In the sample of 96 divisions, there was only a single instance of asymmetric division where one daughter cell contained significantly more fluorescent protein that the other, shown in Figure 4.5.
An earlier study using E.coli [23] observed that the majority of divisions (> 85%) produced daughter cells with a volume difference of less than 5%, and that differences in daughter cell size was the primary cause of asymmetric division. In the single case measured here the GFP was confined to the nucleus. The segmented areas of the daughter nuclei were almost equal (52 & 53 pixels respectively) as were the Hoechst intensities (4600 & 4400 as measured after background subtraction).
Another indicator of asymmetric division could be cell cycle timings, where differentiated cells might have different generation lengths. While the data used here has insufficient cells with multiple divisions, other studies [108] have failed to observe a significant difference in generation times, suggesting that asymmetric divisions are infrequent in-vitro.
Figure 4.5: Example of an asymmetric division (arrowed). Top: GFP fluorescence, Bottom: Nuclei stained using Hoechst. The fluorescence was divided between the daughter cells in the ratio 28:72 for GFP and 51:49 for Hoechst.
4.4
Conclusions
An interactive framework was developed to aid analysis of cell data and this was used to extract fluorescence measurements for a number of cells which were subsequently used to study the partitioning of fluorescent protein during divisions. It was observed that in all but one instance, the protein was equally distributed between daughters and the fluorescence recovery following division was consistent.
Chapter 5
Performance and Validation
of the LineageTracker
Software
This chapter describes the accuracy measurement and performance validation on the segmentation and tracking methods. The tracking validation was per- formed using two manually tracked C2C12 data sets whereas the segmentation validation was based on a combination of manually annotated cell images and a set of synthetic images generated using third-party software.
5.1
A Statistical Analysis of Cell Motility
The C2C12 mesenchymal cells are highly motile muscle precursor cells. Dur- ing migration, cells release chemokines which in turn attract other cells and encourage motion in a particular direction [109]. Cells exhibit a range of behaviours ranging from random motion to migratory travel depending on their local environment.
An analysis of cell motion was performed (similar to [110]) to investigate whether any additional movement parameters could be used to improve the tracking. The manually validated tracked sets (see Section 5.5.1) were analysed. Taking cell motion in a single axis first, the speeds appear to follow a Gaussian distribution (see Figure 5.1). The mean was close to zero (±0.45 for both x- and y-axes), suggesting there was little or no overall drift in any particular direction. The quality of fit (R2>0.99) was good for all the speed
distributions. −200 −10 0 10 20 200 400 600 800 1000
Cell Speed (Y direction)
a) Reference Standard 1: Cell speed in the y direction −200 −10 0 10 20 200 400 600 800 1000 1200
Cell Speed (X direction)
b) Cell speed in the x direction
−200 −10 0 10 20
500 1000 1500 2000
Cell Speed (Y direction)
c) Reference Standard 2: Cell speed in the y direction −200 −10 0 10 20 500 1000 1500 2000
Cell Speed (X direction)
d) Cell speed in the x direction
Figure 5.1: a & b) Cell displacements from Reference Standard 1, with a Gaussian Distribution superimposed. c & d) Cell displacements for Reference Standard 2. Displacements are in pixels per frame, where 1 pixel = 1.36µm.
The peak in Figure 5.1a is offset slightly from the mean value so the Skewness values were calculated for the distributions, which measures the asymmetry around the mean where a normal distribution has skewness of zero. Taking Reference Standard 1, the skewness of the ‘y’ speeds is -0.20, which is of greater magnitude than the skewness of the ‘x’ direction which is <0.01. Since cell motion is not truly random but is influenced by nearby cells, it is expected that the distributions may occasionally deviate from normal.
When a particle moves in two or more axes and the speed in each axis follows a Gaussian distribution, the particle velocities will be given byv=√∆x2+ ∆y2
and will follow a Rayleigh distribution. This is shown in Figures 5.2a & c. The speed distributions were fitted to the Rayleigh distribution formula (given in Equation 5.1) and the quality of fit was calculated.
R(x) = x
v2e −x2
2v (5.1)
Reference Set 1 exhibits a slightly better fit (R2 = 0.96) than Set 2 (R2 =
0.94). Since a true Rayleigh distribution will only be obtained when the x & y components of the velocity are both normally distributed, any slight deviation from normal will be reflected in the quality of fit.
The majority of the cells (>95%) have velocities below 10 pixels per frame. This result could be used to determine the optimum ‘Displacement Parameter’ for the tracking algorithm.
0 5 10 15 20 0 500 1000 1500 Cell Speed No. of cells
a) Cell speeds from Reference Standard 1
0 50 100 150 0 0.005 0.01 0.015 0.02 0.025 Turn Angle Fraction of Cells
b) Change in direction for Reference Stan- dard 1 0 5 10 15 20 0 5000 10000 15000 20000 25000 Cell Speed No. of cells
c) Cell speeds from Reference Standard 2
0 50 100 150 0 0.01 0.02 0.03 0.04 0.05 0.06 Turn Angle Fraction of Cells
d) Change in direction for Reference Stan- dard 2
Figure 5.2: Measurements from Reference Standard data sets. a) Cell speeds fitted with a Rayleigh distribution. b) Change in cell direction. c) Speeds from Reference Standard 2 fitted with a Rayleigh distribution. d) Change in cell direction for Reference Standard 2.
Changes in direction were calculated from 0 to 180◦so a 90◦ angle could be either a left or right turn, as only the degree of turn was considered. The results are displayed in Figure 5.2b & d. The two distributions are quite different, with Set 1 having more cells with lower turn angles than Set 2. A visual
inspection of the two timecourse experiments shows large numbers of cells in Set 1 moving in roughly straight lines. A Gaussian mixture model was applied to the angle distributions and both datasets were found to be built from a mixture of distributions with similar mean values: < 40◦, 50–54◦and a much smaller component at ≥ 60◦, Set 1 had a greater contribution from the lower angle range.
The two angle distributions present in Set 1 suggest that there are two distinct subpopulations where half of the cells have persistent motion whereas the other half are exhibiting a random walk. The cell displacement histograms do not show a similar division, suggesting that cell speed is not dependent on whether the cell is in migration or random motion.
A brief analysis of the cell motion was performed, comparing the motion with random walks, summarised in Figure 5.3, where the mean total displacement squared is plotted against the number of steps.
Motile cells in a uniform environment with no chemoattractants, when viewed from above, will appear to be free to move in 2 dimensions. If the cell could change speed or direction without constraint, the motion would resemble a ‘random walk’ where the cells move in a random direction, with no correlation with previous motion. A traditional random walk with fixed step size and random direction would appear as a straight line on a plot of steps taken (or total distance covered) against distance squared (as shown in Figure 5.3a) whereas straight line motion appears as a quadratic increase as the distance will increase at a constant amount with each step.
The motion of the tracked sets is displayed in Figure 5.3b. The blue trace represents the first Reference Standard and this closely resembles the straight line trace in Figure 5.3a which is consistent with the low turn angles present in that set. The second set, shown in red, starts to deviate from the ‘straight line’ motion and the curve flattens out as cell motion either becomes more random or motion becomes constrained. Mean cell speed drops very slightly during the course of the experiment, a maximum of 3.2 pixels per frame near the start to 2.6 pixels per frame at the end but this would not be sufficient to cause such a drop in the curve. A simulation of a decreasing-step random walk is given in Figure 5.4a where the step size drops from 3.5 to 2.5 pixels per step but this
does not recreate the walk profile from the experiment.
A second simulation increases the angle throughout the run, and this produces a result much closer to the experimental results. One theory for this is that as density increases, persistence decreases since the cells are unable to move in straight lines (supporting data for this is given in Appendix B.5). Reference Standard 1 was of shorter total duration so may not have run for sufficient time to exhibit the same behaviour. An alternative visualisation of the random walk analysis is presented in Appendix B.6 using Distance-Pathlength heatmap plots.