Collective colony growth is optimized by branching pattern formation in Pseudomonas aeruginosa

Abstract

Branching pattern formation is common in many microbes. Extensive studies have focused on addressing how such patterns emerge from local cell-cell and cell-environment interactions. However, little is known about whether and to what extent these patterns play a physiological role. Here, we consider the colonization of bacteria as an optimization problem to find the colony patterns that maximize colony growth efficiency under different environmental conditions. We demonstrate that Pseudomonas aeruginosa colonies develop branching patterns with characteristics comparable to the prediction of modeling; for example, colonies form thin branches in a nutrient-poor environment. Hence, the formation of branching patterns represents an optimal strategy for the growth of Pseudomonas aeruginosa colonies. The quantitative relationship between colony patterns and growth conditions enables us to develop a coarse-grained model to predict diverse colony patterns under more complex conditions, which we validated experimentally. Our results offer new insights into branching pattern formation as a problem-solving social behavior in microbes and enable fast and accurate predictions of complex spatial patterns in branching colonies.

Keywords: bacterial colony; branching pattern; coarse-grained modeling; optimization model; pattern formation.

Figure 1. An optimization model for branched colony growth

1. Pseudomonas aeruginosa colonies develop branching patterns from a symmetric initial shape when growing on swarming media (recipe described in Xavier et al, 2011) in 90‐mm petri dishes. Scale bar: 1 cm.

2. P. aeruginosa colonies expand in 2D when initiating from a point inoculation (left) but form stripe patterns that extend in 1D when initiating from a strip (right). Top: images of colonies growing on swarming media in 90‐mm petri dishes; bottom: schematics of the colony patterns (blue strips represent branches of colonies, and arrows show the directions of branch extension); patterns expanding in 2D can be simplified into bifurcating branches, and patterns expanding in 1D can be simplified to parallel strips. Scale bar: 1 cm.

3. A simple model to describe colony growth with predefined patterns assuming 1D branch extension. The geometry of the colony is given by predefined branch width (W) and density (D) on the vertical direction and the variable, branch length (L), on the horizontal direction (shown in the schematic; blue strips represent branches of a colony growing from one end of a rectangular domain). R is the domain size on the vertical direction. The diffusion and consumption of the nutrient (N) is described by Eq [1], where D_N is the diffusivity and β_N is the consumption rate of nutrient. Eq [2] describes the growth of cells (C), and α_C is the cell growth rate. The cell growth function (f_G) is a function of N and C. The total amount of cell growth in the colony, Ω, is averaged to the total colony area. The elongation rate of branches is obtained based on the assumption that the net expansion of the colony is proportional to the amount of cell growth (Eq [3]), and γ is the efficiency of colony expansion (see Appendix Supplementary Methods for further details).

Source data are available online for this figure.

Figure 2. Branching patterns enable optimal colony growth when nutrient or expansion of the colony is restricted

1. The optimization model implemented with different combinations of branch width and density reveals the optimal colony patterns that yield the highest biomass under different conditions. Colors indicate the total biomass (unit: c.u. mm²; c.u.: cell density unit) at the same time point (t = 24 h). At or beyond the diagonal of the heatmap (where WD = 1), the colony becomes uniform with no branches. (a), (b), and (c) correspond to different initial nutrient concentrations (N ₀) or expansion efficiency (γ) (a: N ₀ = 8 g/l, γ = 7.5 mm/h/c.u.; b: N ₀ = 30 g/l, γ = 7.5 mm/h/c.u.; c: N ₀ = 8 g/l, γ = 25 mm/h/c.u.). Other parameters: D_N = 6 mm²/h;β_N = 160 g/l/h/c.u.; α_C = 0.8/h; K_N = 0.8 g/l; C_m = 0.05 c.u.

2. Pseudomonas colonies with the optimal patterns predicted by the model show higher growth efficiency than the ones with non‐optimal patterns. When nutrient or expansion of the colony is restricted (a: growing on media with 4 g/l casmino acids and 0.5% agar; the experiment was independently replicated twice; n = 5 for wild type and n = 10 for hyperswarmers), wild‐type Pseudomonas PA14 that develop branching patterns grow more efficiently than hyperswarmers that show uniform expansion. On media with higher nutrient concentration (b: with 8 g/l casmino acids and 0.5% agar; the experiment was independently replicated four times; n = 12 for wild‐type and n = 24 for hyperswarmers) or wetter surface (c: with 4 g/l casmino acids and 0.4% agar; the experiment was independently replicated twice; n = 4 for wild type and n = 8 for hyperswarmers), hyperswarmers yield more biomass than the wild type. Images were taken at 20 h after inoculation and are representatives of replicates. One image from Fig 1A is reused in (b). Scale bar: 1 cm.

Source data are available online for this figure.

Figure EV1. Simulations reveal how different conditions lead to different optimal colony patterns

Simulations reveal how the spatial–temporal dynamics of colonies vary depending on the patterns of the colonies and the growth conditions. Left: thin branches (branch width: 2 mm, branch density: 0.10/mm); middle: wide branches (branch width: 5 mm, branch density: 0.04/mm); right: no branches. For different colony patterns, the distributions of cell density (c.u.), nutrient concentration (g/l), and the consumption rate of nutrient (g/l/h) at the same time points (A. t = 20 h, B. t = 18 h, C. t = 9 h) are shown.

1. A

   When nutrient is scarce and the expansion efficiency is low (N ₀ = 8 g/l, γ = 7.5 mm/h/c.u.), nutrient is quickly depleted in the colony‐covered regions and the utilization of nutrient is mainly at the colony front and edges. In this case, the total amount of nutrient utilization is correlated with the length of the colony boundaries, which is higher in colonies with thin branches but minimized in non‐branching colonies.

2. B, C

   When nutrient is abundant (N ₀ = 30 g/l, γ = 7.5 mm/h/c.u.) or the expansion efficiency is high (N ₀ = 8 g/l, γ = 25 mm/h/c.u.), colonies expand before consuming all the nutrient in the area covered by cells. Therefore, the consumption of nutrient is also related to the colony area, which is higher in colonies expanding uniformly.

Figure 3. The optimal colony patterns that maximize biomass accumulation vary with growth conditions

1. Under various sets of initial nutrient concentrations and expansion efficiencies, the optimal branch density and width that yield the highest biomass varies. Colors indicate the total biomass (unit: c.u. mm²) at the same time point (t = 24 h) scaled to the min/max values in each subpanel. At or beyond the diagonal of the heatmap (where WD = 1), the colony becomes uniform with no branches. Other parameters: D_N = 6 mm²/h; β_N = 160 g/l/h/c.u.; α_C = 0.8/h; K_N = 0.8 g/l; C_m = 0.05 c.u.

2. The features of the patterns developed by Pseudomonas colonies under different growth conditions are generally consistent with the optimal patterns predicted by the model. The colony expansion efficiency is modulated by changing the agar density. Images of colonies were taken once colonies reached the plate boundaries or stopped growing. The experiment was independently replicated five times. Images shown are representatives of the replicates. Scale bar: 1 cm.

Source data are available online for this figure.

Figure EV2. Branch width of Pseudomonas colonies increases with increasing nutrient concentration and decreasing agar density

1. The optimal branch widths (mm; shown by numbers and colors in the table) predicted by the optimization model with different combinations of environmental parameters.

2. The average branch widths (mm; shown by numbers and colors in the table) of Pseudomonas colonies under different combinations of growth conditions. The mean branch width of a colony was measured in a semi‐automated manner (described in detail in Methods and Protocols). For each condition, the average of the mean branch widths of 2–4 colonies was shown.

3. The growth conditions of Pseudomonas colonies are divided into four groups: I. Low nutrient concentration (4–8 g/l casamino acids) and high agar density (0.50–0.55%); II. High nutrient concentration (10–16 g/l casamino acids) and high agar density (0.50–0.55%); III. Low nutrient concentration (4–8 g/l casamino acids) and low agar density (0.40–0.45%); and IV. High nutrient concentration (10–16 g/l casamino acids) and low agar density (0.40–0.45%).

4. The average branch width of each group in (C). The sample size is the number of colonies being measured in each group. Error bars show standard deviations. Unpaired, two‐sided t‐tests were used to compare between groups: I versus II: ***P = 0.0004; I versus III: ****P < 0.0001; II versus IV: ****P < 0.0001 (with Welch's correction because their variances are not significantly different); III versus IV: P = 0.1485.

Source data are available online for this figure.

Figure 4. Predicting 2D colony patterns by applying the optimization rule

1. Simulating 2D expansion of branching patterns. The colony starts from a point inoculum (open circle) where branches initiate. Solid lines represent the trajectories of the branch tips (dots). The local branch density at a certain branch tip is given by 1/d, where d is the distance of the branch tip to its nearest neighbor. The given branch width (W) and density (D) determine the shape and bifurcation of branches: The boundary of the colony is at a radius of W/2 from the branch tip trajectories; the branch bifurcates to maintain the local branch density around D. The growth direction of branches follows the local nutrient gradient. Nutrient distribution, cell growth, and branch extension are calculated as described earlier (Fig 1C).

2. The simulated colony patterns capture the general features of the observed patterns of Pseudomonas colonies. For the simulations, under each condition, we implement the optimal W and D that give the maximum biomass to generate the predicted optimal patterns. Other parameters: D_N = 6.951 mm²/h; β_N = 216.9 g/l/h/c.u.; α_C = 1.486/h; K_N = 1.156 g/l; C_m = 0.0548 c.u. In experiments, the colony expansion efficiency is modulated by changing the agar density. Images from Fig 3B are reused. Scale bar: 1 cm.

Source data are available online for this figure.

Figure EV3. Simulated patterns with different combinations of branch widths and densities

Patterns with 2D branch extension generated by the model when different combinations of branch widths and densities are implemented. Other parameters of the model are kept the same (D_N = 9 mm²/h; β_N = 160 g/l/h/c.u.; α_C = 1.2/h; K_N = 0.8 g/l; C_m = 0.05 c.u.; γ = 7.5 mm/h/c.u.; N ₀ = 8 g/l). The patterns are initialized from a disk at the center with uniform initial cell density C ₀ = 0.5 c.u. The boundary of a branch is set by the given branch width, and bifurcations are determined by the given branch density. Each branch extends following the local nutrient gradient. Source data are available online for this figure.

Figure EV4. Simulating diverse colony patterns using the model based on the optimization rule

Varying parameters of the model results in diverse colony patterns with distinct quantitative features, such as thin dendrites and patterns with densely packed fingers. The parameters used for producing these examples are as follows: D_N = 9.326 mm²/h; β_N = 152.9 g/l/h/c.u.; α_C = 1.878/h; K_N = 0.8466 g/l; C_m = 0.06050 c.u.; γ = 7.385 mm/h/c.u.; N ₀ = 4 g/l (left), 5 g/l (middle), or 8 g/l (right). The branch density and width implemented in the model are the optimal ones that result in maximum biomass accumulation efficiency, as found by optimization modeling using the 2D branch extension formulation. Source data are available online for this figure.

Figure 5. Predicting colony patterns under heterogeneous conditions

1. Anti‐nutrient‐gradient growth observed in Pseudomonas colonies (the upper panels) and simulated using the model (the lower panels). Numbers: casamino acid concentration (g/l) on the two ends of the petri dish. The experiment was independently replicated three times, and representative images are shown. Scale bar: 1 cm. Parameters used in the simulations are obtained by systematic screening described in (B): D_N = 5.749 mm²/h; β_N = 195.5 g/l/h/c.u.; α_C = 1.105/h; K_N = 0.6635 g/l; C_m = 0.07890 c.u.; γ = 7.513 mm/h/c.u.

2. Systematic screening for the parameter space that satisfies the growth behaviors of colonies under heterogeneous conditions. ① With a given set of randomized model parameters, we generate heatmaps of biomass under different nutrient concentrations by screening through combinations of branch widths and densities (colors represent biomass accumulation; unit: c.u. mm²); ② we find the optimal patterns under different nutrient concentrations and obtain the mapping between the optimal patterns and the nutrient concentration; ③ with the mapping and the model parameters, we predict the patterns under heterogeneous conditions (e.g., media with a nutrient gradient); ④ to accelerate and enhance the throughput of the screening, we use the simulation data of step ① to train neural networks that allow us to emulate the model and generate more data with great efficiency (yellow arrows).

Source data are available online for this figure.

Figure 6. Predicting colony patterns with various seeding configurations

Simulated and observed colony patterns when colonies initiate from discrete dots, continuous lines, or complex patterns that carry information. Upper rows: initial seeding locations; middle rows: simulations based on the optimization rule; lower rows: images of Pseudomonas colonies inoculated with the corresponding configurations using an automated liquid handling system (0.1 μl cell culture with OD₆₀₀ ~0.2 was dispensed at each spot). The experiment was independently replicated three times, and representative images are shown. Scale bar: 1 cm. In simulations, patterns are initialized from spots with radius of 5 mm and uniform initial cell density C ₀ = 1.6 c.u. The parameters for the simulations are as follows: D_N = 5.749 mm²/h; β_N = 195.5 g/l/h/c.u.; α_C = 1.105/h; K_N = 0.6635 g/l; Cm = 0.07890 c.u.; γ = 4 mm/h/c.u.; N ₀ = 14.5 g/l. Source data are available online for this figure.

Figure EV5. Predicting colony patterns with various seeding configurations and nutrient concentrations

The model is able to predict colony patterns with complex initial configurations and different nutrient concentrations. Top row: the initial seeding locations; lower rows: simulated and observed colony patterns with corresponding initial seeding configurations growing on media with different nutrient concentrations (casamino acid concentration: 4, 8, and 16 g/l from top to bottom). In experiments, multiple colonies were seeded on the same petri dish using a MANTIS automated liquid handler (0.1 μl cell culture with OD₆₀₀ ~0.2 was dispensed at each spot). Images are representatives of 2 replicates. Images from Fig 6 and one panel from Fig 1B are reused in the “Medium nutrient” panels. Scale bar: 1 cm. In simulations, patterns are initialized from spots with a radius = 5 mm and uniform initial cell density C ₀ = 1.6. The parameters for the simulations are as follows: D_N = 5.749 mm²/h; β_N = 195.5 g/l/h/c.u.; α_C = 1.105/h; K_N = 0.6635 g/l; C_m = 0.07890 c.u.; γ = 4 mm/h/c.u.; N ₀ = 8.5 g/l, 14.5 g/l, or 16.5 g/l from top to bottom. The branch density and width implemented are the optimal ones that result in maximum biomass accumulation efficiency, as found by optimization modeling using the 2D branch extension formulation.