How cellular movement determines the collective force generated by the Dictyostelium discoideum slug

Abstract

How the collective motion of cells in a biological tissue originates in the behavior of a collection of individuals, each of which responds to the chemical and mechanical signals it receives from neighbors, is still poorly understood. Here we study this question for a particular system, the slug stage of the cellular slime mold Dictyostelium discoideum (Dd). We investigate how cells in the interior of a migrating slug can effectively transmit stress to the substrate and thereby contribute to the overall motive force. Theoretical analysis suggests necessary conditions on the behavior of individual cells, and computational results shed light on experimental results concerning the total force exerted by a migrating slug. The model predicts that only cells in contact with the substrate contribute to the translational motion of the slug. Since the model is not based specifically on the mechanical properties of Dd cells, the results suggest that this behavior will be found in many developing systems.

Fig. 1.

The figure shows a schematic of a cell and the nomenclature used in the simulations. The labeled vectors a, b, and c indicate the major axes of the ellipsoid. The cell is oriented in the direction of vector a which corresponds with the direction of motion. Each cross-sectional plane ab, ac or bc separates the cell into a plus and minus sides with respect to the normal of the plane. For example the bc plane shown separates the cell into a plus and minus with respect to the a direction denoted a+ and a−, respectively.

Fig. 2.

A schematic of the element of a standard solid, or Kelvin element along each axis of the ellipsoid. It consists of a nonlinear spring in parallel with a Maxwell element, which comprises a linear spring in series with a dashpot.

Fig. 3.

The results of a typical simulation are shown at three different times. At time t = 0 the slug starts out 10 cells long with a square cross-section four cells on a side. A unit on the scale indicated corresponds to 10 μm. The only cells that move are those being simulated by a traveling wave, which is indicated in the foreground by the squares. The wave has a velocity of 30 μm/min, a spatial period of 100 μm and a width of 20 μm in which the cAMP is non-zero. The spatial scale of the stimulus wave is smaller than in a natural wave to accommodate the small size of the slug.

Fig. 4.

The average motive force as a function of the number of the cells in contact with the substrate (a), and as function of the total number of cells in the slug (b). The boxes indicate data from the simulations, the straight lines are linear least-square fits to the data, and the curve in (b) is the least squares fit to a square root function. In all the simulations the slug starts out 10 cells long with square cross-sections that vary from 1 to 15 cells on a side. The cells are stimulated to move by a cAMP wave as described in the text. The motive force is the average motive force for the last 8 min of a 10 min simulation, done with a time-step of 0.002 min.

Fig. 5.

The average motive force as a function of the number of cells in contact with the substrate (a), and as a function of the number of cells in the slug (b), when all cells are moving simultaneously. The figure is plotted in the same manner as Fig. 4 and the simulations use the same parameters. A table of the data is given in Appendix A.

Fig. 6.

The average motive force is plotted vs. the number of cells in contact with the substrate in (a) and vs. the total number of cells in the slug in (b) for simulations in which a portion of the cells are frozen. The data does not follow the square root curve because the length of the slugs during the course of a simulation changes dramatically in some cases. The data is plotted as in Fig. 5 and the simulation parameters are the same. The only difference is that at any time 25% of the cells are frozen for a duration of 6 s and the time step varied between 0.002 and 0.0005 min, depending on the simulation.

Fig. 7.

Data from 250 different simulations are plotted with the vertical axis as the translational force of the slug in nanoNewtons and the horizontal axis as the number of cells in contact with the surface (a), or the total number of cells in the slug (b). Five lines are clear in (b): these correspond to subsets of the data in which the initial height of the slug is held constant, which implies that the volume of the slug and the surface area in contact with the substrate are proportional. All simulations started with a 3D array of cells whose dimensions ranged from 1 to 5 in both dimensions transverse to the length of the slug and 1 to 10 in length. The vertical axis is the average translational force for the last 8 min of a 10-min simulation, done with a time step of 0.002 min. In all the simulations, cells are stimulated to move one direction by a traveling wave. Because the cells are not constantly moving the time averaged force for one cell should be about 8 nN which is consistent with the data shown.

Fig. 8.

The width of slugs is plotted against the length of the slug. The line is the least squares fit to the data, not including the two longest slug data points. The correlation coefficient is 0.93 when these two data points are excluded, and thus a linear relation is very likely (data is taken from Inouye and Takeuchi, 1979).

Fig. 9.

This figure plots a subset of the data from Fig. 7 to illustrate that for some slug dimensions, the data can appear to have a linear relationship, only when a larger data set is considered can the true relationship be clearly seen (Fig. 7). Twenty five of the data points in Fig. 7 are replotted where the transverse cross-section of the slug is a square and the length of the slug ranges from 5 to 10 cells. The plotted line is the least squares fit to a line of the data and has a correlation coefficient of 0.97.

Fig. 10.

The three sketches above show the vectors h_j,i, d_j,i, d_i,j and s_j,i.

Fig. 11.

The figure shows two cells whose membranes are overlapping. The dashed lines represent the ellipsoidal shape of the cells as represented in the model. The solid lines more accurately represent how the cells would locally deform. The cell with the pressure which is smallest in magnitude will deform the most.

Fig. 12.

The creep function for a standard linear solid and the applied force are plotted in (a) and the relaxation function is plotted in (b) with parameters k₁ = 163.8 dyn/cm, k₂ = 147.5 dyn/cm, and μ_a = 123 dyn min/cm. The thin black line in (a) is a scaled plot of the applied force which starts at time 1 with a magnitude of 400 nN and ends at time 12. The thicker gray line in plot (a) is the change in the length of the element (the diameter of the cell). In plot (b) the length of the element (diameter of the cell) at time 1 is fixed at 20 μm (twice its resting length). The graph shows the force with which the element resists the deformation.