The role of mechanical forces in the planar-to-bulk transition in growing Escherichia coli microcolonies

Abstract

Mechanical forces are obviously important in the assembly of three-dimensional multicellular structures, but their detailed role is often unclear. We have used growing microcolonies of the bacterium Escherichia coli to investigate the role of mechanical forces in the transition from two-dimensional growth (on the interface between a hard surface and a soft agarose pad) to three-dimensional growth (invasion of the agarose). We measure the position within the colony where the invasion transition happens, the cell density within the colony and the colony size at the transition as functions of the concentration of the agarose. We use a phenomenological theory, combined with individual-based computer simulations, to show how mechanical forces acting between the bacterial cells, and between the bacteria and the surrounding matrix, lead to the complex phenomena observed in our experiments-in particular the observation that agarose concentration non-trivially affects the colony size at transition. Matching these approaches leads to a prediction for how the friction between the bacteria and the agarose should vary with agarose concentration. Our experimental conditions mimic numerous clinical and environmental scenarios in which bacteria invade soft matrices, as well as shedding more general light on the transition between two- and three-dimensional growth in multicellular assemblies.

Keywords: bacterial biofilm; bacterial microcolony; cell growth; mechanics.

Figure 1.

Experimental set-up. A thin slab of LB–agarose is confined between two microscope slides. The agarose contains a small number of polystyrene beads which act as spacers and ensure that the thickness of the agarose slab is constant at 500 μm. The bacteria are pipetted onto the top of the agarose before it is covered with the upper glass slide, and silicone grease is applied around the agarose to seal the sample. Zoomed-in region: illustration of the forces acting on the colony in our simulations. The microcolony is modelled as a flat disc of fixed height d and variable radius R, which is compressed vertically (force per unit colony area f_z(r)) and radially (force per unit boundary area f_r,edge) by the agarose, and radially by friction between the cells and the agarose (force per unit colony area f_r(r)). (Online version in colour.)

Figure 2.

The transition from two- to three-dimensional growth in E. coli microcolonies. (a,b) Confocal microscopy images from our experiments of a microcolony just before (a) and after (b) the invasion; the microcolony is shaded for depth, showing in dark where invasion has occurred. ‘Up’ in these images is the direction towards the agarose, consistent with the sketch of figure 1. (c,d) Simulations of a microcolony show the same phenomenon: snapshots of the in silico colony are shown before (c) and after (d) invasion; bacteria are shown in red if they have invaded the agarose. (e) An example of our experimental data for microcolony area as a function of time: the area first increases exponentially (straight segment of the plot), but upon invasion the observed area growth rate decreases. The solid line shows the best fit to the initial exponential growth phase, whereas the dashed line shows an exponential fit to the ‘post-invasion’ phase. The red vertical line indicates the invasion time, as determined by the best fit for the intersection of the solid and dashed lines (i.e. fitting with the two segments with a free crossover point). (f) The average microcolony doubling time (circles; left axis), obtained from the pre-invasion area growth rate, is unaffected by agarose concentration. The crosses (right axis) show the average eccentricity of microcolonies at buckling, showing that the colonies are fairly circular. Error bars are standard errors on the mean.

Figure 3.

Simulations using only mechanical forces are able to match experimental observations. (a) The dimensionless distance of invasion from the centre of the colony. The circles show experimental data; markers connected by lines show simulation results where friction is given by equation (3.5) for α = 0.4 (dashed line) and α = 1 (solid line), with the friction coefficient from equation (3.8). (b) The area of the colony at which invasion first takes place. Data markers are the same as in (a). The vertical line marks the agarose concentration C_a = 2.55% which corresponds to the assumed elastic modulus of the bacteria E_b = 375 kPa. Note that this also coincides with C_a = 2.5% at which we observe a peak in the colony size at invasion in our experiments. (c) The density of the bacteria in the colony shows no dependence on the radial position. (d) The mean density of bacteria in the colony increases with agarose concentration. Images correspond to colonies at agarose concentrations of 1.5% (left) and 3.5% (right). The number of colonies analysed for (a,b) is between nine and 24 for each agarose concentration, whereas for (c,d) between six and 15 colonies are analysed for each agarose concentration. There are typically 250 bacteria in the colonies at 1.5% when invasion occurs (approx. eight generations) whereas for 3% there are typically 500 bacteria (approx. nine generations). Simulation results presented in all panels have been averaged over 50 independent runs, and error bars represent standard errors of the mean. (Online version in colour.)

Figure 4.

Invasion does not require chemical factors. (a) Colonies that collide as they grow are shown at three different times (as labelled min.s), and at two focal planes separated by 2.1 µm. At the time of the first image, some initially separated colonies have already collided, but bacterial escape has not yet occurred anywhere. As the collision proceeds, the second layer of cells begins to form at a point denoted by the yellow circle. Invasion thus occurs in the area previously not occupied by bacteria. This implies that invasion is not triggered by chemical cues, because chemical factors secreted by the bacteria could not have accumulated at the collision site. (b,c) Simulations of pairs of colliding colonies. Colonies were seeded from pairs of cells initially located (b) 10 µm and (c) 30 µm apart, and simulations were run until the moment of the invasion transition. The figures show overlays of the results of 50 simulations; the positions of the bacteria at the moment of invasion are shown as green rods, whereas the position at which the invasion transition happens is shown by the red dots. Invasion often (but not always) occurs close to the point where the colonies collide (0,0), rather than at the original centres of the two microcolonies.

Figure 5.

Changing the dependence of the friction coefficient on the agarose concentration affects whether the simulations match the experimental data. In all figures, solid lines correspond to α = 1 and dashed lines to α = 0.4 in equation (3.5). Circles show experimental data. a(i–iii) Simulations with constant friction coefficient (k = 0.7). The panels show (i) the dimensionless buckling distance, which matches the experiments well. (ii) The colony area upon invasion, which does not match the experimental data. (iii) The cell density at the transition, which also matches well. b(i–iii) Simulations with a friction coefficient that is inversely proportional to the agarose concentration (k = 0.7 × 148/E_a). The panels show (i) the dimensionless buckling distance, which again matches well. (ii) The colony area upon invasion, which matches well up to 2.5%. (iii) The cell density, which does not match the experimental data. To fully match the experimental data, we require the friction coefficient to depend on the agarose concentration as described in equation (3.8). (Online version in colour.)

Figure 6.

(a–c) Our experimental data are not reproduced by simulations in which boundary-induced compression of the colony by the agarose dominates over friction. The circles show our experimental data, whereas the solid lines show the results of computer simulations in which radial compression caused by the agarose acting on peripheral cells dominates friction from the substrate. Apart from the density of the bacteria at the transition (c), which agrees with the experimental data for E_b = 750 kPa, neither the area upon invasion (b) nor the reduced distance (a) can reproduce the corresponding data. This suggests that in our experiments, radial compression due to the agarose is less important than radial friction. (d–f) ‘Stokesian’ friction proportional to velocity leads to similar results as velocity-independent friction. The microcolony area upon invasion (e), reduced distance (d) and cell density upon invasion (f) are shown for constant κ (solid line) and κ depending on C_a as in equation (E 1) (dashed line). The circles show our experimental data. (Online version in colour.)

Figure 7.

(a) Plots of the reduced distance of the position of invasion from the colony centre, (b) the microcolony area at the transition (c) and cell density at the invasion transition, for three different forms of the repulsion force: Hertzian (equation (F 1), solid line) with E_b = 375 kPa; equation (F 2) with β = 1, E_b = 200 kPa, k_1.5% = 0.7 (dashed line) and β = 2, E_b = 600 kPa, k_1.5% = 0.7 (dotted line). Unconnected markers are the experimental data. (Online version in colour.)

Figure 8.

Bacterial colony modelled as a rigid punch indenting an elastic medium. (a,b) Distortion of the elastic medium (agarose) caused by the patterned punch (a) and the flat punch (b). The agarose does not penetrate the gaps between the grooves. (c) Total force exerted on the punch by the elastic medium for the two cases, as a function of the punch radius R. The patterning has little effect on the force. h is the height of the punch and E is the Young modulus of the elastic medium. All results were obtained numerically as described in the text. (Online version in colour.)