Single-particle trajectory

Single particle trajectories (SPTs) consist of a collection of successive discrete points causal in time. These trajectories are acquired from images in experimental data. 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 now allow routine collections of thousands of short and long trajectories. These trajectories explore part of a cell, either on the membrane or in 3 dimensions and their paths are critically influenced by the local crowding organization of th ecell , as emphasized in various cell types such as neuronal cells, astrocytes, immune cells and many others.

SPTs in molecular and cellular biology

SPT allowed observing moving particles

SPT allowed observing moving particles at the micrometer scale, to investigate cell membrane organization , but also the cell nucleus dynamics and mRNA production. 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 the shortest events in live tissues.

Once points are acquired, the next step is to reconstruct trajectories. This is done by tracking algorithms that allow connecting the dots of acquired points . a variat of super-resolution mivroscopy called sptPALM is used to detect the dynamical organization of membrane proteins in plant cells, or events of DNA binding by transcription factors in mammalian nucleus. The physical interpretation of the trajectories depends on the quality of the upstream data acquisition. Super-resolution image acquisition and particle tracking are crucial to guarantee high quality data

Redundancy of many short SPTs are necessary to extract physical parameters

The redundancy of many short (SPTs) are necessary to extract physical parameters from empirical data at a molecular level , while long isolated trajectories have been used to extract second order properties of a Brownian motion using mean-square displacement analysis . Some geometrical properties can also be recovered from long trajectories, such as the radius of confinement for a confined Brownian motion . In the context of cellular transport (ameoboid), high resolution motion analysis of long SPTs in micro-fluidic chambers containing obstacles revealed different type of cell motions. Depending on the obstacle density, crawling was found at low density of obstacles and directed and random phases can even be differentiated.

For studying long trajectories, the most common procedure consists in estimating the mean-square displacement (MSD) of the particle. The MSD is proportional to time and allows in theory estimating the diffusion coefficient. It has been widely used in early applications of long but not necessarily redundant single particle trajectories in a biological context. However, the MSD applied to long trajectories suffers from several issues. First, it is not precise in part because the measured points are correlated. Second, it cannot be used to compute any physical diffusion coefficient when trajectories consists of switching episodes for example alternating between free and confined diffusion. 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. To overcome these issues and distinguish between different types !

of motion, a stochastic model of the acquired data is needed.

Physical model

Langevin and Smoluchowski equations as a model of motion :

In the past recent years, challenges of cell membrane reconstruction, structural identification and membrane organization have been addressed following the development of statistical methods based on stochastic models. These models are the Langevin equation, its Smoluchowski limit and associated models that account for additional localization point identification noise. The development of these statistical approaches are based on stochastic models, the deconvolution procedure and the numerical simulations are used to extract biophysical parameters from single particle trajectories data .

Langevin's equation describes a stochastic particle driven by a random force$\Xi$and a field of force (e.g., electrostatic, mechanical, etc...). With an external force F(x,t) , it is written as

$m{\ddot {x}}+\Gamma {\dot {x}}-F(x,t)=\Xi ,$

where m is the mass of the particle and $\Gamma =6\pi a\rho$is the friction coefficient of a diffusing particle, $\rho$ the viscosity and \eta the $\delta$-correlated Gaussian white noise . The force can derive from a potential, $F(x,t)=-U'(x)$ and in that case, the equation takes the form

$m{\frac {d^{2}x}{dt^{2}}}+\Gamma {\frac {dx}{dt}}+\nabla U(x)={\sqrt {2\epsilon \gamma }}\,\eta ,$ where $\epsilon =k_{B}T,$ which represents the energy, $k_{B}$ is the Boltzmann's constant and T the temperature. The Langevin's equation is transformed into the two dimensional stochastic system ${\dot {x}}=v$ $\gamma v=-U^{\prime }(x)+{\sqrt {2\epsilon \gamma }}{\dot {w}}.$ $\gamma =\Gamma /m$ is the dynamical friction coefficient per unit mass. Langevin's equation is used to describe trajectories where inertia or acceleration matters. For example at very short timescales, when a molecule unbinds from a binding site, it escapes the potential well and the inertia term allows the particles to move away from the attractor and thus prevents immediate rebinding that could plague numerical simulations.

In the large friction limit $\gamma \to \infty$ the trajectories x(t) of the Langevin equation converges in probability to these of the Smoluchowski equation

$\gamma {\dot {x}}+U^{\prime }(x)={\sqrt {2\epsilon \gamma }}\,{\dot {w}},$

where ${\dot {w}}(t)$ is \delta-correlated Gaussian white noise. This equation is derived in the case where the diffusion coefficient is constant in space. When it is not case, coarse grained equations (at a coarse spatial resolution) should be derived from molecular consideration. Interpretation of physical forces are not resolved by Ito's vs Stratanovich integral representation.

Physical model equations

For a timescale much longer than elementary molecular events, the position of tracked particles is described by the overdamped limit of the Saffman-Delbruck-Langevin model. The model assumes that the diffusion of a protein or a particle embedded in a membrane surface is generated by a diffusion coefficient D and a field of force F(X,t), according to the overdamped Langevin equation ${\dot {X}}={\frac {F(X(t),t)}{\gamma }}+{\sqrt {2D}}\,{\dot {W}},$where W is a Gaussian white noise and $\gamma$ is the dynamical viscosity. The source of the driving noise is the thermal agitation of the ambient lipid and membrane molecules. However, due to the acquisition timescale of empirical recorded trajectories, which is too low to follow the thermal fluctuations, rapid events are not resolved in the data, and at this coarser spatiotemporal scale, the motion is described by an effective stochastic equation

${\dot {X}}(t)={b}(X(t))+{\sqrt {2}}{B}_{e}(X(t)){\dot {w}}(t),$

where ${b}(X)$ is the drift field and ${B}_{e}$the diffusion matrix. The effective diffusion tensor is given by $D(X)={\frac {1}{2}}B(X)B^{T}(X)$(.^T denotes the transposition). The observed effective diffusion tensor is not necessarily isotropic and can be state-dependent, whereas the friction coefficient $\gamma$ remains constant and the microscopic diffusion coefficient (or tensor) may remain isotropic.

Statistical analysis of these trajectories

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 environment. It is possible to use modeling to construct from diffusion coefficient (or tensor) the confinement or local density of obstacles reflecting the presence of biological objects of different sizes.

Drift and diffusion tensor of a general stochastic process

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. The construction of empirical estimators that serve to recover physical properties from parametric and non-parametric statistics. Retrieving statistical parameters of a diffusion process from one-dimensional time series statistics have been studied using first moment estimators or Bayesian inference. Direct asymptotic of stochastic equations is used to construct direct empirical estimators for recovering drift and diffusion tensor. There are derived from discretizing the stochastic equation. 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.  !

Time-lapse analysis, with a delay of 15 minutes between acquisitions has indeed

revealed slow changes over time of biological significance.

The coarse-grained model is recovered from the conditional moments of the trajectory increments $\Delta X=X(t+\Delta t)-X(t)$:

$a(x)=\lim _{\Delta t\rightarrow 0}{\frac {E[\Delta X(t)\,|\,X(t)=x]}{\Delta t}}$,

$D(x)=\lim _{\Delta t\rightarrow 0}{\frac {E[\Delta X(t)^{T}\Delta X(t)\,|\,X(t)=x]}{2\Delta t}}.$

Here the notation $E[\cdot \,|\,X(t)=x]$means averaging over all trajectories that are at point x at time t. Indeed, the coefficients of the Smoluchowski equation can be statistically estimated at each point x from an infinitely large sample of its trajectories in the neighborhood of the point x at time t.

Empirical estimation

In practice, the expectations for a and D are estimated by finite sample averages and$\Delta t$ is the time-resolution of the recording of the trajectories. Formulas for a and D are approximated at the time step $\Delta t=50ms$, where 200 points falling in any bin is usually enough for the estimation.

To estimate the local drift and diffusion coefficients, the trajectories are first grouped within a small neighbourhood. The field of observation is partitioned into square bins $S(x_{k},r)$of side r and centre $x_{k}$ and the local drift and diffusion are estimated for each of the square. Considering a sample of$N_{t}$ trajectories $\{x^{i}(t_{1}),\dots ,x^{i}(t_{N_{s}})\},$ where $t_{j}$ are the sampling times, the discretization of equation for the drift $a(x_{k})=(a_{x}(x_{k}),a_{y}(x_{k}))$at position $x_{k}$ is

$a_{x}(x_{k})\approx {\frac {1}{N_{k}}}\sum _{j=1}^{N_{t}}\sum _{i=0,{\tilde {x}}_{i}^{j}\in S\left(x_{k},r\right)}^{N_{s}-1}\left({\frac {x_{i+1}^{j}-x_{i}^{j}}{\Delta t}}\right)$

$a_{y}(x_{k})\approx {\frac {1}{N_{k}}}\sum _{j=1}^{N_{t}}\sum _{i=0,{\tilde {x}}_{i}^{j}\in S\left(x_{k},r\right)}^{N_{s}-1}\left({\frac {y_{i+1}^{j}-y_{i}^{j}}{\Delta t}}\right)$

where $N_{k}$is the number of points of trajectory that fall in the square $S(x_{k},r)$.Similarly, the components of the effective diffusion tensor $D(x_{k})$are approximated by the empirical sums

$D_{xx}(x_{k})\approx {\frac {1}{N_{k}}}\sum _{j=1}^{N_{t}}\sum _{i=0,x_{i}\in S(x_{k},r)}^{N_{s}-1}{\frac {(x_{i+1}^{j}-x_{i}^{j})^{2}}{2\Delta t}},$

$D_{yy}(x_{k})\approx {\frac {1}{N_{k}}}\sum _{j=1}^{N_{t}}\sum _{i=0,x_{i}\in S(x_{k},r)}^{N_{s}-1}{\frac {(y_{i+1}^{j}-y_{i}^{j})^{2}}{2\Delta t}}$,

$D_{xy}(x_{k})\approx {\frac {1}{N_{k}}}\sum _{j=1}^{N_{t}}\sum _{i=0,x_{i}\in S(x_{k},r)}^{N_{s}-1}{\frac {(x_{i+1}^{j}-x_{i}^{j})(y_{i+1}^{j}-y_{i}^{j})}{2\Delta t}}.$

The moment estimation requires a large number of trajectories passing through each point of the surface, which fits precisely to the massive data generated by the sptPALM technique on biological samples. Indeed, 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 a moving slinding windows (of the order of 50 to 100 nm). Estimations of the error term in the drift compared to the diffusion coefficient suggest a minimum of 200 points per bin.