Track Analysis

Introduction

This app analyses the track data of multi-particle tracking microrheology and calculates the mean-square displacements (MSDs), the linear viscoelastic moduli (G′ and G″) by the generalized Stokes--Einstein relation (GSER), and the non-Gaussian parameters, etc. ¹

Features:

• supports analysis upon multiple track files
• supports manual optimization for GSER calculation
• supports calculation of the distribution of diffusivity p(D) by Lucy's algorithm²³
• supports 4-point susceptibility calculation

Installation

Unzip and drag the .mlappinstall file to the command window of MATLAB. After the installation it will appear in the app list.

Data preparation

Consider a time-lapse video of M frames. The resolution of the video is X pixels × Y pixels. Each frame is labeled by a positive integer, called "frame ID". Suppose that previous image analysis has provided a result of a number of tracks from this video. Each track is labeled by a positive integer, called "track ID". The information of a track (bearing the same track ID) is provided by the data of x and y positions (in the unit of pixels), each of which corresponds to a distinct frame_ID's. So, each row of a track data consists of the ordered quadruple (x, y, frame ID, track ID). In general, different tracks would start at different frame ID's and last for different number of frames. Also, tracks that skipping some of the frames (i.e. under the same track ID, the frame ID's are not a set of continuous natural numbers) can be handled by this app. An N × 4 matrix each row of which is the quadruple (x, y, frame ID, track ID) is sufficient to completely describe a set of multiple tracks.

The app can open any .mat file that contains an N × 4 matrix named tr. The 4 columns of tr are x positions, y positions, frame ID, and tracks ID in order. See tr_example.mat for an example.

Another app, Particle Tracking, is designed for generating the tr data from time-lapse series.

Within the same tr data, the ordered pairs (frame ID, track ID) must be pair-wise distinct, since no one particle (the same track ID) can appear at two or more positions in the same moment (frame ID).

The app can process multiple '.mat' files. For example, they may be the results from different videos but on the same sample. In this case the track ID and frame ID from the tr of different imported .mat files do not need to be distinct video-wise. In other words, in each of these tr variables, the track ID's and frame ID's can, without exception, be (re-)starting from 1, regardless of the others.

Load data

Click Add... to select .mat files that contain the matrix tr. Multiple selection is supported. Click Add... many times to append more tracks. The files are listed in the table next to the button. Click Remove to remove the currently selected file. The number of tracks contains in each file is also listed in this table.

Setting parameters

The parameters in the Parameters panel should be set before calculation. Their meanings are listed below.

Resolution (µm/pixel)

The conversion factor of the unit of x/y position from pixel to micron. This value is determined by independent calibration of the microscope with a digital camera.

Temperature (°C)

The temperature of the experiment.

Particle radius (μm)

The radius of the probe particles.

Shutter time (s)

The lag time between adjacent steps of each tracks. This should be the same for all loaded tracks.

Calculate

Click Calculate to start the calculation upon the currently loaded tracks. Status that indicating the currently processing quantity will be temporarily shown below the axes. Before finishing the calculation a window will pop up for the user to adjust the parameter for GSER calculation (will be introduced in later sections). After closing this window the calculation finishes. The status label read Done! and the mean-square displacement (MSD) is plotted on the axes.

The GSER pop-up window

During the first calculation or after clicking the GSER button the GSER pop-up window is shown for tuning. GSER calculation can be optimized by a) cutting unwanted portion of the MSD and b) smoothing the MSD curve and its derivative successively by the same smoothing algorithm and setting. By default the MSD is uncut; it and its derivative are smoothed by the moving-average algorithm. In the left figure, the original MSD curve is plotted as blue dots, and the cut and smoothed MSD curve is plotted as a red solid line. In the right figure the resulted G' and G'' are plotted. Change the two slider the set the minimum and maximum Δt to cut out unwanted portion of the MSD. The resulted upper and lower bounds are indicated as black dashed line in the left figure. Change smoothing algorithm by selecting the corresponding radio button in the Smooth algorithm panel. Change the Span (for moving-average and robust lowess algorithms), and Knots (for SLM engine) by the slider next to the panel. After any change both figures will update. Tune until the results are good. Click OK or directly close the GSER pop-up windows will quit this session and confirm the GSER results.

Select plotting data

The Plot data panel allow selection of the following observables to plot.

MSD

The multi-particle mean-square displacements <Δx²>, <Δy²>, and <Δr²>, where Δr² = Δx² + Δy². A black dash line indicates the slope of unity.

Modulus

The storage and loss moduli G′ and G″ calculated by the GSER pop-up window.

alpha_2

The mean single-particle and multi-particle non-Gaussian parameters². The mean single-particle parameter is plotted with its standard error among all tracks.

van Hove

The distribution density of Δx (van Hove function, scattered) and the corresponding Gaussian distribution (solid) estimated from the MSD for the lag time Δt adjusted by the Lag time slider.

The Lucy pop-up window

For each lag time Δt, adjusted by the Lag time slider, the van Hove function can be deconvolve into a distribution of diffusivity P(D) using Lucy's algorithm³. First select the lag time under which you want to calculate P(D), then click the Lucy button to open the Lucy's algorithm pop-up windows.

The algorithm will iterate until the total residue reached the value of Tolerance edit box. But please leave the value being very small at first.

The algorithm will estimate an average diffusion coefficient D from the MSD value of the current lag time, and will deconvolute the van Hove function data into a distribution of D over a range (6 orders of magnitude) around the estimated value. Set the number of bins for such a range in the # bins edit box.

Click Run button to start iteration. The first plot shows the comparison between the experimental data of van Hove function with the reconstructed ones by the iteration algorithm. The second plot shows the resulted probability P(D). The third plot shows the total residue vs number of iterations.

Observe the growth behavior of the total residue until the curve seems to approach to a plateau value (either from below or above). Then pause the iteration by clicking Pause button. Now you can read the total residue data for an estimate of the plateau value. You may want to enter a suitable value of Tolerance at this moment, so that when you click the Resume button the iteration will stop automatically as the total residue reaches the set value, during which you can do other jobs.

Or you may just monitor the goodness of fit in the first plot and click the Stop button once you are satisfied.

The calculated restults will be remembered once you close the pop-up windows. And the Lag time slider bar will show only one minor tick, indicating the lag time for which you have calculated P(D). The reconstructed van Hove function will show in the plot if you select the van Hove option under Plot data and the Lag time slider bar is adjusted to the corresponding lag time.

There can be a set of P(D) data for only one lag time. Everytime you click Lucy button, the old results of P(D) is replaced by new ones.

The $\chi_4$ calculation

The 4-point susceptibilities $\chi_4$ are calculated from the trajectories. The calculation requires Parallel Computing Toolbox. And you should have at least one profile for parallel computing. Please refer to official help of this toolbox. The app check if your environment support parallel computing before started. If so, the option of $\chi_4$ will be enabled.

The theory behind 4-point susceptibility is not discussed here. Please refer to other papers, such as: Karmaker, Dasgupta & Sastry (2016), Rep. Prog. Phys. 79:016601.

Click the button $\chi_4$ to start calculation. If the calculation is successful, the status message is "Done!". Select $\chi_4$ option button and the result will be plotted. The result plotted is the self-part of the 4-point susceptibility, since for multiparticle tracking it is usually the only meaninful result. But by clicking Save ... button a number of results is saved.

The Trajectories popup window

Click the traj. excl. button to open the Trajectories popup window.

In the popup window, single-particle MSD <Δx²>_single (solid) and the corresponding multi-particle counterpart (dash) is plotted in the left axes, with the selected one highlighted. The corresponding trajectory of the selected particle is plotted in the right, bigger axes, with its track id also displayed. You can select a track by clicking on the SP-MSD plot, or from the Track# list box. Double clicking one item in the Track# list box adds the corresponding item to the Exclude: list box, which is the track id's to be excluded. Double clicking one item in the Exclude: list box removes it. Click save or just directly close the popup window to save the excluded tracks information. After that, click the Calculate button immediately to re-calculate the variables based from the excluded tracks. After this calculation, the excluded tracks information is cleared. Clicking the Calculate button again calculates the variable based on the original tracks in the table. Changing the track list in the table also clears the excluded tracks information. You must re-select the excluded tracks if you just change the track list.

Saving data

By clicking the Save... button a folder selection dialogue popup and the current calculated results will be saved into a MATLAB data file named msd_data.mat in the selected folder. If a Lucy calculation has performed an addition MATLAB data file named PD_data_xxx.mat in the same folder, where xxx is the lag time under which the P(D) was calculated. A separated file named chi4.mat will be saved to the selected path.

The msd_data.mat contains the following 6 vairables.

msd

n × 4 matrix of the multi-particle MSDs, where n is the number of lag times. The 4 columns are: Δt (s), <Δx²> (µm²), <Δy²> (µm²), and <Δr²> (µm²), respectively.

msd_i

m × 1 cell array of the single-particle MSD, where m is the total number of tracks processed. Each cell is an n &times 4 matrix with the same column meaning as msd.

G

3-column matrix of the GSER result. The columns are angular frequency ω (rad/s), G′ (Pa), G″ (Pa), respectively.

alpha_2 and alpha_2_i

Multi-particle and mean single-particle non-Gaussian parameters, both of which are 3-column matrices. The 3 columns of alpha_2_i are Δt (s), the mean α₂ among multiple tracks, and the standard error of α₂ among multiple tracks. The 3 columns of alpha_2 are Δt (s), the multi-particle α₂, and the finite-sample variance of the multi-particle α₂.

van_Hove

The displacement Δx distribution density p(Δx), an n &times 2 cell array. The 2 cells in each row are the distribution densities and the bins edges, respectively.

The PD_data_xxx.mat contains the following 3 variables

p_D and P_D

The distribution density and distribution function of diffusivity, respectively. Both are 2-column matrices: D (m²/s), p(D) (s m⁻²) or P(D) (-).

van_Hove_rec

A reconstructed van_Hove function from the calculated P(D) data. It is a 2-column matrix: Δx (micro;m), distribution density p(Δx) (You may check the validity of the latter by comparing van_Hove_rec with the van_Hove data of the same lag time in the msd_data.mat file.

4-point susceptibilities

A separated file named chi4.mat will be saved to the selected path. This file includes the self-, distinct-part and total result of $\chi_4$, as the variables, chi4_s, chi4_d,_chi4_t, as well as those of the overlap functions $Q\left(\Delta t\right)$, as the variables, Q_s, Q_d, Q_t. The first column of these variables are the lag time in second. The values of the $\chi_4$ is by unit m³.

Footnotes

1. This readme.md and also the app are for users familiar with the field of probe microrheology. See for example: Waigh, T. A. "Microrheology of Complex Fluids." Reports on Progress in Physics 68, no. 3 (2005): 685-742. ↩

2. See for example: Hong, W., et al. “Colloidal Probe Dynamics in Gelatin Solution During the Sol–Gel Transition.” Soft Matter 14, no. 19 (2018): 3694-703. ↩ ↩²

3. Lucy, L. B. “An Iterative Technique for the Rectification of Observed Distributions.” The Astronomical Journal 79, no. 6 (1974): 745-54. ↩ ↩²