Homeostasis of cytoplasmic crowding by cell wall fluidization and ribosomal counterions.

In bacteria, algae, fungi, and plant cells, the wall must expand in concert with cytoplasmic biomass production, otherwise cells would experience toxic molecular crowding1,2 or lyse. But how cells achieve expansion of this complex biomaterial in coordination with biosynthesis of macromolecules in the cytoplasm remains unexplained3, although recent works have revealed that these processes are indeed coupled4,5. Here, we report a striking increase of turgor pressure with growth rate in E. coli, suggesting that the speed of cell wall expansion is controlled via turgor. Remarkably, despite this increase in turgor pressure, cellular biomass density remains constant across a wide range of growth rates. By contrast, perturbations of turgor pressure that deviate from this scaling directly alter biomass density. A mathematical model based on cell wall fluidization by cell wall endopeptidases not only explains these apparently confounding observations but makes surprising quantitative predictions that we validated experimentally. The picture that emerges is that turgor pressure is directly controlled via counterions of ribosomal RNA. Elegantly, the coupling between rRNA and turgor pressure simultaneously coordinates cell wall expansion across a wide range of growth rates and exerts homeostatic feedback control on biomass density. This mechanism may regulate cell wall biosynthesis from microbes to plants and has important implications for the mechanism of action of antibiotics6.

Chiral and nematic phases of flexible active filaments.

The emergence of large-scale order in self-organized systems relies on local interactions between individual components. During bacterial cell division, FtsZ-a prokaryotic homologue of the eukaryotic protein tubulin-polymerizes into treadmilling filaments that further organize into a cytoskeletal ring. In vitro, FtsZ filaments can form dynamic chiral assemblies. However, how the active and passive properties of individual filaments relate to these large-scale self-organized structures remains poorly understood. Here we connect single-filament properties with the mesoscopic scale by combining minimal active matter simulations and biochemical reconstitution experiments. We show that the density and flexibility of active chiral filaments define their global order. At intermediate densities, curved, flexible filaments organize into chiral rings and polar bands. An effectively nematic organization dominates for high densities and for straight, mutant filaments with increased rigidity. Our predicted phase diagram quantitatively captures these features, demonstrating how the flexibility, density and chirality of the active filaments affect their collective behaviour. Our findings shed light on the fundamental properties of active chiral matter and explain how treadmilling FtsZ filaments organize during bacterial cell division.

Membrane potential mediates an ancient mechano-transduction mechanism for multi-cellular homeostasis.

Membrane potential is a property of all living cells1. However, its physiological role in non-excitable cells is poorly understood. Resting membrane potential is typically considered fixed for a given cell type and under tight homeostatic control2, akin to body temperature in mammals. Contrary to this widely accepted paradigm, we found that membrane potential is a dynamic property that directly reflects tissue density and mechanical forces acting on the cell. Serving as a quasi-instantaneous, global readout of density and mechanical pressure, membrane potential is integrated with signal transduction networks by affecting the conformation and clustering of proteins in the membrane3,4, as well as the transmembrane flux of key signaling ions5,6. Indeed, we show that important mechano-sensing pathways, YAP, Jnk and p387-121314, are directly controlled by membrane potential. We further show that mechano-transduction via membrane potential plays a critical role in the homeostasis of epithelial tissues, setting tissue density by controlling proliferation and cell extrusion of cells. Moreover, a wave of depolarization triggered by mechanical stretch enhances the speed of wound healing. Mechano-transduction via membrane potential likely constitutes an ancient homeostatic mechanism in multi-cellular organisms, potentially serving as a steppingstone for the evolution of excitable tissues and neuronal mechano-sensing. The breakdown of membrane potential mediated homeostatic regulation may contribute to tumor growth.

A universal mechanism of biomass density homeostasis via ribosomal counterions.

In all growing cells, the cell envelope must expand in concert with cytoplasmic biomass to prevent lysis or molecular crowding. The complex cell wall of microbes and plants makes this challenge especially daunting and it unclear how cells achieve this coordination. Here, we uncover a striking linear increase of cytoplasmic pressure with growth rate in E. coli. Remarkably, despite this increase in turgor pressure with growth rate, cellular biomass density was constant across a wide range of growth rates. In contrast, perturbing pressure away from this scaling directly affected biomass density. A mathematical model, in which endopeptidase-mediated cell wall fluidization enables turgor pressure to set the pace of cellular volume expansion, not only explains these confounding observations, but makes several surprising quantitative predictions that we validated experimentally. The picture that emerges is that changes in turgor pressure across growth rates are mediated by counterions of ribosomal RNA. Profoundly, the coupling between rRNA and cytoplasmic pressure simultaneously coordinates cell wall expansion across growth rates and exerts homeostatic feedback control on biomass density. Because ribosome content universally scales with growth rate in fast growing cells, this universal mechanism may control cell wall biosynthesis in microbes and plants and drive the expansion of ribosome-addicted tumors that can exert substantial mechanical forces on their environment.

Proliferating active matter.

The fascinating patterns of collective motion created by autonomously driven particles have fuelled active-matter research for over two decades. So far, theoretical active-matter research has often focused on systems with a fixed number of particles. This constraint imposes strict limitations on what behaviours can and cannot emerge. However, a hallmark of life is the breaking of local cell number conservation by replication and death. Birth and death processes must be taken into account, for example, to predict the growth and evolution of a microbial biofilm, the expansion of a tumour, or the development from a fertilized egg into an embryo and beyond. In this Perspective, we argue that unique features emerge in these systems because proliferation represents a distinct form of activity: not only do the proliferating entities consume and dissipate energy, they also inject biomass and degrees of freedom capable of further self-proliferation, leading to myriad dynamic scenarios. Despite this complexity, a growing number of studies document common collective phenomena in various proliferating soft-matter systems. This generality leads us to propose proliferation as another direction of active-matter physics, worthy of a dedicated search for new dynamical universality classes. Conceptual challenges abound, from identifying control parameters and understanding large fluctuations and nonlinear feedback mechanisms to exploring the dynamics and limits of information flow in self-replicating systems. We believe that, by extending the rich conceptual framework developed for conventional active matter to proliferating active matter, researchers can have a profound impact on quantitative biology and reveal fascinating emergent physics along the way.

cAMP and the Fibrous Sheath Protein CABYR (Ca2+-Binding Tyrosine-Phosphorylation-Regulated Protein) Is Required for 4D Sperm Movement.

A new life starts with successful fertilization whereby one sperm from a pool of millions fertilizes the oocyte. Sperm motility is one key factor for this selection process, which depends on a coordinated flagellar movement. The flagellar beat cycle is regulated by Ca2+ entry via CatSper, cAMP, Mg2+, ADP and ATP. This study characterizes the effects of these parameters for 4D sperm motility, especially for flagellar movement and the conserved clockwise (CW) path chirality of murine sperm. Therefore, we use detergent-extracted mouse sperm and digital holographic microscopy (DHM) to show that a balanced ratio of ATP to Mg2+ in addition with 18 µM cAMP and 1 mM ADP is necessary for controlled flagellar movement, induction of rolling along the long axis and CW path chirality. Rolling along the sperm's long axis, a proposed mechanism for sperm selection, is absent in sea urchin sperm, lacking flagellar fibrous sheath (FS) and outer-dense fibers (ODFs). In sperm lacking CABYR, a Ca2+-binding tyrosine-phosphorylation regulated protein located in the FS, the swim path chirality is preserved. We conclude that specific concentrations of ATP, ADP, cAMP and Mg2+ as well as a functional CABYR play an important role for sperm motility especially for path chirality.

Multi-ciliated microswimmers-metachronal coordination and helical swimming.

The dynamics and motion of multi-ciliated microswimmers with a spherical body and a small number N (with [Formula: see text]) of cilia with length comparable to the body radius, is investigated by mesoscale hydrodynamics simulations. A metachronal wave is imposed for the cilia beat, for which the wave vector has both a longitudinal and a latitudinal component. The dynamics and motion is characterized by the swimming velocity, its variation over the beat cycle, the spinning velocity around the main body axis, as well as the parameters of the helical trajectory. Our simulation results show that the microswimmer motion strongly depends on the latitudinal wave number and the longitudinal phase lag. The microswimmers are found to swim smoothly and usually spin around their own axis. Chirality of the metachronal beat pattern generically generates helical trajectories. In most cases, the helices are thin and stretched, i.e., the helix radius is about an order of magnitude smaller than the pitch. The rotational diffusion of the microswimmer is significantly smaller than the passive rotational diffusion of the body alone, which indicates that the extended cilia contribute strongly to the hydrodynamic radius. The swimming velocity is found to increase with the cilia number N with a slightly sublinear power law, consistent with the behavior expected from the dependence of the transport velocity of planar cilia arrays on the cilia separation.

The role of thickness inhomogeneities in hierarchical cortical folding.

The mammalian brain cortex is highly folded, with several developmental disorders affecting folding. On the extremes, lissencephaly, a lack of folds in humans, and polymicrogyria, an overly folded brain, can lead to severe mental retardation, short life expectancy, epileptic seizures, and tetraplegia. Not only a specific degree of folding, but also stereotyped patterns, are required for normal brain function. A quantitative model on how and why these folds appear during the development of the brain is the first step in understanding the cause of these conditions. In recent years, there have been various attempts to understand and model the mechanisms of brain folding. Previous works have shown that mechanical instabilities play a crucial role in the formation of brain folds, and that the geometry of the fetal brain is one of the main factors in dictating its folding characteristics. However, modeling higher-order folding, one of the main characteristics of the highly gyrencephalic brain, has not been fully tackled. The simulations presented in this work are used to study the effects of thickness inhomogeneity in the gyrogenesis of the mammalian brain in silico. Finite-element simulations of rectangular slabs are performed. These slabs are divided into two distinct regions, where the outer region mimicks the gray matter, and the inner region the underlying white matter. Differential growth is introduced by growing the top region tangentially, while keeping the underlying region untouched. The brain tissue is modeled as a neo-Hookean hyperelastic material. Simulations are performed with both homogeneous and inhomogeneous cortical thicknesses. Our results show that the homogeneous cortex folds into a single wavelength, as is common for bilayered materials, while the inhomogeneous cortex folds into more complex conformations. In the early stages of development of the inhomogeneous cortex, structures reminiscent of the deep sulci in the brain are obtained. As the cortex continues to develop, secondary undulations, which are shallower and more variable than the structures obtained in earlier gyrification stage emerge, reproducing well-known characteristics of higher-order folding in the mammalian, and particularly the human, brain.

Chiral-filament self-assembly on curved manifolds.

Rod-like and banana-shaped proteins, like BAR-domain proteins and MreB proteins, adsorb on membranes and regulate the membrane curvature. The formation of large filamentous complexes of these proteins plays an important role in cellular processes like membrane trafficking, cytokinesis and cell motion. We propose a simplified model to investigate such curvature-dependent self-assembly processes. Anisotropic building blocks, modeled as trimer molecules, which have a preferred binding site, interact via pair-wise Lennard-Jones potentials. When several trimers assemble, they form an elastic ribbon with an intrinsic curvature and twist, controlled by bending and torsional rigidity. For trimer self-assembly on the curved surface of a cylindrical membrane, this leads to a preferred spatial orientation of the ribbon. We show that these interactions can lead to the formation of helices with several windings around the cylinder. The emerging helix angle and pitch depend on the rigidities and the intrinsic curvature and twist values. In particular, a well-defined and controllable helix angle emerges in the case of equal bending and torsional rigidity. The dynamics of filament growth is characterized by three regimes, in which filament length increases with the power laws tz in time, with z≃ 3/4, z = 1/2, and z = 0 for short, intermediate, and long times, respectively. A comparison with the solutions of the Smoluchowski aggregation equation allows the identification of the underlying mechanism in the short-time regime as a crossover from size-independent to diffusion-limited aggregation. Thus, helical structures, as often observed in biology, can arise by self-assembly of anisotropic and chiral proteins.

Collective dynamics of self-propelled semiflexible filaments.

The collective behavior of active semiflexible filaments is studied with a model of tangentially driven self-propelled worm-like chains. The combination of excluded-volume interactions and self-propulsion leads to several distinct dynamic phases as a function of bending rigidity, activity, and aspect ratio of individual filaments. We consider first the case of intermediate filament density. For high-aspect-ratio filaments, we identify a transition with increasing propulsion from a state of free-swimming filaments to a state of spiraled filaments with nearly frozen translational motion. For lower aspect ratios, this gas-of-spirals phase is suppressed with growing density due to filament collisions; instead, filaments form clusters similar to self-propelled rods. As activity increases, finite bending rigidity strongly effects the dynamics and phase behavior. Flexible filaments form small and transient clusters, while stiffer filaments organize into giant clusters, similarly to self-propelled rods, but with a reentrant phase behavior from giant to smaller clusters as activity becomes large enough to bend the filaments. For high filament densities, we identify a nearly frozen jamming state at low activities, a nematic laning state at intermediate activities, and an active-turbulence state at high activities. The latter state is characterized by a power-law decay of the energy spectrum as a function of wave number. The resulting phase diagrams encapsulate tunable non-equilibrium steady states that can be used in the organization of living matter.

Quantitative modelling of nutrient-limited growth of bacterial colonies in microfluidic cultivation.

Nutrient gradients and limitations play a pivotal role in the life of all microbes, both in their natural habitat as well as in artificial, microfluidic systems. Spatial concentration gradients of nutrients in densely packed cell configurations may locally affect the bacterial growth leading to heterogeneous micropopulations. A detailed understanding and quantitative modelling of cellular behaviour under nutrient limitations is thus highly desirable. We use microfluidic cultivations to investigate growth and microbial behaviour of the model organism Corynebacterium glutamicum under well-controlled conditions. With a reaction-diffusion-type model, parameters are extracted from steady-state experiments with a one-dimensional nutrient gradient. Subsequently, we employ particle-based simulations with these parameters to predict the dynamical growth of a colony in two dimensions. Comparing the results of those simulations with microfluidic experiments yields excellent agreement. Our modelling approach lays the foundation for a better understanding of dynamic microbial growth processes, both in nature and in applied biotechnology.

Human sperm steer with second harmonics of the flagellar beat.

Sperm are propelled by bending waves traveling along their flagellum. For steering in gradients of sensory cues, sperm adjust the flagellar waveform. Symmetric and asymmetric waveforms result in straight and curved swimming paths, respectively. Two mechanisms causing spatially asymmetric waveforms have been proposed: an average flagellar curvature and buckling. We image flagella of human sperm tethered with the head to a surface. The waveform is characterized by a fundamental beat frequency and its second harmonic. The superposition of harmonics breaks the beat symmetry temporally rather than spatially. As a result, sperm rotate around the tethering point. The rotation velocity is determined by the second-harmonic amplitude and phase. Stimulation with the female sex hormone progesterone enhances the second-harmonic contribution and, thereby, modulates sperm rotation. Higher beat frequency components exist in other flagellated cells; therefore, this steering mechanism might be widespread and could inspire the design of synthetic microswimmers.

Spermatozoa as Functional Components of Robotic Microswimmers.

In recent years, the combination of synthetic micro- and nanomaterials with spermatozoa as functional components has led to the development of tubular and helical spermbots - microrobotic devices with potential applications in the biomedical and nanotechnological field. Here, the initial advances in this field are discussed and the use of spermatozoa as functional parts in microdevices elaborated. Besides the potential uses of these hybrid robotic microswimmers, the obstacles along the way are discussed, with suggestions for solutions of the encountered challenges also given.

Generalized Archimedes' principle in active fluids.

We show how a gradient in the motility properties of noninteracting pointlike active particles can cause a pressure gradient that pushes a large inert object. We calculate the force on an object inside a system of active particles with position-dependent motion parameters, in one and two dimensions, and show that a modified Archimedes' principle is satisfied. We characterize the system, both in terms of the model parameters and in terms of experimentally measurable quantities: the spatial profiles of the density, velocity and pressure. This theoretical analysis is motivated by recent experiments, which showed that the nucleus of a mouse oocyte (immature egg cell) moves from the cortex to the center due to a gradient of activity of vesicles propelled by molecular motors; it more generally applies to artificial systems of controlled localized activity.

Forces in inhomogeneous open active-particle systems.

We study the force that noninteracting pointlike active particles apply to a symmetric inert object in the presence of a gradient of activity and particle sources and sinks. We consider two simple patterns of sources and sinks that are common in biological systems. We analytically solve a one-dimensional model designed to emulate higher-dimensional systems, and study a two-dimensional model by numerical simulations. We specify when the particle flux due to the creation and annihilation of particles can act to smooth the density profile that is induced by a gradient in the velocity of the active particles, and find the net resultant force due to both the gradient in activity and the particle flux. These results are compared qualitatively to observations of nuclear motion inside the oocyte, that is driven by a gradient in activity of actin-coated vesicles.

Dynamics of self-propelled filaments pushing a load.

Worm-like filaments, which are propelled by a tangential homogeneous force along their contour, are studied as they push loads of different shapes and sizes. The resulting dynamics is investigated using Langevin dynamics simulations. The effects of size and shape of the load, propulsion strength, and thermal noise are systematically explored. The propulsive force and hydrodynamic friction of the load cause a compression in the filament that results in a buckling instability and versatile motion. Distinct regimes of elongated filaments, curved filaments, beating filaments, and filaments with alternating beating and circular motion are identified, and a phase diagram depending on the propulsion strength and the size of the load is constructed. Characteristic features of the different phases, such as beating frequencies and rotational velocities, are demonstrated to have a power-law dependence on the propulsive force.

Physics of active jamming during collective cellular motion in a monolayer.

Although collective cell motion plays an important role, for example during wound healing, embryogenesis, or cancer progression, the fundamental rules governing this motion are still not well understood, in particular at high cell density. We study here the motion of human bronchial epithelial cells within a monolayer, over long times. We observe that, as the monolayer ages, the cells slow down monotonously, while the velocity correlation length first increases as the cells slow down but eventually decreases at the slowest motions. By comparing experiments, analytic model, and detailed particle-based simulations, we shed light on this biological amorphous solidification process, demonstrating that the observed dynamics can be explained as a consequence of the combined maturation and strengthening of cell-cell and cell-substrate adhesions. Surprisingly, the increase of cell surface density due to proliferation is only secondary in this process. This analysis is confirmed with two other cell types. The very general relations between the mean cell velocity and velocity correlation lengths, which apply for aggregates of self-propelled particles, as well as motile cells, can possibly be used to discriminate between various parameter changes in vivo, from noninvasive microscopy data.

Conformations, hydrodynamic interactions, and instabilities of sedimenting semiflexible filaments.

The conformations and dynamics of semiflexible filaments subject to a homogeneous external (gravitational) field, e.g., in a centrifuge, are studied numerically and analytically. The competition between hydrodynamic drag and bending elasticity generates new shapes and dynamical features. We show that the shape of a semiflexible filament undergoes instabilities as the external field increases. We identify two transitions that correspond to the excitation of higher bending modes. In particular, for strong fields the filament stabilizes in a non-planar shape, resulting in a sideways drift or in helical trajectories. For two interacting filaments, we find the same transitions, with the important consequence that the new non-planar shapes have an effective hydrodynamic repulsion, in contrast to the planar shapes which attract themselves even when their osculating planes are rotated with respect to each other. For the case of planar filaments, we show analytically and numerically that the relative velocity is not necessarily due to a different drag of the individual filaments, but to the hydrodynamic interactions induced by their shape asymmetry.

Self-propelled worm-like filaments: spontaneous spiral formation, structure, and dynamics.

Worm-like filaments that are propelled homogeneously along their tangent vector are studied by Brownian dynamics simulations. Systems in two dimensions are investigated, corresponding to filaments adsorbed to interfaces or surfaces. A large parameter space covering weak and strong propulsion, as well as flexible and stiff filaments is explored. For strongly propelled and flexible filaments, the free-swimming filaments spontaneously form stable spirals. The propulsion force has a strong impact on dynamic properties, such as the rotational and translational mean square displacement and the rate of conformational sampling. In particular, when the active self-propulsion dominates thermal diffusion, but is too weak for spiral formation, the rotational diffusion coefficient has an activity-induced contribution given by v(c)/ξ(P), where v(c) is the contour velocity and ξ(P) the persistence length. In contrast, structural properties are hardly affected by the activity of the system, as long as no spirals form. The model mimics common features of biological systems, such as microtubules and actin filaments on motility assays or slender bacteria, and artificially designed microswimmers.

Giant adsorption of microswimmers: Duality of shape asymmetry and wall curvature.

The effect of shape asymmetry of microswimmers on their adsorption capacity at confining channel walls is studied by a simple dumbbell model. For a shape polarity of a forward-swimming cone, like the stroke-averaged shape of a sperm, extremely long wall retention times are found, caused by a nonvanishing component of the propulsion force pointing steadily into the wall, which grows exponentially with the self-propulsion velocity and the shape asymmetry. A direct duality relation between shape asymmetry and wall curvature is proposed and verified. Our results are relevant for the design microswimmer with controlled wall-adhesion properties. In addition, we confirm that pressure in active systems is strongly sensitive to the details of the particle-wall interactions.