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Introduction

For TraJ a trajectory is just a list of positions:

Each position is defined by

Between every two successive position is a constant timelag . The trajectory is traced over a period of time . These positions could be 1D, 2D and 3D. However, most of the trajectory features are restricted to 2D trajectories. A change in position is called a step. The step length is defined as the euclidian norm . A subtrajectory is a part of a trajectory.

Mean square displacement

A very popular function is the mean squared displacement function and is defined as

For the basic motion types normal diffusion, confined diffusion, anomalous diffusion and directed motion the MSD should theoretically should follow one of these models:

• Normal diffusion:

  

• Anomalous diffusion:

  

• Directed motion:

  

• Confined diffusion

  

D is the diffusion coefficent, the anomalous exponent (< 1), v the velocity, the radius of the confinement and and shape constants.

###Simulation To generate artifical trajectories of the basic motion types TraJ provides the following simulator classes:

• Normal diffusion: FreeDiffusionSimulator.java
• Anomalous subdiffusion: AnomalousDiffusionWMSimulation.java
• Directed motion: ActiveTransportSimulator.java
• Confined diffusion: ConfinedDiffusionSimulator.java

Features

To characterize trajectories TraJ provides several features:

Alpha

Anomalous exponent derived by fitting the MSD values to the function

Asymmetry

The Asymmetry feature a characterizes the asymmetry of the trajectory. It is equal to 0 for circularly symmetric trajectories and 1 for linear trajectories. It was implemented according to Saxton (1993) and is defined as

Whereas and are the eigenvalues of radius of gyration tensor T:

Asymmetry2

Suggested by Saxton (1993) as well, the Asymmetry2 feature is equal to 0 for circularly symmetric trajectories and 1 for linear trajectories:

Asymmetry3:

Similar to the Asymmetry feature but modified by Helmuth et al. (2007). It is defined as

Aspect ratio

Defined as the ratio of the long and short side of the minimum bounding rectangle. It is 1 for very symmetric trajectories and increases for more elongated trajectories.

Boundedness

The boundedness quantifies how much a particle with diffusion coefficient D is restricted by a circular confinement of radius r when it diffuses for a time duration . We define it as follows:

When using it as trajectory feature, we estimate D by the short time diffusion coefficient (only the first two timelags are used) and r by the half of the maximum distance between any two position of the trajectory.

Efficiency

Relates the squared net displacement to the sum of squared step lengths. Based on Helmuth et al. (2007) and defined as:

Elongation

Estimates the the elongation by 1 - S/L where S is the short side and L the long side of the minimum bounding rectangle. When the trajectory consits of colinear points, the elgonation is 1 by default.

Fractal dimension

The fractal path dimension was implemented according Katz et al. (1985) and is defined as:

where L is the total length (sum over all steplengths), n the number of steps and d is the largest distance between any two positions.

It takes between around 1 for directed trajectories, values around 2 and values around 3 for confined or subdiffusion.

Gaussianity

Proposed by Ernst et al. (2014) and defined as:

where

Whereas normal diffusion shows values of 0, other motion types show deviations of zero.

Kurtosis

For this feature we projected the 2D positions on the dominant eigenvector of radius of gyration tensor. The definition of kurtosis is then based on the projected positions :

Mean squared displacement ratio

The MSD ratio characterizes the shape of the MSD curve. We define it as follows:

where . For Brownian motion it is 0, for restricted motion < 0 and for directed motion > 0.

Straightness

Relates the net displacement L to the sum of step lengths and is defined as:

Trappedness

Based on Saxton (1993), the probablity that a particle with diffusion coefficent D and traced for a period of time is trapped into a region is given by:

When using it as trajectory feature, we estimate D by the short time diffusion coefficient (only the first two timelags are used) and r by the half of the maximum distance between any two position of the trajectory.

References:

Ernst, D., Köhler, J. & Weiss, M. Probing the type of anomalous diffusion with single-particle tracking. Phys. Chem. Chem. Phys. 16, 7686–91 (2014).

Helmuth, J. A., Burckhardt, C. J., Koumoutsakos, P., Greber, U. F. & Sbalzarini, I. F. A novel supervised trajectory segmentation algorithm identifies distinct types of human adenovirus motion in host cells. J. Struct. Biol. 159, 347–58 (2007).

Katz, M. J. & George, E. B. Fractals and the analysis of growth paths. Bull. Math. Biol. 47, 273–286 (1985).

Saxton, M. J. Lateral diffusion in an archipelago. Single-particle diffusion. Biophys. J. 64, 1766–1780 (1993).