Distinguishing cell shoving mechanisms

Abstract

Motivated by in vitro time-lapse images of ovarian cancer spheroids inducing mesothelial cell clearance, the traditional agent-based model of cell migration, based on simple volume exclusion, was extended to include the possibility that a cell seeking to move into an occupied location may push the resident cell, and any cells neighbouring it, out of the way to occupy that location. In traditional discrete models of motile cells with volume exclusion such a move would be aborted. We introduce a new shoving mechanism which allows cells to choose the direction to shove cells that expends the least amount of shoving effort (to account for the likely resistance of cells to being pushed). We call this motility rule 'smart shoving'. We examine whether agent-based simulations of different shoving mechanisms can be distinguished on the basis of single realisations and averages over many realisations. We emphasise the difficulty in distinguishing cell mechanisms from cellular automata simulations based on snap-shots of cell distributions, site-occupancy averages and the evolution of the number of cells of each species averaged over many realisations. This difficulty suggests the need for higher resolution cell tracking.

Fig 1. A schematic diagram which distinguishes between simple and smart shoving agents.

Occupied locations are coloured either red or green. Agents which may shove obstructing agents are displayed in red, while the occupied sites which obstruct the possible movements of the shoving cells are displayed in green. a) A simple shoving agent has a choice of four locations to move to, indicated by arrows. It chooses one of these directions at random. b) If the simple shoving agent at location (i, j) chooses to move to location (i + 1, j), it moves to this location, pushing the obstructing cell at (i + 1, j) to location (i + 2, j). c) A smart shoving agent is located at (i, j). d) The smart shoving agent at (i, j) chooses to move to the location that expends the least amount of shoving effort. It will move from (i, j) to (i − 1, j) and will push the obstructing agent at (i − 1, j) to location (i − 2, j). Moving in any other direction would expend more shoving effort than the chosen direction (i.e. it would require pushing two obstructing cells rather than one).

Fig 2. Schematic of the three invasion regions considered.

Invasion regions are displayed in red, whereas a background of resident cells is displayed in green. When considering invasion of red cells into an empty domain, the uniform concentration of the green cells is c_G = 0. When considering invasion of red cells into a non-empty domain, the uniform concentration of the green cells is taken to be c_G = 0.6, unless otherwise stated. The domain has dimensions L_x × L_y = 200 × 20. In a) the red cells enter the lattice from a 2D band in the middle region of the lattice defined by coordinates x ∈ [90, 109] and y ∈ [1, 20]. In b) the red cells enter the lattice from the horizontal midline of the lattice defined by coordinates x ∈ [90, 109] and y = 10. In c) the red cells enter the lattice from upper horizontal boundary of the lattice defined by coordinates x ∈ [90, 109] and y = 20. Homogenous vertical and reflecting horizontal boundary conditions are implemented in all simulations of the agent–based model. In all simulations, invasion of red cells occurs at a rate of 4 attempts per time-step.

Fig 3. Single realisations of the agent–based model displaying snap–shots of cell distributions from invasion of an empty domain from three separate locations (displayed in Fig 2).

Red cells enter the domain at a rate of four attempts per time-step. Simulations are displayed at times t = 50, 250. The first column denotes gentlemen invaders, the middle column denotes simple shover invaders and the third column denotes smart shover invaders.

Fig 4. Site occupancy averages from invasion of an empty domain from three separate locations (displayed in Fig 2).

Red cells enter the domain at a rate of four attempts per time-step. Site occupancies obtained from the agent–based model are averaged over 300 independent realisations at times t = 50, 100, 250. The continuous red curve denotes simple shovers, a broken red curve denotes smart shovers and a continuous blue curve denotes invading gentlemen cells. Arrows denote the direction of cell movement as time increases.

Fig 5. Single realisations of the agent–based model describing invasion of an occupied domain from three separate locations (displayed in Fig 2).

Green resident cells are gentlemen in each scenario (with initial uniform concentration c_G = 0.6). Red cells enter the domain at a rate of four attempts per time-step. Simulations are displayed at times t = 50, 250. The first column represents gentlemen invaders, the middle column represents simple shover invaders and the third column represents smart shover invaders.

Fig 6. Invasion of an occupied domain from the 2D band (displayed in Fig 2(a)).

Green gentlemen cells are present with density c_G = 0.6. Red cells enter the domain at a rate of four attempts per time-step. Site occupancies, obtained from the agent–based model, are averaged over 300 independent realisations at times t = 50, 100, 250. The continuous curves denotes simple shovers, and broken curves denotes smart shovers. Arrows show increasing time.

Fig 7. Invasion of an occupied domain from the horizontal midline (displayed in Fig 2(b)).

Green gentlemen cells are present with density c_G = 0.6. Red cells enter the domain at a rate of four attempts per time-step. The continuous curves denotes simple shovers, and broken curves denotes smart shovers. Site occupancies, obtained from the agent–based model, are averaged over 300 independent realisations at times t = 50, 100, 250. Arrows show increasing time.

Fig 8. Invasion of an occupied domain from the upper horizontal (displayed in Fig 2(c)).

Green gentlemen cells are present with density c_G = 0.6. Red cells enter the domain at a rate of four attempts per time-step. The continuous curves denotes simple shovers, and broken curves denotes smart shovers. Site occupancies, obtained from the agent–based model, are averaged over 300 independent realisations at times t = 50, 100, 250. Arrows show increasing time.

Fig 9. Cell counts, averaged over 300 realisations of the agent–based model, from invasion of an occupied domain (uniformly seeded green gentlemen cells with cell density c_G = 0.6).

Continuous lines denote the scenario where invaders are simple shovers, whereas broken lines denote the scenario where invaders are smart shovers. Red cells enter the domain at a rate of four attempts per time-step. The red lines denotes the number of red cells present, the green lines denote the number of green gentlemen cells present and the black lines denote the total number of cells present.

Fig 10. Averaged simulations of the agent–based model compared with the solutions from their mean-field continuum limit PDEs.

Red cells are simple shovers and green cells are gentlemen. Red cells enter the domain in the 2D band in the centre of the lattice, and invasion occurs at a rate of four attempts per time-step (which corresponds to an entry probability P_entry = 0.01). The initial concentration of green cells is c_G = 0.2. Blue curves are PDE solutions. Simulation results are given at t = 0, 50, 100, 250. Simulations are averaged over 300 realisations of the agent–based model. The red cells shoving the green cells leads to crowding of the green cells to the left and right of the invasion region. Site occupancy exceeds 0.65 for much of the lattice by time t = 250. Arrows show increasing time.