Abstract
In natural settings, microbes tend to grow in dense populations [1–4] where they need to push against their surroundings to accommodate space for new cells. The associated contact forces play a critical role in a variety of population-level processes, including biofilm formation [5–7], the colonization of porous media [8, 9], and the invasion of biological tissues [10–12]. Although mechanical forces have been characterized at the single cell level [13–16], it remains elusive how collective pushing forces result from the combination of single cell forces. Here, we reveal a collective mechanism of confinement, which we call self-driven jamming, that promotes the build-up of large mechanical pressures in microbial populations. Microfluidic experiments on budding yeast populations in space-limited environments show that self-driven jamming arises from the gradual formation and sudden collapse of force chains driven by microbial proliferation, extending the framework of driven granular matter [17–20]. The resulting contact pressures can become large enough to slow down cell growth by delaying the cell cycle in the G1 phase and to strain or even destroy the microenvironment through crack propagation. Our results suggest that self-driven jamming and build-up of large mechanical pressures is a natural tendency of microbes growing in confined spaces, contributing to microbial pathogenesis and biofouling [21–26].
The simulataneous measurement of the physiology and mechanics of microbes is enabled by a microfluidic bioreac-tor [27–30] that we have designed to culture microbes under tightly controlled chemical and mechanical conditions. The setup, shown in Fig. 1A, is optimized for budding yeast (S. cerevisiae). We use this device to measure mechanical forces generated by partially-confined growing populations and the impact of those forces on both the population itself and its micro-environment.
At the beginning of each experiment, we trap a single yeast cell in the growth chamber of the device, which can hold up to about 100 cells. The cells are fed by a continuous flow of culture medium, provided by a narrow set of channels that are impassable for cells.
While cells first proliferate exponentially as in liquid culture, growth dynamics change dramatically once the chamber is filled. At high density, cells move in a stop-and-go manner and increasingly push against the chamber walls. The population develops a contact pressure* that increases over time until it reaches steady state, subject to large fluctuations. Depending on the geometry of the device (Fig. 1B and C), the mean steady-state pressure can reach up to 1 MPa. This pressure is larger than the ≈ 0.2 MPa turgor pressure measured in budding yeast (stationary phase [31]) and much larger than the ≈ 1 mPa needed for the cells to overcome viscous friction (supplementary text).
Both the intermittent flow and pressure build-up are counterintuitive because the outlet channel is wide enough for cells to pass. In principle, excess cells could flow like a liquid out of the chamber. Time lapse movies (here) reveal that blockages in the device stabilize the cell packing and prevent flow. Cells proliferate until a sudden avalanche flushes them through the outlet (Fig. 1D and E). Another jamming event occurs, and the process repeats. These dynamics generate characteristic slow pressure increases followed by sudden pressure drops (Fig. 1C).
Jamming, intermittency and avalanches are familiar aspects of flowing sand, grains or even jelly beans [24]. To test whether the interplay of growth, collective rearrangement, and outflow of cells from the chamber can be explained by the mechanics of granular materials, we set up coarse-grained computer simulations with cells represented as elastic particles that grow exponentially and reproduce by budding. In our simulations, cells move via frictionless overdamped dynamics with repulsive contact interactions between neighbors.
Our simulations indeed reproduce the intermittent dynamics observed in the experiments (Fig. 2A–C). We find that the pressure drops are roughly exponentially-distributed for both experiments and simulations (Fig. 2D) for P > 〈P〉, consistent with hopper flows [32].
Highly intermittent cell flows might reflect spatially heterogeneous mechanical stresses, a hallmark of driven granular materials [17–20]. Assuming that cell shape deformation is indicative of the forces between cells, we developed a non-invasive method to infer these forces (Fig. 2F, supplementary text, and fig. S1). Using this approach, we analyzed microscopy images to determine stress distributions of crowded populations. Both S. cerevisiae experiments and our coarse-grained simulations exhibit disordered cell packings that are stabilized by heterogeneous force networks (Fig. 2F and G). Stress is highly localized along branching “force chains” [17, 18] while adjacent “spectator cells” [33] experience very little mechanical stress.
We find that jamming-induced contact forces can become so large that they feed back on the cell physiology. Indeed, a feedback on both cell shape and the dynamics of cell growth is evident in experiments where we place two devices of different steady state pressures next to one another, as seen in the time lapse movie (here). To quantify the feedback on growth, we estimate the net growth rate, which is the difference between birth and death rate, in our microfluidic bioreactors by measuring mean cell outflow rate at steady state (supplementary text). We find that the growth rate decays roughly exponentially with pressure until growth is undetectable at a stalling pressure of about 1 MPa (Fig. 3C). The stalling pressure, or homeostatic pressure [34], is obtained by using a special device with a “self-closing valve”, in which yeast populations fully confine themselves by the pressure they build up, as seen in Fig. 3A. In this device, the rate of pressure increase gradually decays with pressure until saturation (Fig. 3B). This diminishing return is due to smaller growth rates at higher pressures, and serves as another, dynamical measure for the feedback between contact pressure and growth rate.
Control experiments supported by finite element simulations show that cells are well-fed and viable even at the highest densities suggesting a mechanobiological origin for the reduced growth rates (supplementary text and figs. S3 and S4). As a first step to uncover the mechanistic basis for the feedback, we have found that contact pressure acts to slow down the cell cycle in the G1 phase (Fig. 3D). Specifically the fraction of cells in G1, indicated by subcellular localization of the protein Whi5, increases with decreasing growth rate (fig. S5). This result consistent with a recent study showing a cell cycle arrest in G1 in compressed mammalian cells [35].
Perhaps the most salient consequence of growth-induced pressure is cell shape deformations. While budding yeast cells grown in the absence of mechanical stresses are nearly spherical, we observe that they morph into convex polyhedra as the population pressure becomes growth-limiting (Fig. 1F and G). Close to the stalling pressure, the packing resembles the structure of a dry foam [36], consisting of cells with only flat faces and sharp edges in between, shown in Fig. 2F. The pressure-induced cell shape deformation can be best visualized at the interface between coverslip and cell population: the cell-coverslip contact area increases as the growth-induced pressure increases (Fig. S6). Our simulations further suggest that the cell turgor pressure in the experiments may increase as a function of the growth-induced pressure.
Most microbial cells are sticky [37, 38]. Indeed, while our lab strains of budding yeast have been domesticated to become non-sticky, wild strains can have strong, velcro-like intercellular fiber connections [39]. We find that while these sticky strains develop a very similar maximal pressure as the lab strains do (Fig. 3B), they develop growth-induced pressures under much weaker confinement (Fig. 4A). Our coarse-grained simulations further suggest that attraction between cells can lead to a build up of pressure much larger than expected under a null model of a liquid droplet with surface tension (Fig. 4C and D).
Bacteria and fungi have the ability to colonize a wide range of porous media, including tiny cavities barely larger than their cell size [3, 4]. Our work suggests that self-driven jamming of growing microbes can emerge in these microenvironments as it does in our microfluidic devices if chemical resources are sufficiently abundant. The mechanism underlying self-driven jamming is cell proliferation, thus extending the notion of jamming, which usually result from external sources of driving, such as shear, compression, or gravity [17–20].
The resulting growth-induced forces endow biofilms with the potential to remodel, or even destroy, their micro-environment. This could aid microbes in penetrating the soft tissues of host organisms [10–12], or to invade soil, where most microbes grow in pores of several micro-meter in diameters [3, 4]. At this length scale, it is possible that the growth-induced pressures measured here contribute to straining of even stiff materials. Indeed, when we grow budding yeast populations inside agar gels, we observe the formation and propagation of cracks (Fig. 4D, fig. S8 and time lapse movie here). Thus, just like jamming of granular media can threaten the mechanical integrity of their confinements, which can lead to the bursting of grain silos [32, 40], it could also be an important mechanical aspect of host invasion [10–12] and biofouling [21].
Author Contributions
O.H. designed and supervised the study. M.D., J.H., S.H. and O.H. designed the microfluidic experiments, M.D. and J.H. developed the software and performed experiments. M.D., J.H., and L.H. fabricated devices. M.D. and L.H. performed Comsol simulations, P.G. implemented and performed the mass-spring simulations, and C.S. implemented and performed the coarsegrained simulations. M.D., J.H., C.S., P.G. and O.H. interpreted the data and wrote the manuscript.
Acknowledgments
We would like to thank Jasper Rine, Jeremy Thorner and Liam Holt for helpful discussions. Research reported in this publication was supported by the National Institute Of General Medical Sciences of the National Institutes of Health under Award Number R01GM115851, by a Simons Investigator award from the Simons Foundation (O.H.) and by the German Research Foundation (DFG) in the framework of the SFB 937/A15. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
Footnotes
↵* Note that, because water can flow in and out of cells, hydrostatic pressures are conceptually very different from contact pressures studied here.