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 allow observing moving molecules inside cells to collect statistics

SPT allowed observing moving particles. These trajectories are used to investigate cytoplasm or membrane organization , but also the cell nucleus dynamics, remodeler dynamics or 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 short events occuring 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

Assembling points into a trajectory based on tracking algorithms

Once points are acquired, the next step is to reconstruct a trajectory. This step is done known tracking algorithms to connect the acquired points . Tracking algorithms are based on a physical model of trajectories pertubed by an additive random noise.

Extract physical parameters from redundant SPTs

The redundancy of many short (SPTs) is a key feature to extract biophysical information parameters from empirical data at a molecular level . In constrast, long isolated trajectories have been used to extract information along trajectories, destroying the natural spatial heterogenity associated to the various positions. The main statistical tool is to compute the mean-square displacement (MSD) or second order statistical moment.$<|X(t+\Delta t)-X(t)|^{2}>\sim t^{\alpha }$ (average over realizations), where $\alpha$ is the called the anomalous exponent. For a Brownian motion, $<|X(t+\Delta t)-X(t)|^{2}>=2nDt$, where D is the diffusion coefficient, n is dimension of the space. Some othre properties can also be recovered from long trajectories, such as the radius of confinement for a confined motion. The MSD 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 could be 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. In the context of cellular transport (ameoboid), high resolution motion analysis of long SPTs in micro-fluidic chambers containing obstacles revealed different types of cell motions. Depending on the obstacle density: crawling was found at low density of obstacles and directed motion and random phases can even be differentiated.

Physical model to recover spatial properties from redundant SPTs

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 the elementary molecular collision, the position of a tracked particle is described by the overdamped limit of the Langevin stochatic model. The model assumes that the diffusion of a protein or a particle embedded in a membrane surface or inside the cytoplasm is generated by a constant diffusion coefficient D and a field of force F(X,t), according to the overdamped limit:

${\dot {X}}={\frac {F(X(t),t)}{\gamma }}+{\sqrt {2D}}\,{\dot {W}},$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 or membrane molecules. The acquisition timescale of empirical recorded trajectories is often much lower compared to the thermal fluctuations and rapid events are not resolved in the data. Thus at this coarser spatiotemporal scale, the motion description is replaced 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 can vary in space $D(X)={\frac {1}{2}}B(X)B^{T}(X)$(.^T denotes the transposition). This equation is not derived by assumed. However the diffusion coefficient should smooth enough. Any discontinuity in D should be resolve by a spatial scaling to analyse de source of discontinuty (usually intert obstacles or transition between two medium). The observed effective diffusion tensor is not necessarily isotropic and can be state-dependent, whereas the friction coefficient $\gamma$ remains constant as long as the medium stays the same 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.

References

 This article incorporates text by N. Hoze, D. Holcman available under the CC BY-SA 3.0 license. Also available under the GNU Free Documentation License.