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We have numerically investigated sperm clustering behaviours, modelling cells as superpositions of regularised flow singularities, coarse-grained from experimentally obtained digital microscopy of human sperm, both in watery medium and a highly viscous–weakly elastic, methylcellulose medium. We find that the cell yaw and cell pulling dynamics inhibit clustering in low viscosity media. In contrast clustering is readily visible in simulations modelling sperm within a methylcellulose medium, in line with previous observations that bovine sperm clustering is much more prominent in a rheological polyacrylamide medium. Furthermore, the fine-scale details of sperm flagellar movement substantially impact large-scale collective behaviours, further motivating the need for the digital microscopy and characterization of sperm to understand their dynamics.

In the nearly seven decades since the publication of Alan Turing’s work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reaction–diffusion theory. Some of these developments were nascent in Turing’s paper, and others have been due to new insights from modern mathematical techniques, advances in numerical simulations and extensive biological experiments. Despite such progress, there are still important gaps between theory and experiment, with many examples of biological patterning where the underlying mechanisms are still unclear. Here, we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical two-species reaction–diffusion framework, including evolving and complex manifolds, systems heterogeneous in space and time, and more general reaction-transport equations. While substantial progress has been made in understanding these more complicated models, there are many remaining challenges that we highlight throughout. We focus on the mathematical theory, and in particular linear stability analysis of ‘trivial’ base states. We emphasize important open questions in developing this theory further, and discuss obstacles in using these techniques to understand biological reality. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

Feathers are arranged in a precise pattern in avian skin. They first arise during development in a row along the dorsal midline, with rows of new feather buds added sequentially in a spreading wave. We show that the patterning of feathers relies on coupled fibroblast growth factor (FGF) and bone morphogenetic protein (BMP) signalling together with mesenchymal cell movement, acting in a coordinated reaction-diffusion-taxis system. This periodic patterning system is partly mechanochemical, with mechanical-chemical integration occurring through a positive feedback loop centred on FGF20, which induces cell aggregation, mechanically compressing the epidermis to rapidly intensify FGF20 expression. The travelling wave of feather formation is imposed by expanding expression of Ectodysplasin A (EDA), which initiates the expression of FGF20. The EDA wave spreads across a mesenchymal cell density gradient, triggering pattern formation by lowering the threshold of mesenchymal cells required to begin to form a feather bud. These waves, and the precise arrangement of feather primordia, are lost in the flightless emu and ostrich, though via different developmental routes. The ostrich retains the tract arrangement characteristic of birds in general but lays down feather primordia without a wave, akin to the process of hair follicle formation in mammalian embryos. The embryonic emu skin lacks sufficient cells to enact feather formation, causing failure of tract formation, and instead the entire skin gains feather primordia through a later process. This work shows that a reaction-diffusion-taxis system, integrated with mechanical processes, generates the feather array. In flighted birds, the key role of the EDA/Ectodysplasin A receptor (EDAR) pathway in vertebrate skin patterning has been recast to activate this process in a quasi-1-dimensional manner, imposing highly ordered pattern formation.

The consideration of viscoelasticity within fluid dynamical boundary element methods has traditionally required meshing over the whole flow domain. In turn, a major advantage of the boundary element method is lost, namely the need to consider only surface boundary integrals. Here, using a generalised reciprocal relation and viscoelastic force singularities, a boundary element method is developed for linear viscoelastic flows. We proceed to explore finite-deformation microswimming in a linear Maxwell fluid. We firstly deduce a finite-amplitude generalisation of a previously reported result that the flow field is unchanged between a Newtonian and linear Maxwell fluid for prescribed small-amplitude deformations. Hence Purcell’s theorem holds for a linear Maxwell fluid. We proceed to consider deformation swimming in a linear Maxwell fluid given an external forcing. Boundary scattering trajectories for an exemplar squirmer approaching a surface are observed to exhibit a weak dependence on the Deborah number, while the trajectories of a sperm and monotrichous bacterium near a surface are predicted to be essentially unaffected at moderate Deborah number. In turn, the latter supports the common simplification of using Newtonian Stokes flows for studying flagellate swimming in linear Maxwell media. In addition, the motion of a magnetic helix under the influence of an external magnetic field is considered, and highlights that linear viscoelasticity can significantly impact the propagation of the helix, in turn demonstrating that even linear rheology is important to consider for forced swimmers. Finally, the presented framework requires minimalistic adjustments to Newtonian boundary element codes, enabling rapid implementation, and is more generally applicable, for instance to studies of particle interactions in active linear rheology on the microscale.

Microbes often live in dense communities called biofilms, where competition between strains and species is fundamental to both evolution and community function. Although biofilms are commonly found in soil-like porous environments, the study of microbial interactions has largely focused on biofilms growing on flat, planar surfaces. Here, we use microfluidic experiments, mechanistic models, and game theory to study how porous media hydrodynamics can mediate competition between bacterial genotypes. Our experiments reveal a fundamental challenge faced by microbial strains that live in porous environments: cells that rapidly form biofilms tend to block their access to fluid flow and redirect resources to competitors. To understand how these dynamics influence the evolution of bacterial growth rates, we couple a model of flow–biofilm interaction with a game theory analysis. This investigation revealed that hydrodynamic interactions between competing genotypes give rise to an evolutionarily stable growth rate that stands in stark contrast with that observed in typical laboratory experiments: cells within a biofilm can outcompete other genotypes by growing more slowly. Our work reveals that hydrodynamics can profoundly affect how bacteria compete and evolve in porous environments, the habitat where most bacteria live.

Elucidating pattern forming processes is an important problem in the physical, chemical and biological sciences. Turing’s contribution, after being initially neglected, eventually catalysed a huge amount of work from mathematicians, physicists, chemists and biologists aimed towards understanding how steady spatial patterns can emerge from homogeneous chemical mixtures due to the reaction and diffusion of different chemical species. While this theory has been developed mathematically and investigated experimentally for over half a century, many questions still remain unresolved. This theme issue places Turing’s theory of pattern formation in a modern context, discussing the current frontiers in foundational aspects of pattern formation in reaction-diffusion and related systems. It highlights ongoing work in chemical, synthetic and developmental settings which is helping to elucidate how important Turing’s mechanism is for real morphogenesis, while highlighting gaps that remain in matching theory to reality. The theme issue also surveys a variety of recent mathematical research pushing the boundaries of Turing’s original theory to more realistic and complicated settings, as well as discussing open theoretical challenges in the analysis of such models. It aims to consolidate current research frontiers and highlight some of the most promising future directions. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

Understanding the motility behavior of bacteria in confining microenvironments, in which they search for available physical space and move in response to stimuli, is important for environmental, food industry, and biomedical applications. We studied the motility of five bacterial species with various sizes and flagellar architectures (Vibrio natriegens, Magnetococcus marinus, Pseudomonas putida, Vibrio fischeri, and Escherichia coli) in microfluidic environments presenting various levels of confinement and geometrical complexity, in the absence of external flow and concentration gradients. When the confinement is moderate, such as in quasi-open spaces with only one limiting wall, and in wide channels, the motility behavior of bacteria with complex flagellar architectures approximately follows the hydrodynamics-based predictions developed for simple monotrichous bacteria. Specifically, V. natriegens and V. fischeri moved parallel to the wall and P. putida and E. coli presented a stable movement parallel to the wall but with incidental wall escape events, while M. marinus exhibited frequent flipping between wall accumulator and wall escaper regimes. Conversely, in tighter confining environments, the motility is governed by the steric interactions between bacteria and the surrounding walls. In mesoscale regions, where the impacts of hydrodynamics and steric interactions overlap, these mechanisms can either push bacteria in the same directions in linear channels, leading to smooth bacterial movement, or they could be oppositional (e.g., in mesoscale-sized meandered channels), leading to chaotic movement and subsequent bacterial trapping. The study provides a methodological template for the design of microfluidic devices for single-cell genomic screening, bacterial entrapment for diagnostics, or biocomputation.

Symmetry-breaking instabilities play an important role in understanding the mechanisms underlying the diversity of patterns observed in nature, such as in Turing’s reaction–diffusion theory, which connects cellular signalling and transport with the development of growth and form. Extensive literature focuses on the linear stability analysis of homogeneous equilibria in these systems, culminating in a set of conditions for transport-driven instabilities that are commonly presumed to initiate self-organisation. We demonstrate that a selection of simple, canonical transport models with only mild multistable non-linearities can satisfy the Turing instability conditions while also robustly exhibiting only transient patterns. Hence, a Turing-like instability is insufficient for the existence of a patterned state. While it is known that linear theory can fail to predict the formation of patterns, we demonstrate that such failures can appear robustly in systems with multiple stable homogeneous equilibria. Given that biological systems such as gene regulatory networks and spatially distributed ecosystems often exhibit a high degree of multistability and nonlinearity, this raises important questions of how to analyse prospective mechanisms for self-organisation.

Pattern formation has been extensively studied in the context of evolving (time-dependent) domains in recent years, with domain growth implicated in ameliorating problems of pattern robustness and selection, in addition to more realistic modelling in developmental biology. Most work to date has considered prescribed domains evolving as given functions of time, but not the scenario of concentration-dependent dynamics, which is also highly relevant in a developmental setting. Here, we study such concentration-dependent domain evolution for reaction–diffusion systems to elucidate fundamental aspects of these more complex models. We pose a general form of one-dimensional domain evolution and extend this to N -dimensional manifolds under mild constitutive assumptions in lieu of developing a full tissue-mechanical model. In the 1D case, we are able to extend linear stability analysis around homogeneous equilibria, though this is of limited utility in understanding complex pattern dynamics in fast growth regimes. We numerically demonstrate a variety of dynamical behaviours in 1D and 2D planar geometries, giving rise to several new phenomena, especially near regimes of critical bifurcation boundaries such as peak-splitting instabilities. For sufficiently fast growth and contraction, concentration-dependence can have an enormous impact on the nonlinear dynamics of the system both qualitatively and quantitatively. We highlight crucial differences between 1D evolution and higher-dimensional models, explaining obstructions for linear analysis and underscoring the importance of careful constitutive choices in defining domain evolution in higher dimensions. We raise important questions in the modelling and analysis of biological systems, in addition to numerous mathematical questions that appear tractable in the one-dimensional setting, but are vastly more difficult for higher-dimensional models.

In one of the first examples of how mechanics can inform axonemal mechanism, Machin's study in the 1950s highlighted that observations of sperm motility cannot be explained by molecular motors in the cell membrane, but would instead require motors distributed along the flagellum. Ever since, mechanics and hydrodynamics have been recognised as important in explaining the dynamics, regulation, and guidance of sperm. More recently, the digitisation of sperm videomicroscopy, coupled with numerous modelling and methodological advances, has been bringing forth a new era of scientific discovery in this field. In this review, we survey these advances before highlighting the opportunities that have been generated for both recent research and the development of further open questions, in terms of the detailed characterisation of the sperm flagellum beat and its mechanics, together with the associated impact on cell behaviour. In particular, diverse examples are explored within this theme, ranging from how collective behaviours emerge from individual cell responses, including how these responses are impacted by the local microenvironment, to the integration of separate advances in the fields of flagellar analysis and flagellar mechanics.