Mechanism of branching morphogenesis inspired by diatom silica formation.

The silica-based cell walls of diatoms are prime examples of genetically controlled, species-specific mineral architectures. The physical principles underlying morphogenesis of their hierarchically structured silica patterns are not understood, yet such insight could indicate novel routes toward synthesizing functional inorganic materials. Recent advances in imaging nascent diatom silica allow rationalizing possible mechanisms of their pattern formation. Here, we combine theory and experiments on the model diatom Thalassiosira pseudonana to put forward a minimal model of branched rib patterns-a fundamental feature of the silica cell wall. We quantitatively recapitulate the time course of rib pattern morphogenesis by accounting for silica biochemistry with autocatalytic formation of diffusible silica precursors followed by conversion into solid silica. We propose that silica deposition releases an inhibitor that slows down up-stream precursor conversion, thereby implementing a self-replicating reaction-diffusion system different from a classical Turing mechanism. The proposed mechanism highlights the role of geometrical cues for guided self-organization, rationalizing the instructive role for the single initial pattern seed known as the primary silicification site. The mechanism of branching morphogenesis that we characterize here is possibly generic and may apply also in other biological systems.

Emerging principles of primary cilia dynamics in controlling tissue organization and function.

Primary cilia project from the surface of most vertebrate cells and are key in sensing extracellular signals and locally transducing this information into a cellular response. Recent findings show that primary cilia are not merely static organelles with a distinct lipid and protein composition. Instead, the function of primary cilia relies on the dynamic composition of molecules within the cilium, the context-dependent sensing and processing of extracellular stimuli, and cycles of assembly and disassembly in a cell- and tissue-specific manner. Thereby, primary cilia dynamically integrate different cellular inputs and control cell fate and function during tissue development. Here, we review the recently emerging concept of primary cilia dynamics in tissue development, organization, remodeling, and function.

Novel analytical tools reveal that local synchronization of cilia coincides with tissue-scale metachronal waves in zebrafish multiciliated epithelia.

Motile cilia are hair-like cell extensions that beat periodically to generate fluid flow along various epithelial tissues within the body. In dense multiciliated carpets, cilia were shown to exhibit a remarkable coordination of their beat in the form of traveling metachronal waves, a phenomenon which supposedly enhances fluid transport. Yet, how cilia coordinate their regular beat in multiciliated epithelia to move fluids remains insufficiently understood, particularly due to lack of rigorous quantification. We combine experiments, novel analysis tools, and theory to address this knowledge gap. To investigate collective dynamics of cilia, we studied zebrafish multiciliated epithelia in the nose and the brain. We focused mainly on the zebrafish nose, due to its conserved properties with other ciliated tissues and its superior accessibility for non-invasive imaging. We revealed that cilia are synchronized only locally and that the size of local synchronization domains increases with the viscosity of the surrounding medium. Even though synchronization is local only, we observed global patterns of traveling metachronal waves across the zebrafish multiciliated epithelium. Intriguingly, these global wave direction patterns are conserved across individual fish, but different for left and right noses, unveiling a chiral asymmetry of metachronal coordination. To understand the implications of synchronization for fluid pumping, we used a computational model of a regular array of cilia. We found that local metachronal synchronization prevents steric collisions, i.e., cilia colliding with each other, and improves fluid pumping in dense cilia carpets, but hardly affects the direction of fluid flow. In conclusion, we show that local synchronization together with tissue-scale cilia alignment coincide and generate metachronal wave patterns in multiciliated epithelia, which enhance their physiological function of fluid pumping.

The molecular basis for pore pattern morphogenesis in diatom silica.

Biomineral-forming organisms produce inorganic materials with complex, genetically encoded morphologies that are unmatched by current synthetic chemistry. It is poorly understood which genes are involved in biomineral morphogenesis and how the encoded proteins guide this process. We addressed these questions using diatoms, which are paradigms for the self-assembly of hierarchically meso- and macroporous silica under mild reaction conditions. Proteomics analysis of the intracellular organelle for silica biosynthesis led to the identification of new biomineralization proteins. Three of these, coined dAnk1-3, contain a common protein-protein interaction domain (ankyrin repeats), indicating a role in coordinating assembly of the silica biomineralization machinery. Knocking out individual dank genes led to aberrations in silica biogenesis that are consistent with liquid-liquid phase separation as underlying mechanism for pore pattern morphogenesis. Our work provides an unprecedented path for the synthesis of tailored mesoporous silica materials using synthetic biology.

Synchronization in cilia carpets and the Kuramoto model with local coupling: Breakup of global synchronization in the presence of noise.

Carpets of beating cilia represent a paradigmatic example of self-organized synchronization of noisy biological oscillators, characterized by traveling waves of cilia phase. We present a multi-scale model of a cilia carpet that comprises realistic hydrodynamic interactions between cilia computed for a chiral cilia beat pattern from unicellular Paramecium and active noise of the cilia beat. We demonstrate an abrupt loss of global synchronization beyond a characteristic noise strength. We characterize stochastic transitions between synchronized and disordered dynamics, which generalize the notion of phase slips in pairs of coupled noisy phase oscillators. Our theoretical work establishes a link between the two-dimensional Kuramoto model of phase oscillators with mirror-symmetric oscillator coupling and detailed models of biological oscillators with asymmetric, chiral interactions.

Correction: Quantification of nematic cell polarity in three-dimensional tissues.

[This corrects the article DOI: 10.1371/journal.pcbi.1008412.].

Micromotor-mediated sperm constrictions for improved swimming performance.

Sperm-driven micromotors, consisting of a single sperm cell captured in a microcap, utilize the strong propulsion generated by the flagellar beat of motile spermatozoa for locomotion. It enables the movement of such micromotors in biological media, while being steered remotely by means of an external magnetic field. The substantial decrease in swimming speed, caused by the additional hydrodynamic load of the microcap, limits the applicability of sperm-based micromotors. Therefore, to improve the performance of such micromotors, we first investigate the effects of additional cargo on the flagellar beat of spermatozoa. We designed two different kinds of microcaps, which each result in different load responses of the flagellar beat. As an additional design feature, we constrain rotational degrees of freedom of the cell's motion by modifying the inner cavity of the cap. Particularly, cell rolling is substantially reduced by tightly locking the sperm head inside the microcap. Likewise, cell yawing is decreased by aligning the micromotors under an external static magnetic field. The observed differences in swimming speed of different micromotors are not so much a direct consequence of hydrodynamic effects, but rather stem from changes in flagellar bending waves, hence are an indirect effect. Our work serves as proof-of-principle that the optimal design of microcaps is key for the development of efficient sperm-driven micromotors.

Sperm chemotaxis in marine species is optimal at physiological flow rates according theory of filament surfing.

Sperm of marine invertebrates have to find eggs cells in the ocean. Turbulent flows mix sperm and egg cells up to the millimeter scale; below this, active swimming and chemotaxis become important. Previous work addressed either turbulent mixing or chemotaxis in still water. Here, we present a general theory of sperm chemotaxis inside the smallest eddies of turbulent flow, where signaling molecules released by egg cells are spread into thin concentration filaments. Sperm cells 'surf' along these filaments towards the egg. External flows make filaments longer, but also thinner. These opposing effects set an optimal flow strength. The optimum predicted by our theory matches flow measurements in shallow coastal waters. Our theory quantitatively agrees with two previous fertilization experiments in Taylor-Couette chambers and provides a mechanistic understanding of these early experiments. 'Surfing along concentration filaments' could be a paradigm for navigation in complex environments in the presence of turbulent flow.

Lagrangian mechanics of active systems.

We present a multi-scale modeling and simulation framework for low-Reynolds number hydrodynamics of shape-changing immersed objects, e.g., biological microswimmers and active surfaces. The key idea is to consider principal shape changes as generalized coordinates and define conjugate generalized hydrodynamic friction forces. Conveniently, the corresponding generalized friction coefficients can be pre-computed and subsequently reused to solve dynamic equations of motion fast. This framework extends Lagrangian mechanics of dissipative systems to active surfaces and active microswimmers, whose shape dynamics is driven by internal forces. As an application case, we predict in-phase and anti-phase synchronization in pairs of cilia for an experimentally measured cilia beat pattern.

Quantification of nematic cell polarity in three-dimensional tissues.

How epithelial cells coordinate their polarity to form functional tissues is an open question in cell biology. Here, we characterize a unique type of polarity found in liver tissue, nematic cell polarity, which is different from vectorial cell polarity in simple, sheet-like epithelia. We propose a conceptual and algorithmic framework to characterize complex patterns of polarity proteins on the surface of a cell in terms of a multipole expansion. To rigorously quantify previously observed tissue-level patterns of nematic cell polarity (Morales-Navarrete et al., eLife 2019), we introduce the concept of co-orientational order parameters, which generalize the known biaxial order parameters of the theory of liquid crystals. Applying these concepts to three-dimensional reconstructions of single cells from high-resolution imaging data of mouse liver tissue, we show that the axes of nematic cell polarity of hepatocytes exhibit local coordination and are aligned with the biaxially anisotropic sinusoidal network for blood transport. Our study characterizes liver tissue as a biological example of a biaxial liquid crystal. The general methodology developed here could be applied to other tissues and in-vitro organoids.

Resilience of three-dimensional sinusoidal networks in liver tissue.

Can three-dimensional, microvasculature networks still ensure blood supply if individual links fail? We address this question in the sinusoidal network, a plexus-like microvasculature network, which transports nutrient-rich blood to every hepatocyte in liver tissue, by building on recent advances in high-resolution imaging and digital reconstruction of adult mice liver tissue. We find that the topology of the three-dimensional sinusoidal network reflects its two design requirements of a space-filling network that connects all hepatocytes, while using shortest transport routes: sinusoidal networks are sub-graphs of the Delaunay graph of their set of branching points, and also contain the corresponding minimum spanning tree, both to good approximation. To overcome the spatial limitations of experimental samples and generate arbitrarily-sized networks, we developed a network generation algorithm that reproduces the statistical features of 0.3-mm-sized samples of sinusoidal networks, using multi-objective optimization for node degree and edge length distribution. Nematic order in these simulated networks implies anisotropic transport properties, characterized by an empirical linear relation between a nematic order parameter and the anisotropy of the permeability tensor. Under the assumption that all sinusoid tubes have a constant and equal flow resistance, we predict that the distribution of currents in the network is very inhomogeneous, with a small number of edges carrying a substantial part of the flow-a feature known for hierarchical networks, but unexpected for plexus-like networks. We quantify network resilience in terms of a permeability-at-risk, i.e., permeability as function of the fraction of removed edges. We find that sinusoidal networks are resilient to random removal of edges, but vulnerable to the removal of high-current edges. Our findings suggest the existence of a mechanism counteracting flow inhomogeneity to balance metabolic load on the liver.

Sperm-Driven Micromotors Moving in Oviduct Fluid and Viscoelastic Media.

Biohybrid micromotors propelled by motile cells are fascinating entities for autonomous biomedical operations on the microscale. Their operation under physiological conditions, including highly viscous environments, is an essential prerequisite to be translated to in vivo settings. In this work, a sperm-driven microswimmer, referred to as a spermbot, is demonstrated to operate in oviduct fluid in vitro. The viscoelastic properties of bovine oviduct fluid (BOF), one of the fluids that sperm cells encounter on their way to the oocyte, are first characterized using passive microrheology. This allows to design an artificial oviduct fluid to match the rheological properties of oviduct fluid for further experiments. Sperm motion is analyzed and it is confirmed that kinetic parameters match in real and artificial oviduct fluids, respectively. It is demonstrated that sperm cells can efficiently couple to magnetic microtubes and propel them forward in media of different viscosities and in BOF. The flagellar beat pattern of coupled as well as of free sperm cells is investigated, revealing an alteration on the regular flagellar beat, presenting an on-off behavior caused by the additional load of the microtube. Finally, a new microcap design is proposed to improve the overall performance of the spermbot in complex biofluids.

Chemokinetic Scattering, Trapping, and Avoidance of Active Brownian Particles.

We present a theory of chemokinetic search agents that regulate directional fluctuations according to distance from a target. A dynamic scattering effect reduces the probability to penetrate regions with high fluctuations and thus reduces search success for agents that respond instantaneously to positional cues. In contrast, agents with internal states that initially suppress chemokinesis can exploit scattering to increase their probability to find the target. Using matched asymptotics between the case of diffusive and ballistic search, we obtain analytic results beyond Fox colored noise approximation.

Liquid-crystal organization of liver tissue.

Functional tissue architecture originates by self-assembly of distinct cell types, following tissue-specific rules of cell-cell interactions. In the liver, a structural model of the lobule was pioneered by Elias in 1949. This model, however, is in contrast with the apparent random 3D arrangement of hepatocytes. Since then, no significant progress has been made to derive the organizing principles of liver tissue. To solve this outstanding problem, we computationally reconstructed 3D tissue geometry from microscopy images of mouse liver tissue and analyzed it applying soft-condensed-matter-physics concepts. Surprisingly, analysis of the spatial organization of cell polarity revealed that hepatocytes are not randomly oriented but follow a long-range liquid-crystal order. This does not depend exclusively on hepatocytes receiving instructive signals by endothelial cells, since silencing Integrin-ß1 disrupted both liquid-crystal order and organization of the sinusoidal network. Our results suggest that bi-directional communication between hepatocytes and sinusoids underlies the self-organization of liver tissue.

Body size-dependent energy storage causes Kleiber's law scaling of the metabolic rate in planarians.

Kleiber's law, or the 3/4 -power law scaling of the metabolic rate with body mass, is considered one of the few quantitative laws in biology, yet its physiological basis remains unknown. Here, we report Kleiber's law scaling in the planarian Schmidtea mediterranea. Its reversible and life history-independent changes in adult body mass over 3 orders of magnitude reveal that Kleiber's law does not emerge from the size-dependent decrease in cellular metabolic rate, but from a size-dependent increase in mass per cell. Through a combination of experiment and theoretical analysis of the organismal energy balance, we further show that the mass allometry is caused by body size dependent energy storage. Our results reveal the physiological origins of Kleiber's law in planarians and have general implications for understanding a fundamental scaling law in biology.

Heuristic Modeling and 3D Stereoscopic Visualization of a Chlamydomonas reinhardtii Cell.

The structural modeling and representation of cells is a complex task as different microscopic, spectroscopic and other information resources have to be combined to achieve a three-dimensional representation with high accuracy. Moreover, to provide an appropriate spatial representation of the cell, a stereoscopic 3D (S3D) visualization is favorable. In this work, a structural cell model is created by combining information from various light microscopic and electron microscopic images as well as from publication-related data. At the mesoscopic level each cell component is presented with special structural and visual properties; at the molecular level a cell membrane composition and the underlying modeling method are discussed; and structural information is correlated with those at the functional level (represented by simplified energy-producing metabolic pathways). The organism used as an example is the unicellular Chlamydomonas reinhardtii, which might be important in future alternative energy production processes. Based on the 3D model, an educative S3D animation was created which was shown at conferences. The complete workflow was accomplished by using the open source 3D modeling software Blender. The discussed project including the animation is available from: http://Cm5.CELLmicrocosmos.org.

Critical Point in Self-Organized Tissue Growth.

We present a theory of pattern formation in growing domains inspired by biological examples of tissue development. Gradients of signaling molecules regulate growth, while growth changes these graded chemical patterns by dilution and advection. We identify a critical point of this feedback dynamics, which is characterized by spatially homogeneous growth and proportional scaling of patterns with tissue length. We apply this theory to the biological model system of the developing wing of the fruit fly Drosophila melanogaster and quantitatively identify signatures of the critical point.

Load response of shape-changing microswimmers scales with their swimming efficiency.

External forces acting on a microswimmer can feed back on its self-propulsion mechanism. We discuss this load response for a generic microswimmer that swims by cyclic shape changes. We show that the change in cycle frequency is proportional to the Lighthill efficiency of self-propulsion. As a specific example, we consider Najafi's three-sphere swimmer. The force-velocity relation of a microswimmer implies a correction for a formal superposition principle for active and passive motion.

Renormalization group theory for percolation in time-varying networks.

Motivated by multi-hop communication in unreliable wireless networks, we present a percolation theory for time-varying networks. We develop a renormalization group theory for a prototypical network on a regular grid, where individual links switch stochastically between active and inactive states. The question whether a given source node can communicate with a destination node along paths of active links is equivalent to a percolation problem. Our theory maps the temporal existence of multi-hop paths on an effective two-state Markov process. We show analytically how this Markov process converges towards a memoryless Bernoulli process as the hop distance between source and destination node increases. Our work extends classical percolation theory to the dynamic case and elucidates temporal correlations of message losses. Quantification of temporal correlations has implications for the design of wireless communication and control protocols, e.g. in cyber-physical systems such as self-organized swarms of drones or smart traffic networks.