Learning the space-time phase diagram of bacterial swarm expansion.

Coordinated dynamics of individual components in active matter are an essential aspect of life on all scales. Establishing a comprehensive, causal connection between intracellular, intercellular, and macroscopic behaviors has remained a major challenge due to limitations in data acquisition and analysis techniques suitable for multiscale dynamics. Here, we combine a high-throughput adaptive microscopy approach with machine learning, to identify key biological and physical mechanisms that determine distinct microscopic and macroscopic collective behavior phases which develop as Bacillus subtilis swarms expand over five orders of magnitude in space. Our experiments, continuum modeling, and particle-based simulations reveal that macroscopic swarm expansion is primarily driven by cellular growth kinetics, whereas the microscopic swarming motility phases are dominated by physical cell-cell interactions. These results provide a unified understanding of bacterial multiscale behavioral complexity in swarms.

Bacteria exploit a polymorphic instability of the flagellar filament to escape from traps.

Many bacterial species swim by rotating single polar helical flagella. Depending on the direction of rotation, they can swim forward or backward and change directions to move along chemical gradients but also to navigate their obstructed natural environment in soils, sediments, or mucus. When they get stuck, they naturally try to back out, but they can also resort to a radically different flagellar mode, which we discovered here. Using high-speed microscopy, we monitored the swimming behavior of the monopolarly flagellated species Shewanella putrefaciens with fluorescently labeled flagellar filaments at an agarose-glass interface. We show that, when a cell gets stuck, the polar flagellar filament executes a polymorphic change into a spiral-like form that wraps around the cell body in a spiral-like fashion and enables the cell to escape by a screw-like backward motion. Microscopy and modeling suggest that this propagation mode is triggered by an instability of the flagellum under reversal of the rotation and the applied torque. The switch is reversible and bacteria that have escaped the trap can return to their normal swimming mode by another reversal of motor direction. The screw-type flagellar arrangement enables a unique mode of propagation and, given the large number of polarly flagellated bacteria, we expect it to be a common and widespread escape or motility mode in complex and structured environments.

Using machine learning to predict extreme events in the Hénon map.

Machine Learning (ML) inspired algorithms provide a flexible set of tools for analyzing and forecasting chaotic dynamical systems. We analyze here the performance of one algorithm for the prediction of extreme events in the two-dimensional Hénon map at the classical parameters. The task is to determine whether a trajectory will exceed a threshold after a set number of time steps into the future. This task has a geometric interpretation within the dynamics of the Hénon map, which we use to gauge the performance of the neural networks that are used in this work. We analyze the dependence of the success rate of the ML models on the prediction time T, the number of training samples NT, and the size of the network Np. We observe that in order to maintain a certain accuracy, NT∝exp⁡(2hT) and Np∝exp⁡(hT), where h is the topological entropy. Similar relations between the intrinsic chaotic properties of the dynamics and ML parameters might be observable in other systems as well.

Phase Transition to Large Scale Coherent Structures in Two-Dimensional Active Matter Turbulence.

The collective motion of microswimmers in suspensions induce patterns of vortices on scales that are much larger than the characteristic size of a microswimmer, attaining a state called bacterial turbulence. Hydrodynamic turbulence acts on even larger scales and is dominated by inertial transport of energy. Using an established modification of the Navier-Stokes equation that accounts for the small-scale forcing of hydrodynamic flow by microswimmers, we study the properties of a dense suspension of microswimmers in two dimensions, where the conservation of enstrophy can drive an inverse cascade through which energy is accumulated on the largest scales. We find that the dynamical and statistical properties of the flow show a sharp transition to the formation of vortices at the largest length scale. The results show that 2D bacterial and hydrodynamic turbulence are separated by a subcritical phase transition.

Super Resolution Fluorescence Microscopy and Tracking of Bacterial Flotillin (Reggie) Paralogs Provide Evidence for Defined-Sized Protein Microdomains within the Bacterial Membrane but Absence of Clusters Containing Detergent-Resistant Proteins.

Biological membranes have been proposed to contain microdomains of a specific lipid composition, in which distinct groups of proteins are clustered. Flotillin-like proteins are conserved between pro-and eukaryotes, play an important function in several eukaryotic and bacterial cells, and define in vertebrates a type of so-called detergent-resistant microdomains. Using STED microscopy, we show that two bacterial flotillins, FloA and FloT, form defined assemblies with an average diameter of 85 to 110 nm in the model bacterium Bacillus subtilis. Interestingly, flotillin microdomains are of similar size in eukaryotic cells. The soluble domains of FloA form higher order oligomers of up to several hundred kDa in vitro, showing that like eukaryotic flotillins, bacterial assemblies are based in part on their ability to self-oligomerize. However, B. subtilis paralogs show significantly different diffusion rates, and consequently do not colocalize into a common microdomain. Dual colour time lapse experiments of flotillins together with other detergent-resistant proteins in bacteria show that proteins colocalize for no longer than a few hundred milliseconds, and do not move together. Our data reveal that the bacterial membrane contains defined-sized protein domains rather than functional microdomains dependent on flotillins. Based on their distinct dynamics, FloA and FloT confer spatially distinguishable activities, but do not serve as molecular scaffolds.

Microdomain formation is a general property of bacterial membrane proteins and induces heterogeneity of diffusion patterns.

BACKGROUND: Proteins within the cytoplasmic membrane display distinct localization patterns and arrangements. While multiple models exist describing the dynamics of membrane proteins, to date, there have been few systematic studies, particularly in bacteria, to evaluate how protein size, number of transmembrane domains, and temperature affect their diffusion, and if conserved localization patterns exist. RESULTS: We have used fluorescence microscopy, single-molecule tracking (SMT), and computer-aided visualization methods to obtain a better understanding of the three-dimensional organization of bacterial membrane proteins, using the model bacterium Bacillus subtilis. First, we carried out a systematic study of the localization of over 200 B. subtilis membrane proteins, tagged with monomeric mVenus-YFP at their original gene locus. Their subcellular localization could be discriminated in polar, septal, patchy, and punctate patterns. Almost 20% of membrane proteins specifically localized to the cell poles, and a vast majority of all proteins localized in distinct structures, which we term microdomains. Dynamics were analyzed for selected membrane proteins, using SMT. Diffusion coefficients of the analyzed transmembrane proteins did not correlate with protein molecular weight, but correlated inversely with the number of transmembrane helices, i.e., transmembrane radius. We observed that temperature can strongly influence diffusion on the membrane, in that upon growth temperature upshift, diffusion coefficients of membrane proteins increased and still correlated inversely to the number of transmembrane domains, following the Saffman-Delbrück relation. CONCLUSIONS: The vast majority of membrane proteins localized to distinct multimeric assemblies. Diffusion of membrane proteins can be suitably described by discriminating diffusion coefficients into two protein populations, one mobile and one immobile, the latter likely constituting microdomains. Our results show there is high heterogeneity and yet structural order in the cell membrane, and provide a roadmap for our understanding of membrane organization in prokaryotes.

Front-propagation in bacterial inter-colony communication.

Many bacterial species exchange signaling molecules to coordinate population-wide responses. For this process, known as quorum sensing, the concentration of the respective molecules is crucial. Here, we consider the interaction between spatially distributed bacterial colonies so that the spreading of the signaling molecules in space becomes important. The exponential growth of the signal-producing populations and the corresponding increase in signaling molecule production result in an exponential concentration profile that spreads with uniform speed. The theoretical predictions are supported by experiments with different strains of the soil bacterium Sinorhizobium meliloti that display fluorescence when either producing or responding to the signaling molecules.

Coupling between diffusion and orientation of pentacene molecules on an organic surface.

The realization of efficient organic electronic devices requires the controlled preparation of molecular thin films and heterostructures. As top-down structuring methods such as lithography cannot be applied to van der Waals bound materials, surface diffusion becomes a structure-determining factor that requires microscopic understanding. Scanning probe techniques provide atomic resolution, but are limited to observations of slow movements, and therefore constrained to low temperatures. In contrast, the helium-3 spin-echo (HeSE) technique achieves spatial and time resolution on the nm and ps scale, respectively, thus enabling measurements at elevated temperatures. Here we use HeSE to unveil the intricate motion of pentacene admolecules diffusing on a chemisorbed monolayer of pentacene on Cu(110) that serves as a stable, well-ordered organic model surface. We find that pentacene moves along rails parallel and perpendicular to the surface molecules. The experimental data are explained by admolecule rotation that enables a switching between diffusion directions, which extends our molecular level understanding of diffusion in complex organic systems.

Increasing lifetimes and the growing saddles of shear flow turbulence.

In linearly stable shear flows, turbulence spontaneously decays with a characteristic lifetime that varies with Reynolds number. The lifetime sharply increases with Reynolds number so that a possible divergence marking the transition to sustained turbulence at a critical point has been discussed. We present a mechanism by which the lifetimes increase: in the system's state space, turbulent motion is supported by a chaotic saddle. Inside this saddle a locally attracting periodic orbit is created and undergoes a traditional bifurcation sequence generating chaos. The formed new "turbulent bubble" is initially an attractor supporting persistent chaotic dynamics. Soon after its creation, it collides with its own boundary, by which it becomes leaky and dynamically connected with the surrounding structures. The complexity of the chaotic saddle that supports transient turbulence hence increases by incorporating the remnant of a new bubble. As a a result, the time it takes for a trajectory to leave the saddle and decay to the laminar state is increased. We demonstrate this phenomenon in plane Couette flow and show that characteristic lifetimes vary nonsmoothly and nonmonotonically with Reynolds number.

Complexity of localised coherent structures in a boundary-layer flow.

We study numerically transitional coherent structures in a boundary-layer flow with homogeneous suction at the wall (the so-called asymptotic suction boundary layer ASBL). The dynamics restricted to the laminar-turbulent separatrix is investigated in a spanwise-extended domain that allows for robust localisation of all edge states. We work at fixed Reynolds number and study the edge states as a function of the streamwise period. We demonstrate the complex spatio-temporal dynamics of these localised states, which exhibits multistability and undergoes complex bifurcations leading from periodic to chaotic regimes. It is argued that in all regimes the dynamics restricted to the edge is essentially low-dimensional and non-extensive.

Self-sustained localized structures in a boundary-layer flow.

When a boundary layer starts to develop spatially over a flat plate, only disturbances of sufficiently large amplitude survive and trigger turbulence subcritically. Direct numerical simulation of the Blasius boundary-layer flow is carried out to track the dynamics in the region of phase space separating transitional from relaminarizing trajectories. In this intermediate regime, the corresponding disturbance is fully localized and spreads slowly in space. This structure is dominated by a robust pair of low-speed streaks, whose convective instabilities spawn hairpin vortices evolving downstream into transient disturbances. A quasicyclic mechanism for the generation of offspring is unfolded using dynamical rescaling with the local boundary-layer thickness.

Finite lifetime of turbulence in shear flows.

Generally, the motion of fluids is smooth and laminar at low speeds but becomes highly disordered and turbulent as the velocity increases. The transition from laminar to turbulent flow can involve a sequence of instabilities in which the system realizes progressively more complicated states, or it can occur suddenly. Once the transition has taken place, it is generally assumed that, under steady conditions, the turbulent state will persist indefinitely. The flow of a fluid down a straight pipe provides a ubiquitous example of a shear flow undergoing a sudden transition from laminar to turbulent motion. Extensive calculations and experimental studies have shown that, at relatively low flow rates, turbulence in pipes is transient, and is characterized by an exponential distribution of lifetimes. They also suggest that for Reynolds numbers exceeding a critical value the lifetime diverges (that is, becomes infinitely large), marking a change from transient to persistent turbulence. Here we present experimental data and numerical calculations covering more than two decades of lifetimes, showing that the lifetime does not in fact diverge but rather increases exponentially with the Reynolds number. This implies that turbulence in pipes is only a transient event (contrary to the commonly accepted view), and that the turbulent and laminar states remain dynamically connected, suggesting avenues for turbulence control.

Periodic orbits near onset of chaos in plane Couette flow.

We track the secondary bifurcations of coherent states in plane Couette flow and show that they undergo a periodic doubling cascade that ends with a crisis bifurcation. We introduce a symbolic dynamics for the orbits and show that the ones that exist fall into the universal sequence described by Metropolis, Stein and Stein for unimodal maps. The periodic orbits cover much of the turbulent dynamics in that their temporal evolution overlaps with turbulent motions when projected onto a plane spanned by energy production and dissipation.

Theoretical mechanics: crowd synchrony on the Millennium Bridge.

Soon after the crowd streamed on to London's Millennium Bridge on the day it opened, the bridge started to sway from side to side: many pedestrians fell spontaneously into step with the bridge's vibrations, inadvertently amplifying them. Here we model this unexpected and now notorious phenomenon--which was not due to the bridge's innovative design as was first thought--by adapting ideas originally developed to describe the collective synchronization of biological oscillators such as neurons and fireflies. Our approach should help engineers to estimate the damping needed to stabilize other exceptionally crowded footbridges against synchronous lateral excitation by pedestrians.

Condensate formation and multiscale dynamics in two-dimensional active suspensions.

The collective effects of microswimmers in active suspensions result in active turbulence, a spatiotemporally chaotic dynamics at mesoscale, which is characterized by the presence of vortices and jets at scales much larger than the characteristic size of the individual active constituents. To describe this dynamics, Navier-Stokes-based one-fluid models driven by small-scale forces have been proposed. Here, we provide a justification of such models for the case of dense suspensions in two dimensions (2D). We subsequently carry out an in-depth numerical study of the properties of one-fluid models as a function of the active driving in view of possible transition scenarios from active turbulence to large-scale pattern, referred to as condensate, formation induced by the classical inverse energy cascade in Newtonian 2D turbulence. Using a one-fluid model it was recently shown [M. Linkmann et al., Phys. Rev. Lett 122, 214503 (2019)10.1103/PhysRevLett.122.214503] that two-dimensional active suspensions support two nonequilibrium steady states, one with a condensate and one without, which are separated by a subcritical transition. Here, we report further details on this transition such as hysteresis and discuss a low-dimensional model that describes the main features of the transition through nonlocal-in-scale coupling between the small-scale driving and the condensate.

Chromosome Segregation in Bacillus subtilis Follows an Overall Pattern of Linear Movement and Is Highly Robust against Cell Cycle Perturbations.

Although several proteins have been identified that facilitate chromosome segregation in bacteria, no clear analogue of the mitotic machinery in eukaryotic cells has been identified. In order to investigate if recognizable patterns of segregation exist during the cell cycle, we tracked the segregation of duplicated origin regions in Bacillus subtilis for 60 min in the fastest practically achievable resolution, achieving 10-s intervals. We found that while separation occurred in random patterns, often including backwards movement, overall, segregation of loci near the origins of replication was linear for the entire cell cycle. Thus, the process of partitioning can be best described as directed motion. Simulations with entropy-driven separation of polymers synthesized by two polymerases show sudden bursts of movement and segregation patterns compatible with the observed in vivo patterns, showing that for Bacillus, segregation patterns can be modeled based on entropic forces. To test if obstacles for replication forks lead to an alteration of the partitioning pattern, we challenged cells with chemicals inducing DNA damage or blocking of topoisomerase activity. Both treatments led to a moderate slowing down of separation, but linear segregation was retained, showing that chromosome segregation is highly robust against cell cycle perturbation.IMPORTANCE We have followed the segregation of origin regions on the Bacillus subtilis chromosome in the fastest practically achievable temporal manner, for a large fraction of the cell cycle. We show that segregation occurred in highly variable patterns but overall in an almost linear manner throughout the cell cycle. Segregation was slowed down, but not arrested, by treatment of cells that led to transient blocks in DNA replication, showing that segregation is highly robust against cell cycle perturbation. Computer simulations based on entropy-driven separation of newly synthesized DNA polymers can recapitulate sudden bursts of movement and segregation patterns compatible with the observed in vivo patterns, indicating that for Bacillus, segregation patterns may include entropic forces helping to separate chromosomes during the cell cycle.

Lifetime statistics in transitional pipe flow.

Several experimental and numerical studies have shown that turbulent motions in circular pipe flow near transitional Reynolds numbers may not persist forever, but may decay. We study the properties of these decaying states within direct numerical simulations for Reynolds numbers up to 2200 and in pipes with lengths equal to 5, 9, and 15 times the diameter. We show that the choice of the ensemble of initial conditions affects the short time parts of lifetime distributions, but does not change the characteristic decay rate for long times. Comparing lifetimes for pipes of different length we notice a linear increase in the characteristic lifetime with length, which reproduces the experimental results when extrapolated to 30 diameters, the length of an equilibrium turbulent puff at these Reynolds numbers.

Laminar-turbulent boundary in plane Couette flow.

We apply the iterated edge-state tracking algorithm to study the boundary between laminar and turbulent dynamics in plane Couette flow at Re=400. Perturbations that are not strong enough to become fully turbulent or weak enough to relaminarize tend toward a hyperbolic coherent structure in state space, termed the edge state, which seems to be unique up to obvious continuous shift symmetries. The results reported here show that in cases where a fixed point has only one unstable direction, such as for the lower-branch solution in plane Couette flow, the iterated edge tracking algorithm converges to this state. They also show that the choice of initial state is not critical and that essentially arbitrary initial conditions can be used to find the edge state.

Synchronization, phase locking, and metachronal wave formation in ciliary chains.

Synchronization and wave formation in one-dimensional ciliary arrays are studied analytically and numerically. We develop a simple model for ciliary motion that is complex enough to describe well the behavior of beating cilia but simple enough to study collective effects analytically. Beating cilia are described as phase oscillators moving on circular trajectories with a variable radius. This radial degree of freedom turns out to be essential for the occurrence of hydrodynamically induced synchronization of ciliary beating between neighboring cilia. The transitions to the synchronized and phase-locked state of two cilia and the formation of metachronal waves in ciliary chains with different boundary conditions are discussed.

Spatial arrangement of several flagellins within bacterial flagella improves motility in different environments.

Bacterial flagella are helical proteinaceous fibers, composed of the protein flagellin, that confer motility to many bacterial species. The genomes of about half of all flagellated species include more than one flagellin gene, for reasons mostly unknown. Here we show that two flagellins (FlaA and FlaB) are spatially arranged in the polar flagellum of Shewanella putrefaciens, with FlaA being more abundant close to the motor and FlaB in the remainder of the flagellar filament. Observations of swimming trajectories and numerical simulations demonstrate that this segmentation improves motility in a range of environmental conditions, compared to mutants with single-flagellin filaments. In particular, it facilitates screw-like motility, which enhances cellular spreading through obstructed environments. Similar mechanisms may apply to other bacterial species and may explain the maintenance of multiple flagellins to form the flagellar filament.