A three-dimensional model of myxobacterial aggregation by contact-mediated interactions

Abstract

Myxobacteria provide one of the simplest models of cell-cell interaction and organized cell movement leading to cellular differentiation. When starved, tens of thousands of cells change their movement pattern from outward spreading to inward concentration; they form aggregates that become fruiting bodies. Cells inside fruiting bodies differentiate into round, nonmotile, environmentally resistant spores. Traditionally, cell aggregation has been considered to imply chemotaxis; a long-range cell interaction. However, myxobacterial aggregation is the consequence of direct cell-contact interactions, not chemotaxis. We present here a 3D stochastic lattice-gas cellular automata model of cell aggregation based on local cell-cell contact, and no chemotaxis. We demonstrate that a 3D discrete stochastic model can simulate two stages of cell aggregation. First, a "traffic jam" forms embedded in a field of motile cells. The jam then becomes an aggregation center that accumulates more cells. We show that, at high cell density, cells stream around the traffic jam, generating a 3D hemispherical mound. Later, when the nuclear traffic jam dissolves, the aggregation center becomes a 3D ring of streaming cells.

Fig. 1.

Photographs (taken with a ×16 phase contrast objective lens) of M. xanthus aggregates at 8, 11, and 24 h for the first three frames. The last frame shows an electron micrograph of an early aggregate, a traffic jam. [Reproduced with permission from refs. and (Copyright 1982 and 2004, American Society for Microbiology).]

Fig. 2.

A single myxobacterial cell body (dark dots) with its poles (light dots) as arrays of pixels on a 3D hexagonal lattice contained in an ellipsoid of particular size. The surface is grayed to indicate the cell surface.

Fig. 3.

Traffic jam formation (2D projections of 3D simulations) starting from random initial distribution of cells on the bottom level of the domain 100 × 100 × 3 with the cell density 60% at 0 (a), 4 (b), 15 (c), 32 (d), and 1,370 (e) time steps. (f) An enlarged view of a small section from e, showing the details of cell arrangements. The cells are not drawn to scale but are represented by unit vectors passing through the cell centers. Shades of gray correspond to cell orientations. “S” in f indicates a stream. Cells in streams move parallel to each other in both directions.

Fig. 4.

Simulations of aggregate growth starting from “frozen” jam (a). Some cells start to circulate around jam in both clockwise and counterclockwise directions, forming a motile “skirt” (b), and some of them glide over the jam, where some part of gliding cells is getting “stuck” (c), resulting in a bell-shaped aggregate formation [d (side view)]. (e) Experimental photo of M. xanthus fruiting bodies in a side view (×10 magnification). The cells in a–c are represented by unit vectors passing through the cell center rather than by their real shape and dimensions to visually distinguish one cell from another.

Fig. 5.

Evolution of a 3D bell-shaped aggregate (a) after “unfreezing” bottom jammed cells, into motile 2D toroid-like structure (b).

Fig. 6.

Final configuration of a 3D aggregate as it reaches a hemispherical form by 2,000 time steps. (a) Top view. (b) Side view. Further development does not change the aggregate form, despite of sufficient amount of cells available for further growth. (c) Top view, 4,000 time steps. (d) Side view, 4,000 time steps. The initial cell density is 26%. The rate of cell population increase is 0.19 cells per time step.

Fig. 7.

The average time it takes an aggregate to reach a final hemispherical form versus the rate of cell population increase for the initial cell density of 45%. The graph is in log–log scale. The linear function that fits the data points in a least squares sense is ln(y) = –0.9328 ln(x) + 5.8463.

Fig. 8.

Pattern formation with different cell aspect ratio (cell length/cell diameter) after 200 time steps. (a) Round cells. (b) Cells with aspect ratios of 19:2.4.