Fig 1.
(a) Haematopoietic multi-potent progenitor cells (green) and the local micro-environment in an irradiated bone marrow cavity taken from a time-lapse in vivo imaging experiment (grey: bone; blue and purple: autofluorescence). (b) 3D reconstruction of a MPP trajectory extracted from the same set of experiments.
Fig 2.
Multi-potent progenitor cells display transient non-Gaussian super-diffusion.
(a) Time-averaged mean square displacement trajectories for each cell of track length greater than 3 hours (blue curves) along with the average over all cells (black curve). A linear fit (red curve) yields a super-diffusive exponent of α = 1.22. (b) Curve collapse of the distribution of the re-scaled variable η = δx/Δβ, where δx = x(t + Δ) − x(t), for 6 different lag time intervals Δ. The value of the exponent β is estimated to be 0.72 using a procedure described in Sec. IV.. For standard Fickian diffusion it is expected that α = 1 and β = 0.5.
Fig 3.
Comparison of simulation results and experimental data.
Ensemble averaged TAMSD curves for experimental (black) and simulation (orange) alongside the line of diffusion (red dashed), which is included as a guide to the eye. The simulation data has been extended into a region inaccessible to experimentation in which a crossover to diffusion is observable. Error bars are too small to be visible.
Fig 4.
(a) shows an example coarse-grained track (red) superimposed onto it’s original (blue) in which each subsequent pair of points are separated by 3 minute intervals. Visually it appears as though periods of low motility and persistence are followed by more exploratory periods displaying persistent motion. During periods of lessened motility sampled displacements may not faithfully represented true translocation of entire cell; the coarse-graining procedure described in Sec. IV lessens the effect of this heterogeneity. The displacement length time-series of the original track is shown for the same trajectory in (b).
Fig 5.
Schematic showing the experimental setup used to generate the data used in this article. Bone marrow harvested from transgenic donor mice expressing a fluorophore protein in their haematopoietic cells is purified using fluorescence activated cell sorting (FACS) to produce a sub-set of bone marrow cells highly enriched for multi-potent progenitors (MPPs). These cells are then transplanted into syngeneic recipient mice and imaged using confocal microscopy two days after the transplantation.
Fig 6.
Optimization of scale exponent using Wasserstein distance.
The maximum value of the Wasserstein distance computed pairwise between the empirical cumulative distribution functions of the re-scaled variable η = δx/Δβ where δx = x(t + Δ) − x(t), for each lag time Δ = {3, 6, 15, 30, 60, 120} mins used in the analysis. A clear minimum of this statistic is observed at a value of β = 0.73, which is the value used in Fig 2(b). All cells with track lengths greater than or equal to 120 mins were used in this analysis.
Fig 7.
Non-Gaussian displacement distributions.
x-displacement distributions for six different lag times Δ = {3, 6, 15, 30, 60, 120} mins, where δx = x(t + Δ) − x(t) for all time points t ∈ [0, T − Δ] where T is the length of the track. Cells included in this analysis have track lengths greater than or equal to 120 mins. Maximum likelihood fits for both a generalized Gaussian distribution (red line) and a standard Gaussian distribution (broken black line) are shown. The shape parameter γ for the generalized Gaussian fit is shown along with the lag time Δ in the top left of each plot.
Fig 8.
The non-Gaussianity parameter G(Δ) shown for lag-times of up to one hour. At short times the departure from Gaussian behavior is significant, while longer times show a relaxation to the expected result for Gaussian processes G = 0.
Fig 9.
Curve collapse for y and z displacement distributions.
(a) A repeat of the Fig 2(b) for the re-scaled y-displacement , where we have determined β = 0.76. Figure (b) shows a the same plot for the z-displacement where this time βz = 0.59. The ηy distributions reproduce well the result in Fig 2(b), however the same cannot be said of ηz, which we attribute to sampling bias caused by a limited field of view in the z-direction.
Fig 10.
Time-averaged mean square displacement.
We label the TAMSD of more (θ < 1.5) and less (θ ≥ 1.5) persistent cells with green and blue respectively. The corresponding averages over all cells within the more/less persistent groups are shown by the black triangles/circles, along with least-squares fit lines in red which yield exponents of 1.27 for the group, and 0.78 for
.
Fig 11.
Confinement in the z-direction.
For all cells used in the TAMSD calculation there exists the appearance of a confinement effect in the z-direction due to the limited field of view in this direction. By dividing the cells into two populations based on their average turn angle, we can identify a group of (a) more persistent cells which by chance happen to moving in the xy-plane, and (b) generally less motile cells which do not migrate sufficiently in any direction to leave the field of view.
Fig 12.
a) Schematic of three subsequent points along a coarse grained trajectory. The first point to have at least 7.2 microns displacement from the previous location is included in the coarse-grained trajectory. The distributions of the scaled run lengths (b), scaled turn angles (c), and scaled run time (d) are shown, with their respective distributions of the mean run length per track, etc, shown in the insets.
Fig 13.
Model x-displacement distributions.
Reproduction of Fig 7 for the simulated data. As in Fig 7 we show x-displacement distributions for six different lag times Δ = {3, 6, 15, 30, 60, 120} mins, where δx = x(t + Δ) − x(t) for all time points t ∈ [0, T − Δ] where T is the length of the track. Each plot shows a maximum likelihood fit for both a generalized Gaussian distribution (red line) and a standard Gaussian (broken black line). The shape parameter γ for the generalized Gaussian fit is shown along with the lag time Δ in the top left of each plot. Again in each case we see that the generalized Gaussian provides a superior fit, however, unlike the experimental data the Gaussian provides a progressively better fit with increasing lag time Δ.
Fig 14.
Schematic of superstatistical Bayesian method.
Given an experimental time-series and an observation likelihood function L, which we assume here to be of the form of the Gaussian AR-1 process described in Sec. IVD2, it is possible to compute the posterior distribution of the time-varying activity at and persistence qt parameters using a modified, discretized version of the forward-backward algorithm for hidden Markov models. The posterior parameter distribution is obtained through multiplication of the grids representing the likelihood Lt and the forward and backward priors and
obtained using the recursion relations presented in Sec. IVD3. Point parameter estimates
can be obtained from this posterior, typically through the mean or the mode. The subsequent priors in the forward and backward direction are computed independently using the ad-hoc transformation K.
Fig 15.
Results of Bayesian analysis for example trajectory.
(a) Shows an example MPP trajectory color coded according to the value of the persistence parameter q at that time. Figure (b) shows the time-averaged mean posterior distribution. Figures (c) and (d) show the time evolution of the posterior mean of the activity parameter a and persistence parameter q respectively.
Fig 16.
Results of Bayesian analysis for entire ensemble of MPPs.
2D histogram plot of time varying activity and persistence parameters at and qt collated over all MPP tracks.