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Over the last decade, substantial progress has been made in understanding the topology of quasi-two-dimensional (2-D) non-equilibrium fluid flows driven by ATP-powered microtubules and microorganisms. By contrast, the topology of three-dimensional (3-D) active fluid flows still poses interesting open questions. Here, we study the topology of a spherically confined active flow using 3-D direct numerical simulations of generalized Navier–Stokes (GNS) equations at the scale of typical microfluidic experiments. Consistent with earlier results for unbounded periodic domains, our simulations confirm the formation of Beltrami-like bulk flows with spontaneously broken chiral symmetry in this model. Furthermore, by leveraging fast methods to compute linking numbers, we explicitly connect this chiral symmetry breaking to the entanglement statistics of vortex lines. We observe that the mean of linking number distribution converges to the global helicity, consistent with the asymptotic result by Arnold [InVladimir I. Arnold – Collected Works(ed. A.B. Givental, B.A. Khesin, A.N. Varchenko, V.A. Vassiliev & O.Y. Viro), pp. 357–375. Springer]. Additionally, we characterize the rate of convergence of this measure with respect to the number and length of observed vortex lines, and examine higher moments of the distribution. We find that the full distribution is well described by a k-Gamma distribution, in agreement with an entropic argument. Beyond active suspensions, the tools for the topological characterization of 3-D vector fields developed here are applicable to any solenoidal field whose curl is tangent to or cancels at the boundaries of a simply connected domain. 

Recent advances in high-resolution imaging techniques and particle-based simulation methods have enabled the precise microscopic characterization of collective dynamics in various biological and engineered active matter systems. In parallel, data-driven algorithms for learning interpretable continuum models have shown promising potential for the recovery of underlying partial differential equations (PDEs) from continuum simulation data. By contrast, learning macroscopic hydrodynamic equations for active matter directly from experiments or particle simulations remains a major challenge, especially when continuum models are not known a priori or analytic coarse graining fails, as often is the case for nondilute and heterogeneous systems. Here, we present a framework that leverages spectral basis representations and sparse regression algorithms to discover PDE models from microscopic simulation and experimental data, while incorporating the relevant physical symmetries. We illustrate the practical potential through a range of applications, from a chiral active particle model mimicking nonidentical swimming cells to recent microroller experiments and schooling fish. In all these cases, our scheme learns hydrodynamic equations that reproduce the self-organized collective dynamics observed in the simulations and experiments. This inference framework makes it possible to measure a large number of hydrodynamic parameters in parallel and directly from video data. 

Bacterial biofilms are among the most abundant multicellular structures on Earth and play essential roles in a wide range of ecological, medical, and industrial processes. However, general principles that govern the emergence of biofilm architecture across different species remain unknown. Here, we combine experiments, simulations, and statistical analysis to identify shared biophysical mechanisms that determine early biofilm architecture development at the single-cell level, for the species Vibrio cholerae , Escherichia coli , Salmonella enterica , and Pseudomonas aeruginosa grown as microcolonies in flow chambers. Our data-driven analysis reveals that despite the many molecular differences between these species, the biofilm architecture differences can be described by only 2 control parameters: cellular aspect ratio and cell density. Further experiments using single-species mutants for which the cell aspect ratio and the cell density are systematically varied, and mechanistic simulations show that tuning these 2 control parameters reproduces biofilm architectures of different species. Altogether, our results show that biofilm microcolony architecture is determined by mechanical cell–cell interactions, which are conserved across different species. 

Abstract Multicellular systems, from bacterial biofilms to human organs, form interfaces (or boundaries) between different cell collectives to spatially organize versatile functions 1,2 . The evolution of sufficiently descriptive genetic toolkits probably triggered the explosion of complex multicellular life and patterning 3,4 . Synthetic biology aims to engineer multicellular systems for practical applications and to serve as a build-to-understand methodology for natural systems 5–8 . However, our ability to engineer multicellular interface patterns 2,9 is still very limited, as synthetic cell–cell adhesion toolkits and suitable patterning algorithms are underdeveloped 5,7,10–13 . Here we introduce a synthetic cell–cell adhesin logic with swarming bacteria and establish the precise engineering, predictive modelling and algorithmic programming of multicellular interface patterns. We demonstrate interface generation through a swarming adhesion mechanism, quantitative control over interface geometry and adhesion-mediated analogues of developmental organizers and morphogen fields. Using tiling and four-colour-mapping concepts, we identify algorithms for creating universal target patterns. This synthetic 4-bit adhesion logic advances practical applications such as human-readable molecular diagnostics, spatial fluid control on biological surfaces and programmable self-growing materials 5–8,14 . Notably, a minimal set of just four adhesins represents 4 bits of information that suffice to program universal tessellation patterns, implying a low critical threshold for the evolution and engineering of complex multicellular systems 3,5 .