Self-organization of bacterial communities against environmental pH variation: Controlled chemotactic motility arranges cell population structures in biofilms

As with many living organisms, bacteria often live on the surface of solids, such as foods, organisms, buildings and soil. Compared with dispersive behavior in liquid, bacteria on surface environment exhibit significantly restricted mobility. They have access to only limited resources and cannot be liberated from the changing environment. Accordingly, appropriate collective strategies are necessarily required for long-term growth and survival. However, in spite of our deepening knowledge of the structure and characteristics of individual cells, strategic self-organizing dynamics of their community is poorly understood and therefore not yet predictable. Here, we report a morphological change in Bacillus subtilis biofilms due to environmental pH variations, and present a mathematical model for the macroscopic spatio-temporal dynamics. We show that an environmental pH shift transforms colony morphology on hard agar media from notched ‘volcano-like’ to round and front-elevated ‘crater-like’. We discover that a pH-dependent dose-response relationship between nutritional resource level and quantitative bacterial motility at the population level plays a central role in the mechanism of the spatio-temporal cell population structure design in biofilms.


( )
where M is the maximum value, X M the maximum point, and HWHM the half width at half maximum. The following similar form may also be suitable: Under the following assumptions: • (Random) cell motility takes the first-order (α r ~ 1) form.
• Chemotactic sensitivity follows the Hill equation with Hill coefficient ~ 2 (positively cooperative binding).
• Chemotactic velocity is described by the spatial-gradient of the chemotactic sensitivity multiplied by the cell motility.
we can approximately derive the same form (α c = 2) of the chemotactic motility coefficient:

Population scale model of spatio-temporal growth of bacterial colonies
In our model, we use two functions: the nutrition level N = N(x, t) and the bacterial concentration B = B(x, t), where (x, t) denotes the space and time variables. The nutrition N diffuses in the media and is consumed by bacterial proliferation: where p = p(N, B) denotes the proliferation speed, d and c are the diffusion and consumption rates.
The bacteria B move with the velocity u = u(N, B) and proliferate consuming nutrition: For the proliferation p = p(N, B), we take the following form of the Monod type [51]: In addition to the saturation effect in nutritional increase, it represents bounded proliferation caused by spatial restriction on the two-dimensional medium surface. The maximum growth rate is given by p max . The half speed coefficients are denoted by κ N and κ B . Finally, the moving velocity vector is described as:

Simulations
Numerical calculation was performed by the standard explicit Euler method. We used arbitrary units that were matched to the experimental results. The common parameters were as follows: d = c = 2, p max = 1, κ N = 2, κ B = 1, m r = 0.04, κ r = 0.04, α r = 1, κ c = 10 -5 , α c = 2. The chemotactic coefficient is m c = 0 (pH 7.4) or 5 × 10 -4 (pH 7.0) unless otherwise noted. The random operator Θ was defined by a diagonal matrix. The diagonal components θ 1 and θ 2 were random numbers, taken at each spatio-temporal step, with a symmetric triangular distribution of mean 1 and support [0, 2]. The system size was 400×400 square lattice. The space and time step size was 1 and 0.02, respectively.
The initial data were N(x, 0) = P, B(x, 0) = Σ i I i (x), where This function stands for the inoculation at a designated point x i on the agar medium surface. Here we used function χ = χ(r) defined by χ(r) = 1 if r < 10 and χ(r) = 0 if r ≥ 10. This cut-off function becomes important only when the proliferation speed is much higher than the movement speed. Only for the DBM-like pattern on the nutrient-poor and high-moisture medium (Fig 3E), we used the modified proliferation rate (not essential for DBM pattern formation): This form was taken because the maintenance energy, namely the energy consumed by the upkeep of fundamental bacterial activity, became quite influential in the nutrient-poor situation [52]. Here, the subscript + means the positive part and δ (= 0.2) represents the maintenance cost rate. The parameters were changed: d = 2, c = 0.075, p max = 200, κ N = 5, κ B = 10, m r = 15, κ r = 2.5, α r = 1, m c = 0. In particular, bacteria wildly swarm on surface-moisture rich agar media, and then, collective chemotactic behavior is mostly suppressed [53], which is in agreement with the fact that chemotaxis is not necessary for simulating DBM-like pattern formation.

Front aggregation index
For a quantitative understanding of changing colony patterns, we introduced an index of a spatial structure of cell population. The definition and the calculation method are the following.
First, we prepare a cell population distribution map (e.g., Fig 1E; t = T5 (Tm = 220*m) in simulations). For the results of experiments (5days post-inoculation), we perform the correction of the lowest level (the surrounding agar region) and logarithmic filtering [54]. Then, from the distribution map, we clip a slender rectangular region inscribed in the colony outline including cracks if exist (Fig A). Concerning the present result, for instance, for the length 2R (R is the radius of the colony), the width is 0.1R (~ 2 mm in experiments). To eliminate fluctuations around the inoculation point, we remove the inner rectangular region of the length R and the same width. Then we evaluate the mean cell density M in the outer half region, and the weighted mean cell density M w in the same region. The weight function is where r is the distance from the center, n is an index of front aggregation sharpness (The larger the value of n is, the heavier the weight around colony boundaries is. n = 0 means no weighting. Here we take n = 32). Finally we define a front aggregation index as FAI = M w / M. If the colony is circular and the cell distribution is homogeneous, then FAI = 1. Cell accumulation at growth fronts in crater-like colonies increases the value of FAI, and branching in volcano-like colonies decreases FAI due to the notched space around colony boundaries. Therefore, the index FAI can characterize the morphological change between the volcano-like and the crater-like colonies.