The bank of swimming organisms at the micron scale (BOSO-Micro)

Designing optimal elastic filaments for viscous propulsion

The propulsion of many eukaryotic cells is generated by flagella, flexible slender filaments that are actively oscillating in space and time. The dynamics of these biological appendages have inspired the design of many types of artificial microswimmers. The magnitude of the filament's viscous propulsion depends on the time-varying shape of the filament, and that shape depends in turn on the spatial distribution of the bending rigidity of the filament. In this work, we rigorously determine the relationship between the mechanical (bending) properties of the filament and the viscous thrust it produces using mathematical optimisation. Specifically, by considering a model system (a slender elastic filament with an oscillating slope at its base), we derive the optimal bending rigidity function along the filament that maximises the time-averaged thrust produced by the actuated filament. Instead of prescribing a specific functional form, we use functional optimisation and adjoint-based variational calculus to formally establish the link between the distribution of bending rigidity and propulsion. The optimal rigidities are found to be stiff near the base, and soft near the distal end, with a spatial distribution that depends critically on the constraints used in the optimisation procedure. These findings may guide the optimal design of future artificial swimmers.

Universal Scaling Laws for a Generic Swimmer Model

We introduce a minimal model of a swimmer without body deformation based on force and torque dipoles which allows accurate and efficient 3D Navier-Stokes calculations. Our model can reproduce swimmer propulsion for a large range of Reynolds numbers and generate wake vortices in the inertial regime, reminiscent of the flow generated by the flapping tail of real fish. We perform a numerical exploration of the model from low to high Reynolds numbers and obtain universal laws using scaling arguments. Collecting data from a wide variety of microswimmers, we show that our theoretical scaling laws compare very well with experimental swimming performances across the different hydrodynamic regimes, from Stokes to turbulent flows. The simple design of our generic swimmer model paves the way for efficient large-scale simulations of hundreds of individuals, crucial for understanding collective effects within assemblies of aquatic animals.

Functional morphology of gliding motility in benthic diatoms

Diatoms, a highly successful group of photosynthetic algae, contribute to a quarter of global primary production. Many species are motile, despite having no appendages and a completely rigid cell body. Cells move to seek out nutrients, locate mating partners, and undergo vertical migration. To explore the natural diversity of diatom motility, we perform a comparative study across five common biofilm-forming species. Combining morphological measurements with high-resolution cell tracking, we establish how gliding movements relate to the morphology of the raphe-a specialized slit in the cell wall responsible for motility generation. Our detailed analyses reveal that cells exhibit a rich but species-dependent phenotype, switching stochastically between four stereotyped motility states. We model this behavior and use stochastic simulations to predict how heterogeneity in microscale navigation patterns leads to differences in long-time diffusivity and dispersal. In a representative species, we extend these findings to quantify diatom gliding in complex, naturalistic 3D environments, suggesting that cells may exploit these distinct motility signatures to achieve niche segregation in nature.

Costs and benefits of phytoplankton motility

The motility skills of phytoplankton have evolved and persisted over millions of years, primarily in response to factors such as nutrient and light availability, temperature and viscosity gradients, turbulence, and predation pressure. Phytoplankton motility is broadly categorized into swimming and buoyancy regulation. Despite studies in the literature exploring the motility costs and benefits of phytoplankton, there remains a gap in our integrative understanding of direct and indirect energy expenditures, starting from when an organism initiates movement due to any biophysical motive, to when the organism encounters intracellular and environmental challenges. Here we gather all available pieces of this puzzle from literature in biology, physics, and oceanography to paint an overarching picture of our current knowledge. The characterization of sinking and rising behavior as passive motility has resulted in the concept of sinking and rising internal efficiency being overlooked. We define this efficiency based on any energy dissipation associated with processes of mass density adjustment, as exemplified in structures like frustules and vacuoles. We propose that sinking and rising are active motility processes involving non-visible mechanisms, as species demonstrate active and rapid strategies in response to turbulence, predation risk, and gradients of nutrients, light, temperature, and viscosity. Identifying intracellular buoyancy-regulating dissipative processes offers deeper insight into the motility costs relative to the organism's total metabolic rate.

Biophysics of protist behaviour

Protists, an umbrella term first coined by Ernst Haeckel in 1866, are a vast collection of (primarily unicellular) eukaryotes that are "neither animals nor plants". This basic definition by exclusion has been exercised for centuries, even though recent advances have led to more rigorous taxonomic assignment of various protist groups. Pioneering comparative phylogenetic approaches have been applied to these organisms to reconstruct the deep branches of the eukaryotic tree, revealing essential clues about early eukaryotic evolution. Protists, including amoebae, flagellates, ciliates, and algae, are also vital constituents of global ecosystems, where they appear at the base of food chains, control the relative abundance of other microbes, and participate in global biogeochemical recycling. Due to their typically small size and lack of nervous systems, protists are often associated with the unfortunate label 'primitive'. Yet they exhibit remarkable behavioural sophistication and are able to feed, predate, navigate and interact with their surroundings. Unlike macroscopic animals, many protists reside in a non-intuitive physical regime where viscous forces dominate over inertia, and where they use diverse propulsion and navigation strategies. Interdisciplinary research into these cell-scale phenomena, characterised by a complex interplay of physical forces and mechanical constraints, has significantly advanced the emerging fields of active matter, microhydrodynamics, and non-equilibrium statistical physics. This primer discusses the biophysics of protist behaviour, with a focus on locomotion and feeding. I will highlight the most extensively studied principles and describe some more esoteric behaviours that have not yet been systematically explored.

Learning optimal integration of spatial and temporal information in noisy chemotaxis

We investigate the boundary between chemotaxis driven by spatial estimation of gradients and chemotaxis driven by temporal estimation. While it is well known that spatial chemotaxis becomes disadvantageous for small organisms at high noise levels, it is unclear whether there is a discontinuous switch of optimal strategies or a continuous transition exists. Here, we employ deep reinforcement learning to study the possible integration of spatial and temporal information in an a priori unconstrained manner. We parameterize such a combined chemotactic policy by a recurrent neural network and evaluate it using a minimal theoretical model of a chemotactic cell. By comparing with constrained variants of the policy, we show that it converges to purely temporal and spatial strategies at small and large cell sizes, respectively. We find that the transition between the regimes is continuous, with the combined strategy outperforming in the transition region both the constrained variants as well as models that explicitly integrate spatial and temporal information. Finally, by utilizing the attribution method of integrated gradients, we show that the policy relies on a nontrivial combination of spatially and temporally derived gradient information in a ratio that varies dynamically during the chemotactic trajectories.

Predation in a Microbial World: Mechanisms and Trade-Offs of Flagellate Foraging

Heterotrophic nanoflagellates are the main consumers of bacteria and picophytoplankton in the ocean and thus play a key role in ocean biogeochemistry. They are found in all major branches of the eukaryotic tree of life but are united by all being equipped with one or a few flagella that they use to generate a feeding current. These microbial predators are faced with the challenges that viscosity at this small scale impedes predator-prey contact and that their foraging activity disturbs the ambient water and thus attracts their own flow-sensing predators. Here, I describe some of the diverse adaptations of the flagellum to produce sufficient force to overcome viscosity and of the flagellar arrangement to minimize fluid disturbances, and thus of the various solutions to optimize the foraging-predation risk trade-off. I demonstrate how insights into this trade-off can be used to develop robust trait-based models of microbial food webs.

Methods and Measures for Investigating Microscale Motility

Motility is an essential factor for an organism's survival and diversification. With the advent of novel single-cell technologies, analytical frameworks, and theoretical methods, we can begin to probe the complex lives of microscopic motile organisms and answer the intertwining biological and physical questions of how these diverse lifeforms navigate their surroundings. Herein, we summarize the main mechanisms of microscale motility and give an overview of different experimental, analytical, and mathematical methods used to study them across different scales encompassing the molecular-, individual-, to population-level. We identify transferable techniques, pressing challenges, and future directions in the field. This review can serve as a starting point for researchers who are interested in exploring and quantifying the movements of organisms in the microscale world.

Encounter rates prime interactions between microorganisms

Properties of microbial communities emerge from the interactions between microorganisms and between microorganisms and their environment. At the scale of the organisms, microbial interactions are multi-step processes that are initiated by cell-cell or cell-resource encounters. Quantification and rational design of microbial interactions thus require quantification of encounter rates. Encounter rates can often be quantified through encounter kernels-mathematical formulae that capture the dependence of encounter rates on cell phenotypes, such as cell size, shape, density or motility, and environmental conditions, such as turbulence intensity or viscosity. While encounter kernels have been studied for over a century, they are often not sufficiently considered in descriptions of microbial populations. Furthermore, formulae for kernels are known only in a small number of canonical encounter scenarios. Yet, encounter kernels can guide experimental efforts to control microbial interactions by elucidating how encounter rates depend on key phenotypic and environmental variables. Encounter kernels also provide physically grounded estimates for parameters that are used in ecological models of microbial populations. We illustrate this encounter-oriented perspective on microbial interactions by reviewing traditional and recently identified kernels describing encounters between microorganisms and between microorganisms and resources in aquatic systems.

Microswimmers in vortices: dynamics and trapping

Biological and artificial microswimmers often self-propel in external flows of vortical nature; relevant examples include algae in small-scale ocean eddies, spermatozoa in uterine peristaltic flows and bacteria in microfluidic devices. A recent experiment has shown that swimming bacteria in model vortices are expelled from the vortex all the way to a well-defined depletion zone (A. Sokolov and I. S. Aranson, Rapid expulsion of microswimmers by a vortical flow. Nat. Commun., 2016, 7, 11114). In this paper, we propose a theoretical model to investigate the dynamics of elongated microswimmers in elementary vortices, namely active particles in two- and three-dimensional rotlets. A deterministic model first reveals the existence of bounded orbits near the centre of the vortex and unbounded orbits elsewhere. We further discover a conserved quantity of motion that allows us to map the phase space according to the type of the orbit (bounded vs unbounded). We next introduce translational and rotational noise into the system. Using a Fokker-Planck formalism, we quantify the quality of trapping near the centre of the vortex by examining the probability of escape and the mean time of escape from the region of deterministically bounded orbits. We finally show how to use these findings to formulate a prediction for the radius of the depletion zone, which compares favourably with the experiments (A. Sokolov and I. S. Aranson, Rapid expulsion of microswimmers by a vortical flow. Nat. Commun., 2016, 7, 11114).

Predicting the locations of force-generating dyneins in beating cilia and flagella

Cilia and flagella are slender cylindrical organelles whose bending waves propel cells through fluids and drive fluids across epithelia. The bending waves are generated by dynein motor proteins, ATPases whose force-generating activity changes over time and with position along the axoneme, the motile structure within the cilium. A key question is: where, in an actively beating axoneme, are the force-generating dyneins located? Answering this question is crucial for determining which of the conformational states adopted by the dynein motors generate the forces that bend the axoneme. The question is difficult to answer because the flagellum contains a large number of dyneins in a complex three-dimensional architecture. To circumvent this complexity, we used a molecular-mechanics approach to show how the bending moments produced by single pairs of dynein motors work against elastic and hydrodynamic forces. By integrating the individual motor activities over the length of the axoneme, we predict the locations of the force-generating dyneins in a beating axoneme. The predicted location depends on the beat frequency, the wavelength, and the elastic and hydrodynamic properties of the axoneme. To test these predictions using cryogenic electron microscopy, cilia with shorter wavelengths, such as found in Chlamydomonas, are more suitable than sperm flagella with longer wavelengths because, in the former, the lag between force and curvature is less dependent on the specific mechanical properties and experimental preparation.

Chemically active filaments: analysis and extensions of slender phoretic theory

Autophoretic microswimmers self-propel via surface interactions with a surrounding solute fuel. Chemically-active filaments are an exciting new microswimmer design that augments traditional autophoretic microswimmers, such as spherical Janus particles, with extra functionality inherent to their slender filament geometry. Slender Phoretic Theory (SPT) was developed by Katsamba et al. to analyse the dynamics of chemically-active filaments with arbitrary three-dimensional shape and chemical patterning. SPT provides a line integral solution for the solute concentration field and slip velocity on the filament surface. In this work, we exploit the generality of SPT to calculate a number of new, non-trivial analytical solutions for slender autophoretic microswimmers, including a general series solution for phoretic filaments with arbitrary geometry and surface chemistry, a universal solution for filaments with a straight centreline, and explicit solutions for some canonical shapes useful for practical applications and benchmarking numerical code. Many common autophoretic particle designs include discrete jumps in surface chemistry; here we extend our SPT to handle such discontinuities, showing that they are regularised by a boundary layer around the jump. Since our underlying framework is linear, combinations of our results provide a library of analytic solutions that will allow researchers to probe the interplay of activity patterning and shape.

Self-organized swimming with odd elasticity

We theoretically investigate self-oscillating waves of an active material, which were recently introduced as a nonsymmetric part of the elastic moduli, termed odd elasticity. Using Purcell's three-link swimmer model, we reveal that an odd-elastic filament at low Reynolds number can swim in a self-organized manner and that the time-periodic dynamics are characterized by a stable limit cycle generated by elastohydrodynamic interactions. Also, we consider a noisy shape gait and derive a swimming formula for a general elastic material in the Stokes regime with its elasticity modulus being represented by a nonsymmetric matrix, demonstrating that the odd elasticity produces biased net locomotion from random noise.

Separation of Heterotrophic Microalgae Crypthecodinium cohnii by Dielectrophoresis

Microalgae constitute an abundant source of poly-unsaturated fatty acids which are applied in various biotechnological fields such as pharmaceuticals and food supplement. Separating microalgae cells with respect to their lipid content would establish a relevant at-line analytical technique. The present study demonstrates an electrical approach for the separation of the lipid-producing microalgae Crypthecodinium cohnii using the effect of dielectrophoresis (DEP) in a microfluidic flow cell. Microalgae were cultivated for 8 days, while cell growth was characterized by optical density, dry cell weight, glucose concentration and lipid content via fluorescence microscopy. The size distribution of cells during cultivation was thoroughly investigated, since the DEP force scales with cell volume, but also depends on lipid content via cell electrophysiological constants. Thus, the challenge was to deconvolute one separation effect from the other, while the electrical cell constants of C. cohnii are not known yet. The DEP-dependent separation was realized by slanted top-bottom electrodes with the flowing cell suspension between them. Turning on the voltage deflected the cells from their initial path as determined by the streaming and thus changed their direction of flow. The separation efficiency of DEP was tested for various electrical field strengths and its performance was determined by quantitative analysis of optical and fluorescence videos. It could be shown for all size groups that the most lipid-containing cells were always subject to DEP separation and that the method is thus not only suitable for process analysis, but also for strain selection of the most productive cell lines.

Quantification of Motility in Bacillus subtilis at Temperatures Up to 84°C Using a Submersible Volumetric Microscope and Automated Tracking

We describe a system for high-temperature investigations of bacterial motility using a digital holographic microscope completely submerged in heated water. Temperatures above 90°C could be achieved, with a constant 5°C offset between the sample temperature and the surrounding water bath. Using this system, we observed active motility in Bacillus subtilis up to 66°C. As temperatures rose, most cells became immobilized on the surface, but a fraction of cells remained highly motile at distances of >100 μm above the surface. Suspended non-motile cells showed Brownian motion that scaled consistently with temperature and viscosity. A novel open-source automated tracking package was used to obtain 2D tracks of motile cells and quantify motility parameters, showing that swimming speed increased with temperature until ∼40°C, then plateaued. These findings are consistent with the observed heterogeneity of B. subtilis populations, and represent the highest reported temperature for swimming in this species. This technique is a simple, low-cost method for quantifying motility at high temperatures and could be useful for investigation of many different cell types, including thermophilic archaea.

Clustering of fast gyrotactic particles in low-Reynolds-number flow

Systems of particles in turbulent flows exhibit clustering where particles form patches in certain regions of space. Previous studies have shown that motile particles accumulate inside the vortices and in downwelling regions, while light and heavy non-motile particles accumulate inside and outside the vortices, respectively. While strong clustering is generated in regions of high vorticity, clustering of motile particles is still observed in fluid flows where vortices are short-lived. In this study, we investigate the clustering of fast swimming particles in a low-Reynolds-number turbulent flow and characterize the probability distributions of particle speed and acceleration and their influence on particle clustering. We simulate gyrotactic swimming particles in a cubic system with homogeneous and isotropic turbulent flow. Here, the swimming velocity explored is relatively faster than what has been explored in other reports. The fluid flow is produced by conducting a direct numerical simulation of the Navier-Stokes equation. In contrast with the previous results, our results show that swimming particles can accumulate outside the vortices, and clustering is dictated by the swimming number and is invariant with the stability number. We have also found that highly clustered particles are sufficiently characterized by their acceleration, where the increase in the acceleration frequency distribution of the most clustered particles suggests a direct influence of acceleration on clustering. Furthermore, the acceleration of the most clustered particles resides in acceleration values where a cross-over in the acceleration PDFs are observed, an indicator that particle acceleration generates clustering. Our findings on motile particles clustering can be applied to understanding the behavior of faster natural or artificial swimmers.

Instability caused swimming of ferromagnetic filaments in pulsed field

Magnetic filaments driven by external magnetic field are an interesting topic of research in-terms of the possible bio-medical applications. In this paper, we investigated the applicability of using ferromagnetic filaments as micro swimmers both experimentally and numerically. It was found that applying a pulse wave field profile with a duty cycle of 30[Formula: see text] induced experimentally observable swimming, which is similar to the breast stroke of micro algae. Good agreement with numerical simulations was found. Moreover, for stable continuous swimming, an initial filament shape is required to avoid transition to the structurally preferred non-swimming S-like mode.

The bank of swimming organisms at the micron scale (BOSO-Micro)

Unicellular microscopic organisms living in aqueous environments outnumber all other creatures on Earth. A large proportion of them are able to self-propel in fluids with a vast diversity of swimming gaits and motility patterns. In this paper we present a biophysical survey of the available experimental data produced to date on the characteristics of motile behaviour in unicellular microswimmers. We assemble from the available literature empirical data on the motility of four broad categories of organisms: bacteria (and archaea), flagellated eukaryotes, spermatozoa and ciliates. Whenever possible, we gather the following biological, morphological, kinematic and dynamical parameters: species, geometry and size of the organisms, swimming speeds, actuation frequencies, actuation amplitudes, number of flagella and properties of the surrounding fluid. We then organise the data using the established fluid mechanics principles for propulsion at low Reynolds number. Specifically, we use theoretical biophysical models for the locomotion of cells within the same taxonomic groups of organisms as a means of rationalising the raw material we have assembled, while demonstrating the variability for organisms of different species within the same group. The material gathered in our work is an attempt to summarise the available experimental data in the field, providing a convenient and practical reference point for future studies.

2D vs 3D tracking in bacterial motility analysis

<abstract> <p>Digital holographic microscopy provides the ability to observe throughout a large volume without refocusing. This capability enables simultaneous observations of large numbers of microorganisms swimming in an essentially unconstrained fashion. However, computational tools for tracking large 4D datasets remain lacking. In this paper, we examine the errors introduced by tracking bacterial motion as 2D projections vs. 3D volumes under different circumstances: bacteria free in liquid media and bacteria near a glass surface. We find that while XYZ speeds are generally equal to or larger than XY speeds, they are still within empirical uncertainties. Additionally, when studying dynamic surface behavior, the Z coordinate cannot be neglected.</p> </abstract>