Dynamics of single human embryonic stem cells and their pairs: a quantitative analysis

Numerous biological approaches are available to characterise the mechanisms which govern the formation of human embryonic stem cell (hESC) colonies. To understand how the kinematics of single and pairs of hESCs impact colony formation, we study their mobility characteristics using time-lapse imaging. We perform a detailed statistical analysis of their speed, survival, directionality, distance travelled and diffusivity. We confirm that single and pairs of cells migrate as a diffusive random walk for at least 7 hours of evolution. We show that the presence of Cell Tracer significantly reduces hESC mobility. Our results open the path to employ the theoretical framework of the diffusive random walk for the prognostic modelling and optimisation of the growth of hESC colonies. Indeed, we employ this random walk model to estimate the seeding density required to minimise the occurrence of hESC colonies arising from more than one founder cell and the minimal cell number needed for successful colony formation. Our prognostic model can be extended to investigate the kinematic behaviour of somatic cells emerging from hESC differentiation and to enable its wide application in phenotyping of pluripotent stem cells for large scale stem cell culture expansion and differentiation platforms.

. Summary of parameters acquired for single cells cultured in the absence and presence of Cell Tracer. The entries represent the mean values with their standard errors and the spread within the sample given by the standard deviation (SD) of individual measurements around the mean. Migration velocities and step lengths and were calculated by averaging the displacements between images taken at 15 min intervals. Step length in y ( m) 3.5  0.1 3.5 2.1  0.1 2.5 Time to first division (hr) div 7  1 5 10  2 6.5 Supplementary Table T2   Table T2. Parameters characterising the migration of hESC pairs, both in the absence and in the presence of Cell Tracer. For each parameter, we present its mean value and standard deviation, as well as the spread given by the standard deviation of the individual measurements from the mean.

Supplementary Section S1: Directionality
Directionality, (or the straightness index) is a simple and convenient parameter to quantify an isotropic random walk, as employed by Li et al 1 . The displacement of the cell at a time , measured along the straight line from the starting point is = √[ ( ) − ,0 ] 2 + [ ( ) − ,0 ] 2 (with the cell identifier) and the total distance traversed during the time is denoted . The directionality of the cell migration is then defined as = / , and its values lie in the range 0   1. If the cell moves along a straight path, we have = , and the directionality has its maximum value, = 1. If, however, the cell follows a long and tortuous trajectory, then is much larger than , and the directionality is low, ≈ 0. Thus, the directionality quantifies how tangled and convoluted the cell's trajectory is. This quantity is closely related to the tortuosity, similarly characterising the shape of convoluted trajectories 2 . While the directionality may not the most useful characteristic of trajectories 3 , we use it to retain comparability with earlier work on cell kinematics 3 . In particular, the directionality depends on the number of steps taken in the random walk. However, the unstained and stained cells move with similar correlation times, performing similar number of steps per unit time; this allows us to compare their trajectories in real time. It would not be difficult to describe the trajectories in terms of the number of random walk steps, but such a description would be less intuitive.
For a two-dimensional isotropic random walk, with steps of a length , the average displacement from the starting point increases with the number of steps as = √ , where = / is the number of steps in time . Meanwhile, the total distance traversed is = . Then the average directionality of an isotropic random walk varies with time as 11 decreasing towards zero as the number of steps , or time , increases. The reduction of the average directionality with time in inverse proportion to the square root of time elapsed since the start of the migration is a diagnostic property of an isotropic random walk. Note that Equation (1) gives the averaged directionality; the displacement and directionality for a single walker may deviate significantly due to the probabilistic nature of the walk.
To further confirm that the cell migration for unstained cells is consistent, on average, with the theory of isotropic random walk, we consider the average directionality (averaged over all 26 single hESCs) versus time, shown in Figure S1a. Up to around 7 hours there is a systematic decrease in the averaged directionality from unity to low values, in qualitative agreement with the random walk behaviour. To ascertain the functional form of this decay, the data is plotted on log-log axes (inset of Figure S1a). The prominent straight-line behaviour during this time indicates that the directionality decays as a powerlaw with time, and a straight-line least-squares fit (not constrained to go through any particular point) gives ̅ ( ) = (0.50 ± 0.02) −0.44±0.04 . The scaling with time is close to the −1/2 dependence characteristic of the isotropic random walk, Equation (1). Beyond 7 hours, the evolution of the average directionality changes its character and deviates from the 1/√ random walk behaviour; a similar deviation was noted in the plot of mean-square displacement versus time in Figure 3.
The averaged directionality in the presence of Cell Tracer, shown in Figure S1b, also indicates the systematic decrease over time, characteristic of an isotropic random walk. The least-squares fit is ̅ ( ) = (0.61 ± 0.05) −0.50±0.04 , which is also close to the −1/2 scaling characteristic of the diffusive motion.
The directionality of the unstained pair centroid motion, shown in Figure S2a, confirms that the pair as a whole can be described as an unbiased random walk at a good level of accuracy over the observation time range. The least-squares fit for unstained cells ̅ ( ) = (0.49 ± 0.02) −0.42±0.02 . The pair centroid motion of Cell Tracer stained cells is also consistent with a random walk: the least-squares fit is ̅ ( ) = (0.54 ± 0.06) −0.56±0.06 , shown in Figure S2b.

Supplementary Figure S1
Supplementary Figure

Supplementary Figure S3
Supplementary Figure S3: Histograms of division times, with bin widths of 5 hours in each case, for the single hESCs in the absence (blue) and presence of Cell Tracer (cross hatched in red). The Kolmogorov-Smirnov two-sample test confirms that the two distributions are distinct, suggesting that the Cell Tracer treatment affects significantly the ability of the cells to divide. The Mann-Whitney U test also confirms the two distributions are distinct ( < 0.05).

Supplementary Figure S4
Supplementary Figure S4: The scatter plot of the step lengths and at each time frame (every 15 minutes) for cells (a) without Cell Tracer and (b) with Cell Tracer. Together with the low crosscorrelation coefficient between the two variables discussed in the main text, the lack of any pronounced correlation between and [except perhaps the rare events with large values of in Panel (a)] suggests the isotropy of the random walk. According to the Kolmogorov-Smirnov and Mann-Whitney U tests, there is no evidence to distinguish between the distributions of and . The Pearson productmoment correlation coefficient of and is as small as 0.22, confirming the steps in the and directions are uncorrelated.

Supplementary Figure S5
Supplementary Figure S5: (a) The speed of the pair centroid in the absence of Cell Tracer for the Type A (red) and Type B (blue) pairs: (i) the median speeds, with error bars representing the upper and lower quartiles, and (ii) the corresponding probability densities of the centroid speeds. Horizontal lines in (i) indicate the average across the entire category. (b): as in Panels (a) but for the relative speed within a pair. According to Kolmogorov-Smirnov and Mann-Whitney U tests the probability distributions for the Type A and Type B relative speeds are different.