Single-particle trajectories (SPTs) consist of a collection of successive discrete points causal in time.
In the context of cell biology, the trajectories are obtained by the transient activation by a laser of small dyes attached to a moving molecule.
Molecules can now by visualized based on recent super-resolution microscopy, which allow routine collections of thousands of short and long trajectories.
Due to the constant improvement of the instrumentation, the spatial resolution is continuously decreasing, reaching now values of approximately 20 nm, while the acquisition time step is usually in the range of 10 to 50 ms to capture short events occurring in live tissues.
A variant of super-resolution microscopy called sptPALM is used to detect the local and dynamically changing organization of molecules in cells, or events of DNA binding by transcription factors in mammalian nucleus.
Super-resolution image acquisition and particle tracking are crucial to guarantee a high quality data[5][6][7] Once points are acquired, the next step is to reconstruct a trajectory.
[8] Tracking algorithms are based on a physical model of trajectories perturbed by an additive random noise.
The redundancy of many short (SPTs) is a key feature to extract biophysical information parameters from empirical data at a molecular level.
[12] The MSD has been widely used in early applications of long but not necessarily redundant single-particle trajectories in a biological context.
At low spatiotemporal resolution of the observed trajectories, the MSD behaves sublinearly with time, a process known as anomalous diffusion, which is due in part to the averaging of the different phases of the particle motion.
For example, at very short timescales, when a molecule unbinds from a binding site or escapes from a potential well[15] and the inertia term allows the particles to move away from the attractor and thus prevents immediate rebinding that could plague numerical simulations.
For a timescale much longer than the elementary molecular collision, the position of a tracked particle is described by a more general overdamped limit of the Langevin stochastic model.
Indeed, if the acquisition timescale of empirical recorded trajectories is much lower compared to the thermal fluctuations, rapid events are not resolved in the data.
Thus at this coarser spatiotemporal scale, the motion description is replaced by an effective stochastic equation where
The observed effective diffusion tensor is not necessarily isotropic and can be state-dependent, whereas the friction coefficient
The development of statistical methods are based on stochastic models, a possible deconvolution procedure applied to the trajectories.
Numerical simulations could also be used to identify specific features that could be extracted from single-particle trajectories data.
[16] The goal of building a statistical ensemble from SPTs data is to observe local physical properties of the particles, such as velocity, diffusion, confinement or attracting forces reflecting the interactions of the particles with their local nanometer environments.
It is possible to use stochastic modeling to construct from diffusion coefficient (or tensor) the confinement or local density of obstacles reflecting the presence of biological objects of different sizes.
Several empirical estimators have been proposed to recover the local diffusion coefficient, vector field and even organized patterns in the drift, such as potential wells.
[17] The construction of empirical estimators that serve to recover physical properties from parametric and non-parametric statistics.
The models and the analysis assume that processes are stationary, so that the statistical properties of trajectories do not change over time.
In practice, this assumption is satisfied when trajectories are acquired for less than a minute, where only few slow changes may occur on the surface of a neuron for example.
Non stationary behavior are observed using a time-lapse analysis, with a delay of tens of minutes between successive acquisitions.
To estimate the local drift and diffusion coefficients, trajectories are first grouped within a small neighbourhood.
are approximated by the empirical sums The moment estimation requires a large number of trajectories passing through each point, which agrees precisely with the massive data generated by the a certain types of super-resolution data such as those acquired by sptPALM technique on biological samples.
The exact inversion of Lagenvin's equation demands in theory an infinite number of trajectories passing through any point x of interest.
In practice, the recovery of the drift and diffusion tensor is obtained after a region is subdivided by a square grid of radius r or by moving sliding windows (of the order of 50 to 100 nm).
Algorithms based on mapping the density of points extracted from trajectories allow to reveal local binding and trafficking interactions and organization of dynamic subcellular sites.
The method is based on spatiotemporal segmentation to detect local architecture and boundaries of high-density regions for domains measuring hundreds of nanometers.