Pattern formation and traveling waves in myxobacteria: theory and modeling

Exotic swarming dynamics of high-dimensional swarmalators

Swarmalators are oscillators that can swarm as well as sync via a dynamic balance between their spatial proximity and phase similarity. Swarmalator models employed so far in the literature comprise only one-dimensional phase variables to represent the intrinsic dynamics of the natural collectives. Nevertheless, the latter can indeed be represented more realistically by high-dimensional phase variables. For instance, the alignment of velocity vectors in a school of fish or a flock of birds can be more realistically set up in three-dimensional space, while the alignment of opinion formation in population dynamics could be multidimensional, in general. We present a generalized D-dimensional swarmalator model, which more accurately captures self-organizing behaviors of a plethora of real-world collectives by self-adaptation of high-dimensional spatial and phase variables. For a more sensible visualization and interpretation of the results, we restrict our simulations to three-dimensional spatial and phase variables. Our model provides a framework for modeling complicated processes such as flocking, schooling of fish, cell sorting during embryonic development, residential segregation, and opinion dynamics in social groups. We demonstrate its versatility by capturing the maneuvers of a school of fish, qualitatively and quantitatively, by a suitable extension of the original model to incorporate appropriate features besides a gallery of its intrinsic self-organizations for various interactions. We expect the proposed high-dimensional swarmalator model to be potentially useful in describing swarming systems and programmable and reconfigurable collectives in a wide range of disciplines, including the physics of active matter, developmental biology, sociology, and engineering.

Order parameter dynamics in complex systems: From models to data

Collective ordering behaviors are typical macroscopic manifestations embedded in complex systems and can be ubiquitously observed across various physical backgrounds. Elements in complex systems may self-organize via mutual or external couplings to achieve diverse spatiotemporal coordinations. The order parameter, as a powerful quantity in describing the transition to collective states, may emerge spontaneously from large numbers of degrees of freedom through competitions. In this minireview, we extensively discussed the collective dynamics of complex systems from the viewpoint of order-parameter dynamics. A synergetic theory is adopted as the foundation of order-parameter dynamics, and it focuses on the self-organization and collective behaviors of complex systems. At the onset of macroscopic transitions, slow modes are distinguished from fast modes and act as order parameters, whose evolution can be established in terms of the slaving principle. We explore order-parameter dynamics in both model-based and data-based scenarios. For situations where microscopic dynamics modeling is available, as prototype examples, synchronization of coupled phase oscillators, chimera states, and neuron network dynamics are analytically studied, and the order-parameter dynamics is constructed in terms of reduction procedures such as the Ott-Antonsen ansatz, the Lorentz ansatz, and so on. For complicated systems highly challenging to be well modeled, we proposed the eigen-microstate approach (EMP) to reconstruct the macroscopic order-parameter dynamics, where the spatiotemporal evolution brought by big data can be well decomposed into eigenmodes, and the macroscopic collective behavior can be traced by Bose-Einstein condensation-like transitions and the emergence of dominant eigenmodes. The EMP is successfully applied to some typical examples, such as phase transitions in the Ising model, climate dynamics in earth systems, fluctuation patterns in stock markets, and collective motion in living systems.

Mathematical modeling of mechanosensitive reversal control in Myxococcus xanthus

Adjusting motility patterns according to environmental cues is important for bacterial survival. Myxococcus xanthus, a bacterium moving on surfaces by gliding and twitching mechanisms, modulates the reversal frequency of its front-back polarity in response to mechanical cues like substrate stiffness and cell-cell contact. In this study, we propose that M. xanthus's gliding machinery senses environmental mechanical cues during force generation and modulates cell reversal accordingly. To examine our hypothesis, we expand an existing mathematical model for periodic polarity reversal in M. xanthus, incorporating the experimental data on the intracellular dynamics of the gliding machinery and the interaction between the gliding machinery and a key polarity regulator. The model successfully reproduces the dependence of cell reversal frequency on substrate stiffness observed in M. xanthus gliding. We further propose reversal control networks between the gliding and twitching motility machineries to explain the opposite reversal responses observed in wild type M. xanthus cells that possess both motility mechanisms. These results provide testable predictions for future experimental investigations. In conclusion, our model suggests that the gliding machinery in M. xanthus can function as a mechanosensor, which transduces mechanical cues into a cell reversal signal.

Soliton approximation in continuum models of leader-follower behavior

Complex biological processes involve collective behavior of entities (bacteria, cells, animals) over many length and time scales and can be described by discrete models that track individuals or by continuum models involving densities and fields. We consider hybrid stochastic agent-based models of branching morphogenesis and angiogenesis (new blood vessel creation from preexisting vasculature), which treat cells as individuals that are guided by underlying continuous chemical and/or mechanical fields. In these descriptions, leader (tip) cells emerge from existing branches and follower (stalk) cells build the new sprout in their wake. Vessel branching and fusion (anastomosis) occur as a result of tip and stalk cell dynamics. Coarse graining these hybrid models in appropriate limits produces continuum partial differential equations (PDEs) for endothelial cell densities that are more analytically tractable. While these models differ in nonlinearity, they produce similar equations at leading order when chemotaxis is dominant. We analyze this leading order system in a simple quasi-one-dimensional geometry and show that the numerical solution of the leading order PDE is well described by a soliton wave that evolves from vessel to source. This wave is an attractor for intermediate times until it arrives at the hypoxic region releasing the growth factor. The mathematical techniques used here thus identify common features of discrete and continuum approaches and provide insight into general biological mechanisms governing their collective dynamics.

Cell behaviors underlying Myxococcus xanthus aggregate dispersal

IMPORTANCE

Understanding the processes behind bacterial biofilm formation, maintenance, and dispersal is essential for addressing their effects on health and ecology. Within these multicellular communities, various cues can trigger differentiation into distinct cell types, allowing cells to adapt to their specific local environment. The soil bacterium Myxococcus xanthus forms biofilms in response to starvation, marked by cells aggregating into mounds. Some aggregates persist as spore-filled fruiting bodies, while others disperse after initial formation for unknown reasons. Here, we use a combination of cell tracking analysis and computational simulations to identify behaviors at the cellular level that contribute to aggregate dispersal. Our results suggest that cells in aggregates actively determine whether to disperse or persist and undergo a transition to sporulation based on a self-produced cue related to the aggregate size. Identifying these cues is an important step in understanding and potentially manipulating bacterial cell-fate decisions.

Unidirectional and phase-gated signaling synchronizes murine presomitic mesoderm cells

Oscillator systems achieve synchronization when oscillators are coupled. The presomitic mesoderm is a system of cellular oscillators, where coordinated genetic activity is necessary for proper periodic generation of somites. While Notch signaling is required for the synchronization of these cells, it is unclear what information the cells exchange and how they react to this information to align their oscillatory pace with that of their neighbors. Combining mathematical modeling and experimental data, we found that interaction between murine presomitic mesoderm cells is controlled by a phase-gated and unidirectional coupling mechanism and results in deceleration of their oscillation pace upon Notch signaling. This mechanism predicts that isolated populations of well-mixed cells synchronize, revealing a stereotypical synchronization in the mouse PSM and contradicting expectations from previously applied theoretical approaches. Collectively, our theoretical and experimental findings reveal the underlying coupling mechanisms of the presomitic mesoderm cells and provide a framework to quantitatively characterize their synchronization.

Formation and emergent dynamics of spatially organized microbial systems

Spatial organization is the norm rather than the exception in the microbial world. While the study of microbial physiology has been dominated by studies in well-mixed cultures, there is now increasing interest in understanding the role of spatial organization in microbial physiology, coexistence and evolution. Where studied, spatial organization has been shown to influence all three of these aspects. In this mini review and perspective article, we emphasize that the dynamics within spatially organized microbial systems (SOMS) are governed by feedbacks between local physico-chemical conditions, cell physiology and movement, and evolution. These feedbacks can give rise to emergent dynamics, which need to be studied through a combination of spatio-temporal measurements and mathematical models. We highlight the initial formation of SOMS and their emergent dynamics as two open areas of investigation for future studies. These studies will benefit from the development of model systems that can mimic natural ones in terms of species composition and spatial structure.

Collective Motion and Pattern Formation in Phase-Synchronizing Active Fluids

Many active particles, such as swimming micro-organisms or motor proteins, do work on their environment by going though a periodic sequence of shapes. Interactions between particles can lead to synchronization of their duty cycles. Here, we study the collective dynamics of a suspension of active particles coupled through hydrodynamics. We find that at high enough density the system transitions to a state of collective motion by a mechanism that is distinct from other instabilities in active matter systems. Second, we demonstrate that the emergent nonequilibrium states feature stationary chimera patterns in which synchronized and phase-isotropic regions coexist. Third, we show that in confinement, oscillatory flows and robust unidirectional pumping states exist, and can be selected by choice of alignment boundary conditions. These results point toward a new route to collective motion and pattern formation and could guide the design of new active materials.

Diverse behaviors in non-uniform chiral and non-chiral swarmalators

We study the emergent behaviors of a population of swarming coupled oscillators, dubbed swarmalators. Previous work considered the simplest, idealized case: identical swarmalators with global coupling. Here we expand this work by adding more realistic features: local coupling, non-identical natural frequencies, and chirality. This more realistic model generates a variety of new behaviors including lattices of vortices, beating clusters, and interacting phase waves. Similar behaviors are found across natural and artificial micro-scale collective systems, including social slime mold, spermatozoa vortex arrays, and Quincke rollers. Our results indicate a wide range of future use cases, both to aid characterization and understanding of natural swarms, and to design complex interactions in collective systems from soft and active matter to micro-robotics.

Flagellar Motor Transformed: Biophysical Perspectives of the Myxococcus xanthus Gliding Mechanism

Many bacteria move on solid surfaces using gliding motility, without involvement of flagella or pili. Gliding of Myxococcus xanthus is powered by a proton channel homologous to the stators in the bacterial flagellar motor. Instead of being fixed in place and driving the rotation of a circular protein track like the flagellar basal body, the gliding machinery of M. xanthus travels the length of the cell along helical trajectories, while mechanically engaging with the substrate. Such movement entails a different molecular mechanism to generate propulsion on the cell. In this perspective, we will discuss the similarities and differences between the M. xanthus gliding machinery and bacterial flagellar motor, and use biophysical principles to generate hypotheses about the operating mechanism, efficiency, sensitivity to control, and mechanosensing of M. xanthus gliding.

Formation, collective motion, and merging of macroscopic bacterial aggregates

Chemotactic bacteria form emergent spatial patterns of variable cell density within cultures that are initially spatially uniform. These patterns are the result of chemical gradients that are created from the directed movement and metabolic activity of billions of cells. A recent study on pattern formation in wild bacterial isolates has revealed unique collective behaviors of the bacteria Enterobacter cloacae. As in other bacterial species, Enterobacter cloacae form macroscopic aggregates. Once formed, these bacterial clusters can migrate several millimeters, sometimes resulting in the merging of two or more clusters. To better understand these phenomena, we examine the formation and dynamics of thousands of bacterial clusters that form within a 22 cm square culture dish filled with soft agar over two days. At the macroscale, the aggregates display spatial order at short length scales, and the migration of cell clusters is superdiffusive, with a merging acceleration that is correlated with aggregate size. At the microscale, aggregates are composed of immotile cells surrounded by low density regions of motile cells. The collective movement of the aggregates is the result of an asymmetric flux of bacteria at the boundary. An agent-based model is developed to examine how these phenomena are the result of both chemotactic movement and a change in motility at high cell density. These results identify and characterize a new mechanism for collective bacterial motility driven by a transient, density-dependent change in motility.

Bacteria growing and swimming in soft agar often aggregate to form elaborate spatial patterns. Here we examine the patterns formed by the bacteria Enterobacter cloacae. An unusual behavior of these bacteria is the collective movement of cells after the initial aggregation into a tiny spot. Despite the majority of the cells within an aggregate being immotile at any point in the time, the flux of cells entering and leaving the aggregate, as motility is lost and regained in individual cells, led to a net, collective movement of the aggregate. These spots sometimes run into each other and combine. By looking at the cells within these spots under a microscope, we find that cells within each spot stop swimming. The process of switching back and forth between swimming and not swimming causes the movement and fusion of the spots. A numerical simulation shows that the migration and merging of these spots can be expected if the cells swim towards regions of space with high concentrations of attractant molecules and stop swimming in locations crowded with many cells. This work identifies a novel process through which populations of bacteria cooperate and control the movement of large groups of cells.

Collective behavior of swarmalators on a ring

We study the collective behavior of swarmalators, generalizations of phase oscillators that both sync and swarm, confined to move on a one-dimensional (1D) ring. This simple model captures the essence of movement in two or three dimensions, but has the benefit of being solvable: most of the collective states and their bifurcations can be specified exactly. The model also captures the behavior of real-world swarmalators which swarm in quasi-1D rings such as bordertaxic vinegar eels and sperm.

Emergent Myxobacterial Behaviors Arise from Reversal Suppression Induced by Kin Contacts

A wide range of biological systems, from microbial swarms to bird flocks, display emergent behaviors driven by coordinated movement of individuals. To this end, individual organisms interact by recognizing their kin and adjusting their motility based on others around them. However, even in the best-studied systems, the mechanistic basis of the interplay between kin recognition and motility coordination is not understood. Here, using a combination of experiments and mathematical modeling, we uncover the mechanism of an emergent social behavior in Myxococcus xanthus. By overexpressing the cell surface adhesins TraA and TraB, which are involved in kin recognition, large numbers of cells adhere to one another and form organized macroscopic circular aggregates that spin clockwise or counterclockwise. Mechanistically, TraAB adhesion results in sustained cell-cell contacts that trigger cells to suppress cell reversals, and circular aggregates form as the result of cells’ ability to follow their own cellular slime trails. Furthermore, our in silico simulations demonstrate a remarkable ability to predict self-organization patterns when phenotypically distinct strains are mixed. For example, defying naive expectations, both models and experiments found that strains engineered to overexpress different and incompatible TraAB adhesins nevertheless form mixed circular aggregates. Therefore, this work provides key mechanistic insights into M. xanthus social interactions and demonstrates how local cell contacts induce emergent collective behaviors by millions of cells.

IMPORTANCE In many species, large populations exhibit emergent behaviors whereby all related individuals move in unison. For example, fish in schools can all dart in one direction simultaneously to avoid a predator. Currently, it is impossible to explain how such animals recognize kin through brain cognition and elicit such behaviors at a molecular level. However, microbes also recognize kin and exhibit emergent collective behaviors that are experimentally tractable. Here, using a model social bacterium, we engineer dispersed individuals to organize into synchronized collectives that create emergent patterns. With experimental and mathematical approaches, we explain how this occurs at both molecular and population levels. The results demonstrate how the combination of local physical interactions triggers intracellular signaling, which in turn leads to emergent behaviors on a population scale.

Quantification of Myxococcus xanthus Aggregation and Rippling Behaviors: Deep-Learning Transformation of Phase-Contrast into Fluorescence Microscopy Images

Myxococcus xanthus bacteria are a model system for understanding pattern formation and collective cell behaviors. When starving, cells aggregate into fruiting bodies to form metabolically inert spores. During predation, cells self-organize into traveling cell-density waves termed ripples. Both phase-contrast and fluorescence microscopy are used to observe these patterns but each has its limitations. Phase-contrast images have higher contrast, but the resulting image intensities lose their correlation with cell density. The intensities of fluorescence microscopy images, on the other hand, are well-correlated with cell density, enabling better segmentation of aggregates and better visualization of streaming patterns in between aggregates; however, fluorescence microscopy requires the engineering of cells to express fluorescent proteins and can be phototoxic to cells. To combine the advantages of both imaging methodologies, we develop a generative adversarial network that converts phase-contrast into synthesized fluorescent images. By including an additional histogram-equalized output to the state-of-the-art pix2pixHD algorithm, our model generates accurate images of aggregates and streams, enabling the estimation of aggregate positions and sizes, but with small shifts of their boundaries. Further training on ripple patterns enables accurate estimation of the rippling wavelength. Our methods are thus applicable for many other phenotypic behaviors and pattern formation studies.

Binary mixtures of locally coupled mobile oscillators

Synchronized behavior in a system of coupled dynamic objects is a fascinating example of an emerged cooperative phenomena which has been observed in systems as diverse as a group of insects, neural networks, or networks of computers. In many instances, however, the synchronization is undesired because it may lead to system malfunctioning, as in the case of Alzheimer's and Parkinson's diseases, for example. Recent studies of static networks of oscillators have shown that the presence of a small fraction of so-called contrarian oscillators can suppress the undesired network synchronization. On the other hand, it is also known that the mobility of the oscillators can significantly impact their synchronization dynamics. Here, we combine these two ideas-the oscillator mobility and the presence of heterogeneous interactions-and study numerically binary mixtures of phase oscillators performing two-dimensional random walks. Within the framework of a generalized Kuramoto model, we introduce two phase-coupling schemes. The first one is invariant when the types of any two oscillators are swapped, while the second model is not. We demonstrate that the symmetric model does not allow for a complete suppression of the synchronized state. However, it provides means for a robust control of the synchronization timescale by varying the overall number density and the composition of the mixture and the strength of the off-diagonal Kuramoto coupling constant. Instead, the asymmetric model predicts that the coherent state can be eliminated within a subpopulation of normal oscillators and evoked within a subpopulation of the contrarians.

Linking single-cell decisions to collective behaviours in social bacteria

Social bacteria display complex behaviours whereby thousands of cells collectively and dramatically change their form and function in response to nutrient availability and changing environmental conditions. In this review, we focus on Myxococcus xanthus motility, which supports spectacular transitions based on prey availability across its life cycle. A large body of work suggests that these behaviours require sensory capacity implemented at the single-cell level. Focusing on recent genetic work on a core cellular pathway required for single-cell directional decisions, we argue that signal integration, multi-modal sensing and memory are at the root of decision making leading to multicellular behaviours. Hence, Myxococcus may be a powerful biological system to elucidate how cellular building blocks cooperate to form sensory multicellular assemblages, a possible origin of cognitive mechanisms in biological systems. This article is part of the theme issue 'Basal cognition: conceptual tools and the view from the single cell'.

Emergent behavior in an adversarial synchronization and swarming model

We consider a red-versus-blue coupled synchronization and spatial swarming (i.e., swarmalator) model that incorporates attraction and repulsion terms and an adversarial game of phases. The model exhibits behaviors such as spontaneous emergence of tactical manoeuvres of envelopment (e.g., flanking, pincer, and envelopment) that are often proposed in military theory or observed in nature. We classify these states based on a large set of features such as spatial densities, synchronization between clusters, and measures of cluster distances. These features are used to study the influence of coupling parameters on the expected presence of these states and the-sometimes sharp-transitions between them.

The ups and downs of S. aureus nasal carriage

Staphylococcus aureus infections are a growing concern worldwide due to the increasing number of strains that exhibit antibiotic resistance. Recent studies have indicated that some percentage of people carry the bacteria in the nasal cavity and therefore are at a higher risk of subsequent, and more serious, infections in other parts of the body. However, individuals carrying the infection can be classified as only intermittent carriers versus persistent carriers, being able to eliminate the bacteria and later colonized again. Using a model of bacterial colonization of the anterior nares, we investigate oscillatory patterns related to intermittent carriage of S. aureus. Following several studies using global sensitivity analysis techniques, various insights into the model's behaviour were made including interacting effects of the bacteria's growth rate and movement in the mucus, suggesting parameter connections associated with biofilm-like behaviour. Here the bacterial growth rate and bacterial movement are explicitly connected, leading to expanded oscillatory behaviour in the model. We suggest possible implications that this oscillatory behaviour can have on the definition of intermittent carriage and discuss differences in the bacterial virulence dependent upon individual host health. Furthermore, we show that connecting the bacterial growth and movement also expands the region of the parameter space for which the bacteria are able to survive and persist.

Swarm Robotic Behaviors and Current Applications

In swarm robotics multiple robots collectively solve problems by forming advantageous structures and behaviors similar to the ones observed in natural systems, such as swarms of bees, birds, or fish. However, the step to industrial applications has not yet been made successfully. Literature is light on real-world swarm applications that apply actual swarm algorithms. Typically, only parts of swarm algorithms are used which we refer to as basic swarm behaviors. In this paper we collect and categorize these behaviors into spatial organization, navigation, decision making, and miscellaneous. This taxonomy is then applied to categorize a number of existing swarm robotic applications from research and industrial domains. Along with the classification, we give a comprehensive overview of research platforms that can be used for testing and evaluating swarm behavior, systems that are already on the market, and projects that target a specific market. Results from this survey show that swarm robotic applications are still rare today. Many industrial projects still rely on centralized control, and even though a solution with multiple robots is employed, the principal idea of swarm robotics of distributed decision making is neglected. We identified mainly following reasons: First of all, swarm behavior emerging from local interactions is hard to predict and a proof of its eligibility for applications in an industrial context is difficult to provide. Second, current communication architectures often do not match requirements for swarm communication, which often leads to a system with a centralized communication infrastructure. Finally, testing swarms for real industrial applications is an issue, since deployment in a productive environment is typically too risky and simulations of a target system may not be sufficiently accurate. In contrast, the research platforms present a means for transforming swarm robotics solutions from theory to prototype industrial systems.

Biophysics at the coffee shop: lessons learned working with George Oster

Over the past 50 years, the use of mathematical models, derived from physical reasoning, to describe molecular and cellular systems has evolved from an art of the few to a cornerstone of biological inquiry. George Oster stood out as a pioneer of this paradigm shift from descriptive to quantitative biology not only through his numerous research accomplishments, but also through the many students and postdocs he mentored over his long career. Those of us fortunate enough to have worked with George agree that his sharp intellect, physical intuition, and passion for scientific inquiry not only inspired us as scientists but also greatly influenced the way we conduct research. We would like to share a few important lessons we learned from George in honor of his memory and with the hope that they may inspire future generations of scientists.

Contrarian compulsions produce exotic time-dependent flocking of active particles

Animals having a tendency to align their velocities to an average of those of their neighbors may flock as illustrated by the Vicsek model and its variants. If, in addition, they feel a systematic contrarian trend, the result may be a time periodic adjustment of the flock or period doubling in time. These exotic phases are predicted from kinetic theory and numerically found in a modified two-dimensional Vicsek model of self-propelled particles. Numerical simulations demonstrate striking effects of alignment noise on the polarization order parameter measuring particle flocking: maximum polarization length is achieved at an optimal nonzero noise level. When contrarian compulsions are more likely than conformist ones, nonuniform polarized phases appear as the noise surpasses threshold.

A statistical physics view of swarming bacteria

Bacterial swarming is a collective mode of motion in which cells migrate rapidly over surfaces, forming dynamic patterns of whirls and jets. This review presents a physical point of view of swarming bacteria, with an emphasis on the statistical properties of the swarm dynamics as observed in experiments. The basic physical principles underlying the swarm and their relation to contemporary theories of collective motion and active matter are reviewed and discussed in the context of the biological properties of swarming cells. We suggest a paradigm according to which bacteria have optimized some of their physical properties as a strategy for rapid surface translocation. In other words, cells take advantage of favorable physics, enabling efficient expansion that enhances survival under harsh conditions.

Agent-Based Modeling Reveals Possible Mechanisms for Observed Aggregation Cell Behaviors

Myxococcus xanthus is a soil bacterium that serves as a model system for biological self-organization. Cells form distinct, dynamic patterns depending on environmental conditions. An agent-based model was used to understand how M. xanthus cells aggregate into multicellular mounds in response to starvation. In this model, each cell is modeled as an agent represented by a point particle and characterized by its position and moving direction. At low agent density, the model recapitulates the dynamic patterns observed by experiments and a previous biophysical model. To study aggregation at high cell density, we extended the model based on the recent experimental observation that cells exhibit biased movement toward aggregates. We tested two possible mechanisms for this biased movement and demonstrate that a chemotaxis model with adaptation can reproduce the observed experimental results leading to the formation of stable aggregates. Furthermore, our model reproduces the experimentally observed patterns of cell alignment around aggregates.

Ring states in swarmalator systems

Synchronization is a universal phenomenon, occurring in systems as disparate as Japanese tree frogs and Josephson junctions. Typically, the elements of synchronizing systems adjust the phases of their oscillations, but not their positions in space. The reverse scenario is found in swarming systems, such as schools of fish or flocks of birds; now the elements adjust their positions in space, but without (noticeably) changing their internal states. Systems capable of both swarming and synchronizing, dubbed swarmalators, have recently been proposed, and analyzed in the continuum limit. Here, we extend this work by studying finite populations of swarmalators, whose phase similarity affects both their spatial attraction and repulsion. We find ring states, and compute criteria for their existence and stability. Larger populations can form annular distributions, whose density we calculate explicitly. These states may be observable in groups of Japanese tree frogs, ferromagnetic colloids, and other systems with an interplay between swarming and synchronization.

Counter-propagating wave patterns in a swarm model with memory

Hyperbolic transport-reaction equations are abundant in the description of movement of motile organisms. Here, we focus on a system of four coupled transport-reaction equations that arises from an age-structuring of a species of turning individuals. By modeling how the behavior depends on the time since the last reversal, we introduce a memory effect. The highlight consists of the explicit construction and characterization of counter-propagating traveling waves, patterns which have been observed in bacterial colonies. Stability analysis reveals conditions for the wave formation as well as pulsating-in-time spatially constant solutions.

Dissipative structures in biological systems: bistability, oscillations, spatial patterns and waves

The goal of this review article is to assess how relevant is the concept of dissipative structure for understanding the dynamical bases of non-equilibrium self-organization in biological systems, and to see where it has been applied in the five decades since it was initially proposed by Ilya Prigogine. Dissipative structures can be classified into four types, which will be considered, in turn, and illustrated by biological examples: (i) multistability, in the form of bistability and tristability, which involve the coexistence of two or three stable steady states, or in the form of birhythmicity, which involves the coexistence between two stable rhythms; (ii) temporal dissipative structures in the form of sustained oscillations, illustrated by biological rhythms; (iii) spatial dissipative structures, known as Turing patterns; and (iv) spatio-temporal structures in the form of propagating waves. Rhythms occur with widely different periods at all levels of biological organization, from neural, cardiac and metabolic oscillations to circadian clocks and the cell cycle; they play key roles in physiology and in many disorders. New rhythms are being uncovered while artificial ones are produced by synthetic biology. Rhythms provide the richest source of examples of dissipative structures in biological systems. Bistability has been observed experimentally, but has primarily been investigated in theoretical models in an increasingly wide range of biological contexts, from the genetic to the cell and animal population levels, both in physiological conditions and in disease. Bistable transitions have been implicated in the progression between the different phases of the cell cycle and, more generally, in the process of cell fate specification in the developing embryo. Turing patterns are exemplified by the formation of some periodic structures in the course of development and by skin stripe patterns in animals. Spatio-temporal patterns in the form of propagating waves are observed within cells as well as in intercellular communication. This review illustrates how dissipative structures of all sorts abound in biological systems.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)'.

Epithelial vertex models with active biochemical regulation of contractility can explain organized collective cell motility

Collective motions of groups of cells are observed in many biological settings such as embryo development, tissue formation, and cancer metastasis. To effectively model collective cell movement, it is important to incorporate cell specific features such as cell size, cell shape, and cell mechanics, as well as active behavior of cells such as protrusion and force generation, contractile forces, and active biochemical signaling mechanisms that regulate cell behavior. In this paper, we develop a comprehensive model of collective cell migration in confluent epithelia based on the vertex modeling approach. We develop a method to compute cell-cell viscous friction based on the vertex model and incorporate RhoGTPase regulation of cortical myosin contraction. Global features of collective cell migration are examined by computing the spatial velocity correlation function. As active cell force parameters are varied, we found rich dynamical behavior. Furthermore, we find that cells exhibit nonlinear phenomena such as contractile waves and vortex formation. Together our work highlights the importance of active behavior of cells in generating collective cell movement. The vertex modeling approach is an efficient and versatile approach to rigorously examine cell motion in the epithelium.

A gated relaxation oscillator mediated by FrzX controls morphogenetic movements in Myxococcus xanthus

Dynamic control of cell polarity is of critical importance for many aspects of cellular development and motility. In Myxococcus xanthus, MglA, a G protein, and MglB, its cognate GTPase-activating protein, establish a polarity axis that defines the direction of movement of the cell and that can be rapidly inverted by the Frz chemosensory system. Although vital for collective cell behaviours, how Frz triggers this switch has remained unknown. Here, we use genetics, imaging and mathematical modelling to show that Frz controls polarity reversals via a gated relaxation oscillator. FrzX, which we identify as a target of the Frz kinase, provides the gating and thus acts as the trigger for reversals. Slow relocalization of the polarity protein RomR then creates a refractory period during which another switch cannot be triggered. A secondary Frz output, FrzZ, decreases this delay, allowing rapid reversals when required. Thus, this architecture results in a highly tuneable switch that allows a wide range of reversal frequencies.

Oscillators that sync and swarm

Synchronization occurs in many natural and technological systems, from cardiac pacemaker cells to coupled lasers. In the synchronized state, the individual cells or lasers coordinate the timing of their oscillations, but they do not move through space. A complementary form of self-organization occurs among swarming insects, flocking birds, or schooling fish; now the individuals move through space, but without conspicuously altering their internal states. Here we explore systems in which both synchronization and swarming occur together. Specifically, we consider oscillators whose phase dynamics and spatial dynamics are coupled. We call them swarmalators, to highlight their dual character. A case study of a generalized Kuramoto model predicts five collective states as possible long-term modes of organization. These states may be observable in groups of sperm, Japanese tree frogs, colloidal suspensions of magnetic particles, and other biological and physical systems in which self-assembly and synchronization interact.

Collective self-organized behavior can be observed in a variety of systems such as colloids and microswimmers. Here O’Keeffe et al. propose a model of oscillators which move in space and tend to synchronize with neighboring oscillators and outline five types of collective self-organized states.

Wavenumber selection in coupled transport equations

We study mechanisms for wavenumber selection in a minimal model for run-and-tumble dynamics. We show that nonlinearity in tumbling rates induces the existence of a plethora of traveling- and standing-wave patterns, as well as a subtle selection mechanism for the wavenumbers of spatio-temporally periodic waves. We comment on possible implications for rippling patterns observed in colonies of myxobacteria.

Mesoscale pattern formation of self-propelled rods with velocity reversal

We study self-propelled particles with velocity reversal interacting by uniaxial (nematic) alignment within a coarse-grained hydrodynamic theory. Combining analytical and numerical continuation techniques, we show that the physics of this active system is essentially controlled by the reversal frequency. In particular, we find that elongated, high-density, ordered patterns, called bands, emerge via subcritical bifurcations from spatially homogeneous states. Our analysis reveals further that the interaction of bands is weakly attractive and, consequently, bands fuse upon collision in analogy with nonequilibrium nucleation processes. Moreover, we demonstrate that a renormalized positive line tension can be assigned to stable bands below a critical reversal rate, beyond which they are transversally unstable. In addition, we discuss the kinetic roughening of bands as well as their nonlinear dynamics close to the threshold of transversal instability. Altogether, the reduction of the multiparticle system onto the dynamics of bands provides a unified framework to understand the emergence and stability of nonequilibrium patterns in this self-propelled particle system. In this regard, our results constitute a proof of principle in favor of the hypothesis in microbiology that velocity reversal of gliding rod-shaped bacteria regulates the transitions between various self-organized patterns observed during the bacterial life cycle.

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.

Myxobacteria: Moving, Killing, Feeding, and Surviving Together

Myxococcus xanthus, like other myxobacteria, is a social bacterium that moves and feeds cooperatively in predatory groups. On surfaces, rod-shaped vegetative cells move in search of the prey in a coordinated manner, forming dynamic multicellular groups referred to as swarms. Within the swarms, cells interact with one another and use two separate locomotion systems. Adventurous motility, which drives the movement of individual cells, is associated with the secretion of slime that forms trails at the leading edge of the swarms. It has been proposed that cellular traffic along these trails contributes to M. xanthus social behavior via stigmergic regulation. However, most of the cells travel in groups by using social motility, which is cell contact-dependent and requires a large number of individuals. Exopolysaccharides and the retraction of type IV pili at alternate poles of the cells are the engines associated with social motility. When the swarms encounter prey, the population of M. xanthus lyses and takes up nutrients from nearby cells. This cooperative and highly density-dependent feeding behavior has the advantage that the pool of hydrolytic enzymes and other secondary metabolites secreted by the entire group is shared by the community to optimize the use of the degradation products. This multicellular behavior is especially observed in the absence of nutrients. In this condition, M. xanthus swarms have the ability to organize the gliding movements of 1000s of rods, synchronizing rippling waves of oscillating cells, to form macroscopic fruiting bodies, with three subpopulations of cells showing division of labor. A small fraction of cells either develop into resistant myxospores or remain as peripheral rods, while the majority of cells die, probably to provide nutrients to allow aggregation and spore differentiation. Sporulation within multicellular fruiting bodies has the benefit of enabling survival in hostile environments, and increases germination and growth rates when cells encounter favorable conditions. Herein, we review how these social bacteria cooperate and review the main cell–cell signaling systems used for communication to maintain multicellularity.

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.

Mechanism for Collective Cell Alignment in Myxococcus xanthus Bacteria

Myxococcus xanthus cells self-organize into aligned groups, clusters, at various stages of their lifecycle. Formation of these clusters is crucial for the complex dynamic multi-cellular behavior of these bacteria. However, the mechanism underlying the cell alignment and clustering is not fully understood. Motivated by studies of clustering in self-propelled rods, we hypothesized that M. xanthus cells can align and form clusters through pure mechanical interactions among cells and between cells and substrate. We test this hypothesis using an agent-based simulation framework in which each agent is based on the biophysical model of an individual M. xanthus cell. We show that model agents, under realistic cell flexibility values, can align and form cell clusters but only when periodic reversals of cell directions are suppressed. However, by extending our model to introduce the observed ability of cells to deposit and follow slime trails, we show that effective trail-following leads to clusters in reversing cells. Furthermore, we conclude that mechanical cell alignment combined with slime-trail-following is sufficient to explain the distinct clustering behaviors observed for wild-type and non-reversing M. xanthus mutants in recent experiments. Our results are robust to variation in model parameters, match the experimentally observed trends and can be applied to understand surface motility patterns of other bacterial species.

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.

Describing Myxococcus xanthus aggregation using Ostwald ripening equations for thin liquid films

When starved, a swarm of millions of Myxococcus xanthus cells coordinate their movement from outward swarming to inward coalescence. The cells then execute a synchronous program of multicellular development, arranging themselves into dome shaped aggregates. Over the course of development, about half of the initial aggregates disappear, while others persist and mature into fruiting bodies. This work seeks to develop a quantitative model for aggregation that accurately simulates which will disappear and which will persist. We analyzed time-lapse movies of M. xanthus development, modeled aggregation using the equations that describe Ostwald ripening of droplets in thin liquid films, and predicted the disappearance and persistence of aggregates with an average accuracy of 85%. We then experimentally validated a prediction that is fundamental to this model by tracking individual fluorescent cells as they moved between aggregates and demonstrating that cell movement towards and away from aggregates correlates with aggregate disappearance. Describing development through this model may limit the number and type of molecular genetic signals needed to complete M. xanthus development, and it provides numerous additional testable predictions.

Continuum modeling of clustering of myxobacteria

In this paper we develop a continuum theory of clustering in ensembles of self-propelled inelastically colliding rods with applications to collective dynamics of common gliding bacteria Myxococcus Xanthus. A multiphase hydrodynamic model that couples densities of oriented and isotropic phases is described. This model is used for the analysis of an instability that leads to spontaneous formation of directionally moving dense clusters within initially dilute isotropic "gas" of myxobacteria. Numerical simulations of this model confirm the existence of stationary dense moving clusters and also elucidate the properties of their collisions. The results are shown to be in a qualitative agreement with experiments.

The mechanistic basis of Myxococcus xanthus rippling behavior and its physiological role during predation

Myxococcus xanthus cells self-organize into periodic bands of traveling waves, termed ripples, during multicellular fruiting body development and predation on other bacteria. To investigate the mechanistic basis of rippling behavior and its physiological role during predation by this Gram-negative soil bacterium, we have used an approach that combines mathematical modeling with experimental observations. Specifically, we developed an agent-based model (ABM) to simulate rippling behavior that employs a new signaling mechanism to trigger cellular reversals. The ABM has demonstrated that three ingredients are sufficient to generate rippling behavior: (i) side-to-side signaling between two cells that causes one of the cells to reverse, (ii) a minimal refractory time period after each reversal during which cells cannot reverse again, and (iii) physical interactions that cause the cells to locally align. To explain why rippling behavior appears as a consequence of the presence of prey, we postulate that prey-associated macromolecules indirectly induce ripples by stimulating side-to-side contact-mediated signaling. In parallel to the simulations, M. xanthus predatory rippling behavior was experimentally observed and analyzed using time-lapse microscopy. A formalized relationship between the wavelength, reversal time, and cell velocity has been predicted by the simulations and confirmed by the experimental data. Furthermore, the results suggest that the physiological role of rippling behavior during M. xanthus predation is to increase the rate of spreading over prey cells due to increased side-to-side contact-mediated signaling and to allow predatory cells to remain on the prey longer as a result of more periodic cell motility.

Myxococcus xanthus cells collectively move on solid surfaces and reorganize their colonies in response to environmental cues. Under some conditions, cells exhibit an intriguing form of collective motility by self-organizing into bands of travelling alternating-density waves termed ripples. These waves are distinct from the waves originating from Turing instability in diffusion-reaction systems, as these counter-traveling waves do not annihilate but appear to pass through each other. Here we developed a new mathematical model of rippling behavior based on a recently observed contact signaling mechanism – cells that make side-to-side contacts can signal one another to reverse. We hypothesize that this signaling is enhanced by the presence of prey-associated macromolecules and compare modeling predictions with experimentally observed waves generated on E. coli prey cells. The model predicts a modified relationship between the wavelength and individual predatory cell motility parameters and provides a physiological role for rippling during predation. We show that ripples allow predatory cells to increase the rate of their spreading to quickly envelope the prey, and subsequently to decrease their random drift to remain in the prey region for longer. These and other predictions are confirmed by the experimental observations.

Traveling waves in actin dynamics and cell motility

Much of current understanding of cell motility arose from studying steady treadmilling of actin arrays. Recently, there have been a growing number of observations of a more complex, non-steady, actin behavior, including self-organized waves. It is becoming clear that these waves result from activation and inhibition feedbacks in actin dynamics acting on different scales, but the exact molecular nature of these feedbacks and the respective roles of biomechanics and biochemistry are still unclear. Here, we review recent advances achieved in experimental and theoretical studies of actin waves and discuss mechanisms and physiological significance of wavy protrusions.

Macroscopic model of self-propelled bacteria swarming with regular reversals

Periodic reversals in the direction of motion in systems of self-propelled rod-shaped bacteria enable them to effectively resolve traffic jams formed during swarming and maximize the swarming rate of the colony. In this paper, a connection is established between a microscopic one-dimensional cell-based stochastic model of reversing nonoverlapping bacteria and a macroscopic nonlinear diffusion equation describing the dynamics of cellular density. Boltzmann-Matano analysis is used to determine the nonlinear diffusion equation corresponding to the specific reversal frequency. Stochastic dynamics averaged over an ensemble is shown to be in very good agreement with the numerical solutions of this nonlinear diffusion equation. Critical density p(0) is obtained such that nonlinear diffusion is dominated by the collisions between cells for the densities p>p(0). An analytical approximation of the pairwise collision time and semianalytical fit for the total jam time per reversal period are also obtained. It is shown that cell populations with high reversal frequencies are able to spread out effectively at high densities. If the cells rarely reverse, then they are able to spread out at lower densities but are less efficient at spreading out at higher densities.

Identification and localization of Myxococcus xanthus porins and lipoproteins

Myxococcus xanthus DK1622 contains inner (IM) and outer membranes (OM) separated by a peptidoglycan layer. Integral membrane, β-barrel proteins are found exclusively in the OM where they form pores allowing the passage of nutrients, waste products and signals. One porin, Oar, is required for intercellular communication of the C-signal. An oar mutant produces CsgA but is unable to ripple or stimulate csgA mutants to develop suggesting that it is the channel for C-signaling. Six prediction programs were evaluated for their ability to identify β-barrel proteins. No program was reliable unless the predicted proteins were first parsed using Signal P, Lipo P and TMHMM, after which TMBETA-SVM and TMBETADISC-RBF identified β-barrel proteins most accurately. 228 β-barrel proteins were predicted from among 7331 protein coding regions, representing 3.1% of total genes. Sucrose density gradients were used to separate vegetative cell IM and OM fractions, and LC-MS/MS of OM proteins identified 54 β-barrel proteins. Another class of membrane proteins, the lipoproteins, are anchored in the membrane via a lipid moiety at the N-terminus. 44 OM proteins identified by LC-MS/MS were predicted lipoproteins. Lipoproteins are distributed between the IM, OM and ECM according to an N-terminal sorting sequence that varies among species. Sequence analysis revealed conservation of alanine at the +7 position of mature ECM lipoproteins, lysine at the +2 position of IM lipoproteins, and no noticable conservation within the OM lipoproteins. Site directed mutagenesis and immuno transmission electron microscopy showed that alanine at the +7 position is essential for sorting of the lipoprotein FibA into the ECM. FibA appears at normal levels in the ECM even when a +2 lysine is added to the signal sequence. These results suggest that ECM proteins have a unique method of secretion. It is now possible to target lipoproteins to specific IM, OM and ECM locations by manipulating the amino acid sequence near the +1 cysteine processing site.

Quantifying aggregation dynamics during Myxococcus xanthus development

Under starvation conditions, a swarm of Myxococcus xanthus cells will undergo development, a multicellular process culminating in the formation of many aggregates called fruiting bodies, each of which contains up to 100,000 spores. The mechanics of symmetry breaking and the self-organization of cells into fruiting bodies is an active area of research. Here we use microcinematography and automated image processing to quantify several transient features of developmental dynamics. An analysis of experimental data indicates that aggregation reaches its steady state in a highly nonmonotonic fashion. The number of aggregates rapidly peaks at a value 2- to 3-fold higher than the final value and then decreases before reaching a steady state. The time dependence of aggregate size is also nonmonotonic, but to a lesser extent: average aggregate size increases from the onset of aggregation to between 10 and 15 h and then gradually decreases thereafter. During this process, the distribution of aggregates transitions from a nearly random state early in development to a more ordered state later in development. A comparison of experimental results to a mathematical model based on the traffic jam hypothesis indicates that the model fails to reproduce these dynamic features of aggregation, even though it accurately describes its final outcome. The dynamic features of M. xanthus aggregation uncovered in this study impose severe constraints on its underlying mechanisms.

Hyperbolic and kinetic models for self-organized biological aggregations and movement: a brief review

We briefly review hyperbolic and kinetic models for self-organized biological aggregations and traffic-like movement. We begin with the simplest models described by an advection-reaction equation in one spatial dimension. We then increase the complexity of models in steps. To this end, we begin investigating local hyperbolic systems of conservation laws with constant velocity. Next, we proceed to investigate local hyperbolic systems with density-dependent speed, systems that consider population dynamics (i.e., birth and death processes), and nonlocal hyperbolic systems. We conclude by discussing kinetic models in two spatial dimensions and their limiting hyperbolic models. This structural approach allows us to discuss the complexity of the biological problems investigated, and the necessity for deriving complex mathematical models that would explain the observed spatial and spatiotemporal group patterns.

Cell flexibility affects the alignment of model myxobacteria

Myxobacteria are social bacteria that exhibit a complex life cycle culminating in the development of multicellular fruiting bodies. The alignment of rod-shaped myxobacteria cells within populations is crucial for development to proceed. It has been suggested that myxobacteria align due to mechanical interactions between gliding cells and that cell flexibility facilitates reorientation of cells upon mechanical contact. However, these suggestions have not been based on experimental or theoretical evidence. Here we created a computational mass-spring model of a flexible rod-shaped cell that glides on a substratum periodically reversing direction. The model was formulated in terms of experimentally measurable mechanical parameters, such as engine force, bending stiffness, and drag coefficient. We investigated how cell flexibility and motility engine type affected the pattern of cell gliding and the alignment of a population of 500 mechanically interacting cells. It was found that a flexible cell powered by engine force at the rear of the cell, as suggested by the slime extrusion hypothesis for myxobacteria motility engine, would not be able to glide in the direction of its long axis. A population of rigid reversing cells could indeed align due to mechanical interactions between cells, but cell flexibility impaired the alignment.

Gliding motility revisited: how do the myxobacteria move without flagella?

In bacteria, motility is important for a wide variety of biological functions such as virulence, fruiting body formation, and biofilm formation. While most bacteria move by using specialized appendages, usually external or periplasmic flagella, some bacteria use other mechanisms for their movements that are less well characterized. These mechanisms do not always exhibit obvious motility structures. Myxococcus xanthus is a motile bacterium that does not produce flagella but glides slowly over solid surfaces. How M. xanthus moves has remained a puzzle that has challenged microbiologists for over 50 years. Fortunately, recent advances in the analysis of motility mutants, bioinformatics, and protein localization have revealed likely mechanisms for the two M. xanthus motility systems. These results are summarized in this review.

Myxobacteria, polarity, and multicellular morphogenesis

Myxobacteria are renowned for the ability to sporulate within fruiting bodies whose shapes are species-specific. The capacity to build those multicellular structures arises from the ability of M. xanthus to organize high cell-density swarms, in which the cells tend to be aligned with each other while constantly in motion. The intrinsic polarity of rod-shaped cells lays the foundation, and each cell uses two polar engines for gliding on surfaces. It sprouts retractile type IV pili from the leading cell pole and secretes capsular polysaccharide through nozzles from the trailing pole. Regularly periodic reversal of the gliding direction was found to be required for swarming. Those reversals are generated by a G-protein switch which is driven by a sharply tuned oscillator. Starvation induces fruiting body development, and systematic reductions in the reversal frequency are necessary for the cells to aggregate rather than continue to swarm. Developmental gene expression is regulated by a network that is connected to the suppression of reversals.

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.