Qualitative assessment of trajectory data
Cell trajectory data, describing the motion of 80 randomly-chosen 3T3 fibroblast cells [34] (Fig. 1
b) under a range of gradients, EF=0,100,200 and 400 mV/mm, within the experimental apparatus (Fig. 1
a, c-d) are analysed [35]. Since 3T3 fibroblast cells are known to migrate towards the cathode in these types of experiments [35], the Cartesian coordinate axes are aligned so that the positive x-direction points towards the cathode (Fig. 1
c-d). We note that there is no gradient in the y-direction (Fig. 1
c-d).
The data involves recording the initial position of each trajectory, (x
′(0),y
′(0)) and the position of each cell every half-hour over a two hour interval, giving: (x
′(0.5),y
′(0.5)), (x
′(1),y
′(1)), (x
′(1.5),y
′(1.5)) and (x
′(2),y
′(2)). Using this data, we shift the coordinate system for each trajectory so that the initial location of the cell is at the origin, giving (x(t),y(t))=(x
′(t)−x
′(0),y
′(t)−y
′(0)). Plots showing (x(2),y(2)) for 80 trajectories under four different gradients are shown in Fig. 2. The scatter plot in Fig. 2
a, under the action of no gradient, shows an approximately symmetric distribution of the end points of the trajectories. In this case the trajectories extend no further than approximately 40 μm away from the origin. Since these trajectories appear to follow no particular preferred direction, this cells seem to undergo an unbiased migration process. In comparison, the scatter plot in Fig. 2
b shows that there is some drift in the positive x-direction when the cells move under the action of a gradient. Despite the fact that there is an obvious drift in the positive x-direction in Fig. 2
b, there remains some randomness in the distribution of (x(2),y(2)). Therefore, under the action of the electric field, these 3T3 fibroblast cells move with both a directed and an undirected component. Comparing results in Fig. 2
b-d confirms that the drift in the positive x-direction increases with the increasing electric field, and there appears to be some randomness in the distribution of cells regardless of the strength of the electric field. To provide more information about the roles of directed and undirected motion in these experiments, we will now interpret this data using a biased random walk model that is related to an advection-diffusion equation.
Quantitative assessment of trajectory data
We first quantify the directed component of the motility depicted in Fig. 2. Estimates of the drift velocity are obtained, in both the x and y directions, for each of the 80 trajectories, under the four different gradient conditions. These data are presented as histograms in Fig. 3. Results in Fig. 3
a-b characterize the estimates of v
x
and v
y
when there is no gradient, and averaging these 80 estimates gives us an approximation of the average drift velocity in each direction. This gives 〈v
x
〉=−1μm/h and 〈v
y
〉=−1μm/h. Therefore, the average drift velocity in both directions is approximately zero, as we anticipate intuitively by inspecting the data in Fig. 2
a. Results in Fig. 3
c-h show estimates of v
x
and v
y
for EF=100,200 and 400 mV/mm, respectively. In each case we see that 〈v
y
〉≈0μm/h, which is consistent with the experimental design since there is no gradient in the y direction (Fig. 1
c-d). In contrast, estimates of 〈v
x
〉 increase with EF, as we have 〈v
x
〉=−1,9,14 and 25 μm/h when EF=0,100,200 and 400 mV/mm, respectively. In addition to characterizing the mean drift velocities, 〈v
x
〉 and 〈v
y
〉, the data in the histograms in Fig. 3
a-h show how the individual estimates of v
x
and v
y
are distributed for each of the 80 trajectories considered. A qualitative assessment of these distributions indicates that, for each value of the EF, estimates of v
x
and v
y
are approximately symmetrically distributed about the mean. Furthermore, the spread about the mean appears to be approximately constant for each value of the EF.
Given our estimates of 〈v
x
〉 and 〈v
y
〉 (Fig. 3
a-h), we now estimate the diffusivity coefficients, D
x
and D
y
, for each experiment. Results showing estimates of D
x
and D
y
under the application of no gradient are summarised in Fig. 3
i-j. Averaging our estimates across the 80 trajectories we obtain 〈D
x
〉=59 μm2/h and 〈D
y
〉=50μm2/h for the experiments in which there is no gradient. The magnitude of these estimates of cell diffusivity are consistent with previous estimates 3T3 fibroblast cells obtained using single cell trajectory data [36, 37]. Additional estimates of D
x
and D
y
, and 〈D
x
〉 and 〈D
y
〉 are shown in Fig. 3
k-p for cell migration under the influence of gradients of 100, 200 and 400 mV/mm, respectively. For each of these data sets we have 〈D
x
〉≈〈D
y
〉, indicating that the random motility coefficient is isotropic. Furthermore, unlike our estimates of 〈v
x
〉, our estimates of 〈D
x
〉 and 〈D
y
〉 appear not to depend on the electric field.
Relationship between the applied gradient, cell diffusivities and drift velocities
To further explore the relationships between D
x
, D
y
, v
x
, v
y
and the applied gradient, we calculate the sample mean and sample standard deviation for each of the 16 histograms in Fig. 3. Results in Fig. 4 show 〈v
x
〉, 〈v
y
〉, 〈D
x
〉 and 〈D
y
〉, each plotted as a function of the electric field. The plots show the variation in the average transport coefficients with the EF. In addition, the variability in the estimates of the average transport coefficients is indicated by the error bars. The error bars indicate the sample mean plus or minus one sample standard deviation.
Results in Fig. 4
a-b show 〈v
x
〉 and 〈v
y
〉 as a function of the EF. As we anticipate, 〈v
x
〉 increases with EF whereas 〈v
y
〉≈0 for all EF considered. To examine the putative relationship between 〈v
x
〉 and EF, and between 〈v
y
〉 and EF, we perform an unconstrained linear regression. The coefficient of determination for the 〈v
x
〉 data is very high, r
2=0.98, suggesting that the linear relationship between 〈v
x
〉 and EF provides a good explanation of the variability. In contrast, the coefficient of determination for 〈v
y
〉 is very low, r
2=0.00, suggesting that the null hypothesis is valid and there is no relationship between 〈v
y
〉 and EF. In summary, these results imply that a linear relationship between 〈v
x
〉 and EF is consistent with the observed data. To match the drift term in Eq. (1) with the advection-diffusion (Eq. (6)) we require that v
x
=χ(S)∂
S/∂
x. Since our data is consistent with a linear relationship between v
x
and the applied gradient, ∂
S/∂
x, it appears that a constant tactic sensitivity function, χ(S)=χ, provides the simplest explanation of our experimental results.
Results in Fig. 4
c-d show 〈D
x
〉 and 〈D
y
〉 as a function of EF. Visually, we see no discernible trend in the data for different values of EF. This visual interpretation is consistent with the fact that we obtain a small coefficient for each of the linear regressions in Fig. 4
c-d. Therefore, it is reasonable to assume that the cell diffusivities appear to be independent of the electric field. If we accept this assumption and further average the data in Fig. 3
i-p in each direction we obtain overall estimates of 〈D
x
〉=48μm2/h and 〈D
y
〉=46μm2/h. Again, this suggests that the diffusion of 3T3 fibroblast cells is approximately isotropic since we have D
x
≈D
y
, across all the experimental conditions considered.
Now that we have summarised the estimates of the directed and undirected components of cell migration in the experiments, we can quantify the relative roles in terms of the dimensionless Peclet number [38],
$$ Pe = \frac{v L}{D}, $$
(3)
where v is the drift velocity, D is the diffusivity and L is a relevant length-scale, which here we will take to be the cell diameter of fibroblast cells, L≈25μm [37]. The Peclet number is a measure of the time scale of advection to the time scale of diffusion [38]. When Pe≪1, undirected diffusive transport dominates, when Pe≫1, directed transport dominates, and when Pe≈1 to two mechanisms are in balance. Comparing estimates of the drift velocity and the diffusivity in the x-direction suggests that our experiments deal with a range of Peclet numbers from Pe≈0 when EF=0 mV/mm to Pe≈10 when EF=400 mV/mm. Therefore, our experimental data covers a wide range of transport conditions ranging from purely undirected, diffusive transport to highly directed, advection-dominant conditions.
To summarise our findings, results in Fig. 4 suggest that 〈v
x
〉 increases linearly with EF, whereas the data suggests that the other transport coefficients, 〈v
y
〉, 〈D
x
〉 and 〈D
y
〉, appear to be independent of EF. Guided by these results, we assume that 〈v
x
〉 increases linearly with EF, and that the other transport coefficients are independent of EF. Comparing the results in Fig. 1
c and d also allows us to also consider whether there is any possible relationship between the transport coefficients and the electric potential. Repeating the process of plotting our estimates of the four transport coefficients as a function of the electric potential (not shown) suggests that there is no obvious trends in the data. Furthermore, linear regressions between each transport coefficient and the associated value of the electric potential reveals a low coefficient of determination, r
2<1. Therefore, based on the data, we assume that the transport coefficients appear to be independent of the electric potential in these experiments.