Two continuum models for the spreading of myxobacteria swarms

Mechanisms for bacterial gliding motility on soft substrates

The motility mechanism of certain prokaryotes has long been a mystery, since their motion, known as gliding, involves no external appendages. The physical principles behind gliding still remain poorly understood. Using myxobacteria as an example of such organisms, we identify here the physical principles behind gliding motility and develop a theoretical model that predicts a 2-regime behavior of the gliding speed as a function of the substrate stiffness. Our theory describes the elasto-capillary-hydrodynamic interactions between the membrane of the bacteria, the slime it secretes, and the soft substrate underneath. Defining gliding as the horizontal translation under zero net force, we find the 2-regime behavior is due to 2 distinct mechanisms of motility thrust. On mildly soft substrates, the thrust arises from bacterial shape deformations creating a flow of slime that exerts a pressure along the bacterial length. This pressure in conjunction with the bacterial shape provides the necessary thrust for propulsion. On very soft substrates, however, we show that capillary effects must be considered that lead to the formation of a ridge at the slime-substrate-air interface, thereby creating a thrust in the form of a localized pressure gradient at the bacterial leading edge. To test our theory, we perform experiments with isolated cells on agar substrates of varying stiffness and find the measured gliding speeds in good agreement with the predictions from our elasto-capillary-hydrodynamic model. The mechanisms reported here serve as an important step toward an accurate theory of friction and substrate-mediated interactions between bacteria proliferating in soft media.

Colony Expansion of Socially Motile Myxococcus xanthus Cells Is Driven by Growth, Motility, and Exopolysaccharide Production

Myxococcus xanthus, a model organism for studies of multicellular behavior in bacteria, moves exclusively on solid surfaces using two distinct but coordinated motility mechanisms. One of these, social (S) motility is powered by the extension and retraction of type IV pili and requires the presence of exopolysaccharides (EPS) produced by neighboring cells. As a result, S motility requires close cell-to-cell proximity and isolated cells do not translocate. Previous studies measuring S motility by observing the colony expansion of cells deposited on agar have shown that the expansion rate increases with initial cell density, but the biophysical mechanisms involved remain largely unknown. To understand the dynamics of S motility-driven colony expansion, we developed a reaction-diffusion model describing the effects of cell density, EPS deposition and nutrient exposure on the expansion rate. Our results show that at steady state the population expands as a traveling wave with a speed determined by the interplay of cell motility and growth, a well-known characteristic of Fisher’s equation. The model explains the density-dependence of the colony expansion by demonstrating the presence of a lag phase–a transient period of very slow expansion with a duration dependent on the initial cell density. We propose that at a low initial density, more time is required for the cells to accumulate enough EPS to activate S-motility resulting in a longer lag period. Furthermore, our model makes the novel prediction that following the lag phase the population expands at a constant rate independent of the cell density. These predictions were confirmed by S motility experiments capturing long-term expansion dynamics.

Collective motility is a key mechanism bacteria use to self-organize into multicellular structures and to adapt to various environments. An important example of such behavior is social (S) motility in the gram-negative bacterium Myxococcus xanthus. S-motile cells are restricted to movement in groups and do not move as individual cells. S-motility is powered by type IV pili (TFP)–multi-subunit filaments, which extrude from the cell poles, adhere to the substrate and retract, pulling the cell forward. TFP retraction or adhesion is suggested to be triggered by extracellular exopolysaccharides (EPS) deposited by cells on the substrate. As individual cells synthesize both pili and EPS, it is unclear why S-motile cells only exhibit group movement. Moreover, the experimentally observed initial cell-density dependence of S-motility remains unexplained. To understand these phenomena, we developed a mathematical model for the colony expansion of S-motile cells. Our model hypothesizes that the EPS level regulates the TFP activity that initiates collective cell movements. With this assumption, the model quantitatively matches the density-dependent expansion rate. Moreover, the model predicts two phases during colony expansion: an initial density-dependent lag phase with a slow expansion rate, followed by a faster expansion phase with a density-independent rate. These model predictions were confirmed by long-term colony expansion experiments.

Wavelength selection of rippling patterns in myxobacteria

Rippling patterns of myxobacteria appear in starving colonies before they aggregate to form fruiting bodies. These periodic traveling cell density waves arise from the coordination of individual cell reversals, resulting from an internal clock regulating them and from contact signaling during bacterial collisions. Here we revisit a mathematical model of rippling in myxobacteria due to Igoshin et al. [Proc. Natl. Acad. Sci. USA 98, 14913 (2001)PNASA60027-842410.1073/pnas.221579598 and Phys. Rev. E 70, 041911 (2004)PLEEE81539-375510.1103/PhysRevE.70.041911]. Bacteria in this model are phase oscillators with an extra internal phase through which they are coupled to a mean field of oppositely moving bacteria. Previously, patterns for this model were obtained only by numerical methods, and it was not possible to find their wave number analytically. We derive an evolution equation for the reversal point density that selects the pattern wave number in the weak signaling limit, shows the validity of the selection rule by solving numerically the model equations, and describes other stable patterns in the strong signaling limit. The nonlocal mean-field coupling tends to decohere and confine patterns. Under appropriate circumstances, it can annihilate the patterns leaving a constant density state via a nonequilibrium phase transition reminiscent of destruction of synchronization in the Kuramoto model.

Short-range guiding can result in the formation of circular aggregates in myxobacteria populations

Myxobacteria are social bacteria that upon starvation form multicellular fruiting bodies whose shape in different species can range from simple mounds to elaborate tree-like structures. The formation of fruiting bodies is a result of collective cell movement on a solid surface. In the course of development, groups of flexible rod-shaped cells form streams and move in circular or spiral patterns to form aggregation centers that can become sites of fruiting body formation. The mechanisms of such cell movement patterns are not well understood. It has been suggested that myxobacterial development depends on short-range contact-mediated interactions between individual cells, i.e. cell aggregation does not require long-range signaling in the population. In this study, by means of a computational mass-spring model, we investigate what types of short-range interactions between cells can result in the formation of streams and circular aggregates during myxobacterial development. We consider short-range head-to-tail guiding between individual cells, whereby movement direction of the head of one cell is affected by the nearby presence of the tail of another cell. We demonstrate that stable streams and circular aggregates can arise only when the trailing cell, in addition to being steered by the tail of the leading cell, is able to speed up to catch up with it. It is suggested that necessary head-to-tail interactions between cells can arise from physical adhesion, response to a diffusible substance or slime extruded by cells, or pulling by motility engine pili. Finally, we consider a case of long-range guiding between cells and show that circular aggregates are able to form without cells increasing speed. These findings present a possibility to discriminate between short-range and long-range guiding mechanisms in myxobacteria by experimentally measuring distribution of cell speeds in circular aggregates.

Myxobacteria are social bacteria that upon starvation form multicellular fruiting bodies whose shape in different species can range from simple mounds to elaborate tree-like structures. The formation of fruiting bodies is a result of collective cell movement on a solid surface. Since collective cell motility during biological morphogenesis is also common in higher organisms, myxobacteria serve as a relatively simple model organism to study multicellular movement, organization and development. In the course of myxobacterial development, groups of flexible rod-shaped cells form streams and move in circular or spiral patterns to form aggregation centers that can become sites of fruiting body formation. The mechanisms of such cell movement patterns are not well understood. In this study, by means of a computational mass-spring model, we demonstrate that the formation of streams and circular aggregates during myxobacterial development can be explained by short-range head-to-tail guiding between individual cells, whereby movement direction of the head of one cell is affected by the nearby presence of the tail of another cell. We suggest that such interactions between cells can result from physical adhesion, response to a diffusible substance or slime extruded by cells, or the action of cell motility engine.

Self-organization in bacterial swarming: lessons from myxobacteria

When colonizing surfaces, many bacteria are able to self-organize into an actively expanding biofilm, in which millions of cells move smoothly and orderly at high densities. This phenomenon is known as bacterial swarming. Despite the apparent resemblance to patterns seen in liquid crystals, the dynamics of bacterial swarming cannot be explained by theories derived from equilibrium statistical mechanics. To understand how bacteria swarm, a central question is how order emerges in dense and initially disorganized populations of bacterial cells. Here we briefly review recent efforts, with integrated computational and experimental approaches, in addressing this question.

Dynamic-energy-budget-driven fruiting-body formation in myxobacteria

We develop an interacting particle model to simulate the life cycle of myxobacteria, which consists of two main stages--the swarming stage and the development (fruiting body formation) stage. As experiments have shown that the phase transition from swarming to development stage is triggered by starvation, we incorporate into the simulation a system of ordinary differential equations (ODEs) called the dynamic energy budget, which controls the uptake and use of energy by individuals. This inclusion successfully automates the phase transition in our simulation. Only one parameter, namely, the food density, controls the entire simulation of the life cycle.

Submicron trenches reduce the Pseudomonas fluorescens colonization rate on solid surfaces

Bacterial adhesion and spreading on biomaterials are considered key features of pathogenicity. Roughness and topography of the substrate have been reported to affect bacterial adhesion, but little is known about their effect on spreading. Submicron row and channel tuning with bacterial diameter (S2) were designed to test bacterial motility on these surfaces. Random nanometer-sized structures (S1) were used as controls. Optical microscopy and AFM were employed to detect biological and surface pattern details in the micro- and nanoscale, respectively. Results showed that motility strategies (flagella orientation, elongation, aggregation in rafts, formation of network structures, and development of a bacterial frontier) were affected by the presence of submicropatterns. Importantly, the rate of bacterial spreading on S2 was significantly reduced and influenced by the orientation of the submicropatterns. Consequently, submicroengineered substrates could be employed as a tool to downgrade bacterial colonization. Such patterns could impact on the design of proper engineered structures to control biofilm spreading on solid surfaces.

Social interactions in myxobacterial swarming

Swarming, a collective motion of many thousands of cells, produces colonies that rapidly spread over surfaces. In this paper, we introduce a cell-based model to study how interactions between neighboring cells facilitate swarming. We chose to study Myxococcus xanthus, a species of myxobacteria, because it swarms rapidly and has well-defined cell–cell interactions mediated by type IV pili and by slime trails. The aim of this paper is to test whether the cell contact interactions, which are inherent in pili-based S motility and slime-based A motility, are sufficient to explain the observed expansion of wild-type swarms. The simulations yield a constant rate of swarm expansion, which has been observed experimentally. Also, the model is able to quantify the contributions of S motility and A motility to swarming. Some pathogenic bacteria spread over infected tissue by swarming. The model described here may shed some light on their colonization process.

Many bacteria are able to spread rapidly over the surface using a strategy called swarming. When the cells cover a surface at high density and compete with each other for nutrients, swarming permits them to maintain rapid growth at the swarm edge. Swarming with flagella has been investigated for many years, and much has been learned about its regulation. Nevertheless, its choreography, which is somewhat related to the counterflow of pedestrians on a city sidewalk, has remained elusive. It is the bacterial equivalent of dancing toward the exit in a crowd of moving bodies that usually are in close contact. Myxococcus xanthus expands its swarms at 1.6 μm/min, about a third the speed of individual cells gliding over the same surface. Each cell has pilus engines at its front end and slime secretion engines at its rear. Using the known mechanics of these engines and the ways they are coordinated, we have developed a cell-based model to study the role of social interactions in bacterial swarming. The model is able to quantify the contributions of individual motility engines to swarming. It also shows that microscopic social interactions help to form the ordered collective motion observed in swarms.