Fig 1.
Diagram of the components of the model.
To fully capture complex biological process of cell motility one needs to couple membrane patterning by reaction–diffusion–advection network to forces which induce cytosolic and membrane flows as well as facilitate shape change and motility in general. The coupling loop is closed by allowing the generated flows and geometry of the cell to influence reaction–diffusion–advection system.
Fig 2.
Schematic representation of the reactions of the model.
Arrows indicate the direction of the reaction with governing constants for that reaction annotated beside them. Cytosolic substrate binds to the membrane becoming the activator. Membrane bound activator can form a complex with the cytosolic inhibitor. The cycle ends when the complex dissociates, releasing substrate and inhibitor back into the cytosol.
Fig 3.
Flow induced by protrusive coupling.
Cytosolic and cortex flows, in the centre-of-mass reference frame of the cell, during movement under the influence of only protrusive coupling for and
. The black arrow marks the polarity direction (and, consequently, the direction of motion as well as the position of the patch of activator). Red lines represent streamlines inside the cytosol. The colour scale for both plots is shown in the middle. Left panel shows cytosolic flow and right panel presents cortex flow induced by protrusive coupling.
Fig 4.
Constrictive and dispersive flow.
The flow induced in the cell without protrusion force () (A) for a positive surface tension coupling
, and (B) for a negative coupling
. In both panels the patch of activator is located on the right side of cell and the snapshot has been taken in the initial phase of the simulation, where the effect of Marangoni flow is the best visible. In both panels the flow field is presented with arrows and streamlines are denoted with red lines. The magnitude of velocity is also shown with the colour code as presented on the scale bar in the middle of panels. Top and bottom graphs show the cytosolic and the cortex flows, respectively. (C) The effect of Marangoni flow on the profile of activator concentration along the cortex. Purple arrows refer to positive surface tension coupling while orange ones represent negative surface tension coupling. (D) Mean period of the rotation or oscillation RD dynamics as a function of surface tension coupling. Colour marks the type of initial concentration profile for each simulation. Markers represent the observed limit cycle RD pattern obtained after 1300 s of simulation time. Mean period was averaged over 900˗1300 of simulation time. Higher values of surface tension coupling speed up the RD dynamics and favour oscillation pattern. After the threshold of
is passed, limit cycle solution of simulations initiated with rotation pattern becomes oscillation. We note that the snapshots present the flow in early stages of simulation, where the flow have not managed to deform the shape of cell significantly.
Fig 5.
Characteristic motility modes and shape phenotypes.
Colour codes denote the concentration of activator in the cortex (as shown by scale bars). Trajectory of the cell geometric centre is plotted with red colour for (A) stationary cells, (B) persistent runner cells, (C) alternating runner cells and the dependence of their time evolution of elongation and polarity. Vertical black lines denote moments in which the plotted snapshots were taken. (D) Circular runner cell’s trajectories and the associated velocity autocorrelation functions. (E) Run-and-turn cell trajectory is plotted separately on the right side and four circles mark the points in which snapshots plotted on the left side were taken. Videos showing the time evolution of presented cells, are provided in the Supporting information Section. The parameters for these simulations are given in Table D in S1 Appendix.
Fig 6.
Phase diagrams for limit cycle motility phenotypes as a function of coupling strength (A) for cells initiated with a rotational concentration profile and (B) for cells initiated with an oscillatory concentration profile. Colour represents the motility class as shown above the panels, while marker represents characteristic shape of the cell. (C) Legend specifying qualitatively the characteristic shapes and patterns observed in the simulations. Quantitative analysis of simulated cells is presented in Fig 7.
Fig 7.
Cell characteristics as a function of cytoplasmic and cortical flow coupling constants and
.
The parameters range is the same as for the phase diagram graphs in Fig 6. Red lines denote borders of regions with different motility patterns. Left panels present data for cells initiated with a rotational concentration profile, while on the right cells were initiated with an oscillatory concentration profile. The magnitude for each quantity is denoted with the colour code, as presented in the colour bar above each diagram. (A) The average speed of cell calculated for simulation time 900–1000s. (B) Characteristic length of motion. The length is defined as the circumference of the trajectory for circular runners, length of a segment between two turning points for alternating runners and run-and-turn cells. It is set to zero for stationary cells and infinite for persistent runners. (C) Activator concentration at its peak in the membrane pattern. (D) Width of the pattern of activator concentration relative to the cell circumference size. Properties of the membrane patterns shown in C and D were extracted 1 000 s after the onset of the simulation, or at the closest maximum for the oscillatory or run and turn pattern.