Differential growth of wrinkled biofilms

Biofilms are antibiotic-resistant bacterial aggregates that grow on moist surfaces and can trigger hospital-acquired infections. They provide a classical example in biology where the dynamics of cellular communities may be observed and studied. Gene expression regulates cell division and differentiation, which affect the biofilm architecture. Mechanical and chemical processes shape the resulting structure. We gain insight into the interplay between cellular and mechanical processes during biofilm development on air-agar interfaces by means of a hybrid model. Cellular behavior is governed by stochastic rules informed by a cascade of concentration fields for nutrients, waste, and autoinducers. Cellular differentiation and death alter the structure and the mechanical properties of the biofilm, which is deformed according to Föppl–Von Kármán equations informed by cellular processes and the interaction with the substratum. Stiffness gradients due to growth and swelling produce wrinkle branching. We are able to reproduce wrinkled structures often formed by biofilms on air-agar interfaces, as well as spatial distributions of differentiated cells commonly observed with B. subtilis.

Figure 1

(a) B. subtilis biofilm grown on a 1.5% agar surface in a biofilm-inducing medium. Photograph taken after four days. Reprinted with the permission of Cambridge University Press from Chai et al. [14]. (b) In silico film showing a corona of radial wrinkles emerging from a central core.

Figure 2

(a) B. subtilis biofilm grown on the air-solid surface of agar gel containing water and nutrients. Reprinted with the permission of Cambridge University Press from Wilking et al. [15]. (b) In silico film showing successive wrinkle branching.

Figure 3

Schematic representation of a biofilm. (a) Biofilm without water network, (b) biofilm with water channels. For computational purposes, space is partitioned in a grid of tiles of size $a,a$ being the average length of a bacterium. Each tile may be filled with air, water, agar, or biofilm biomass (either a bacterium or ECM). This grid is used to discretize the continuum concentration and displacement fields, and to track the status of individual bacteria, which constitute a fraction of the total biofilm biomass.

Figure 4

Flow chart of the hybrid model.

Figure 5

Directional signaling in B. subtilis. When a threshold concentration of ComX is reached, a percentage of the population of cells expresses genes that induce a change in their phenotype, becoming surfactin producers. Surfactin may signal other cells to produce extracellular matrix. Starving cells become inert spores.

Figure 6

Expansion of a circular biofilm seed driven only by nutrient consumption. The initial seed is formed by two layers of cells with a diameter of 40 cells. Only division and spread mechanisms are involved. Snapshots taken every 100 steps. The initial nutrient concentration in the substratum is $C=3K_n$ and the ratio $\frac{k_n{a}^2}{D_n{K}_n}=8$.

Figure 7

Slices of a biofilm showing the cell type distribution during early stages of the differentiation process. All the slices correspond to the same biofilm and are taken at the same time. The evolution started from an initial seed with a diameter of 60 cells and variable thickness, represented by three small peaks. Dead cells appear first where the initial peaks where located. In later stages, inert cells proliferate at the top whereas normal cells are restricted to the bottom and the edges. Each colored box is one cell. Parameter values: $C=K_n,\frac{k_n{a}^2}{D_n{K}_n}=0.01,\frac{k_w{a}^2}{D_w{K}_w}=0.001,\frac{k_{cx}{a}^2}{D_{cx}{K}_{cx}}=0.01,\frac{k_s{a}^2}{D_s{K}_s}=0.8,{\gamma }_w=0.005,{\gamma }_d=0.00001,{\gamma }_s=0.001,{\gamma }_{e_1}=0.00001,{\gamma }_{e_2}=0.2$.

Figure 8

Connected network of wrinkles switching from an intricate core to radial branches, formed in a homogeneous circular film when a circular front of residual compression stresses of magnitude ${\varepsilon }_0=0.1$ expands at constant velocity. Parameter values: $\nu =0.5,E=25\text{kPa},{\nu }_v=0.45,{\mu }_v=0$ and $\gamma =16.$ $dx=0.1h$, where $h$ is the film thickness before wrinkling.

Figure 9

(a) Biofilm-substratum system deformed following out-of-plane displacements $\xi$ predicted by Eqs. (21) and (22) for residual stresses expanding as a radial compression front. (b) Bottom slice of (a) displaying the arrangement of biofilm and substratum. (c) Intermediate slice of (a) showing the allocation of dead cells. (d), (e) Accumulation of absorbed water in slice (c) triggered by the presence of weakened dead areas after 1 and 40 steps of the water absorption process, respectively.

Figure 10

Residual stresses ${\varepsilon }_{xx}^0$ for the biofilm seeds depicted in (a)–(c). (d)–(f) represent their values after one step of the stochastic division and spread process, with $C=2K_n$ and $\frac{k_n{a}^2}{D_n{K}_n}=0.01$. Spatial variations related to thickness are identifiable in (g)–(i), obtained averaging a few trials of the stochastic procedure. These fields approach their average values (about $-0.1)$. A single trial already gives the average value used in rough approximations by constants. (j)–(l) show the effect of including dead spots in the biofilm seed. Localized depressions are observed. (m)–(o) reproduce the residual stresses generated by the water absorption scheme when $\Pi =E/3$. The average values are about $-0.2$. ${\varepsilon }_{yy}^0$ resembles ${\varepsilon }_{xx}^0$ in all cases. ${\varepsilon }_{xy}^0$ is about $3–2$ times smaller.

Figure 11

(a) Initial biofilm seed. (b)–(f) Residual stresses ${\varepsilon }_{xx}^0$ averaged after 100 trials of the stochastic processes at steps 4, 8, 10, 11, 12. The average compression is $-0.132,-0.1425,-0.146,-0.1437,-0.124$, respectively. ${\varepsilon }_{yy}^0$ has a similar structure and average. ${\varepsilon }_{xy}^0$ takes smaller values and is neglected. (g)–(k) Vertical displacements. The height mark $dx$ corresponds to one cell. Parameter values are $\gamma =8,\nu =0.5,E=25\text{kPa},{\nu }_v=0.45$, and ${\mu }_v=0$.

Figure 12

Snapshots of the formation of wrinkles for ${\varepsilon }_0=0.1$. (a)–(d) Images taken at times $280\frac{T}{\tau },560\frac{T}{\tau },840\frac{T}{\tau },1120\frac{T}{\tau }$. An initial random state coarsens to produce a connected wrinkle network that opens up into radial branches as the compressed region spreads. The biofilm has Poisson ratio $\nu =0.5$ and Young modulus $E=25\text{kPa}$. The Poisson ratio and rubbery modulus of the substratum are ${\nu }_v=0.45$ and ${\mu }_v=0$. $T$ and $\tau$ are defined in Eq. (25). The ratio of the in-plane spatial scale to the out-of-plane spatial scale is set to $\gamma =16.$ The equations are nondimensionalized so that the dimensionless biofilm thickness becomes $\widehat{h}=1$. The time step is $dt=\frac{0.14}{\tau }d{x}^2$. The spatial step is $dx=0.1\widehat{h}$.

Figure 13

Wrinkled structures obtained varying the residual stresses and the elastic moduli for a smaller film with lower Poisson's ratio. (a) Transition from a harder central region with larger Young modulus to a softer outer corona: radial branches in the corona are separated from the core by rings. (b) Uniform residual compression in the central region with localized sinks due to dead spots plus radially increasing compression in the outer corona: small wrinkles anchored by the dead spots in the center merge with larger radial wrinkles in the outer corona. (c) Residual compression increasing slowly with distance to the center: labyrinths formed in the center become radial branches in the outer corona. Parameter values: same as previous figures except $\nu =0.3$ and $\gamma =8$. $dx$ is the spatial step.