptM is bold-scriptM = ∫ 0 τ th normald τ bold-scriptN e false( bold-scriptL − bold-scriptN false) τ The normalization factor Z and the initial, p in , and final, p f , vectors take burst selection into account and involve the threshold number of photons, N th , and the threshold time, τ th .…”

“…The performance of burstML was evaluated for a single species labeled with one fluorophore. 36 The method demonstrated accurate estimation of photon count rates even when utilizing the isotropic spot approximation. However, the accuracy of the extracted diffusion coefficients improved when both axial and lateral coordinates were discretized.…”

“…The vector p 0 in eq has elements 4π q j 2 and appears due to the entropic factor in the spherically symmetric problem, 4π r 2 . The discretized diffusion operator (which includes the entropic factor 4π q 2 ) is a tridiagonal matrix, .25ex2ex L j ± 1 j = D a 2 q j ± 1 / 2 2 q j 2 δ 2 = 1 τ d a x y 2 a 2 q j ± 1 / 2 2 q j 2 δ 2 , L j j = prefix− false( L j + 1 j + L j − 1 j false) , with L 01 = L j + 1 j = 0 , which corresponds to the reflecting boundary at q = 0 and q = q max . These boundary conditions lead to scriptL 11 = prefix− scriptL 21 and L j j = prefix− L j…”

“…For the sake of simplicity, we assume that the species/states do not change on the time scale of burst duration. To accurately treat the selection bias, we employ a recently developed photon-by-photon maximum likelihood method , (termed burstML) that explicitly considers burst selection and allows one to infer species brightness and diffusion coefficient, in addition to FRET efficiency and population. The burstML method is tested using simulated photons from diffusing molecules of two kinds and is compared with colorML , which assumes equal species brightness and diffusivity.…”