Submitted / under review

Differential-activity driven instabilities in biphasic active matter

Christoph A. Weber, Chris H. Rycroft and L. Mahadevan.


Even simple active systems can show a plethora of intriguing phenomena and often we find complexity where we would have expected simplicity. One striking example is the occurrence of a quiescent or absorbing state with frozen fluctuations that at first sight seems to be impossible for active matter driven by the incessant input of energy. While such states were reported for externally driven systems through macroscopic shear or agitation, the investigation of frozen active states in inherently active systems like cytoskeletal suspensions or active gels is still at large. Using high-density motility assay experiments, we demonstrate that frozen steady states can arise in active systems if active transport is coupled to growth processes. The experiments are complemented by agent-based simulations which identify the coupling between self-organization, growth, and mechanical properties to be responsible for the pattern formation process.

Pili mediated intercellular forces shape heterogeneous bacterial microcolonies prior to multicellular differentiation

Wolfram Poenisch, Khaled Alzurqa, Kelly Eckenrode, Hadi Nasrollahi, Christoph A. Weber, Vasily Zaburdaev, Nicolas Biais.


Microcolonies are aggregates of a few dozen to a few thousand cells exhibited by many bacteria. The formation of microcolonies is a crucial step towards the formation of more mature bacterial communities known as biofilms, but also marks a significant change in bacterial physiology. Within a microcolony, bacteria forgo a single cell lifestyle for a communal lifestyle hallmarked by high cell density and physical interactions between cells potentially leading to differentiation. It is thus crucial to understand how initially identical single cells start to behave differently while assembling in these tight communities. Here we show that cells in the microcolonies formed by the human pathogen Neisseria gonorrhoeae (Ng) present differential motility behaviors within an hour upon colony formation. Observation of merging microcolonies and tracking of single cells within microcolonies reveal a heterogeneous motility behavior: cells close to the surface of the microcolony exhibit a much higher motility compared to cells towards the center. Numerical simulations of a biophysical model for the microcolonies at the single cell level of detail suggest that mechanical forces exerted by the bacterial cells are sufficient to generate the observed heterogeneous motility. Further corroborating this idea, bacteria lacking the ability to exert forces on their surroundings segregate on the outside of microcolonies as predicted by the model. This emergence of differential behavior within a multicellular microcolony of otherwise identical cells is thus mainly of mechanical origin and is likely the first step toward further bacterial differentiation and ultimately mature biofilms.

Switching Droplet Positions by Concentration Gradients

Samuel Krüger, Christoph A. Weber, Jens-Uwe Sommer, and Frank Jülicher.


Here we investigate how droplet position can be controlled using concentration profiles of a regulator that influences phase separation. We consider a mean field model of a ternary mixture where a concentration gradient of a regulator is imposed by an external potential. We show that novel first order phase transition exists that controls droplet position in a discontinuous manner. Such a droplet switch in concentration gradients could be relevant for the spatial organization of biological cells and provides a control mechanism for droplets in microfluidic systems.


Droplet ripening in concentration gradients

Christoph A. Weber, Chiu Fan Lee, and Frank Jülicher.


Living cells use phase separation and concentration gradients to organize chemical compartments in space. Here, we present a theoretical study of droplet dynamics in gradient systems. We derive the corresponding growth law of droplets and find that droplets exhibit a drift velocity and position dependent growth. As a consequence, the dissolution boundary moves through the system, thereby segregating droplets to one end. We show that for steep enough gradients, the ripening leads to a transient arrest of droplet growth that is induced by a narrowing of the droplet size distribution.


New Journal of Physics, 19, 5, Pp. 053021.


Binary Mixtures of Particles with Different Diffusivities Demix

Simon N. Weber, Christoph A. Weber, and Erwin Frey


The influence of size differences, shape, mass, and persistent motion on phase separation in binary mixtures has been intensively studied. Here we focus on the exclusive role of diffusivity differences in binary mixtures of equal-sized particles. We find an effective attraction between the less diffusive particles, which are essentially caged in the surrounding species with the higher diffusion constant. This effect leads to phase separation for systems above a critical size: A single close-packed cluster made up of the less diffusive species emerges. Experiments for testing our predictions are outlined.


Phys. Rev. Lett., 116, Pp. 058301.

Growth and Division of Active Droplets Provides a Model for Protocells

David Zwicker, Rabea Seyboldt, Christoph A. Weber, Anthony A. Hyman, and Frank Jülicher


It has been proposed that during the early steps in the origin of life, small droplets could have formed via the segregation of molecules from complex mixtures by phase separation. These droplets could have provided chemical reaction centres. However, whether these droplets could divide and propagate is unclear. Here we examine the behaviour of droplets in systems that are maintained away from thermodynamic equilibrium by an external supply of energy. In these systems, droplets grow by the addition of droplet material generated by chemical reactions. Surprisingly, we find that chemically driven droplet growth can lead to shape instabilities that trigger the division of droplets into two smaller daughters. Therefore, chemically active droplets can exhibit cycles of growth and division that resemble the proliferation of living cells. Dividing active droplets could serve as a model for prebiotic protocells, where chemical reactions in the droplet play the role of a prebiotic metabolism.


 Nature Physics, doi:10.1038/nphys3984, Pp. 1745–2481

Multiscale modeling of bacterial colonies: how pili mediate the dynamics of single cells and cellular aggregates

Wolfram Pönisch, Christoph A Weber, Guido Juckeland, Nicolas Biais, and Vasily Zaburdaev


Neisseria gonorrhoeae is the causative agent of one of the most common sexually transmitted diseases, gonorrhea. Over the past two decades there has been an alarming increase of reported gonorrhea cases where the bacteria were resistant to the most commonly used antibiotics thus prompting for alternative antimicrobial treatment strategies. The crucial step in this and many other bacterial infections is the formation of microcolonies, agglomerates consisting of up to several thousands of cells. The attachment and motility of cells on solid substrates as well as the cell–cell interactions are primarily mediated by type IV pili, long polymeric filaments protruding from the surface of cells. While the crucial role of pili in the assembly of microcolonies has been well recognized, the exact mechanisms of how they govern the formation and dynamics of microcolonies are still poorly understood. Here, we present a computational model of individual cells with explicit pili dynamics, force generation and pili–pili interactions. We employ the model to study a wide range of biological processes, such as the motility of individual cells on a surface, the heterogeneous cell motility within the large cell aggregates, and the merging dynamics and the self-assembly of microcolonies. The results of numerical simulations highlight the central role of pili generated forces in the formation of bacterial colonies and are in agreement with the available experimental observations. The model can quantify the behavior of multicellular bacterial colonies on biologically relevant temporal and spatial scales and can be easily adjusted to include the geometry and pili characteristics of various bacterial species. Ultimately, the combination of the microbiological experimental approach with the in silico model of bacterial colonies might provide new qualitative and quantitative insights on the development of bacterial infections and thus pave the way to new antimicrobial treatments.


New Journal of Physics, 19, 1, Pp. 015003

Polar Positioning of Phase-Separated Liquid Compartments in Cells Regulated by an mRNA Competition Mechanism

Shambaditya Saha, Christoph A. Weber, Marco Nousch, Omar Adame-Arana, Carsten Hoege, Marco Y. Hein, Erin Osborne-Nishimura, Julia Mahamid, Marcus Jahnel, Louise Jawerth, ...., Frank Jülicher and Anthony A. Hyman


P granules are non-membrane-bound RNA-protein compartments that are involved in germline development in C. elegans. They are liquids that condense at one end of the embryo by localized phase separation, driven by gradients of polarity proteins such as the mRNA-binding protein MEX-5. To probe how polarity proteins regulate phase separation, we combined biochemistry and theoretical modeling. We reconstitute P granule-like droplets in vitro using a single protein PGL-3. By combining in vitro reconstitution with measurements of intracellular concentrations, we show that competition between PGL-3 and MEX-5 for mRNA can regulate the formation of PGL-3 droplets. Using theory, we show that, in a MEX-5 gradient, this mRNA competition mechanism can drive a gradient of P granule assembly with similar spatial and temporal characteristics to P granule assembly in vivo. We conclude that gradients of polarity proteins can position RNP granules during development by using RNA competition to regulate local phase separation.


Cell, 166, 6, Pp. 1572–1584


Formation and dissolution of bacterial colonies

Christoph A. Weber, Yen Ting Lin, Nicolas Biais, and Vasily Zaburdaev


Many organisms form colonies for a transient period of time to withstand environmental pressure. Bacterial biofilms are a prototypical example of such behavior. Despite significant interest across disciplines, physical mechanisms governing the formation and dissolution of bacterial colonies are still poorly understood. Starting from a kinetic description of motile and interacting cells we derive a hydrodynamic equation for their density on a surface, where most of the kinetic coefficients are estimated from experimental data for N. gonorrhoeae bacteria. We use it to describe the formation of multiple colonies with sizes consistent with experimental observations. Finally, we show how the changes in the cell-to-cell interactions lead to the dissolution of the bacterial colonies. The successful application of kinetic theory to a complex far from equilibrium system such as formation and dissolution of living bacterial colonies potentially paves the way for the physical quantification of the initial stages of biofilm formation.


Phys. Rev. E, 92, Pp. 032704.

Polar pattern formation in driven filament systems requires non-binary particle collisions

Ryo Suzuki, Christoph A. Weber, Erwin Frey, and Andreas R Bausch.


From the self-organization of the cytoskeleton to the synchronous motion of bird flocks, living matter has the extraordinary ability to behave in a concerted manner. The Boltzmann equation for self-propelled particles is frequently used in silico to link a system's meso- or macroscopic behaviour to the microscopic dynamics of its constituents. But so far such studies have relied on an assumption of simplified binary collisions owing to a lack of experimental data suggesting otherwise. We report here experimentally determined binary-collision statistics by studying a recently introduced molecular system, the high-density actomyosin motility assay. We demonstrate that the alignment induced by binary collisions is too weak to account for the observed ordering transition. The transition density for polar pattern formation decreases quadratically with filament length, indicating that multi-filament collisions drive the observed ordering phenomenon and that a gas-like picture cannot explain the transition of the system to polar order. Our findings demonstrate that the unique properties of biological active-matter systems require a description that goes well beyond that developed in the framework of kinetic theories.


Nature Physics, 11, 10, Pp. 839–843

Random bursts determine dynamics of active filaments

Christoph A. Weber, Ryo Suzuki, Volker Schaller, Igor S. Aranson, Andreas R. Bausch, and Erwin Frey


Constituents of living or synthetic active matter have access to a local energy supply that serves to keep the system out of thermal equilibrium. The statistical properties of such fluctuating active systems differ from those of their equilibrium counterparts. Using the actin filament gliding assay as a model, we studied how nonthermal distributions emerge in active matter. We found that the basic mechanism involves the interplay between local and random injection of energy, acting as an analog of a thermal heat bath, and nonequilibrium energy dissipation processes associated with sudden jump-like changes in the system's dynamic variables. We show here how such a mechanism leads to a nonthermal distribution of filament curvatures with a non-Gaussian shape. The experimental curvature statistics and filament relaxation dynamics are reproduced quantitatively by stochastic computer simulations and a simple kinetic model.


Proceedings of the National Academy of Sciences, 112, 34, Pp. 10703-10707


Defect-Mediated Phase Transitions in Active Soft Matter

Christoph A. Weber, Christopher Bock, and Erwin Frey


How do topological defects affect the degree of order in active matter? To answer this question we investigate an agent-based model of self-propelled particles, which accounts for polar alignment and short-ranged repulsive interactions. For strong alignment forces we find collectively moving polycrystalline states with fluctuating networks of grain boundaries. In the regime where repulsive forces dominate, the fluctuations generated by the active system give rise to quasi-long-range transitional order, but unlike the thermal system without creating topological defects.


Phys. Rev. Lett., 112, Pp. 168301

Liquid-liquid phase separation in biology

Anthony A. Hyman, Christoph A. Weber, and Frank Jülicher


Cells organize many of their biochemical reactions in non-membrane compartments. Recent evidence has shown that many of these compartments are liquids that form by phase separation from the cytoplasm. Here we discuss the basic physical concepts necessary to understand the consequences of liquid-like states for biological functions.


Annual review of cell and developmental biology, 30, Pp. 39–58

Numerical Treatment of the Boltzmann Equation for Self-Propelled Particle Systems

Florian Thüroff, Christoph A. Weber, and Erwin Frey


Kinetic theories constitute one of the most promising tools to decipher the characteristic spatiotemporal dynamics in systems of actively propelled particles. In this context, the Boltzmann equation plays a pivotal role, since it provides a natural translation between a particle-level description of the system's dynamics and the corresponding hydrodynamic fields. Yet, the intricate mathematical structure of the Boltzmann equation substantially limits the progress toward a full understanding of this equation by solely analytical means. Here, we propose a general framework to numerically solve the Boltzmann equation for self-propelled particle systems in two spatial dimensions and with arbitrary boundary conditions. We discuss potential applications of this numerical framework to active matter systems and use the algorithm to give a detailed analysis to a model system of self-propelled particles with polar interactions. In accordance with previous studies, we find that spatially homogeneous isotropic and broken-symmetry states populate two distinct regions in parameter space, which are separated by a narrow region of spatially inhomogeneous, density-segregated moving patterns. We find clear evidence that these three regions in parameter space are connected by first-order phase transitions and that the transition between the spatially homogeneous isotropic and polar ordered phases bears striking similarities to liquid-gas phase transitions in equilibrium systems. Within the density-segregated parameter regime, we find a novel stable limit-cycle solution of the Boltzmann equation, which consists of parallel lanes of polar clusters moving in opposite directions, so as to render the overall symmetry of the system's ordered state nematic, despite purely polar interactions on the level of single particles.


Phys. Rev. X, 4, Pp. 041030


Critical Assessment of the Boltzmann Approach to Active Systems

Florian Thüroff, Christoph A. Weber, and Erwin Frey


Generic models of propelled particle systems posit that the emergence of polar order is driven by the competition between local alignment and noise. Although this notion has been confirmed employing the Boltzmann equation, the range of applicability of this equation remains elusive. We introduce a broad class of mesoscopic collision rules and analyze the prerequisites for the emergence of polar order in the framework of kinetic theory. Our findings suggest that a Boltzmann approach is appropriate for weakly aligning systems but is incompatible with experiments on cluster forming systems.


Phys. Rev. Lett., 111, Pp. 190601

Long-Range Ordering of Vibrated Polar Disks

C. A. Weber, T. Hanke, J. Deseigne, S. Léonard, O. Dauchot, E. Frey, and H. Chaté


Vibrated polar disks have been used experimentally to investigate collective motion of driven particles, where fully ordered asymptotic regimes could not be reached. Here we present a model reproducing quantitatively the single, binary, and collective properties of this granular system. Using system sizes not accessible in the laboratory, we show in silico that true long-range order is possible in the experimental system. Exploring the model?s parameter space, we find a phase diagram qualitatively different from that of dilute or pointlike particle systems.


Phys. Rev. Lett., 110, Pp. 208001

Role of particle conservation in self-propelled particle systems

Christoph A. Weber, Florian Thüroff, and Erwin Frey


Actively propelled particles undergoing dissipative collisions are known to develop a state of spatially distributed coherently moving clusters. For densities larger than a characteristic value, clusters grow in time and form a stationary well-ordered state of coherent macroscopic motion. In this work we address two questions. (i) What is the role of the particles' aspect ratio in the context of cluster formation, and does the particle shape affect the system's behavior on hydrodynamic scales? (ii) To what extent does particle conservation influence pattern formation? To answer these questions we suggest a simple kinetic model permitting us to depict some of the interaction properties between freely moving particles and particles integrated in clusters. To this end, we introduce two particle species: single and cluster particles. Specifically, we account for coalescence of clusters from single particles, assembly of single particles on existing clusters, collisions between clusters and cluster disassembly. Coarse graining our kinetic model, (i) we demonstrate that particle shape (i.e. aspect ratio) shifts the scale of the transition density, but does not impact the instabilities at the ordering threshold and (ii) we show that the validity of particle conservation determines the existence of a longitudinal instability, which tends to amplify density heterogeneities locally, and in turn triggers a wave pattern with wave vectors parallel to the axis of macroscopic order. If the system is in contact with a particle reservoir, this instability vanishes due to a compensation of density heterogeneities.


New Journal of Physics, 15, 4, Pp. 045014


Collective motion in actively propelled particle systems is triggered on the very local scale by nucleation of coherently moving units consisting of just a handful of particles. These units grow and merge over time, ending up in a long-range ordered, coherently moving state. So far, there exists no bottom-up understanding of how the microscopic dynamics and interactions between the constituents are related to the system's ordering instability. In this paper, we study a class of models for propelled colloids allowing an explicit treatment of the microscopic details of the collision process. Specifically, the model equations are Newtonian equations of motion with separate force terms for particles' driving, dissipation, and interaction forces. Focusing on dilute particle systems, we analyze the binary scattering behavior for these models and determine, based on the microscopic dynamics, the corresponding collision rule, i.e., the mapping of precollisional velocities and impact parameter on postcollisional velocities. By studying binary scattering we also find that the considered models for active colloids share the same principle for parallel alignment: The first incoming particle (with respect to the center of collision) is aligned to the second particle as a result of the encounter. This behavior distinctively differs from alignment in nondriven dissipative gases. Moreover, the obtained collision rule lends itself as a starting point to apply kinetic theory for propelled particle systems in order to determine the phase boundary to a long-range ordered, coherently moving state. The microscopic origin of the collision rule offers the opportunity to quantitatively scrutinize the predictions of kinetic theory for propelled particle systems through direct comparison with multiparticle simulations. We identify local precursor correlations at the onset of collective motion to constitute the essential determinant for a qualitative and quantitative validity of kinetic theory. In conclusion, our "renormalized" approach clearly indicates that the framework of kinetic theory is flexible enough to accommodate the complex behavior of soft active colloids and allows a bottom-up understanding of how the microscopic dynamics of binary collisions relates to the system's behavior on large length and time scales.


Phys. Rev. E, 88, Pp. 052309


Nucleation-induced transition to collective motion in active systems

Christoph A. Weber, Volker Schaller, Andreas R. Bausch, and Erwin Frey


While the existence of polar ordered states in active systems is well established, the dynamics of the self-assembly processes are still elusive. We study a lattice gas model of self-propelled elongated particles interacting through excluded volume and alignment interactions, which shows a phase transition from an isotropic to a polar ordered state. By analyzing the ordering process we find that the transition is driven by the formation of a critical nucleation cluster and a subsequent coarsening process. Moreover, the time to establish a polar ordered state shows a power-law divergence.


Phys. Rev. E, 86, Pp. 030901


Chemotactic cell trapping in controlled alternating gradient fields

Börn Meier, Alejandro Zielinski, Christoph A. Weber, Delphine Arcizet, Simon Youssef, Thomas Franosch, Joachim O. Rädler, and Doris Heinrich


Directed cell migration toward spatio-temporally varying chemotactic stimuli requires rapid cytoskeletal reorganization. Numerous studies provide evidence that actin reorganization is controlled by intracellular redistribution of signaling molecules, such as the PI4,5P2/PI3,4,5P3 gradient. However, exploring underlying mechanisms is difficult and requires careful spatio-temporal control of external chemotactic stimuli. We designed a microfluidic setup to generate alternating chemotactic gradient fields for simultaneous multicell exposure, greatly facilitating statistical analysis. For a quantitative description of intracellular response dynamics, we apply alternating time sequences of spatially homogeneous concentration gradients, reorienting on timescales down to a few seconds. Dictyostelium discoideum amoebae respond to gradient switching rates below 0.02 Hz by readapting their migration direction. For faster switching, cellular repolarization ceases and is completely stalled at 0.1 Hz. In this chemotactically trapped cell state, external stimuli alternate faster than intracellular feedback is capable to respond by onset of directed migration. To investigate intracellular actin cortex rearrangement during gradient switching, we correlate migratory cell response with actin repolymerization dynamics, quantified by a fluorescence distribution moment of the GFP fusion protein LimEcc. We find two fundamentally different cell polarization types and we could reveal the role of PI3-Kinase for cellular repolarization. In the early aggregation phase, PI3-Kinase enhances the capability of D. discoideum cells to readjust their polarity in response to spatially alternating gradient fields, whereas in aggregation competent cells the effect of PI3-Kinase perturbation becomes less relevant.


Proceedings of the National Academy of Sciences, 108, 28, Pp. 11417-11422

Frozen steady states in active systems

Volker Schaller, Christoph A. Weber, Benjamin Hammerich, Erwin Frey, and Andreas R. Bausch


Even simple active systems can show a plethora of intriguing phenomena and often we find complexity where we would have expected simplicity. One striking example is the occurrence of a quiescent or absorbing state with frozen fluctuations that at first sight seems to be impossible for active matter driven by the incessant input of energy. While such states were reported for externally driven systems through macroscopic shear or agitation, the investigation of frozen active states in inherently active systems like cytoskeletal suspensions or active gels is still at large. Using high-density motility assay experiments, we demonstrate that frozen steady states can arise in active systems if active transport is coupled to growth processes. The experiments are complemented by agent-based simulations which identify the coupling between self-organization, growth, and mechanical properties to be responsible for the pattern formation process.


Proceedings of the National Academy of Sciences, 108, 48, Pp. 19183-19188

Polar pattern formation: hydrodynamic coupling of driven filaments

Volker Schaller, Christoph A. Weber, Erwin Frey, and Andreas R. Bausch


How order can emerge spontaneously from a disordered system has always fascinated scientists from numerous disciplines. Especially in active systems like flocks animals, self-propelled microorganisms or the cytoskeleton, a unifying understanding of the pattern formation remains elusive. This is attributed to the inherent complexity of most model systems that prevents a thorough identification of the fundamental mechanisms that are responsible for the intriguing self-organizing phenomena in active systems. Here we show that long ranged hydrodynamic interactions play a crucial role in the pattern forming mechanisms in the high density motility assay, a precisely controllable minimal model system consisting of highly concentrated filaments that are driven on the nanoscale. Stability and size of the patterns depend on long ranged hydrodynamic interactions that are self-induced by the coherently moving filaments. The hydrodynamic interactions not only influence the spatial and temporal scale of the patterns but also affect the dynamics of a particular cluster in close proximity to confining boundaries or other surrounding clusters.


Soft Matter, 7, Pp. 3213–3218


Polar patterns of driven filaments

Volker Schaller, Christoph A. Weber, Christine Semmrich, Erwin Frey, and Andreas R Bausch


The emergence of collective motion exhibited by systems ranging from flocks of animals to self-propelled microorganisms to the cytoskeleton is a ubiquitous and fascinating self-organization phenomenon. Similarities between these systems, such as the inherent polarity of the constituents, a density-dependent transition to ordered phases or the existence of very large density fluctuations suggest universal principles underlying pattern formation. This idea is followed by theoretical models at all levels of description: micro- or mesoscopic models directly map local forces and interactions using only a few, preferably simple, interaction rules, and more macroscopic approaches in the hydrodynamic limit rely on the systems' generic symmetries. All these models characteristically have a broad parameter space with a manifold of possible patterns, most of which have not yet been experimentally verified. The complexity of interactions and the limited parameter control of existing experimental systems are major obstacles to our understanding of the underlying ordering principles. Here we demonstrate the emergence of collective motion in a high-density motility assay that consists of highly concentrated actin filaments propelled by immobilized molecular motors in a planar geometry. Above a critical density, the filaments self-organize to form coherently moving structures with persistent density modulations, such as clusters, swirls and interconnected bands. These polar nematic structures are long lived and can span length scales orders of magnitudes larger than their constituents. Our experimental approach, which offers control of all relevant system parameters, complemented by agent-based simulations, allows backtracking of the assembly and disassembly pathways to the underlying local interactions. We identify weak and local alignment interactions to be essential for the observed formation of patterns and their dynamics. The presented minimal polar-pattern-forming system may thus provide new insight into emerging order in the broad class of active fluids and self-propelled particles.


Nature, 467, 7311, Pp. 73–77