The interplay of adhesion, friction, and nutrient availability in modulating biofilm wrinkling behavior

Abstract

Wrinkled patterns in biofilms arise from buckling instabilities triggered by stresses that accumulate as growth is constrained by a stationary substrate. While nutrient availability, friction, and adhesion each influence wrinkling, their combined effects remain poorly understood. Here, we address this gap using a lattice-network model of biofilm morphogenesis. Under constant nutrient supply, wrinkles initiate at the center, where stresses are highest and isotropic, regardless of the level of friction or adhesion. Stronger adhesion delays wrinkling and decouples the length scale governing the buckling instability from the overall biofilm size. Heterogeneous adhesion lowers the critical stress by triggering wrinkles in weakly adhered regions, with the effect modulated by friction and the average adhesion. Under nonuniform nutrient supply, our model predicts that wrinkle initiation shifts from the center to the edge as initial nutrient availability decreases, a transition we experimentally validate using E. coli biofilms.

Introduction

Biofilms—surface-attached bacterial communities embedded in a self-produced extracellular matrix^1,2—represent the most common microbial lifestyle on Earth³. As such, they are widespread across medical, industrial, and natural settings, impacting human health, technology, and the environment in both beneficial and detrimental ways. For example, biofilms promote corrosion on pipelines and ship hulls⁴, and when growing on medical devices like heart valves and joint prosthetics, contribute to persistent infections⁵. On the other hand, biofilms aid in breaking down organic matter in bioreactors for water treatment⁶ and help metabolize pollutants like oil spills and heavy metals in natural environments⁷.

Biofilms exhibit complex morphological behaviors that shape their broader ecological and technological roles. As bacteria within the biofilm consume nutrients, they grow, divide, and produce extracellular matrix (ECM), generating compressive stresses that drive the expansion of the biofilm. This expansion is resisted by frictional forces, which are the macroscopic manifestation of stick-slip molecular-level interactions between the biopolymers in the ECM and the stationary substrate⁸. Once the compressive stress exceeds a critical threshold, a mechanical buckling instability is triggered, giving rise to wrinkles that delaminate from the substrate if the adhesion energy is overcome. This process, and the roles of friction, adhesion, and nutrient availability, are illustrated in Fig. 1.

Fig. 1: Illustration of the mechanical buckling instability leading to wrinkles.

The biofilm (green) and substrate (orange) are shown, along with the main governing factors: nutrient uptake and growth, friction, and adhesion.

Understanding the mechanisms that shape biofilm wrinkled morphologies has been the subject of extensive research^9,10,11 because of the critical role wrinkles play in regulating nutrient¹², oxygen¹³, and bacterial¹⁴ and phage¹⁵ transport. Previous experimental studies have elucidated the influence of substrate stiffness on biofilm wrinkling. For example, biofilms on stiffer substrates have been shown to exhibit shorter wrinkle wavelengths compared to those grown on softer substrates¹⁶. Moreover, while on stiffer substrates wrinkling originates at the center and propagates outward, on softer substrates (often associated with lower friction), wrinkles initiate at the biofilm edge and propagate inward⁸.

Unfortunately, characterizing the spatial distribution of mechanical stress or disentangling the individual contributions of friction and adhesion to the wrinkling behavior experimentally remains extremely challenging. To address those limitations, researchers have turned to computational models based on continuum mechanics and originally developed to study mechanical instabilities in abiotic thin-film systems^17,18. Individual or agent-based models are also popular for simulating bacterial communities¹⁹; however, while they can capture cellular-level behavior, they are not suited for modeling the macroscopic time and length scales relevant to the morphological evolution of biofilms.

Using a continuum model, Zhang et al. demonstrated that a broad range of experimentally observed wrinkled morphologies could be reproduced by prescribing anisotropic strain profiles to selected regions of a circular thin film^20,21. In their study, the biofilm was modeled as a multilayer system, where growth was limited to a top layer obeying the Föppl-von Kármán (FvK) equations^16,20, cohesively bound to a softer, intermediate layer, in turn fixed to a rigid substrate. The multilayer architecture was originally justified by experimental observations of stratified biofilm structures²². However, the underlying assumptions that growth is confined to a thin top layer and that delamination occurs within the biofilm may not broadly apply. For instance, in most experiments, biofilms are grown on agar substrates^8,9,14,23, where nutrients are supplied from below, rendering the assumption of growth limited to an upper layer implausible. Moreover, in multilayer models, delamination is assumed to occur within the biofilm, between the top two layers. While Yan et al.¹⁶ demonstrated that such internal delamination can occur in V. cholerae, their experiments were conducted under fully submerged conditions, which differ from the drier environments common in most other studies. As a case in point, a more recent study has shown that P. aeruginosa biofilms lack a stratified structure and delaminate directly from the substrate²³, contradicting the aforementioned assumptions. Another limitation of the multilayer model is that it restricts biofilm expansion to small elastic deformations of the intermediate layer, which prevents it from capturing the large-scale expansion observed in real biofilms.

More recently, Fei et al.⁸ developed a continuum model that, for the first time, accounted for nutrient diffusion and uptake alongside realistic expansion dynamics by including frictional forces between the biofilm and the substrate⁸. The evolution of wrinkling was not explicitly modeled but instead approximated using axisymmetric solutions within a coarse-grained framework that minimized a Landau–Ginzburg-type free energy, with two order parameters representing wrinkle amplitude and pattern (straight vs. zigzag). Their results demonstrated that both friction (modeled as proportional to the substrate stiffness) and nonuniform growth are key determinants of the initial wrinkled morphology. Notably, they identified a transition in the wrinkling mode. At low friction, radial wrinkles initiate at the biofilm edge and propagate inward, while at high friction, disordered wrinkles first emerge at the center. While the model, once calibrated, achieved good quantitative agreement with experiments and offered valuable insight into the early wrinkling stages, it did not systematically address the role of adhesion or the abundance of nutrients on the wrinkling behavior.

Although adhesion is widely acknowledged as a key factor in biofilm wrinkling, it remains relatively understudied. In a clever setup, where compressive stresses in the biofilm were induced through lateral confinement rather than frictional forces, Geisel et al.²³ showed that strong adhesion can even suppress wrinkling in P. aeruginosa biofilms. Yet in computational models, adhesion is often treated as constant and always uniform, if delamination is considered at all. This is a major oversight as both friction and adhesion stem from the same ECM–substrate interactions yet play opposing roles—friction promotes stress buildup and thus wrinkling, while adhesion resists delamination, delaying or even inhibiting wrinkle formation. As a result, understanding how friction, adhesion and its possibly heterogeneous distribution, and nutrient availability jointly shape the biofilm wrinkling behavior remains a critical challenge, and current continuum modeling frameworks do not fully capture this interplay.

Here, we present a novel computational model of biofilm morphogenesis that allows independent control of friction and adhesion, while explicitly capturing the three-dimensional, time-resolved morphological evolution of the biofilm in response to nutrient diffusion and uptake. We begin by systematically varying friction and adhesion under a uniform nutrient supply. In this regime, biofilms consistently wrinkle at the center, where compressive stresses are highest and isotropic. Increasing adhesion raises the stress required for wrinkling but delays its onset to larger biofilm radii, which contradicts the relationship between critical stress and system size predicted by Euler buckling theory, and suggests that increasing adhesion induces the decoupling between the biofilm size and the length scale governing the buckling instability. Introducing spatial heterogeneity in adhesion lowers the critical buckling stress by localizing wrinkles in weakly adhered regions, though, interestingly, the effect depends strongly on both average adhesion and friction. We then investigate the coupled effects of nonuniform nutrient supply, friction, and adhesion on the wrinkling mode. We find that under abundant nutrient conditions or at high friction, growth remains nearly uniform in the biofilm and consequently wrinkles consistently form at the center first. For low initial nutrient concentrations or at low friction, early nutrient depletion at the center halts growth in that region, causing radial wrinkles to instead initiate at the nutrient-rich outer edge. We experimentally confirm in E. coli the transition in wrinkling mode with changes in initial nutrient concentration predicted by the model.

Results

A brief overview of the computational model is provided here, with a detailed description available in the Methods section. The biofilm is modeled as a two-dimensional triangular spring-lattice network embedded in three-dimensional space, where each node corresponds to a microscopic hexagonal domain of the biofilm. Nodes interact with their neighbors through harmonic potentials that account for both in-plane stretching and out-of-plane bending. The biofilm is treated as a linear elastic solid with uniform and time-invariant mechanical properties. Adhesion is modeled using a combination of Lennard-Jones and Yukawa potentials, both defined as functions of the vertical distance between the biofilm and a substrate that is not explicitly represented. Node dynamics are governed by the overdamped Langevin equation, with friction implemented through nodal diffusivity.

The biomass production follows Monod kinetics and thus depends on the local nutrient availability. Growth is modeled by increasing the equilibrium length of the springs connecting neighboring nodes, in a manner consistent with the amount of biomass produced. The network topology remains fixed during the simulation, and no nodes are added or removed. Nutrient concentration is computed by solving a reaction-diffusion equation on a square grid aligned with the substrate, where the sink term represents nutrients consumed by the biofilm. We assume that nodes delaminated from the substrate lose access to nutrients and stop growing, mimicking experimental conditions such as growth on agar plates, where nutrients are supplied through the substrate.

Effect of adhesion and friction under constant nutrient supply

In the simulations presented in this section, nutrient supply was held constant to isolate the individual effects of adhesion and friction on biofilm wrinkling, without confounding influences from nonuniform growth. The friction and adhesion values studied span over two orders of magnitude, covering the full range of potential responses. In the following, we refer to friction qualitatively as “high” or “low” for clarity, however, figure captions and legends explicitly report the diffusivity parameter ${{\rm{D}}}_{{\rm{b}}}^{\mathrm{on}}$, which is inversely related to the friction that nodes in contact with the substrate experience, and the actual input to our model (see the Methods section for more details).

The archetypal wrinkling behavior of a biofilm under constant nutrient supply, which is consistent across all tested friction and adhesion values, is shown in Fig. 2. As mentioned in the introduction, as the biofilm consumes nutrients and grows, frictional forces restrict its expansion, causing compressive stresses to build up. Figure 2a shows the distribution along the biofilm radius of radial and circumferential or hoop stresses (${{\rm{\sigma }}}_{{\rm{r}}}$ and ${{\rm{\sigma }}}_{{\rm{\theta }}}$, respectively) at the onset of buckling (just before the first wrinkles appear). Section III of the Supplementary Information (SI) contains details for the calculation of Cauchy stress, an inherently continuum quantity, from our discrete model. We observe both ${{\rm{\sigma }}}_{{\rm{r}}}$ and ${{\rm{\sigma }}}_{{\rm{\theta }}}$ are heterogeneous, highest near the center and decreasing toward the edge, with ${{\rm{\sigma }}}_{{\rm{r}}}$ decaying faster than ${{\rm{\sigma }}}_{{\rm{\theta }}}$. If one defines stress anisotropy as $\varphi =\left({{{\rm{\sigma }}}_{{\rm{\theta }}}-{\rm{\sigma }}}_{{\rm{r}}}\right)/\left({{{\rm{\sigma }}}_{{\rm{\theta }}}+{\rm{\sigma }}}_{{\rm{r}}}\right)$, then $\varphi =0$ at the center, while $\varphi =1$ at the edge, indicative of pure hoop stresses in that region. The stress distribution in Fig. 2a qualitatively agrees with previous continuum models of biofilm morphogenesis⁸. The corresponding distribution of strains, calculated under plane stress conditions (${\sigma }_{z}=0$), are shown in Fig. 2b.

Fig. 2: Archetypal wrinkling behavior of a biofilm under constant nutrient supply.

a Radial (${{\rm{\sigma }}}_{{\rm{r}}}$) and circumferential (${{\rm{\sigma }}}_{{\rm{\theta }}}$) stress distributions, and (b) radial (${{\rm{\varepsilon }}}_{{\rm{r}}}$) and circumferential (${{\rm{\varepsilon }}}_{{\rm{\theta}}}$) strain distributions, both plotted just prior to buckling as a function of normalized radial distance (r/R), where R is the biofilm radius. c Time evolution of the maximum compressive principal stress (${{\rm{\sigma }}}_{\max }$). d Biofilm expansion rate (R) over time. Hollow circles in panels (c, d) mark the onset of buckling. (e–h) Snapshots illustrating the formation and evolution of wrinkles. Color bars indicate the height of the biofilm above the plane of the substrate. All results shown correspond to a simulation with intermediate friction and adhesion values within the studied ranges (${D}_{{\rm{b}}}^{\mathrm{on}}=36\,{\mathrm{nm}}^{2}\,{{\rm{h}}}^{-1}$ and ${\rm{\gamma }}=5\,{\rm{\mu }}{\rm{J}}\,{{\rm{m}}}^{-2}$).

The maximum compressive principal stress in the system (${{\rm{\sigma }}}_{\max }$), shown in Fig. 2c, increases steadily until reaching the onset of buckling, indicated by a hollow circle. After the formation of the initial wrinkles, ${{\rm{\sigma }}}_{\max }$ drops abruptly. We define the peak value of ${{\rm{\sigma }}}_{\max }$ during the simulation as the critical stress (${{\rm{\sigma }}}_{{\rm{cr}}}$). Before reaching ${{\rm{\sigma }}}_{{\rm{cr}}}$, the biofilm expands radially at a rate, $\dot{R}$, which increases monotonically but nonlinearly over time, peaking, similarly to ${{\rm{\sigma }}}_{\max }$, near the onset of buckling (Fig. 2d). The decrease in expansion rate following wrinkling arises from two factors: first, growth shifts from primarily in-plane expansion to increasing the amplitude of the wrinkles; second, the formcation of delaminated wrinkles reduces the contact area of the biofilm with the substrate, limiting nutrient uptake and thereby slowing further growth.

Snapshots illustrating the evolution of the wrinkled morphology are shown in Fig. 2e–h. At the onset of buckling, compressive stresses peak at the center where they are nearly isotropic (Fig. 2a), which triggers the formation of dome-like blisters as shown in Fig. 2e. As the biofilm keeps growing, the initial blisters at the center start to coalesce forming an irregular pattern, while new blisters continue to form further away from the center (Fig. 2f). The coalesced blisters eventually propagate outward as radial wrinkles (Fig. 2g). Ultimately, the wrinkled morphology exhibits a central, delaminated region without a well-defined pattern, surrounded by radial wrinkles that may bifurcate as they extend outward, maintaining a relatively constant wrinkle frequency toward the biofilm edge (Fig. 2h).

While the qualitative wrinkling behavior remains consistent across all tested values of friction and adhesion under constant nutrient supply, the specific values of friction and adhesion significantly influence the expansion rate of the biofilms (discussed in Section IV of the SI and illustrated in Fig. SI.1), as well as the magnitude of ${{\rm{\sigma }}}_{{\rm{cr}}}$ and the size of the biofilm at the onset of buckling, which we dub critical radius (${R}_{\mathrm{cr}}$). Classical buckling theory predicts that ${\sigma }_{{\rm{cr}}}$ scales inversely with the system size as ${\sigma }_{{\rm{cr}}} \sim {R}^{-2}$ ²⁴. The challenge in growing biofilms, is that the system size is not constant, but it continuously increases over time, and the distribution of stress is not trivial. Therefore, whether the classical scaling (i.e., ${\sigma }_{\mathrm{cr}} \sim {{R}_{\mathrm{cr}}}^{-2}$) holds for biofilms, or how friction and adhesion might alter or disrupt it, remains an open question.

Figure 3 shows ${{\rm{\sigma }}}_{{\rm{cr}}}$ versus ${R}_{\mathrm{cr}}$ for all combinations of friction (increasing from yellow to blue) and adhesion (increasing with symbol size). Before interpreting the data in Fig. 3, it is important to recognize that ${R}_{\mathrm{cr}}$ cannot be smaller than the initial biofilm radius (${R}_{{\rm{o}}}$). While the effect of this constraint is negligible when ${R}_{0}\,\ll \,{R}_{\mathrm{cr}}$ (which is the case for low-friction conditions), at high friction, where stresses build up rapidly and buckling occurs early on, ${R}_{{\rm{o}}}$ imposes an artificial lower bound on ${R}_{\mathrm{cr}}$. As shown in Fig. SI.2 (Section V of the SI), reducing ${R}_{{\rm{o}}}$ extends the scaling relationship to arbitrarily smaller ${R}_{\mathrm{cr}}$ values, confirming that the apparent breakdown in scaling at high friction observed in Fig. 3 is not a physical effect and should not be misinterpreted as such.

Fig. 3: Scaling of critical stress with biofilm size at the onset of buckling.

Various frictional forces are indicated by color (increasing from yellow to blue), and adhesion energies by symbol size (larger symbols correspond to higher values: ${\rm{\gamma }}=0.5,\,4.6,\,9.2,\,23,\,46\,{\rm{\mu }}{\rm{J}}\,{{\rm{m}}}^{-2}$). Data for ${\rm{\gamma }}\le 0.5\,{\rm{\mu }}{\rm{J}}\,{{\rm{m}}}^{-2}$ are not shown, as all such cases converge to the same values of critical stress and critical radius. All simulations shown in this figure were initialized with a biofilm radius of ${R}_{0}=1250\,{\rm{\mu }}{\rm{m}}$.

The data in Fig. 3 reveals two distinct regimes in the scaling relationship ${\sigma }_{\mathrm{cr}} \sim {{R}_{\mathrm{cr}}}^{\alpha }$ with respect to adhesion and friction. When adhesion is held constant (solid lines), increasing friction (from yellow to blue) results in higher ${{\rm{\sigma }}}_{{\rm{cr}}}$ and smaller ${R}_{\mathrm{cr}}$, meaning that biofilms wrinkle earlier, at smaller sizes and higher stresses. The direction of the scaling is consistent with classical buckling theory, where smaller systems require higher critical stress to buckle. At low adhesion (smallest symbol size), the simulations even closely match the classical scaling exponent, $\alpha =-2.$ This agreement becomes even clearer in Fig. SI.2, where the bias from the initial biofilm size is eliminated. Interestingly, as adhesion increases, ${{\rm{\sigma }}}_{{\rm{cr}}}$ and ${R}_{\mathrm{cr}}$ as a function of friction become increasingly uncorrelated as evidenced by the progressively flatter slopes of the solid curves in Fig. 3 with increasing adhesion ($\alpha \to 0$ in as $\gamma \to \infty$). This observation suggests that the biofilm size (i.e., ${R}_{\mathrm{cr}}$) no longer governs the buckling instability, which is instead controlled by a shorter, localized length scale.

Consistent with the notion that localized length scales govern the buckling instability at higher adhesion, we find that increasing adhesion leads to a higher initial wrinkle count, reflecting more localized deformation (blue to green curves in Fig. 4a). This effect becomes less pronounced with increasing friction. For instance, at ${D}_{{\rm{b}}}^{\mathrm{on}}=360\,{\rm{n}}{{\rm{m}}}^{2}\,{{\rm{h}}}^{-1}$, the wrinkle count increases nearly eightfold from the lowest to the highest adhesion studied, whereas at higher friction, ${D}_{{\rm{b}}}^{\mathrm{on}}=36\,{\rm{n}}{{\rm{m}}}^{2}$, the increase is more modest, about twofold. Snapshots illustrating these behaviors are shown in Fig. 4b–e.

Fig. 4: Adhesion promoting localized wrinkling modes.

a Number of initial wrinkles as a function of friction for different adhesion energies (increasing from blue to green). b–e Snapshots taken shortly after wrinkling, comparing low and high friction (top vs. bottom) and low and high adhesion (left vs. right). Specific parameter values are (b) ${D}_{{\rm{b}}}^{\mathrm{on}}=360\,{\rm{n}}{{\rm{m}}}^{2}\,{{\rm{h}}}^{-1};\,{\rm{\gamma }}=0.5{\rm{\mu }}{\rm{J}}{\,{\rm{m}}}^{-2}$, (c) ${D}_{{\rm{b}}}^{\mathrm{on}}=360\,{\rm{n}}{{\rm{m}}}^{2}\,{{\rm{h}}}^{-1};\,{\rm{\gamma }}=46{\rm{\mu }}{\rm{J}}{\,{\rm{m}}}^{-2}$, (d) ${D}_{{\rm{b}}}^{\mathrm{on}}=36\,{\rm{n}}{{\rm{m}}}^{2}\,{{\rm{h}}}^{-1};\,{\rm{\gamma }}=0.5\,{\rm{\mu }}{\rm{J}}{\,{\rm{m}}}^{-2}$, and (e) ${D}_{{\rm{b}}}^{\mathrm{on}}=36\,{\rm{n}}{{\rm{m}}}^{2}\,{{\rm{h}}}^{-1};\,{\rm{\gamma }}=46\,{\rm{\mu }}{\rm{J}}{\,{\rm{m}}}^{-2}$.

At constant friction, increasing adhesion (dashed lines connecting symbols of the same color in Fig. 3) leads to both higher ${\sigma }_{{\rm{cr}}}$ and larger ${R}_{{\rm{cr}}}$, yielding a positive scaling exponent ($\alpha > 0$), which directly contradicts, even qualitatively, the scaling predicted by classical buckling theory. Interestingly, our simulations consistently produce a robust scaling exponent of $\alpha \approx 7/2$, irrespective of the level of friction. The origin of this specific scaling exponent remains unexplained and highlights a compelling direction for future theoretical work.

While we treat adhesion and friction as independent parameters to systematically explore their individual roles, these two properties, both rooted in the same underlying physico-chemical interactions between the biofilm and the substrate, are likely to be positively correlated in real biofilms. However, in the absence of quantitative experimental data relating adhesion and friction in biofilms, enforcing a specific coupling would have been speculative at best. Moreover, mediated by structural or biochemical mechanisms, some biological systems exhibit nontrivial relationships between adhesion and friction²⁵. As future experiments provide a clearer understanding of how adhesion and friction are coupled in biofilms, our framework could be used to define biologically relevant trajectories through the parameter space mapped in Fig. 3.

Effect of adhesion heterogeneity under constant nutrient supply

Biofilm adhesion to substrates is inherently heterogeneous, stemming from spatial variations in extracellular matrix production and other biological factors. These heterogeneities can have a pronounced impact on the wrinkling response. For instance, experiments have shown that regions associated with cell death often serve as nucleation sites for localized delamination⁹. However, capturing spatial variations in adhesion strength within continuum models remains challenging, leaving the role of adhesion heterogeneity in biofilm mechanics largely unexplored.

In this section, we take advantage of the ease with which heterogeneities can be implemented in our modeling framework. Adhesion heterogeneity is introduced by assigning an adhesion energy to each node in the biofilm, randomly sampled from a Gaussian distribution, $N(\bar{\gamma },{s}_{\gamma })$, where $\bar{\gamma }$ is the average adhesion, and ${s}_{\gamma }$ is the standard deviation. The degree of heterogeneity is quantified using the coefficient of variation, ${{CV}}_{\gamma }={s}_{\gamma }/\bar{\gamma }$, which we systematically study from 0 (uniform adhesion) to 1 (individual adhesion values are, on average, as far from the mean as the mean is from zero). We explore four representative cases combining high and low friction (${D}_{{\rm{b}}}^{\mathrm{on}}=36$ and 360 nm² h⁻¹) and high and low average adhesion ($\bar{\gamma }=4.6$ and $46\,{\rm{\mu }}{\rm{J}}\cdot {{\rm{m}}}^{-2}$), capturing the full spectrum of wrinkling behaviors observed in our simulations.

As shown in Fig. 5a, increasing adhesion heterogeneity promotes biofilm wrinkling by lowering the critical stress required to trigger the wrinkle formation. However, the sensitivity of the normalized critical stress ${\sigma }_{c{\rm{r}}}^{* }$ (defined relative to the value for homogeneous adhesion), depends strongly on both the mean adhesion $\bar{\gamma }$ and the level of friction. When friction is high or $\bar{\gamma }$ is low, heterogeneity has only a modest effect, with ${\sigma }_{c{\rm{r}}}^{* }$ decreasing by less than 20% at ${{CV}}_{\gamma }=1$. Notably, at high friction, the influence of adhesion heterogeneity remains largely unchanged across an order-of-magnitude variation in $\bar{\gamma }$ (blue vs. red squares). The strongest effect of adhesion heterogeneity appears in the case with low friction and high adhesion (blue circles), where ${\sigma }_{c{\rm{r}}}^{* }$ drops sharply, by nearly 40% at ${{CV}}_{\gamma }\approx 0.4$, before plateauing. Beyond a certain level of heterogeneity, ${\sigma }_{c{\rm{r}}}^{* }$, and hence the wrinkling behavior, becomes insensitive to further increases in adhesion heterogeneity.

Fig. 5: Effect of adhesion heterogeneity on the critical stress.

a Critical stress normalized by that for the case of homogeneous adhesion, ${{\rm{\sigma }}}_{{\rm{cr}}}^{* }$, as a function of the coefficient of variation, ${{\rm{CV}}}_{{\rm{\gamma }}}$, of the distribution of adhesion. b, c Snapshots showing initial wrinkling patterns for the uniform adhesion case (${{\rm{CV}}}_{{\rm{\gamma }}}=0$) and a heterogeneous adhesion case (${{\rm{CV}}}_{{\rm{\gamma }}}=0.2$), respectively, both with $\bar{{\rm{\gamma }}}=46\,{\rm{\mu }}{\rm{J}}\,{{\rm{m}}}^{-2}$ and ${D}_{{\rm{b}}}^{\mathrm{on}}=360\,{\rm{n}}{{\rm{m}}}^{2}\,{{\rm{h}}}^{-1}$. The homogeneous case shows wrinkles distributed across the biofilm center, while the heterogeneous one displays significantly fewer wrinkles, localized in regions with weak adhesion.

Adhesion heterogeneity has minimal impact on the overall wrinkling morphology. The only consistent difference is a slight reduction in the initial wrinkle count under heterogeneous adhesion, as wrinkles preferentially nucleate in regions of locally weaker adhesion (Fig. 5b, c). A quantitative analysis of the correlation between local adhesion values and wrinkle formation is provided in Section VI and Fig. SI.3 of the SI. Overall, the spatial onset of wrinkling is governed by the interplay between regions of high compressive stress (near the biofilm center under uniform nutrient conditions), and those of low adhesion (randomly assigned).

Effect of nonuniform nutrient supply

In this section, we examine how nonuniform nutrient availability, arising from finite nutrient diffusion, affects the wrinkling behavior of biofilms. In previous sections, we assumed constant nutrient availability, representing a large substrate relative to the biofilm size (effectively an infinite reservoir) where nutrient diffusion greatly outpaces consumption. This simplification allowed us to isolate the effects of adhesion and friction. However, in more realistic settings, nutrient diffusion often occurs on timescales comparable to or slower than bacterial consumption, leading to spatiotemporal nutrient gradients and, consequently, nonuniform biofilm growth⁸.

To validate our framework under nonuniform nutrient conditions, we revisited a previously reported transition in the initial wrinkling location driven by changes in substrate friction⁸. In Fig. 6, we show that our model indeed reproduces the friction-dependent morphological patterns observed by Fei et al.⁸. At high friction (which correspond to high-agar substrates in an experimental setting), wrinkles initiate at the center (Fig. 6a), where compressive stresses are highest and nearly isotropic (Fig. 6b). Conversely, at lower friction, radial wrinkles first appear at the biofilm edge (Fig. 6d), triggered by circumferential stresses (Fig. 6e). The corresponding radial distribution of strains for each case are shown in Fig. 6c, f. The agreement between our model and prior experimental and modeling efforts supports the validity of our modeling framework.

Fig. 6: Friction-induced transition in the wrinkling mode.

Snapshots and stress distributions are shown for two cases: (a–c) high friction (${D}_{{\rm{b}}}^{\mathrm{on}}=36\,{\rm{n}}{{\rm{m}}}^{2}\,{{\rm{h}}}^{-1}$) and (d–f) lower friction (${D}_{{\rm{b}}}^{\mathrm{on}}=180\,{\rm{n}}{{\rm{m}}}^{2}\,{{\rm{h}}}^{-1}$). Panels (a) and (d) show the biofilm surface morphology soon after the initial wrinkles form, while (b) and (e), and (c) and (f) display the corresponding stress and strain distributions, respectively. All results shown in this figure correspond to simulations where ${\rm{\gamma }}=9.2\,{\rm{\mu }}{\rm{J}}\,{{\rm{m}}}^{-2}$ and ${C}_{0}=1\,\mathrm{pg}\,{\rm{\mu }}{{\rm{m}}}^{-2}$.

Beyond the previously characterized friction-induced transition, we also identify a new transition in wrinkling mode driven by changes in the initial nutrient concentration, ${C}_{0}$ (Fig. 7). When ${C}_{0}$ is above a certain threshold, wrinkles first appear at the center and later propagate outward, a pattern similar to the constant nutrient supply scenario. However, below this threshold, depletion of nutrients at the biofilm center halts growth before the critical buckling stress can be reached in that region (Fig. 7a). Meanwhile, nutrients remain accessible to an outer annulus of the biofilm, which continues to expand, building up circumferential stress (Fig. 7b). When the stress eventually reaches a critical value, radial wrinkles form at the biofilm edge (Fig. 7d), which gradually coarsen and propagate inward (Fig. 7e).

Fig. 7: Nutrient-induced transition in the wrinkling mode.

a Radial distribution of nutrient concentration at different time stamps. Radial distribution of radial and circumferential stresses (b) and strains (c). Snapshots illustrating the initial wrinkles (d) and the wrinkled pattern at some later time (e). All results shown in this figure correspond to simulations where: ${\rm{\gamma }}=9.2\,{\rm{\mu }}{\rm{J}}\,{{\rm{m}}}^{-2}$; ${D}_{{\rm{b}}}^{\mathrm{on}}=36\,{\rm{n}}{{\rm{m}}}^{2}\,{{\rm{h}}}^{-1}$, and ${C}_{0}=0.8\,\mathrm{pg}\,{\rm{\mu }}{{\rm{m}}}^{-2}$.

The wrinkling mode, whether “edge-first” or “center-first”, significantly affects the expansion rate of the biofilm and the time that it takes to reach the critical stress (${t}_{{\rm{cr}}}$), two factors that together determine ${R}_{\mathrm{cr}}$. As shown in Fig. 8a, the average expansion rate, which we define as ${\dot{R}}_{{\rm{avg}}}=\left({R}_{{\rm{cr}}}-{R}_{0}\right)/{t}_{{\rm{cr}}}$, increases monotonically, but non-linearly, with ${C}_{0}$. This trend arises from both the faster growth of individual biofilm domains with access to abundant nutrients and from a larger biofilm contact area having access to nutrients. The eventual saturation of ${\dot{R}}_{{\rm{avg}}}$ with ${C}_{0}$ is consistent with Monod kinetics (inset of Fig. 8a). Figure 8b shows that ${t}_{{\rm{cr}}}$ also depends strongly on ${C}_{0}$, decreasing steeply within the “edge-first” regime and exhibiting a sharp, potentially discontinuous, drop at the transition to the “center-first” regime, beyond which it remains nearly constant. Together, the trends in ${\dot{R}}_{{\rm{avg}}}$ and ${t}_{{\rm{cr}}}$ with ${C}_{0}$ account for the counterintuitive dependence of ${R}_{\mathrm{cr}}$ on ${C}_{0}$ for each wrinkling mode (Fig. 8c). For low ${C}_{0}$, within the “edge-first” regime, ${\dot{R}}_{{\rm{avg}}}$ increases much more rapidly than the decrease in ${t}_{{\rm{cr}}}$, leading to a sharp increase in ${R}_{\mathrm{cr}}$ with increasing ${C}_{0}$. In contrast, within the “center-first” regime, both ${\dot{R}}_{{\rm{avg}}}$ and ${t}_{{\rm{cr}}}$ quickly plateau with increasing ${C}_{0}$, resulting in a slight decrease of ${R}_{\mathrm{cr}}$ followed by a nearly constant value. This plateauing behavior reflects the onset of nutrient saturation in the Monod law, beyond which further increases in ${C}_{0}$ no longer affect the growth dynamics. Slight deviations from these trends are likely due to statistical fluctuations, as the results shown are from individual simulation runs.

Fig. 8: Effect of initial nutrient concentration on biofilm growth dynamics and the onset of buckling.

a Average expansion rate, ${\dot{R}}_{{\rm{avg}}}=\left({R}_{{\rm{cr}}}-{R}_{0}\right)/{t}_{{cr}}$ as a function of ${C}_{0}$. (b) ${t}_{{\rm{cr}}}$ vs. ${C}_{0}$. (c) ${R}_{{cr}}$ vs. ${C}_{0}$. Markers are colored to distinguish the two wrinkling regimes with blue corresponding to “edge-first” and red to “center-first”. The dashed lines serve only as visual guides. All results shown in this figure correspond to simulations where ${\rm{\gamma }}=9.2\,{\rm{\mu }}{\rm{J}}\,{{\rm{m}}}^{-2}$ and ${D}_{{\rm{b}}}^{\mathrm{on}}=36\,{\rm{n}}{{\rm{m}}}^{2}\,{{\rm{h}}}^{-1}$.

The results in Fig. 8 highlight that the when and how of the wrinkling instability are not solely determined by the biofilm radius. Instead, they arise from a complex interplay between growth dynamics and the spatiotemporal evolution of stress, shaped by nonuniform nutrient availability, and modulated by friction and adhesion interactions.

We experimentally validate the nutrient-induced transition in wrinkling mode predicted by the model, by conducting biofilm growth experiments using E. coli (strain AR3110) cultured overnight in Luria-Bertani (LB, Invitrogen) medium and inoculated onto salt-free LB (SFLB) agar plates. Biofilm development was monitored from day 1 to 5 using an optical microscope equipped with a 28 °C incubation chamber. To probe the effect of nutrient availability on initial wrinkling morphology, SFLB plates were prepared at different nutrient concentrations: 0.5X, 1X, 2X, 3X, and 4X. The agar content remained constant across all conditions. Additional experimental details are provided in the Methods section.

Figure 9 shows that at low nutrient concentration (1X), wrinkles initiate at the colony edge (Fig. 9a), while at high nutrient concentration (2X), they first emerge at the center (Fig. 9b), confirming the model predictions of a nutrient-induced transition in wrinkling mode. Representative snapshots from simulations corresponding to each wrinkling mode are shown in Fig. 9c, d. It is worth emphasizing that no model parameters were fitted to the experimental system.

Fig. 9: Transition in the initial wrinkling morphology with nutrient concentration.

a, b Experimental results for E. coli biofilms for nutrients concentration 1X SFLB and 2X SFLB, respectively. c, d Simulation results for ${C}_{0}=0.8\,\mathrm{pg}\,{\rm{\mu }}{{\rm{m}}}^{-2}$ and ${C}_{0}=1\,\mathrm{pg}\,{\rm{\mu }}{{\rm{m}}}^{-2}$, respectively (${\rm{\gamma }}=9.2\,{\rm{\mu }}{\rm{J}}\,{{\rm{m}}}^{-2}$; ${D}_{{\rm{b}}}^{\mathrm{on}}=36\,{\rm{n}}{{\rm{m}}}^{2}\,{{\rm{h}}}^{-1}$). c, d $1{\rm{X}}$.

We considered and ruled out several possible confounding factors responsible for the transition in the wrinkling mode observed experimentally. Because the agar content was fixed across experiments, the mechanical properties of the substrate should remain mostly unchanged and therefore changes in friction or adhesion cannot explain the observed transition. While elevated nutrient levels could, in principle, reduce the osmotic pressure associated with the production of extracellular polymeric substances (EPS), which might influence the spreading dynamics of the biofilm²⁶, there is no clear mechanism by which this could account for the shift in wrinkling modes, and any resulting effects are unlikely to be qualitative. We also tested whether nutrient concentration affects gene expression patterns that could alter the mechanical properties of the biofilm by modulating cellular phenotypes, particularly those related to matrix production and motility. Matrix-producing cells secrete extracellular polymers that grant stiffness (i.e., solid-like properties) to the biofilm, while motile cells produce surfactants that promote spreading and reduce cohesion between bacteria. Shifts in the balance between these phenotypes could, in principle, impact the mechanical behavior of the biofilm and thereby influence the onset and mode of wrinkling. To test this, we performed additional fluorescence microscopy experiments using an E. Coli strain engineered with three reporters: constitutive expression, matrix production, and motility. Radial fluorescence profiles (Fig. SI.4) show that from 0.5X to 2X, which covers the nutrient range corresponding to the transition in wrinkling modes, the motility-to-matrix expression ratio remains nearly unchanged, with significant phenotypic shifts observed only at higher concentrations (≥3X SFLB). These findings rule out changes in gene expression as a relevant factor in the observed transition in wrinkling behavior.

Given that both nutrients and friction can alone cause the transition between wrinkling modes, their combined effect warrants closer examination. To this end, we analyze the interplay between adhesion, friction, and nutrient availability in determining the transition between wrinkling modes by systematically mapping two-dimensional slices of this three-dimensional parameter space. Namely, at fixed adhesion, we constructed a phase diagram as a function of friction and initial nutrient concentration (Fig. 10a), and at fixed friction, we constructed a second phase diagram varying adhesion energy and nutrient concentration (Fig. 10b). To quantitatively characterize the transition between wrinkling modes, we introduce a continuous order parameter, $\psi$, that captures the radial location of the first wrinkles. To compute the order parameter, we analyze a snapshot taken shortly after wrinkling. At each radial distance from the center, we calculate the standard deviation of vertical positions along the azimuth. The order parameter is defined as the radial location of the maximum in this profile, normalized by the biofilm radius. By construction, $\psi \approx0$ corresponds to the “center-first” mode, and $\psi \approx1$ corresponds to the “edge-first” mode, and intermediate values indicate initial wrinkle formation at intermediate radial positions between the center and the edge of the biofilm.

Fig. 10: Interplay of friction, adhesion, and initial nutrient concentration in the transition between “edge-first” (blue) and “center-first” (red) wrinkling modes.

a, b Phase diagrams showing the value of an order parameter $\psi$, which quantifies the radial location of the first wrinkles, ranging from 0 for the “center-first” mode to 1 for the “edge-first” mode. a $\psi$ as a function of ${D}_{{\rm{b}}}^{\mathrm{on}}$ and ${C}_{o}$ at fixed adhesion energy $\gamma =9.2\,{\rm{\mu }}{\rm{J}}\,{{\rm{m}}}^{-2}$. (b) $\psi$ as a function of $\gamma$ and ${C}_{o}$ at fixed friction ${D}_{{\rm{b}}}^{\mathrm{on}}=180\,{\mathrm{nm}}^{2}\,{{\rm{h}}}^{-1}$. c $\psi$ as a function of ${C}_{o}$ for ${D}_{{\rm{b}}}^{\mathrm{on}}=180\,{\mathrm{nm}}^{2}\,{{\rm{h}}}^{-1}$, and $\gamma =9.2\,{\rm{\mu }}{\rm{J}}\,{{\rm{m}}}^{-2}$.

The phase diagram in Fig. 10a shows that increasing friction (i.e., decreasing ${D}_{{\rm{b}}}^{\mathrm{on}}$) shifts the transition between wrinkling modes toward lower nutrient concentrations. This trend reflects the fact that higher friction slows biofilm expansion, and smaller biofilms require less nutrients to sustain growth at the center. At very low nutrient concentrations, the “edge-first” mode dominates regardless of friction, consistent with the limit where only bacteria at the edge remains active due to arrested growth in the central region of the biofilm. In contrast, Fig. 10b shows that increasing adhesion has the opposite effect. Higher adhesion delays the onset of wrinkling, allowing the biofilm to grow larger before wrinkling. As a result, higher nutrient concentrations are needed to maintain growth at the center and accumulate sufficient compressive stress in that region to trigger the “center-first” wrinkling mode.

To evaluate whether the transition in wrinkling mode is continuous or discontinuous, we analyzed the order parameter $\psi$ as a function of ${C}_{o}$, using a finer resolution than that employed in the phase diagrams of Fig. 10a, b, for a representative case where ${D}_{{\rm{b}}}^{\mathrm{on}}=180\,{\mathrm{nm}}^{2}\,{{\rm{h}}}^{-1}$, and $\gamma =9.2\,{\rm{\mu }}{\rm{J}}\,{{\rm{m}}}^{-2}$. As shown in Fig. 10c, $\psi$ changes steeply but continuously, with intermediate values corresponding to cases where wrinkles first emerge in an annular region rather than strictly at the center or edge (see snapshot in the inset of Fig. 10c). This behavior suggests a smooth crossover between wrinkling modes rather than a sharp, discontinuous transition.

The model predicts a single wrinkling mode for any given set of conditions, an outcome also confirmed experimentally (Fig. 9a, b and Movies SI.1,2). Simultaneous wrinkling at both the center and edge would require unrealistically precise parameter tuning such that the critical stress is reached at both locations simultaneously. Interestingly, the model also predicts the possibility of wrinkling at intermediate radial locations (Fig. 10c), suggesting a continuous crossover between “edge-first” and “center-first” modes. While this intermediate behavior has not yet been observed experimentally, likely due to the narrow parameter range over which it occurs and the challenge of independently tuning adhesion and friction, we consider it a robust outcome of the model. As shown in Movie SI.1, real biofilms can develop more complex morphologies at later stages, which are beyond the scope of our model. Capturing such dynamics would require a substantially more advanced framework incorporating environment-dependent gene expression, spatially and temporally heterogeneous mechanical properties, and improved representations of biomass expansion over extended timescales.

Discussion

In this study, we developed a lattice-network model of biofilm morphogenesis that incorporates biofilm-substrate interactions through friction and adhesion (both uniform and heterogeneous) as well as nutrient diffusion and uptake while explicitly resolving the three-dimensional morphological evolution of the biofilm.

Under constant supply of nutrients, we identified opposing effects of friction and adhesion on the onset of biofilm wrinkling. Increasing friction promotes the buildup of compressive stress, leading to earlier wrinkling at smaller radii and higher critical stresses, which is consistent with classical buckling theory. In contrast, increasing adhesion stabilizes the biofilm against delamination, raising the critical stress required for wrinkling, but also delaying the onset of the buckling instability, leading to biofilms that reach larger sizes before wrinkling. This behavior, which may seem counterintuitive from the perspective of classical buckling theory, can be explained by a decoupling between the overall biofilm size and the characteristic length scale that governs the buckling instability. As adhesion increases, we observe a higher initial wrinkle count, supporting of the fact that the instability is controlled by a shorter length scale than the overall biofilm size. At high enough adhesion, buckling is no longer dictated by the global geometry of the biofilm but rather by a local response.

We found that introducing spatial heterogeneity in adhesion, under constant nutrient supply, can substantially lower the critical stress required for biofilm wrinkling by promoting buckling in regions of weak local adhesion. However, this effect strongly depends on both friction and mean adhesion. At high friction or low mean adhesion, biofilms are less sensitive to heterogeneity, showing only modest reductions in critical stress compared to the homogeneous case. The effect is also nonlinear. Initial increases in heterogeneity produce sharp decreases in critical stress but further increases yield diminishing returns.

Our results suggest potential strategies by which biofilms might exploit adhesion heterogeneity to modulate their wrinkling response. Specifically, biofilms displaying low friction, but high average adhesion, would exhibit the greatest sensitivity to spatial variations in adhesion, such that even modest heterogeneity could substantially reduce the critical stress required for wrinkling. This raises the possibility that biofilms could use heterogeneity to trigger mechanical instabilities locally without globally compromising their attachment to the substrate, thereby maintaining resistance to external stresses such as hydrodynamic shear. In contrast, biofilms displaying high friction or low average adhesion would be far less responsive to adhesion heterogeneity, indicating that these traits may promote mechanical stability on irregular or heterogeneous substrates. It is worth noting that while adhesion and friction are likely positively correlated, the exact relationship remains unknown and represents an interesting direction for future research.

We found that the wrinkling morphology is primarily governed by nutrient availability. Under constant nutrient supply, the wrinkling mechanism remained consistent across more than two orders of magnitude variation in both friction and adhesion. Wrinkles, in the form of blisters, first emerge at the biofilm center, where compressive stresses are highest and nearly isotropic. These initial blisters then coalesce and propagate outward as radial wrinkles. Under nonuniform nutrient conditions, we discovered a previously unreported transition in the wrinkling mechanism, which we confirmed experimentally in E. coli. At high initial nutrient levels, biofilm growth remains approximately uniform, and the wrinkling mechanism resembles that observed under constant nutrient supply. In contrast, at low nutrient levels, nutrient depletion in the center halts growth preventing further stress accumulation in that region. Continued growth at the nutrient-rich edge eventually leads to the formation of radial wrinkles at the edge, which then propagate inward. We systematically ruled out potential confounding factors for the observed transition in wrinkling mode, confirming that it arises from changes in nutrient availability rather than shifts in substrate mechanics or gene expression. Phase diagrams constructed from simulations reveal that increasing friction shifts the transition between wrinkling modes toward lower nutrient concentrations, while increasing adhesion shifts it toward higher values. A continuous order parameter quantifying wrinkle location further showed that this transition between wrinkling modes, while sharp, is smooth rather than discontinuous.

The nutrient-dependent transition in wrinkling mode leads us to hypothesize that biofilms may adopt distinct wrinkling strategies, each serving a different functional purpose. Under nutrient-rich conditions, wrinkles initiate at the center and propagate outward, possibly facilitating the dispersal of bacteria near the center to colonize new environments. In contrast, under nutrient-limited conditions, wrinkles emerge at the periphery and move inward. These inward-propagating wrinkles may act as channels to enhance nutrient infiltration into the starved interior, consistent with previous observations that biofilm wrinkling facilitates fluid transport and nutrient delivery^12,27.

Overall, our work offers a mechanistic understanding of how the interplay between friction, adhesion, and nutrient availability—including spatial heterogeneity in adhesion—governs biofilm wrinkling. We hypothesize that natural biofilms may exploit these physical features to optimize their morphological response under varying environmental conditions. Whether biofilms actively regulate their adhesion and frictional interactions with the substrate, or exploit environmental nutrient conditions to modulate wrinkling, remains an open question requiring experimental validation.

Methods

Overview of the computational model

We model the biofilm as a two-dimensional spring-lattice network embedded in three-dimensional space, where each node represents a microscopic spatial domain. Nodes interact with their neighbors through potentials that capture both in-plane elasticity and out-of-plane bending stiffness. Another potential that depends only on the vertical distance between each node and a plane substrate (which is not modeled explicitly) captures the adhesion of the biofilm to the substrate. Node dynamics are governed by an overdamped Langevin equation, with the viscous damping modulated to account for frictional forces between the biofilm and the substrate. The spatial dimensions of each domain grow over time driven by biomass production, which depends on the local availability of nutrients. Nutrients are calculated by solving a reaction-diffusion equation on a grid aligned with the substrate, with nutrient uptake proportional to biomass production.

Our model, presented in detail in the following sections, is aimed at understanding how friction, adhesion, and nutrient availability interact to shape the qualitative wrinkling behavior of bacterial biofilms. Given the wide variability across bacterial species and experimental conditions, we select model parameters from physiologically reasonable ranges (see Tables SI.1–SI.5), without attempting to fit any specific experimental outcome. We chose to prioritize mechanistic understanding over quantitative prediction because our model, like all continuum models developed so far, does not account for gene expression. This omission is critical because gene expression governs the heterogenous distribution of phenotypes, like matrix producers or motile cells, within the biofilm²⁸. As a result, the local constitutive response of the biofilm can span a wide spectrum, from fluid-like regions dominated by motile cells to viscoplastic regions rich in ECM. This spatiotemporal variability in mechanical properties plays a key role in shaping how and where mechanical instabilities emerge. Without incorporating these effects of gene expression, attempts to quantitatively reproduce the wrinkling behavior of real biofilms are inherently limited.

Discretization Scheme

The biofilm is modeled as a 2D triangular spring-lattice network embedded in 3D space (Fig. 11), thereby neglecting any variation or processes across its thickness. Each node in the network represents a microscopic hexagonal spatial domain with an associated undeformed length scale, ${l}_{0}$, which grows over time, and which is much larger than an individual bacterium $\sim 1-5\,{\rm{\mu }}{\rm{m}}$. The nutrients diffuse in a separate square lattice fixed on the plane of the substrate ($z=0$), with a distance between grid points, $c$.

Fig. 11

Illustration of the discretization scheme for both biofilm and substrate.

Interaction potentials

Each node in the spring-lattice network interacts with its nearest neighbors and with the substrate. The potential energy of the system, ${\rm{V}}\left({\bf{r}}\right)$, is a function of the node positions and can be expressed as:

$$V\left({\bf{r}}\right)={V}_{\mathrm{bond}}\left({\bf{r}}\right)+{V}_{\mathrm{bend}}\left({\bf{r}}\right)+{V}_{\mathrm{adh}}\left(z\right)$$

(1)

where ${V}_{\mathrm{bond}}$ and ${V}_{\mathrm{bend}}$, are the stretching and bending contributions to the potential energy, respectively, and ${V}_{\mathrm{adh}}$ represents the adhesive interactions between the nodes and the substrate, which only depend on their vertical distance to the substrate, $z$. We define ${V}_{\mathrm{bond}}$ and ${V}_{\mathrm{bend}}$ as:

$${V}_{\mathrm{bond}}=\mathop{\sum }\limits_{{ij}=1}^{{n}_{{\rm{b}}}}\frac{1}{2}{k}_{{\rm{b}}}{\left({l}^{{ij}}-{l}_{o}^{{ij}}\right)}^{2}$$

(2)

$${V}_{\mathrm{bend}}=\mathop{\sum }\limits_{i=1}^{{n}_{\theta }}\frac{1}{2}{k}_{{\rm{\theta }}}{\left({\theta }_{i}-\pi \right)}^{2}$$

(3)

where ${n}_{{\rm{b}}}$ is the number of bonds in the system, ${k}_{{\rm{b}}}$ is the spring constant of the bond potential, ${l}^{{ij}}$ and ${l}_{o}^{{ij}}$ are the deformed and undeformed lengths, respectively, of the bond between nodes $i$ and $j$ (Fig. 12a), defined as the average of the characteristic length scales of the domains associated with nodes $i$ and $j$, i.e., ${l}_{o}^{{ij}}=\frac{1}{2}\left({l}_{o}^{i}+{l}_{o}^{j}\right)$. ${{\rm{n}}}_{\theta }$ is the number of improper angles in the system, ${k}_{{\rm{\theta }}}$ is the spring constant for bending interactions, and ${\theta }_{i}$ is the value of the improper angle associated with bond $i$, as illustrated in Fig. 12b. The derivation of ${{\rm{k}}}_{{\rm{b}}}$ and ${{\rm{k}}}_{{\rm{\theta }}}$ from the continuum elastic properties of the biofilm is given in Section II of the SI. Consistent with previous models^8,16,20,21, we assume that the constitutive response of the biofilm is uniform in space and time, remaining unchanged with variations in thickness or density during deformation.

Fig. 12: Interaction potentials.

a In-plane stretching, (b) out-of-plane bending, (c) adhesion energy corresponding to $\gamma =9.6\,\mu {\rm{J}}\,{{\rm{m}}}^{-2}$.

As illustrated in Fig. 12c, we model the adhesive interactions between each node in the biofilm and the substrate as the sum of a Lennard-Jones and a Yukawa potential:

$${V}_{\mathrm{adh}}\left(z\right)=4{\rm{\varepsilon }}\left[{\left(\frac{{\rm{\sigma }}}{z}\right)}^{12}-{\left(\frac{{\rm{\sigma }}}{z}\right)}^{6}\right]+{\rm{B}}{\rm{\varepsilon }}\frac{{e}^{-{z}^{* }}}{{z}^{* }}$$

(4)

where ${\rm{\varepsilon }}$ is the depth of the potential well, ${\rm{\sigma }}$ is the distance at which the potential is zero, B is an adimensional constant that controls the height of an activation barrier, and ${z}^{* }=z/\left({\rm{C}}{\rm{\sigma }}\right)$ where C is another adimensional constant that controls the distance between the minimum and the maximum of the potential energy function. The potential energy has a minimum at an equilibrium distance at half of the thickness of the biofilm, ${h}_{0}$, and becomes negligible for distances greater than ${{\rm{z}}}_{{\rm{cut}}}$. The adhesion energy, $\gamma$, reported throughout the paper is calculated as:

$$\gamma =\frac{\max \left({{\rm{V}}}_{{\rm{adh}}}\right)-\min ({{\rm{V}}}_{{\rm{adh}}})}{{A}_{0}}$$

(5)

where ${A}_{0}$ is the initial area of the biofilm domains. To maintain a constant energy per unit area as the contact area of the domains grow over time, we need to scale the potential energy associated to each node as:

$${V}_{\mathrm{adh}}\left(z,t\right)={V}_{\mathrm{adh}}(z,0)\cdot {\left(\frac{l(t)}{{l}_{0}(0)}\right)}^{2}$$

(6)

Reaction-diffusion of nutrients

The concentration of nutrients on the substrate is calculated by integrating a reaction-diffusion equation numerically using the Euler first-order explicit method under periodic boundary conditions (PBCs) on the 2D rectangular lattice defined on the substrate:

$$\frac{\partial C}{\partial t}={D}_{c}{\nabla }^{2}C-\frac{{\rho }_{o}{h}_{o}}{{\rm{Y}}}\mu \left(C\right)$$

(7)

where $C$ denotes the nutrient concentration at substrate position $\left(x,y\right)$ and time $t$, ${D}_{c}$ is the nutrient diffusivity, ${\rho }_{o}$ is the reference biofilm density, ${h}_{o}$ is the undeformed biofilm thickness, Y is a yield coefficient converting nutrient mass to biomass, and $\mu \left(C\right)$ follows Monod kinetics:

$$\mu \left(C\right)={{\rm{\mu }}}_{\max }\cdot \frac{C}{{K}_{{\rm{s}}}+C}\cdot \Theta \left(z\right)$$

(8)

where ${{\rm{\mu }}}_{\max }$ is the maximum growth rate, ${K}_{{\rm{s}}}$ is the value of $C$ at which $\mu /{{\rm{\mu }}}_{\max }=0.5$, and the switching function $\Theta \left(z\right)$ ensures that any domain of the biofilm detached from the substrate, and thus lacking access to nutrients, does not grow or uptake nutrients:

$$\Theta \left(z\right)=\left\{\begin{array}{c}1,\,\mathrm{if}\,z < {{\rm{z}}}_{\mathrm{cut}}\\ 0,\,\mathrm{if}\,z\ge {{\rm{z}}}_{\mathrm{cut}}\end{array}\right.$$

(9)

For the sink term in Eq. (7) to be strictly valid, the contact area between each biofilm domain and the substrate must be equal to its undeformed area. Deviations from this assumption, would underestimate the nutrient uptake rate by a factor: ${\left({l}_{o}/l\right)}^{2}$.

Biomass production and growth

Biomass production depends on the local nutrient supply and encompasses cell growth, proliferation, ECM production, and any other related process, without distinguishing among them.

We assume that biomass is conserved between the undeformed and deformed configurations:

$${m}_{\mathrm{bio}}\propto {l}^{2}\cdot h\cdot \rho ={{l}_{{\rm{o}}}}^{2}\cdot {h}_{{\rm{o}}}\cdot {\rho }_{o}$$

(10)

where $h$ is the thickness of the biofilm, $l$ is the deformed length scale of the domain, and the subindices $o$ refers to the undeformed state. The proportionality $\propto$ allows us to omit constant prefactors related to the specific geometry of the hexagonal domains, which cancel out between deformed and undeformed configurations.

We also assume that all cells within a domain grow, and at the same rate:

$$\frac{{\rm{d}}{m}_{\mathrm{bio}}}{{\rm{d}}t}={m}_{\mathrm{bio}}\cdot \mu \left(C\right)$$

(11)

With $\mu \left(C\right)$ defined in Eq. (8). Substituting Eq. (10) into Eq. (11) results into:

$$\frac{{\rm{d}}{l}_{0}}{{\rm{d}}t}=\frac{1}{2}\cdot {l}_{0}\cdot \mu \left(C\right)$$

(12)

where $C$ in this equation is the nutrient concentration at the substrate grid point nearest to the node at the center of the domain. This approximation is reasonable as long as the nutrient concentration gradients are low over the length scale of the domains.

Equations of motion

We simulate the motion of the biofilm nodes by numerically integrating the overdamped Langevin equation for each node $i$:

$$\frac{{\rm{d}}{{\bf{r}}}_{i}\left(t\right)}{{\rm{d}}t}=-\frac{{D}_{{\rm{b}},i}\left(t\right)}{{k}_{B}T}{\nabla }_{{{\bf{r}}}_{i}}V\left({\bf{r}}\right)+\sqrt{2{D}_{b,i}}{{\boldsymbol{\eta }}}_{i}\left(t\right)$$

(13)

where ${{\bf{r}}}_{i}\left(t\right)$ and ${D}_{b,i}\left(t\right)$ are the position and diffusivity of node $i$, ${k}_{B}T$ is the thermal energy, $V\left({\bf{r}}\right)$ is the total potential energy of the system as a defined in Eq. (1). The term ${\nabla }_{{{\bf{r}}}_{i}}V\left({\bf{r}}\right)$ is the gradient of $V\left({\bf{r}}\right)$ with respect to ${{\bf{r}}}_{i}$, and ${{\boldsymbol{\eta }}}_{i}\left(t\right)$ is a Gaussian white noise term (i.e., zero mean and unit variance). The simulations are performed at room temperature, $T=298\,{\rm{K}}$, and the numerical integration is carried out using the Euler–Maruyama scheme with a time step $\Delta t=0.05{\rm{s}}$.

The diffusivity of each node is not constant but evolves over time as the biofilm domain associated with a particular node grows. Specifically (and omitting the subscript $i$ for clarity),

$${D}_{{\rm{b}}}\left(t\right)=\frac{kT}{m\left(t\right)\zeta \left(t\right)}$$

(14)

where $m\left(t\right)$ is the mass and $\zeta \left(t\right)$ the damping coefficient (units of inverse time) of the biofilm domain. We assume the damping is caused by molecular-level stick–slip interactions coarse-grained over the size of the domain. Assuming a constant areal density of such interactions, the damping is expected to increase proportionally to the deformed contact area of the biofilm domain:

$$\zeta \left(t\right)=\zeta \left(0\right){\left(\frac{l(t)}{l(0)}\right)}^{2}=\zeta \left(0\right){\left(\frac{l(t)}{{l}_{0}(0)}\right)}^{2}$$

(15)

Because as biomass is conserved between the undeformed and undeformed configurations, and all the cells within a domain grow at the same rate, regardless of the deformed configuration of such domain, the mass of the domain grows proportionally to the undeformed area:

$$m\left(t\right)=m\left(0\right){\left(\frac{{l}_{0}(t)}{{l}_{0}(0)}\right)}^{2}$$

(16)

Substituting Eqs. (15) and (16) into Eq. (14), the time-dependent diffusivity becomes:

$${D}_{{\rm{b}}}\left(t\right)={D}_{{\rm{b}}}\left(0\right){\left[\frac{{l}_{0}(t)l(t)}{{l}_{0}^{2}\left(0\right)}\right]}^{-2}$$

(17)

In addition, the diffusivity of a node depends on whether the node is in contact with the substrate ($z < {{\rm{z}}}_{{\rm{cut}}}$), or delaminated ($z\ge {{\rm{z}}}_{{\rm{cut}}}$). For nodes that are delaminated from the substrate, stick-slip interactions are no longer present. Therefore, the reference diffusivity ${{\rm{D}}}_{{\rm{b}}}\left(0\right)$ for delaminated nodes, which we define as ${D}_{{\rm{b}}}^{\mathrm{off}}$, is significantly higher than the diffusivity of nodes in contact with the substrate, ${D}_{{\rm{b}}}^{\mathrm{on}}$, i.e., ${D}_{{\rm{b}}}^{\mathrm{off}}\gg {D}_{{\rm{b}}}^{\mathrm{on}}$. Moreover, the time dependence of the diffusivity for delaminated nodes originates solely from the increase in mass. The full expression for nodal diffusivity then becomes

$${D}_{b}=\left\{\begin{array}{c}{D}_{{\rm{b}}}^{\mathrm{on}}{\left[\frac{l\left(t\right){l}_{0}\left(t\right)}{{l}_{0}^{2}\left(0\right)}\right]}^{-2},\,\mathrm{if}\,z < {{\rm{z}}}_{\mathrm{cut}}\\ {D}_{{\rm{b}}}^{\mathrm{off}}{\left[\frac{{l}_{0}\left(t\right)}{{l}_{0}(0)}\right]}^{-2},\,\mathrm{if}\,z\ge {{\rm{z}}}_{\mathrm{cut}}\end{array}\right.$$

(18)

Bacterial strains

We used Escherichia coli AR3110, a strain known to produce wrinkled biofilms on salt-free LB agar, and a derivative triple fluorescent reporter strain engineered for this study to track gene expression related to matrix production and motility. The construction of the reporter strain involved lambda red recombineering and multicomponent assembly of promoter–reporter–terminator cassettes into the rbsAR locus. Details of the genetic engineering protocol, including plasmids, cloning steps, and primer sequences, are provided in Section VIII of the SI and Table SI.6.

Bacterial culture and biofilm imaging

Both wild-type and triple reporter strains were cultured overnight at 37 °C in Luria-Bertani (LB, Invitrogen) medium. Following overnight growth, 1 µL of the culture was inoculated onto salt-free LB (SF-LB) agar plates to facilitate biofilm development¹¹. Biofilm formation was monitored using a Zeiss Axio Zoom.V16 stereo microscope equipped with a Zeiss PlanApo Z 0.5X /0.125 FWD 114 mm objective lens, within an incubation chamber at 28 °C. Bright-field and fluorescence images were acquired either as time-stamped single images or as part of a time-lapse series captured at 40-minute intervals, starting on day 1 of biofilm development and continuing for a minimum of 5 days. Fluorescent intensities representing matrix- and motility-specific expression were normalized to the intensity of a fluorescent reporter driven by the constitutive promoter, which served as a proxy for cellular density. To investigate the effect of nutrient availability on biofilm formation, SF-LB agar plates with different concentrations of SF-LB (0.5X, 1X, 2X, 3X and 4X) were prepared.