Recovering a stochastic process from super-resolution noisy ensembles of single-particle trajectories

Recovering a stochastic process from noisy ensembles of single-particle trajectories is resolved here using the coarse-grained Langevin equation as a model. The massive redundancy contained in single-particle tracking data allows recovering local parameters of the underlying physical model. We use several parametric and nonparametric estimators to compute the first and second moments of the process, to recover the local drift, its derivative, and the diffusion tensor, and to deconvolve the instrumental from the physical noise. We use numerical simulations to also explore the range of validity for these estimators. The present analysis allows defining what can exactly be recovered from statistics of super-resolution microscopy trajectories used for characterizing molecular trafficking underlying cellular functions.

Figure 1

An observed trajectory (grey dashed line) is obtained as the sum of physical trajectories (black line) with an additional instrumental noise. At constant time intervals, the physical trajectory is subsampled (black circles). Instrumental noise perturbs the exact localization, and observed points (grey stars) are positioned in a neighborhood of physical ones. The present method is to recover the physical trajectories from noisy observations.

Figure 2

Diffusion coefficients are estimated for (a, b) a Brownian motion (BM) and (c, d) an Ornstein-Uhlenbeck process (OU) and for various values of the signal-to-noise ratio (SNR) represented in a log-10 scale. Trajectories were simulated using Euler's scheme and subsampled so that the observed trajectories contain 10 000 points, and position noise $\sigma$ was subsequently added to each point of the trajectories. The diffusion coefficient $D_{\mathrm{\Delta }t}$ is estimated using formula (A4). Black dots represent $\tilde{D}=D_{\mathrm{\Delta }t}-\frac{{\sigma }^2}{\mathrm{\Delta }t}$ and the continuous lines bounding the grey area represent $\tilde{D}\pm std$. The diffusion coefficient is $D=1$ and for the OU process the drift is $a\left(x\right)=-2x$. Variations of the SNR, defined as $\frac{D}{\frac{{\sigma }^2}{\mathrm{\Delta }t}}$, are obtained for fixed position noise (a, c) or fixed sampling time (b, d). In (a) and (c), the positional noise is fixed at $\sigma =0.1$, while the sampling rate $\mathrm{\Delta }t$ is varying. In (b) and (d), the sampling time is fixed to $\mathrm{\Delta }t=0.001$ and the position noise $\sigma$ is varying.

Figure 3

(a) Trajectory of a one-dimensional OU process, generated using Euler's scheme (blue curve), and the observed trajectory (red curve) is obtained by subsampling at $\mathrm{\Delta }t=0.1$ and with an additional position noise of standard deviation $\sigma =0.05$ (SNR=40). The other parameters are fixed to $D=1, \lambda =2$, and $\mu =0$. The observed trajectories contain 10 000 points. (b) Estimation of the local drift using Eq. (A3) (black dots), and comparison with the analytical formulas (22) (red) and (30) (blue). (c) Estimation of the local diffusion coefficient using Eq. (A4) (black dots) and comparison with the analytical formulas (24) (red) and (31) (blue).

Figure 4

The maximum-likelihood estimators presented in Secs. 5a (blue) and 5b (red) for an OU process with $D=1, \mu =1$, and various values of $\lambda$ are compared. From left to right, the plotted estimations are $\tilde{\mu }-\mu$ (a, d, g), $\tilde{\lambda }-\lambda$ (b, e, h), and $\tilde{D}-D$ (c, f, i). The observation time step is $\mathrm{\Delta }t=0.1$ and from top to bottom, $\sigma =\sqrt{0.1}$ [(a–c) SNR=1], $\sigma =\sqrt{0.01}$ [(d–f) SNR=10], and $\sigma =\sqrt{0.001}$ [(g–i) SNR=100]. The black line is the mean of the estimation for trajectories of 500 points, and the colored part indicates $\pm$ the standard deviation.