To justify the complexity of the Models A and B, let us first consider a simpler model. Consider a discrete-time model of random motion where cells are allowed to change direction to a uniformly chosen random variable on [0, 2π) with probability 0.2 for each time step; we refer to this as Model R. We compare this random motion Model R with a the neighbor-dependent random model, Model B, where, for each time step with probability 0.2, cells are allowed to change direction to an angle chosen uniformly from [0, 2π), but only if there is a neighboring cell within 5 pixels distance. In , we compare the random model, shown in , to Model B, where a directional change in movement depends on the presence of a neighbor within D = 5 pixels. Clearly, Model B, which is a rather simple neighbor-dependent model, yields results that better match the experimental data, compared with the random model. This is manifested in the cells in that tend to align along the boundary (similar to what is experimentally observed).
Figure 5 (a)–(c) Random motion model (Model R) and (d)–(f) Random motion with neighbor dependence model (Model B) on changing direction for N = 250 cells at times t=0, 350, and 1000 time steps. The red circle around the cell in the upper left hand (more ...)
We quantify this behavior in . To do this, we partitioned the 100 pixel radius circle into 169 equal area sections. The sections are labeled in . For each model, we then identified the numbered partition for each cell in space and times from start to 1000 time steps. This data was then compiled into a histogram counting how many cells fall into each section for each model. In this way, each cell is counted 1001 times. In , we show a histogram of the cell locations for the random motion Model R. We observe that the cells are relatively uniformly distributed in space-time. In contrast, in , we show a histogram of the cell locations over each time step for the neighbor-dependent random motion, Model B. In this case, we observe that cells are gathering on the boundary, partitions 122 through 169, with approximately twice the probability of them falling in the interior of the circle, partitions 1 through 121. This ratio is likely parameter dependent. Qualitatively, this agrees with the experimental observations.
Figure 6 (a) Partitioning the domain into 169 equal area sections. The numbers represent partition number labels. (b) A histogram of the number of cells in each partition over time for the random motion model (Model R). (c) A histogram of the number of cells in (more ...)
While this analysis helps to further understand the dot patterns presented in , we do not perform such a study on all of the simulations. The example we provide does illustrate the difference between random motion and the patterns of the motions we study. This pattern of motion has not been previously studied.
We now consider a few simulations comparing Model A, the local-interactions model, from section 3.2, to Model B, the neighbor-dependent random movement model, from section 3.3.
In , we see a comparison of Models A and B in which cells are constrained to a circle with a diameter of 200. We compare the effect of varying interaction distance and consider the cases of D = 5 and D = 10. The length scale of interaction distance D is shown in the upper left corner of each picture around a stationary cell. This cell, contained in a circle of radius D, is not included in any of the simulations. Each simulation is run with N = 250 cells and can be seen at t = 0, 50, and 350 increments. The initial conditions, seen in 7(a), 7(d), 7(g), and 7(j), are identical in all simulations. We observe that for the local interaction model in 7(c) and the random neighbor-dependent motion model in 7(i), the same cells tend to stay around the border, similar to what is seen in experiments. We recall that there is no mechanism that is embedded into the model that forces cells to stay on the boundary of the domain once the boundary is reached. Indeed, occasionally, individual particles return back into the domain. The cells appear to be slightly more uniformly distributed in 7(i) than in 7(c). In 7(f), Model A with D = 10 yields cells aggregating in small clumps of 5 to 15 cells. The simulated cells in appear to be distributed uniformly with little cohesion to the boundary. In fact, the motion observed in the simulation depicted in 7(j)–(l) is very similar to the random motion model from because the chosen surface density and interaction distance are such that, on average, each cell has 1.5 neighbors, effectively removing the neighbor-dependence from this model.
Figure 7 (a)–(f) Model A; (g)–(l) Model B. N = 250 particles are constrained to a circle of diameter 200. Shown are simulation results at times t = 0, 50, 350. The interaction distance is D = 5 for (a)–(c), (g)–(i) and D = 10 for (more ...)
In , we offer a quantitative comparison of Models A and B for D = 5. We partition the domain into 169 equal-area sections and count the total number of particles that are present in each cell over 1000 time steps. We observe that for Model A, in , the boundary, partitions 122 through 169, has more cells on it with larger variability than for Model B (shown in ). We suspect that this is due to the formation of aggregations on the boundary. As previously stated, cells have been experimentally observed to collect along the boundary.
The boundary conditions in simulate a setup that is similar to the experimental setup focusing on short-time dynamics of cells. In experiments lasting relatively short periods of time (on the order of hours), cells are confined to a given area due to the properties of the underlying surface. However, in experiments lasting longer periods of time (on the order of days), cells overcome the underlying surface and, in the absence of a directional global light source, spread out on the surface of the agarose. In contrast, we consider large-time simulations of cells that are free to move anywhere in 2
. Such simulations are shown in . The initial conditions are identical to the initial circular drop of cells seen in . The local-interactions model, Model A, is shown in with D
= 5 and with D
= 10. The random neighbor-dependent model, Model B, is shown in with D
= 5 and with D
= 10. All results are shown at t
= 1000. Clearly, in , the majority of cells have dispersed. stands different. In this case, of Model A, with the longer interaction distance, few aggregations still remain after these rather long-time simulations, in spite of the particles not being confined to the domain that is shown in the figure.
Figure 8 (a), (b) Model A; (c), (d) Model B. Both models are simulated in 2 at t = 1000 for N = 250 particles. The interaction distance is D = 5 for (a), (c) and D = 10 for (b), (d). The initial conditions and parameters a, b are identical to those used (more ...)
To complement the previous cases, we also consider periodic boundary conditions. In , we show results obtained from simulations of cells on a 240 × 240 region with periodic boundary conditions. The same initial conditions as in are used. The results are shown at time t = 3000. Model A is shown in with D = 5 and in with D = 10. Model B is shown in with D = 5 and in with D = 10. In these snapshots, aggregations are visible in for D = 10 in Model A. There appears to be slightly more white space in than in . In order to quantitatively compare this clumping effect seen in , we show a histogram of how many neighbors are found within a distance of 5 or 10 pixels of each cell in . For Model A with D = 5, we observe in that within 5 pixels of each cell, there are more cells with two or more neighbors than in Model B shown in . In comparing the same histograms, there are also slightly more cells with no neighbors for Model B than for Model A. Within 10 pixels, the phenomenon is slightly reversed; that is, within 10 pixels, in there are more cells with no neighbors for Model A than for Model B, in . This trend reversal implies that the aggregates seen in simulations of Model A may be tighter, as there are more isolated cells, meaning the other cells must be packed into a slightly smaller space.
(a), (b) Model A; (c), (d) Model B. Periodic boundary conditions; t = 1000; N = 250 particles. The interaction distance is D = 5 for (a), (c) and D = 10 for (b), (d). The initial conditions and parameters a, b are identical to those in .
Figure 10 Histogram illustrating how many cells have 0 to 10 neighbors within 5 or 10 pixels. Subfigures (a), (b) correspond to the simulation of Model A with D = 5 in . Subfigures (c), (d) correspond to the simulation of Model B with D = 5 in (more ...)
In , the positions of five cells, with identical initial conditions and the same model and interaction distance conditions as in , are illustrated for t = 1000. These trajectories are plotted on top of the final locations of the other 245 cells at time t = 1000. Note that the distance travelled by the particles is longer in both models when the local interaction distance is smaller.
Figure 11 The same five cells traced over 1000 time steps for (a), (b) Model A, and (c), (d) Model B. Periodic boundary conditions; t = 1000; N = 250 particles. The interaction distance is D = 5 for (a), (c) and D = 10 for (b), (d). The initial conditions and parameters (more ...)
We further explore properties of Model A by considering multiple runs with the same initial conditions and setup. In , we consider six different runs, all with an interaction distance D = 10 pixels and N = 250 cells at t = 1000 time steps. Note that in comparing the six images the resulting distribution of cells and the size and number of aggregations are similar, but there do not appear to be any stable aggregations which routinely form in the same location. In fact, the aggregations which form move over time as well as dissociate and form new aggregations.
Figure 12 (a)–(f) Six realizations of Model A with the same initial conditions and parameters a and b, using D = 10 and N = 250 cells at times t = 1000 time steps. While precise locations of aggregations vary, the aggregations sizes and number are similar. (more ...)
Varying the interaction distance D
, we can gain further insight about the properties of this model. In , we consider three different interaction distances D
= 10, 20, and 40 at t
= 1000 time steps with the same initial conditions but different boundary conditions. In , cells are constrained to a circle, whereas in , cells are simply restricted to 2
(note that in these cases not all cells are necessarily visible in the viewing window). In comparing the results for D
= 10 and D
= 20, we observe that the distribution and number and size of aggregations of cells is very similar. For D
= 40, even the locations of the aggregations is almost identical. This is typical for simulations where D
= 40; however, it is also possible for three aggregations of cells to form. The positions of the four aggregations and the distances between aggregations are generally slightly different. In both cases, all cells for D
= 40 typically end up in an aggregation.
Figure 13 A study on the effect of interaction distance D. We consider (a), (d) D = 10, (b), (e) D = 20, and (c), (f) D = 40 with the same initial conditions and same parameter a, b and N = 250 cells at time t=1000. For (a)–(c), the cells are constrained (more ...)